![From Segmented Images to Good Quality
Meshes Using Delaunay Refinement
Jean-Daniel Boissonnat1
, Jean-Philippe Pons2
, and Mariette Yvinec1
1
INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93,
06902 Sophia-Antipolis Cedex, France
Jean-Daniel.Boissonnat@sophia.inria.fr
2
CSTB, 290 route des Lucioles, BP 209, 06904 Sophia-Antipolis Cedex, France
Jean-Philippe.Pons@cstb.fr
Abstract. This paper surveys Delaunay-based meshing techniques for
curved objects, and their application in medical imaging and in computer
vision to the extraction of geometric models from segmented images. We
show that the so-called Delaunay refinement technique allows to mesh
surfaces and volumes bounded by surfaces, with theoretical guarantees
on the quality of the approximation, from a geometrical and a topologi-
cal point of view. Moreover, it offers extensive control over the size and
shape of mesh elements, for instance through a (possibly non-uniform)
sizing field. We show how this general paradigm can be adapted to pro-
duce anisotropic meshes, i.e. meshes elongated along prescribed direc-
tions. Lastly, we discuss extensions to higher dimensions, and especially
to space-time for producing time-varying 3D models. This is also of inter-
est when input images are transformed into data points in some higher
dimensional space as is common practice in machine learning.
1 Introduction
Motivation. The ubiquity of digital imaging in scientific research and in indus-
try calls for automated tools to extract high-level information from raster rep-
resentations (2D, 3D, or higher-dimensional rectilinearly-sampled scalar/vector
fields), the latter often being not directly suitable for analysis and interpreta-
tion. Notably, the computerized creation of geometric models from digital images
plays a crucial role in many medical imaging applications.
A precondition for extracting geometry from images is usually to partition
image pixels (voxels) into multiple regions of interest. This task, known as im-
age segmentation, is a central long-standing problem in image processing and
computer vision. Doing a review of this area is out of the scope of this paper.
Let us only mention that it is a highly ill-posed problem due to various per-
turbing factors such as noise, occlusions, missing parts, cluttered data, etc. The
interested reader may refer to e.g. [1] for a specific survey on segmentation of
medical images.
This paper focuses on a step posterior to image segmentation: the automatic
generation of discrete geometric representations from segmented images, such
F. Nielsen (Ed.): ETVC 2008, LNCS 5416, pp. 13–37, 2009.
c
Springer-Verlag Berlin Heidelberg 2009](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-25-320.jpg)
![14 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
as surface meshes representing boundaries between different regions of inter-
est, or volume meshes of their interior. This step is determinant in numer-
ous applications. For instance, in medicine, an increasing number of numerical
simulations of physical or physiological processes call for geometric models of
anatomical structures: electroencephalography (EEG) and magnetoencephalog-
raphy (MEG), image-guided neurosurgery, electromagnetic modeling, blood flow
simulation, etc.
However, this topic has attracted less interest than image segmentation so far.
As a result, reliable fully-automated tools for the unstructured discretization of
segmented images, and in particular of medical datasets, are still lacking. So
that simplistic or low-quality geometric models are still of wide use in some ap-
plications. For example, in electromagnetic modeling, such as specific absorption
rate studies, finite element methods (FEM) on unstructured grids conforming
to anatomical structures would be desirable; but due to the difficulty of produc-
ing such models, most numerical simulations so far have been conducted using
finite difference methods on rectilinear grids, although the poor definition of tis-
sue boundaries (stair-casing effect) strongly limits their accuracy. Similarly, in
the EEG/MEG source localization problem using the boundary element method
(BEM), simplistic head models consisting of a few nested tissue layers remain
more popular than realistic models featuring multiple junctions.
The generation of geometric models from segmented images presents many
challenges. The output must fulfill many requirements in terms of geometric ac-
curacy and topological correctness, smoothness, number, type, size and shape of
mesh elements, in order to obtain acceptable results and make useful predictions,
avoid instabilities in the simulations, or reduce the overall processing time. No-
tably, the conditioning of stiffness matrices in FEM directly depends on the sizes
and shapes of the elements. Another example is image-guided neurosurgery, for
which real-time constraints impose strong limitations on the complexity of the
geometric brain model being dynamically registered onto the patient anatomy.
Grid-based methods. Commonly used techniques do not meet the aforemen-
tioned specifications. The most popular technique for producing surface meshes
from raster data is undoubtedly the marching cubes algorithm, introduced by
Lorensen and Cline [2]. Given a scalar field sampled on a rectilinear grid, the
marching cubes algorithm efficiently generates a triangular mesh of an isosurface
by tessellating each cubic cell of the domain according to a case table constructed
off-line.
Unfortunately, this technique, as well as its many subsequent variants, typ-
ically produces unnecessarily large meshes (at least one triangle per boundary
voxel) of very low quality (lots of skinny triangles). This may be acceptable for
visualization purposes, but not for further numerical simulations. In order to ob-
tain suitable representations, the resulting meshes often have to be regularized,
optimized and decimated, while simultaneously controlling the approximation
accuracy and preserving some topological properties, such as the absence of self-
intersections. Sometimes, good tetrahedral meshes of the domains bounded by](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-26-320.jpg)
![From Segmented Images to Good Quality Meshes 15
the marching cubes surfaces also have to be generated. Most of the time, these
tasks are overconstrained.
Recently, the interest in grid-based techniques has been renewed by a few
methods with theoretical guarantees. Plantiga and Vegter [3] propose an algo-
rithm to mesh implicit surfaces with guaranteed topology, based on an adaptive
octree subdivision controlled by interval arithmetic. But in its current form, this
algorithm is relevant to closed-form expressions, not to sampled data.
The recent algorithm of Labelle and Shewchuck [4] fills an isosurface with a
uniformly sized tetrahedral mesh whose dihedral angles are bounded between
10.7◦
and 164.8◦
. The algorithm is very fast, numerically robust, and easy to
implement because, like the marching cubes algorithm, it generates tetrahedra
from a small set of precomputed stencils. Moreover, if the isosurface is a smooth
2-manifold with bounded curvature, and the tetrahedra are sufficiently small,
then the boundary of the mesh is guaranteed to be a geometrically and topo-
logically accurate approximation of the isosurface. However, this algorithm lacks
flexibility: notably, it is limited to uniform surface meshes, and isotropic surface
and volume meshes.
Delaunay-based methods. This paper surveys Delaunay-based meshing tech-
niques for curved objects. It is recognized as one of the most powerful techniques
for generating surface and volume meshes with theoretical guarantees on the
quality of the approximation, from a geometrical and topological point of view.
Moreover, it offers extensive control over the size and shape of mesh elements, for
instance through a (possibly non-uniform) sizing field. It also allows to mesh sev-
eral domains simultaneously. Recent extensions show that this general paradigm
can be adapted to produce anisotropic meshes, i.e. meshes elongated along pre-
scribed directions, as well as meshes in higher dimensions.
In this paper, we show how Delaunay-based meshing can be applied in medi-
cal imaging and in computer vision to the extraction of meshes from segmented
images, with all the desired specifications. The rest of the paper is organized
as follows. We first introduce the notion of restricted Delaunay triangulation
in Section 2. We then show how to mesh surfaces (Section 3) and volumes
bounded by surfaces (Section 4) using the so-called Delaunay refinement tech-
nique. Anisotropic meshes are discussed in Section 5. Lastly, we tackle extensions
of Delaunay refinement to higher dimensions (Section 6), and especially to space-
time for producing time-varying 3D models. This is also of interest when input
images are transformed into data points in some higher dimensional space as is
common practice in machine learning.
2 Restricted Delaunay Triangulations
In this section, we recall the definitions of Voronoi diagrams and Delaunay tri-
angutions, and their generalization known as power (or Laguerre) diagrams and
weighted Delaunay (or regular) triangulations. We then introduce the concept
of restricted Delaunay triangulation which is central in this paper.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-27-320.jpg)
![16 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
2.1 Voronoi Diagrams and Delaunay Triangulations
Voronoi diagrams are versatile structures which encode proximity relationships
between objects. They are particularly relevant to perform nearest neighbor
search and motion planning (e.g. in robotics), and to model growth processes
(e.g. crystal growth in materials science). Delaunay triangulations, which are
geometrically dual to Voronoi diagrams, are a classical tool in the field of mesh
generation and mesh processing due to their optimality properties.
In the sequel, we call k-simplex the convex hull of k + 1 affinely independent
points. For example, a 0-simplex is a point, a 1-simplex is a line segment, a
2-simplex is a triangle and a 3-simplex is a tetrahedron.
Let E = {p1, . . . , pn} be set of points in Rd
, called sites. Note that in this
paper, we are mainly interested in d = 3, except in Section 6, where the case d 3
is studied. The Voronoi region, or Voronoi cell, denoted by V (pi), associated to
a point pi ∈ E is the region formed by points that are closer to pi than to all
other sites in E:
V (pi) = {x ∈ Rd
: ∀j, x − pi ≤ x − pj}.
V (pi) is the intersection of n − 1 half-spaces bounded by the bisector planes
of segments [pipj], j
= i. V (pi) is therefore a convex polyhedron, possibly un-
bounded. The Voronoi diagram of E, denoted by Vor(E), is the subdivision of
space induced by the Voronoi cells V (p1), . . . , V (pn).
See Fig. 1 for a two-dimensional example of a Voronoi diagram. In two di-
mensions, the edges shared by two Voronoi cells are called Voronoi edges and
the points shared by three Voronoi cells are called Voronoi vertices. Similarly,
in three dimensions, we term Voronoi facets, edges and vertices the geometric
objects shared by respectively two, three and four Voronoi cells, respectively.
The Voronoi diagram is the collection of all these k-dimensional objects, with
0 ≤ k ≤ d, which we call Voronoi faces. In particular, note that Voronoi cells
V (pi) correspond to d-dimensional Voronoi faces.
To simplify the presentation and without real loss of generality, we will assume
in the sequel that E does not contain any subset of d + 2 points that lie on a
same hypersphere. We say that the points of E are then in general position.
The Delaunay triangulation of E, noted Del(E), is the geometric dual of
Vor(E), and can be described as an embedding of the nerve1
of Vor(E). The
nerve of Vor(E) is the abstract simplicial complex that contains a simplex
σ = (pi0 , . . . , pik
) iff V (pi0 ) ∩ . . . ∩ V (pik
)
= ∅. Specifically, if k + 1 Voronoi cells
have a non-empty intersection, this intersection constitutes a (d−k)-dimensional
face f of Vor(E). The convex hull of the associated k + 1 sites constitutes a k-
dimensional simplex in the Delaunay triangulation and this simplex is the dual
of face f. In 3D, the dual of a Delaunay tetrahedron is the Voronoi vertex that
coincides with the circumcenter of the tetrahedron, the dual of a Delaunay facet
is a Voronoi edge, the dual of a Delaunay edge is a Voronoi facet, and the dual
of a Delaunay vertex pi is the Voronoi cell V (pi). See Fig. 1.
1
The notion of nerve of a covering is a basic concept in algebraic topology [5].](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-28-320.jpg)
![18 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
triangulation of Σ. Note that because some spheres of Σ may have an empty
power cell, the set of vertices in the weighted Delaunay triangulation is only a
subset of the centers of the σi.
It is a remarkable fact that weighted Delaunay triangulations can be computed
almost as fast as non weighted Delaunay triangulations. An efficient implemen-
tation of both types of triangulations can be found in the Cgal library [6,7]. It
is robust to degenerate configurations and floating-point errors through the use
of exact geometric predicates.
2.3 Restricted Delaunay Triangulations
We introduce the concept of restricted Delaunay triangulation which, as the con-
cept of Delaunay triangulation, is related to the notion of nerve. Given a subset
Ω ⊂ Rd
and a set E of points, we call Delaunay triangulation of E restricted
to Ω, and note Del|Ω(E), the subcomplex of Del(E) composed of the Delaunay
simplices whose dual Voronoi faces intersect Ω. We refer to Fig. 2 to illustrate
this concept in 2D. Fig. 2 (left) shows a Delaunay triangulation restricted to a
curve C, which is composed of the Delaunay edges whose dual Voronoi edges
intersect C. Fig. 2 (right) shows the Delaunay triangulation of the same set of
points restricted to the region R bounded by the curve C. The restricted triangu-
lation is composed of the Delaunay triangles whose circumcenters are contained
in R. For an illustration in R3
, consider a region O bounded by a surface S and
a sample E, the Delaunay triangulation restricted to S, Del|S(E), is composed
of the Delaunay facets in Del(E) whose dual Voronoi edges intersect S while the
Delaunay triangulation restricted to O, Del|O(E), is made of those tetrahedra
in Del(E) whose circumcenters belong to O.
The attentive reader may have noticed that in both cases of Figure 2, the
restricted Delaunay triangulation forms a good approximation of the object.
Actually, this is a general property of the restricted Delaunay triangulation. It
can be shown that, under some assumptions, and especially if E is a sufficiently
dense sample of a smooth surface S, Del|S(E) is a good approximation of S, both
in a topological and in a geometric sense. Specifically, Del|S(E) is a triangulated
surface that is isotopic to S; the isotopy moves the points by a quantity that
becomes arbitrarily small when the density increases; in addition, normals of S
of can be consistently approximated from Del|S(E).
Before stating precise results, we define what “sufficiently dense” means. The
definition is based on the notion of medial axis. In the rest of the paper, S will
denote a closed smooth surface of R3
.
Definition 1 (Medial axis). The medial axis of a surface S is the closure of
the set of points with at least two closest points on S.
Definition 2 (lfs). The local feature size at a point x on a surface S, noted lfs(x),
is the distance from x to the medial axis of S. We write lfs(S) = infx∈S lfs(x).
It can be shown that lfs(x) does not exceed the reach of S at x, denoted by
rch(x). The reach at x is defined as the radius of the largest open ball tangent](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-30-320.jpg)
![From Segmented Images to Good Quality Meshes 19
Fig. 2. The Voronoi diagram (in red) and the Delaunay triangulation (in blue) of a
sample of red points on a planar closed curve C (in black). On the left: the edges of
the Voronoi diagram and of the Delaunay triangulation that are restricted to the curve
are in bold lines. On the right: the triangles belonging to the Delaunay triangulation
of the sample restricted to the domain bounded by C are in blue.
Fig. 3. The medial axis of a planar curve (only the portion inside the domain bounded
by the curve is shown). The thin curves are parallel to the boundary of the domain.
to S at x whose interior does not contain any point of S. Plainly, rch(x) cannot
exceed the smallest radius of curvature at x and can be strictly less at points
where the thickness of the object bounded by S is small. As shown by Federer [8],
the local feature size of a smooth surface object is bounded away from 0.2
The following notion of ε-sample has been proposed by Amenta and Bern in
their seminal paper on surface reconstruction [9].
2
In fact, Federer proved the stronger result that the local feature size is bounded
away from 0 as soon as S belongs to the class C1,1
of surfaces that admit a normal
at each point and whose normal field is Lipschitz. This class is larger than the class
of C2
surfaces and includes surfaces whose curvature may be discontinuous at some
points. An example of a surface that is C1,1
but not C2
is the offset of a cube.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-31-320.jpg)
![20 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
Fig. 4. A surface Delaunay ball whose center is a candidate for being inserted in E
Definition 3 (ε-sample). Let ε 1 and S be a smooth surface. We say that
a finite point set E ⊂ S is an ε-sample of S if any point x of S is at distance at
most ε lfs(x) from a point of E.
The notion of ε-sample is not very handy since it requires that any point of the
surface is close to a sample point. A more convenient notion of sample, called
loose ε-sample, only requires a finite set of points of S to be close to the sample
set [10]. More precisely, consider the Voronoi edges of Vor(E) that intersect
S. We require that each such intersection point is close to the sample set. By
definition, these Voronoi edges are dual to the facets of Del|S(E). An intersection
point of such an edge with S is thus the center of a so-called surface Delaunay
ball, i.e. a ball circumscribing a facet of Del|S(E) and centered on the surface S
(see Fig. 4).
Definition 4 (Loose ε-sample). Let ε 1 be a constant and S be a smooth
surface. A point set E ⊂ S is a loose ε-sample of S if Del|S(E) has a vertex on
each connected component of S and if, in addition, any surface Delaunay ball
B(cf , rf ) circumscribing a facet f of Del|S(E) is such that rf εlfs(cf ).
The following theorem states that, for sufficiently dense samples, Del|S(E) is a
good approximation of S.
Theorem 1. If E is a loose ε-sample of a smooth compact surface S, with
ε 0.12, then the restriction of the orthogonal projection πS : R3
M(S) → S,
induces an isotopy that maps Del|S(E) to S. The isotopy does not move the
points of Del|S(E) by more than O(ε2
). The angle between the normal to a facet
f of Del|S(E) and the normals to S at the vertices of f is O(ε).
Weaker variants of this theorem have been proved by Amenta and Bern [9] and
Boissonnat and Oudot [10]. Cohen-Steiner and Morvan have further shown that
one can estimate the tensor of curvatures from Del|S(E) [11].](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-32-320.jpg)
![From Segmented Images to Good Quality Meshes 21
3 Surface Sampling and Meshing
In this section, we show how the concept of restricted Delaunay triangulation
can be used to mesh smooth surfaces. The algorithm is proven to terminate
and to construct good-quality meshes, while offering bounds on the accuracy of
the original boundary approximation and on the size of the output mesh. The
refinement process is controlled by highly customizable quality and size criteria
on triangular facets. A notable feature of this algorithm is that the surface needs
only to be known through an oracle that, given a line segment, detects whether
the segment intersects the surface and, in the affirmative, returns an intersection
point. This makes the algorithm useful in a wide variety of contexts and for a
large class of surfaces.
The paradigm of Delaunay refinement has been first proposed by Ruppert for
meshing planar domains [12]. The meshing algorithm presented in this section
is due to Chew [13,14].
3.1 Delaunay Refinement for Meshing Surfaces
Let S be a surface of R3
. If we know a loose ε-sample E of S, with ε 0.12,
then, according to Theorem 1, the restricted Delaunay triangulation Del|S(E)
is a good approximation of S. In this section, we present an algorithm that can
construct such a sample and the associated restricted Delaunay triangulation.
We restrict the presentation to the case of smooth, compact and closed surfaces.
Hence, lfs(S) = infx∈S lfs(x) 0.
The algorithm is greedy. It inserts points one by one and maintains the current
set E, the Delaunay triangulation Del(E) and its restriction Del|S(E) to S.
Let ψ be a function defined over S such that
∀x ∈ S, 0 ψinf ≤ ψ(x) ≤ εlfs(x).
where ψmin = infx∈S ψ(x). Function ψ will control the sampling density and is
called the sizing field.
The shape quality of the mesh facets is controlled through their radius-edge
ratio, where the radius-edge ratio of a facet is the ratio between the circumradius
of the facet and the length of its shortest edge. We define a bad facet as a facet
f of Del|S(E) that:
– either has a too big surface Delaunay ball Bf = B(cf , rf ), meaning that
rf ψ(cf ),
– or is badly shaped, meaning that its radius-edge ratio ρ is such that ρ β
for a constant β ≥ 1.
Bad facets will be removed from the mesh by inserting the centers of their
surface Delaunay balls, The algorithm is initialized with a (usually small) set of
points E0 ⊂ S. Three points per connected component of S are sufficient. Then
the algorithm maintains, in addition to Del(E) and Del|S(E), a list of bad facets
and, as long as there remain bad facets, applies the following procedure](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-33-320.jpg)
![From Segmented Images to Good Quality Meshes 23
Fig. 5. Meshing an isosurface in a 3D image of the brain
at x, which is a global parameter and therefore difficult to estimate. However,
if the sample is too sparse with respect to the object thickness, the restricted
Delaunay triangulation is likely to be non manifold and/or to have boundaries.
The algorithm can check on the fly that Del|S(E) is a triangulated surface with
no boundary by checking that each edge in the restricted triangulation is incident
to two facets, and that the link of each vertex (i.e. the boundary of the union of
the facets incident to the vertex) is a simple polygon.
The issue of estimating lfs can also be circumvented by using a multiscale
approach that has been first proposed in the context of manifold reconstruc-
tion [15,16]. We slightly modify the algorithm so as to insert at each step the
candidate point that is furthest from the current sample. This will guarantee
that the sample remains roughly uniform through the process. If we let the algo-
rithm insert points, the topology of the triangulated surface maintained by the
algorithm may well change. Consider, for instance, the case of an isosurface in
a noisy image, say the brain in Fig. 5. Depending on the sampling density, the
topology of the surface may be a topological sphere (which the brain is indeed)
or a sphere with additional handles due to noise. Accordingly, the algorithm will
produce intermediate meshes of different topologies approximating surfaces of
various lfs. Since the changes of topology can be detected by computing at each
step the Betti numbers of the current triangulated surface, we can output the
various surfaces and the user can decide what is the best one.
The surface meshing algorithm is available in the open source library Cgal
[7]. Fig. 5 shows a result on a medical image. A thorough discussion of the
implementation of the algorithm and other experimental results can be found
in [14].](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-35-320.jpg)
![24 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
4 Meshing Volumes with Curved Boundaries
Let O be an object of R3
bounded by a surface S. The meshing algorithm of the
previous section constructs the 3D Delaunay triangulation of the sample E and
extracts from Del(E) the restricted Delaunay triangulation Del|S(E). Hence,
the algorithm constructs a 3D triangulation T of O as well as a polyhedral
surface approximating S. However, since the algorithm does not insert points
inside O, the aspect ratio of the tetrahedra of T cannot be controlled. If further
computations are to be performed, it is then mandatory to improve the shape
of the tetrahedra by sampling also the interior of O.
We present in this section a modification of the Delaunay-based surface mesher
of the previous section due to Oudot et al. [17]. This algorithm samples the inte-
rior and the boundary of the object at the same time so as to obtain a Delaunay
refinement volume mesher. Delaunay refinement removes all badly shaped tetra-
hedra except the so-called slivers. A special postprocessing is required to remove
those slivers.
4.1 3D Mesh Refinement Algorithm
The algorithm is still a greedy algorithm that builds a sample E while main-
taining the Delaunay triangulations Del(E) and its restrictions Del|O(E) and
Del|S(E) to the object O and its bounding surface S.
The sampling density is controlled by a function ψ(x) defined over O called
the sizing field. Using constant α, β and γ, we define two types of bad elements.
As above, a facet f of Del|S(E) is considered as bad if
– either it has a too big surface Delaunay ball Bf = B(cf , rf ), i.e. rf αψ(cf ),
– or it is badly shaped, meaning that its radius-edge ratio ρf is such that
ρf β.
A tetrahedron t of Del|O is considered as bad if
– either its circumradius rt is too big, i.e. rt ψ(ct)
– or it is badly shaped, meaning that its radius-edge ratio ρt is such that
ρt γ. The radius-edge ratio ρt of a tetrahedron t is the ratio between the
circumradius and the length of the shortest edge.
The algorithm uses two basic procedures, refine facet(f), which has been
defined in Section 3 and the following procedure refine tet(t).
refine tet(t)
1. insert in E the center ct of the ball circumscribing t
2. update Del(E), Del|S(E), Del|O(E) and the lists of bad elements.
The algorithm is initialized as the surface meshing algorithm. Then it applies
the following refinement rules in order, Rule 2 being applied only when Rule 1
can no longer be applied.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-36-320.jpg)
![From Segmented Images to Good Quality Meshes 25
Rule 1. If Del|S(E) contains a facet f which has a vertex in O S or is bad,
refine facet(f)
Rule 2. If there is a bad tetrahedron t ∈ Del|O(E)
1. compute the center ct of the circumscribing ball
2. if ct is included in the surface Delaunay ball of some facet f ∈ Del|S(E),
refine facet(f)
3. else refine tet(t).
It is proved in [17] that, for appropriate choices of parameters α, β and γ, the
algorithm terminates. Upon termination, Del|S(E) = Del|S(E ∩S) and DelO(E)
is a 3D-triangulation isotopic to O.
4.2 Sliver Removal
While Delaunay refinement techniques can be proven to generate tetrahedra
with a good radius-edge ratio, they may create flat tetrahedra of a special type
called slivers. A sliver is a tetrahedron whose four vertices lie close to a plane
and whose projection to that plane is a quadrilateral with no short edge. Slivers
have a good radius-edge ratio but a poor radius-radius ratio (ratio between the
circumradius and the radius of the largest contained sphere). Unfortunately, the
latter measure typically influences the numerical conditioning of finite element
methods. Slivers occur for example if one computes the Delaunay triangulation
of points on a regular grid (slightly pertubed to avoid degeneracies). Each square
in the grid can be circumscribed by an empty ball and is therefore a sliver of the
triangulation.
Fig. 6. A sliver
Two techniques are known to remove slivers from volume meshes. One consists
of a post-processing step called sliver exudation [18]. This step does not include
any new vertex in the mesh, nor does it move any of them. Each vertex is assigned
a weight and the Delaunay triangulation is turned into a weighted Delaunay
triangulation. The weights are carefully chosen so that no vertex disappear from
the mesh, nor any change occurs in the boundary facets (i. e. the facets of
Del|S(E)). Within these constraints, the weight of each vertex is optimized in
turn to maximize the minimum dihedral angles of the tetrahedra incident to that
vertex. Although the guaranteed theoretical bound on dihedral angles is known](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-37-320.jpg)
![26 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec
Fig. 7. The new point to be inserted is taken from the grey disk centered at the
circumcenter of the bad element τ but not in the black annulus to prevent the creation
of slivers
to be miserably low, this algorithm is quite efficient in practice at removing
slivers.
Another technique, due to Li [19], avoids the appearance of small slivers in
the mesh by relaxing the choice of the refinement points of a bad element (tetra-
hedron or boundary facet). The new points are no longer inserted at the cir-
cumcenters of Delaunay balls or surface Delaunay balls but in small picking
regions around those circumcenters. Within such a picking region, we further
avoid inserting points that would create slivers. (see Fig. 7).
4.3 Implementation
We present two results in Fig. 8 on both uniform and non uniform sizing fields.
The uniform model is an approximation of an isosurface in a 3D medical image.
The initial mesh of the surface had 33,012 vertices while the second step of the
algorithm added 53,762 new vertices in the interior of the object and 2,471 new
vertices on its boundary. The total CPU time was 20s on a Pentium IV (1.7
GHz). A thorough discussion of the implementation of the algorithm and other
experimental results can be found in [17,20]. The algorithm will be soon available
in the open source library Cgal [7].
4.4 Meshing of Multi-label Datasets
The above method seamlessly extends to the case of non-binary partitions, so
that it can be applied to the generation of high quality geometric models with
multiple junctions from multi-label datasets, frequently encountered in medical
applications.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-38-320.jpg)
![with a full orchestra. I mean soon to publish six preludes and fugues, two of
which you have already seen; this is the sort of life I like to lead, but not that
of an intendant. How vexatious it is, that at the close of such well-spent days
we cannot all assemble together to enjoy each other’s society![18]
I enclose my translation of “Alexander’s Feast;” you must read it aloud to
the family in the evening, and in various passages where the rhymes are
rugged or deficient, if you will let me have your amendments I shall be
grateful. One stipulation, however, I must make, that Ramler, or rather, I
should say, the English text, should not be sacrificed. Apropos, since then I
have once more mounted Pegasus, and translated Lord Byron’s poem, the
first strophe of which, by Theremin, is incomprehensible, and the second
false. I find, however, that my lines halt a little; perhaps, some evening, you
may discover something better.
Schlafloser Augensonne, heller Stern!
Der du mit thränenvollem Schein, unendlich fern,
Das Dunkel nicht erhellst, nur besser zeigst,
O wie du ganz des Glücks Erinn’rung gleichst!
So funkelt längst vergangner Freuden Licht,
Es scheint, doch wärmt sein matter Schimmer nicht,
Der wache Gram erspäht die Nachtgestalt,
Hell, aber fern, klar—aber ach! wie kalt!
The poem is very sentimental, and I think I should have set it to music
repeatedly in G sharp minor or B major, (but, at all events, with no end of
sharps,) had it not occurred to me that the music of Löwe pleases you and
Fanny; so this prevents my doing so, and there is an end of it, and of my
letter also. Adieu, love me as ever.—Your
Felix.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-40-320.jpg)
![To Pastor Bauer, Beszig.
Düsseldorf, January 12th, 1835.
[About a proposal as to some words for sacred music.]
... What I do not understand is the purport—musical, dramatic, or
oratorical, or whatever you choose to call it—that you have in view. What
you mention on the subject—the time before John, and then John himself,
till the appearance of Christ—is to my mind equally conveyed in the word
‘Advent,’ or the birth of Christ. You are aware, however, that the music must
represent one particular moment, or a succession of moments; and how you
intend this to be done you do not say. Actual church music,—that is, music
during the Evangelical Church service, which could be introduced properly
while the service was being celebrated,—seems to me impossible; and this,
not merely because I cannot at all see into which part of the public worship
this music can be introduced, but because I cannot discover that any such
part exists. Perhaps you have something to say which may enlighten me on
the subject.... But even without any reference to the Prussian Liturgy, which
at once cuts off everything of the kind, and will neither remain as it is nor go
further, I do not see how it is to be managed that music in our Church should
form an integral part of public worship, and not become a mere concert,
conducive more or less to piety. This was the case with Bach’s “Passion;” it
was sung in church as an independent piece of music, for edification. As for
actual church music, or, if you like to call it so, music for public worship, I
know none but the old Italian compositions for the Papal Chapel, where,
however, the music is a mere accompaniment, subordinate to the sacred
functions, co-operating with the wax candles and the incense, etc. If it be this
style of church music that you really mean, then, as I said, I cannot discover
the connecting link which would render it possible to employ it. For an
oratorio, one principal subject must be adopted, or the progressive history of
particular persons, otherwise the object would not be sufficiently defined;
for if all is to be only contemplative with reference to the coming of Christ,
then this theme has already been more grandly and beautifully treated in
Handel’s “Messiah,” where he begins with Isaiah, and, taking the Birth as a
central point, closes with the Resurrection.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-44-320.jpg)
![To Felix Mendelssohn Bartholdy, from his Father.
[19]
Berlin, March 10th, 1835.
This is the third letter I have written to you this week, and if this goes on,
reading my letters will become a standing article in the distribution of the
budget of your time; but you must blame yourself for this, as you spoil me
by your praise. I at once pass to the musical portion of your last letter.
Your aphorism, that every room in which Sebastian Bach is sung is
transformed into a church, I consider peculiarly appropriate; and when I
once heard the last movement of the piece in question, it made a similar
impression on myself; but I own I cannot overcome my dislike to figured
chorales in general, because I cannot understand the fundamental idea on
which they are based, especially where the contending parts are maintained
in an equal balance of power. For example, in the first chorus of the
“Passion,”—where the chorale forms only a more important and consistent
part of the basis; or where, as in the above-mentioned movement of the
cantata (if I remember it rightly, having only heard it once), the chorale
represents the principal building, and the individual parts only the
decorations,—I can better comprehend the purpose and the conception; but
not so certainly where the figure, in a certain manner, carries out variations
on the theme. No liberties ought ever assuredly to be taken with a chorale.
Its highest purpose is, that the congregation should sing it in all its purity to
the accompaniment of the organ; all else seems to me idle and inappropriate
for a church.
At Fanny’s last morning’s music the motett of Bach, “Gottes Zeit ist die
allerbeste Zeit,” and your “Ave Maria,” were sung by select voices. A long
passage in the middle of the latter, as well as the end also, appeared to me
too learned and intricate to accord with the simple piety, and certainly
genuine catholic spirit, which pervades the rest of the music. Rebecca
remarked that there was some confusion in the execution of those very
passages which I considered too intricate; but this only proves that I am an
ignoramus, but not that the conclusion is not too abstrusely modulated. With
regard to Bach, the composition in question seems to me worthy of the](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-49-320.jpg)
![To his Father.
Düsseldorf, April 3rd, 1835.
Dear Father,
I am delighted to hear that you are satisfied with the programme of the
Cologne Musical Festival. I shall not be able to play the organ for
“Solomon,” as it must stand in the background of the orchestra and
accompany almost every piece, the choruses and other performers here being
accustomed to constant beating of time. I must therefore transcribe the whole
of the organ part in the manner in which I think it ought to be played, and the
cathedral organist there, Weber, will play it; I am told he is a sound musician
and first-rate player. This is all so far well, and only gives me the great
labour of transcribing, as I wish to have the performance as perfect as
possible. I have had a good deal of trouble too with the “Morgengesang,”[20]
as there is much in it that requires alteration, owing to the impossibility of
executing it as written, with the means we have here. In doing so, however,
it again caused me extreme pleasure, especially the stars, the moon, the
elements, and the whole of the admirable finale. At the words “und schlich
in dieser Nacht,” etc., it becomes so romantic and poetical, that each time I
hear it I feel more touched and charmed; it therefore gratifies me to be of
any use to so noble a man. The Comité were very much surprised when I
maintained that it was a fine composition, and scarcely would consent to
have it, but at that moment they were in a mood to be persuaded to anything.
I would also have insisted on their giving an overture of Bach’s, if I had not
dreaded too strong a counter-revolution. There is to be nothing of mine;
therefore (from gratitude, I presume) they persist that my “admirable
likeness” shall appear and be published by Whitsunday, a project from which
I gallantly defend myself, refusing either to sit or stand for the purpose,
having a particular objection to such pretensions.
You must be well aware that your presence at the festival would not only
be no gêne to me, but on the contrary, would cause me first to feel true joy
and delight in my success. Allow me to take this opportunity to say to you,
that the approbation and enjoyment of the public, to which I am certainly
very sensible, only causes me real satisfaction when I can write to tell you of
it, because I know it rejoices you, and one word of praise from you is more](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-55-320.jpg)
![truly precious to me, and makes me happier, than all the publics in the world
applauding me in concert; and thus to see you among the audience, would be
the dearest of all rewards to me for my labours.
My oratorio[21] is to be performed in Frankfort in November, so Schelble
writes to me; and much as I should like you to hear it soon, still I should
prefer your hearing it first next year, at the Musical Festival. Before
decidedly accepting the proposal, I have stipulated to wait till after the
performance at Frankfort, that I may judge whether it be suitable for the
festival; but should this prove to be the case, as I hope and wish it may, it
will have a much finer effect there, and besides it is the festival that you like,
and Whitsunday instead of November; and above all, I shall then know
whether it pleases you or not, on which point I feel by no means sure.
I cannot close this letter without speaking of the heavenly weather that
delights us here. Light balmy air and sunshine, and a profusion of green, and
larks! To-day I rode through the forest, and stopped for at least a quarter of
an hour to listen to the birds, who in the deep solitude were fluttering about
incessantly and warbling.—Your
Felix.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-56-320.jpg)
![To the Herr Regierungs-Secretair Hixte, Cologne.
Düsseldorf, May 18th, 1835.
Sir,
I thank you much for the kind letter you have gratified me by addressing
to me. The idea which you communicate in it is very flattering for me, and
yet I confess that I feel a certain degree of dislike to do what you propose,
and for a long time past I have entertained this feeling. It is now so very
much the fashion for obscure or commonplace people to have their likeness
given to the public, in order to become more known, and for young
beginners to do so at first starting in life, that I have always had a dread of
doing so too soon. I do not wish that my likeness should be taken, until I
have accomplished something to render me more worthy, according to my
idea, of such an honour. This, however, not being yet the case, I beg to defer
such a compliment till I am more deserving of it; but receive my best thanks
for the friendly good-nature with which you made me this offer.[22]—I am,
etc.,
Felix Mendelssohn Bartholdy.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-59-320.jpg)
![capitally; the violins attacking it with a degree of vehemence that quite
startled me and delighted the publicus. The following pieces, an air in E
major of Weber, a violin concerto by Spohr, and the introduction to “Ali
Baba” did not go so well; the one rehearsal was not sufficient, and they were
often unsteady; but, on the other hand, Beethoven’s B flat symphony, which
formed the second part, was splendidly given, so that the Leipzigers shouted
with delight at the close of each movement. I never in any orchestra saw
such zeal and excitement; they listened like—popinjays, Zelter would say.
After the concert I received, and offered in turn, a mass of
congratulations: first the orchestra, then the Thomas School collegians (who
are capital fellows, and go to college, and are dismissed so punctually that I
have promised them an order); then came Moscheles, with a Court suite of
dilettanti, then two editors of musical papers, and so on. Moscheles’ concert
is on Friday, and I am to play his piece for two pianos[23] with him, and he is
to play my new pianoforte-concerto. My “Hebrides” have also contrived to
creep into the concert. This afternoon Moscheles, Clara Wieck, and I, play
Sebastian Bach’s triple concerto in D minor. How amiable Moscheles is
towards myself, how cordially he is interested in my situation here, how it
delights me that he is so satisfied with it, how he plays my rondo in E flat to
my great admiration, and far better than I originally conceived it, and how
we dine together every forenoon in his hotel, and every evening drink tea
and have music in mine,—all this you can imagine for yourself, for you
know him,—especially you, dear Father. These are pleasant days; and if I
have not much leisure to work, I mean to make up for it hereafter, and shall
derive as much benefit from it then as now.
My first concert caused me no perturbation, dear Mother, but to my
shame I confess, that I never felt so embarrassed at the moment of appearing
as on that occasion; I believe it arose from our long correspondence and
treaty on the subject, and I had never before seen a concert of the kind. The
locality and the lights confused me. Now farewell all. May you be well and
happy, and pray write to me very often.—Your
Felix.](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-62-320.jpg)
![To Pastor Julius Schubring, Dessau.
Leipzig, December 6th, 1835.
Dear Schubring,
You have no doubt heard of the heavy stroke that has fallen on my happy
life and those dear to me.[24] It is the greatest misfortune that could have
befallen me, and a trial that I must either strive to bear up against, or must
utterly sink under. I say this to myself after the lapse of three weeks, without
the acute anguish of the first days, but I now feel it even more deeply; a new
life must now begin for me, or all must be at an end,—the old life is now
severed. For our consolation and example, our Mother bears her loss with
the most wonderful composure and firmness; she comforts herself with her
children and grandchildren, and thus strives to hide the chasm that never can
be filled up. My Brother and Sisters do what they can to fulfil their duties
better than ever, the more difficult they have become. I was ten days in
Berlin, that by my presence my Mother should at least be surrounded by her
whole family; but I need scarcely tell you what these days were; you know it
well, and no doubt you thought of me in that dark hour. God granted to my
Father the prayer that he had often uttered; his end was as peaceful and
quiet, and as sudden and unexpected as he desired. On Wednesday, the 18th,
he was surrounded by all his family, went to bed late the same evening,
complained a little early on Thursday, and at half-past eleven his life was
ended. The physicians can give his malady no name. It seems that my
grandfather Moses died in a similar manner,—so my uncle told us,—at the
same age, without sickness, and in a calm and cheerful frame of mind. I do
not know whether you are aware that more especially for some years past,
my Father was so good to me, so thoroughly my friend, that I was devoted to
him with my whole soul, and during my long absence I scarcely ever passed
an hour without thinking of him; but as you knew him in his own home with
us, in all his kindliness, you can well realize my state of mind. The only
thing that now remains is to do one’s duty, and this I strive to accomplish
with all my strength, for he would wish it to be so if he were still present,
and I shall never cease to endeavour to gain his approval as I formerly did,
though I can no longer enjoy it. When I delayed answering your letter, I little
thought that I should have to answer it thus; let me thank you for it now, and](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-63-320.jpg)
![etc., to the close in D minor. Our second violin player, an old musician, said
to me afterwards, when he met me in the passage, that he had heard it
played in the same Hall by Mozart himself, but since that day he had heard
no one introduce such good cadenzas as I did yesterday, which gave me
very great pleasure.
Do you know Handel’s “Coronation Anthem”? It is most singular. The
beginning is one of the finest which not only Handel, but any man, ever
composed; and all the remainder, after the first short movement, horridly
dry and commonplace. The performers could not master it, but are certainly
far too busy to grieve much about that.
Many persons here consider “Melusina” to be my best overture; at all
events, it is the most deeply felt; but as to the fabulous nonsense of the
musical papers, about red coral and green sea monsters, and magic palaces,
and deep seas, this is stupid stuff, and fills me with amazement. But now I
take my leave of water for some time to come, and must see how things are
going on elsewhere.[25] I received to-day a letter from Düsseldorf, with the](https://image.slidesharecdn.com/12627-250604222340-fb218aff/85/Lecture-Notes-in-Computer-Science-1st-Edition-by-Springer-ISBN-72-320.jpg)
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- 4. Lecture Notes in Computer Science 5416 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany
- 5. Frank Nielsen (Ed.) Emerging Trends in Visual Computing LIX Fall Colloquium, ETVC 2008 Palaiseau, France, November 18-20, 2008 Revised Invited Papers 1 3
- 6. Volume Editor Frank Nielsen Ecole Polytechnique, LIX Route de Saclay, 91128 Palaiseau Cedex, France E-mail: nielsen@lix.polytechnique.fr and Sony Computer Science Laboratories, Inc. 3-14-13 Higashi Gotanda 3F, 141-0022 Shinagawa-ku, Tokyo, Japan E-mail: Frank.Nielsen@acm.org Library of Congress Control Number: Applied for CR Subject Classification (1998): I.4, I.5, I.2.10, I.3.3, I.3.5, I.3.7, I.2.6, F.2, G.1.2 LNCS Sublibrary: SL 6 – Image Processing, Computer Vision, Pattern Recognition, and Graphics ISSN 0302-9743 ISBN-10 3-642-00825-9 Springer Berlin Heidelberg New York ISBN-13 978-3-642-00825-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12612574 06/3180 5 4 3 2 1 0
- 7. Preface ETVC 2008, the fall colloquium of the computer science department (LIX) of the École Polytechnique, held in Palaiseau, France, November 18-20, 2008, focused on the Emerging Trends in Visual Computing. The colloquium gave scientists the opportunity to sketch a state-of-the-art picture of the mathematical foundations of visual computing. We were delighted to invite and welcome the following distinguished speakers to ETVC 2008 (listed in alphabetical order): – Shun-ichi AMARI (Mathematical Neuroscience Laboratory, Brain Science Institute, RIKEN, Wako-Shi, Japan): Information Geometry and Its Applications – Tetsuo ASANO (School of Information Science, Japan Advanced Institute of Science and Technology, JAIST, Japan): Constant-Working-Space Algo- rithms for Image Processing – Francis BACH (INRIA/ENS, France): Machine Learning and Kernel Meth- ods for Computer Vision – Frédéric BARBARESCO (Thales Air Systems, France): Applications of In- formation Geometry to Radar Signal Processing – Michel BARLAUD (I3S CNRS, University of Nice-Sophia-Antipolis, Poly- tech’Nice & Institut Universitaire de France, France): Image Retrieval via Kullback Divergence of Patches of Wavelets Coefficients in the k-NN Framework – Jean-Daniel BOISSONNAT (GEOMETRICA, INRIA Sophia-Antipolis, France): Certified Mesh Generation – Pascal FUA (EPFL, CVLAB, Switzerland): Recovering Shape and Motion from Video Sequences – Markus GROSS (Department of Computer Science, Institute of Scientific Computing, Swiss Federal Institute of Technology Zurich, ETHZ, Switzer- land): 3D Video: A Fusion of Graphics and Vision – Xianfeng David GU (State University of New York at Stony Brook, USA): Discrete Curvature Flow for Surfaces and 3-Manifolds – Leonidas GUIBAS (Computer Science Department, Stanford University, USA): Detection of Symmetries and Repeated Patterns in 3D Point Cloud Data – Sylvain LAZARD (VEGAS, INRIA LORIA Nancy, France): 3D Visibility and Lines in Space
- 8. VI Preface – Stéphane MALLAT (École Polytechnique, Centre de Mathématiques Ap- pliquées (CMAP), France): Sparse Geometric Super-Resolution – Hiroshi MATSUZOE (Department of Computer Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, NITECH, Japan): Computational Geometry from the Viewpoint of Affine Differential Geometry – Dimitris METAXAS (Computational Biomedicine Imaging and Modeling Center, CBMI, Rutgers University, USA): Unifying Subspace and Distance Metric Learning with Bhattacharyya Coefficient for Image Classification – Frank NIELSEN (LIX, École Polytechnique, Paris, France & Sony Com- puter Science Laboratories Inc., Tokyo, Japan): Computational Geometry in Dually Flat Spaces: Theory, Applications and Perspectives – Richard NOCK (CEREGMIA, University of Antilles-Guyane, France): The Intrinsic Geometries of Learning – Nikos PARAGIOS (École Centrale de Paris, ECP, Paris, France): Procedural Modeling of Architectures: Towards Large Scale Visual Reconstruction – Xavier PENNEC (ASCLEPIOS, INRIA Sophia-Antipolis, France): Statis- tical Computing on Manifolds for Computational Anatomy – Ramesh RASKAR (MIT Media Lab, USA): Computational Photography: Epsilon to Coded Imaging – Cordelia SCHMID (LEAR, INRIA Grenoble, France): Large-Scale Object Recognition Systems – Gabriel TAUBIN (Division of Engineering, Brown University, USA): Shape from Depth Discontinuities – Baba VEMURI (CISE Dept., University of Florida, USA): Information- Theoretic Algorithms for Diffusion Tensor Imaging – Suresh VENKATASUBRAMANIAN (School of Computing, University of Utah, USA): Non-standard Geometries and Data Analysis – Martin VETTERLI (School of Computer and Communication Sciences, EPFL, Switzerland): Sparse Sampling: Variations on a Theme by Shannon – Jun ZHANG (Department of Psychology, University of Michigan, USA): Information Geometry: Duality, Convexity and Divergences Invited speakers were encouraged to submit a state-of-the-art chapter on their research area. The review process was carried out by members of the Program Committee and other reviewers. We would like to sincerely thank the contribut- ing authors and thank the reviewers for the careful feedback that helped the authors prepare their camera-ready papers. Videos of the lectures synchronized with slides are available from www.videolectures.net
- 9. Preface VII We were very pleased to welcome all the 150+ participants to ETVC 2008. For those who did not attend, we hope the chapters of this publication provide a good snapshot of the current research status in visual computing. December 2008 Frank Nielsen Group picture of the participants at ETVC 2008 (November 19, 2008)
- 10. Organization Frank Nielsen (Program Chair) Evelyne Rayssac (Secretary) Corinne Poulain (Secretary) Philippe Baptiste (Financial Advisor) Jean-Marc Steyaert (Scientific Advisor) Luca Castelli Aleardi (Photographer) Referees S. Boltz F. Chazal B. Lévy A. André F. Hetroy R. Keriven F. Nielsen R. Nock T. Nakamura S. Oudot S. Owada M. Pauly A. Vigneron Sponsoring Institutions We gratefully acknowledge the following institutions for their generous support: – CNRS – DIGITEO – École Polytechnique – Groupe de Recherche Informatique & Mathématique (GdR IM) – University of Antilles-Guyane, CEREGMIA Department
- 11. Table of Contents Geometric Computing Abstracts of the LIX Fall Colloquium 2008: Emerging Trends in Visual Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Frank Nielsen From Segmented Images to Good Quality Meshes Using Delaunay Refin ement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Jean-Daniel Boissonnat, Jean-Philippe Pons, and Mariette Yvinec Information Geometry and Applications Discrete Curvature Flows for Surfaces and 3-Manifolds . . . . . . . . . . . . . . . 38 Xiaotian Yin, Miao Jin, Feng Luo, and Xianfeng David Gu Information Geometry and Its Applications: Convex Function and Dually Flat Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Shun-ichi Amari Computational Geometry from the Viewpoint of Affine Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Hiroshi Matsuzoe Interactions between Symmetric Cone and Information Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Frederic Barbaresco Clustering Multivariate Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . 164 Frank Nielsen and Richard Nock Computer Graphics and Vision Intrinsic Geometries in Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Richard Nock and Frank Nielsen Shape from Depth Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Gabriel Taubin, Daniel Crispell, Douglas Lanman, Peter Sibley, and Yong Zhao Computational Photography: Epsilon to Coded Photography . . . . . . . . . . 238 Ramesh Raskar
- 12. XII Table of Contents Unifying Subspace and Distance Metric Learning with Bhattacharyya Coefficient for Image Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Qingshan Liu and Dimitris N. Metaxas Information Retrieval Constant-Working-Space Algorithms for Image Processing. . . . . . . . . . . . . 268 Tetsuo Asano Sparse Multiscale Patches for Image Processing . . . . . . . . . . . . . . . . . . . . . . 284 Paolo Piro, Sandrine Anthoine, Eric Debreuve, and Michel Barlaud Medical Imaging and Computational Anatomy Recent Advances in Large Scale Image Search . . . . . . . . . . . . . . . . . . . . . . . 305 Herve Jegou, Matthijs Douze, and Cordelia Schmid Information Theoretic Methods for Diffusion-Weighted MRI Analysis . . . 327 Angelos Barmpoutis and Baba C. Vemuri Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Xavier Pennec Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
- 13. Abstracts of the LIX Fall Colloquium 2008: Emerging Trends in Visual Computing Frank Nielsen Ecole Polytechnique, Palaiseau, France Sony CSL, Tokyo, Japan Abstract. We list the abstracts of the distinguished speakers that par- ticipated to the 2008 LIX fall colloquium. Leonidas GUIBAS Computer Science Department, Stanford University, USA Detection of Symmetries and Repeated Patterns in 3D Point Cloud Data Digital models of physical shapes are becoming ubiquitous in our economy and life. Such models are sometimes designed ab initio using CAD tools, but more and more often they are based on existing real objects whose shape is acquired using various 3D scanning technologies. In most instances, the original scanner data is just a set, but a very large set, of points sampled from the surface of the object. We are interested in tools for understanding the local and global structure of such large-scale scanned geometry for a variety of tasks, including model completion, reverse engineering, shape comparison and retrieval, shape editing, inclusion in virtual worlds and simulations, etc. This talk will present a number of point-based techniques for discovering global structure in 3D data sets, including partial and approximate symmetries, shared parts, repeated patterns, etc. It is also of interest to perform such structure discovery across multiple data sets distributed in a network, without actually ever bring them all to the same host. Xianfeng David GU State University of New York at Stony Brook, USA Discrete Curvature Flow for Surfaces and 3-Manifolds This talk introduce the concepts, theories and algorithms for discrete curvature flows for surfaces with arbitrary topologies. Discrete curvature flow for hyperbolic 3-manifolds with geodesic boundaries are also explained. Curvature flow method can be used to design Riemannian metrics by prescribed curvatures, and applied for parameterization in graphics, shape registration and comparison in vision and brain mapping in medical imaging, spline construction in computer aided geometric design, and many other engineering fields. F. Nielsen (Ed.): ETVC 2008, LNCS 5416, pp. 1–12, 2009. c Springer-Verlag Berlin Heidelberg 2009
- 14. 2 F. Nielsen Jean-Daniel BOISSONNAT GEOMETRICA, INRIA Sophia-Antipolis, France Certified Mesh Generation Given a domain D, the problem of mesh generation is to construct a simplicial complex that approximates D in both a topological and a geometrical sense and whose elements satisfy various constraints such as size, aspect ratio or anisotropy. The talk will cover some recent results on triangulating surfaces and volumes by Delaunay refinement, anisotropic mesh generation and surface reconstruction. Applications in medical images, computer vision and geology will be discussed. Baba VEMURI CISE Dept., University of Florida, USA Information-Theoretic Algorithms for Diffusion Tensor Imaging Concepts from Information Theory have been used quite widely in Image Processing, Computer Vision and Medical Image Analysis for several decades now. Most widely used concepts are that of KL-divergence, minimum descrip- tion length (MDL), etc. These concepts have been popularly employed for image registration, segmentation, classification etc. In this chapter we review several methods, mostly developed by our group at the Center for Vision, Graphics Medical Imaging in the University of Florida, that glean concepts from Informa- tion Theory and apply them to achieve analysis of Diffusion-Weighted Magnetic Resonance (DW-MRI) data. This relatively new MRI modality allows one to non-invasively infer axonal connectivity patterns in the central nervous system. The focus of this chapter is to review automated image analysis techniques that allow us to automatically segment the region of interest in the DWMRI im- age wherein one might want to track the axonal pathways and also methods to reconstruct complex local tissue geometries containing axonal fiber crossings. Implementation results illustrating the algorithm application to real DW-MRI data sets are depicted to demonstrate the effectiveness of the methods reviewed. Xavier PENNEC ASCLEPIOS, INRIA Sophia-Antipolis, France Statistical Computing on Manifolds for Computational Anatomy Computational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. The goal is not only to model the normal variations among a popula- tion, but also discover morphological differences between normal and pathological populations, and possibly to detect, model and classify the pathologies from struc- tural abnormalities. Applications are very important both in neuroscience, to min- imize the influence of the anatomical variability in functional group analysis, and in medical imaging, to better drive the adaptation of generic models of the anatomy (atlas) into patient-specific data (personalization). However, understanding and
- 15. Abstracts of the LIX Fall Colloquium 2008 3 modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics and computational meth- ods on objects that do not belong to standard Euclidean spaces. We investigate in this chapter the Riemannian metric as a basis for developing generic algorithms to compute on manifolds. We show that few computational tools derived from this structure can be used in practice as the atoms to build more complex generic algo- rithms such as mean computation, Mahalanobis distance, interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational frame- work is illustrated with the joint estimation and anisotropic smoothing of diffusion tensor images and with the modeling of the brain variability from sulcal lines. Cordelia SCHMID LEAR, INRIA Grenoble, France Large-Scale Object Recognition Systems This paper introduces recent methods for large scale image search. State-of-the- art methods build on the bag-of-features image representation. We first analyze bag-of-features in the framework of approximate nearest neighbor search. This shows the sub-optimality of such a representation for matching descriptors and leads us to derive a more precise representation based on 1) Hamming embedding (HE) and 2) weak geometric consistency constraints (WGC). HE provides binary signatures that refine the matching based on visual words. WGC filters matching descriptors that are not consistent in terms of angle and scale. HE and WGC are integrated within the inverted file and are efficiently exploited for all images, even in the case of very large datasets. Experiments performed on a dataset of one million of images show a significant improvement due to the binary signature and the weak geometric consistency constraints, as well as their efficiency. Estimation of the full geometric transformation, i.e., a re-ranking step on a short list of images, is complementary to our weak geometric consistency constraints and allows to further improve the accuracy. Pascal FUA EPFL, CVLAB, Swiss Recovering Shape and Motion from Video Sequences In recent years, because cameras have become inexpensive and ever more preva- lent, there has been increasing interest in video-based modeling of shape and motion. This has many potential applications in areas such as electronic pub- lishing, entertainment, sports medicine and athletic training. It, however, is an inherently difficult task because the image-data is often incomplete, noisy, and ambiguous. In our work, we focus on the recovery of deformable and articulated 3D motion from single video sequences. In this talk, I will present the models we
- 16. 4 F. Nielsen have developed for this purpose and demonstrate the applicability of our tech- nology for Augmented Reality and human body tracking purposes. Finally, I will present some open research issues and discuss our plans for future developments. Ramesh RASKAR MIT Media Lab, USA Computational Photography: Epsilon to Coded Imaging Computational photography combines plentiful computing, digital sensors, mod- ern optics, actuators, and smart lights to escape the limitations of traditional cameras, enables novel imaging applications and simplifies many computer vision tasks. However, a majority of current Computational Photography methods in- volve taking multiple sequential photos by changing scene parameters and fusing the photos to create a richer representation. The goal of Coded Computational Photography is to modify the optics, illumination or sensors at the time of cap- ture so that the scene properties are encoded in a single (or a few) photographs. We describe several applications of coding exposure, aperture, illumination and sensing and describe emerging techniques to recover scene parameters from coded photographs. Dimitris METAXAS Computational Biomedicine Imaging and Modeling Center, CBMI, Rutgers Uni- versity, USA Unifying Subspace and Distance Metric Learning with Bhattacharyya Coefficient for Image Classification In this talk, we propose a unified scheme of subspace and distance metric learning under the Bayesian framework for image classification. According to the local distribution of data, we divide the k-nearest neighbors of each sample into the intra-class set and the inter-class set, and we aim to learn a distance metric in the embedding subspace, which can make the distances between the sample and its intra-class set smaller than the distances between it and its inter-class set. To reach this goal, we consider the intra-class distances and the inter-class distances to be from two different probability distributions respectively, and we model the goal with minimizing the overlap between two distributions. Inspired by the Bayesian classification error estimation, we formulate the objective function by minimizing the Bhattachyrra coefficient between two distributions. We further extend it with the kernel trick to learn nonlinear distance metric. The power and generality of the proposed approach are demonstrated by a series of experiments on the CMU-PIE face database, the extended YALE face database, and the COREL-5000 nature image database. Nikos PARAGIOS Ecole Centrale de Paris, ECP, Paris, France
- 17. Abstracts of the LIX Fall Colloquium 2008 5 Procedural Modeling of Architectures: Towards Large Scale Visual Reconstruction Three-dimensional content is a novel modality used in numerous domains like navigation, post production cinematography, architectural modeling and ur- ban planning. These domains have benefited from the enormous progress has been made on 3D reconstruction from images. Such a problem consists of build- ing geometric models of the observed environment. State of the art methods can deliver excellent results in a small scale but suffer from being local and cannot be considered in a large scale reconstruction process since the assumption of recov- ering images from multiple views for an important number of buildings is rather unrealistic. On the other hand several efforts have been made in the graphics community towards content creation with city engines. Such models are purely graphics-based and given a set of rules (grammars) as well as dictionary of ar- chitectures (buildings) can produce virtual cities. Such engines could become far more realistic through the use of actual city models as well as knowledge of build- ing architectures. Developing 3D models/rules/grammars that are image-based and coupling these models with actual observations is the greatest challenge of urban modeling. Solving the large-scale geometric modeling problem from min- imal content could create novel means of world representation as well as novel markets and applications. In this talk, we will present some preliminary results on large scale modeling and reconstruction through architectural grammars. Gabriel TAUBIN Division of Engineering, Brown University, USA Shape from Depth Discontinuities We propose a new primal-dual framework for representation, capture, processing, and display of piecewise smooth surfaces, where the dual space is the space of oriented 3D lines, or rays, as opposed to the traditional dual space of planes. An image capture process detects points on a depth discontinuity sweep from a camera moving with respect to an object, or from a static camera and a moving object. A depth discontinuity sweep is a surface in dual space composed of the time-dependent family of depth discontinuity curves span as the camera pose describes a curved path in 3D space. Only part of this surface, which includes silhouettes, is visible and measurable from the camera. Locally convex points deep inside concavities can be estimated from the visible non-silhouette depth discontinuity points. Locally concave point laying at the bottom of concavities, which do not correspond to visible depth discontinuities, cannot be estimated, resulting in holes in the reconstructed surface. A first variational approach to fill the holes, based on fitting an implicit function to a reconstructed oriented point cloud, produces watertight models.We describe a first complete end-to-end system for acquiring models of shape and appearance.We use a single multi-flash camera and turntable for the data acquisition and represent the scanned objects as point clouds, with each point being described by a 3-D location, a surface normal, and a Phong appearance model.
- 18. 6 F. Nielsen Shun-ichi AMARI Mathematical Neuroscience Laboratory, Brain Science Institute, RIKEN, Wako- Shi, Japan Information Geometry and Its Applications Information geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its duality as a spe- cial but important part. Here, we begin with a convex function, and construct a dually flat manifold. The manifold possesses a Riemannian metric, two types of geodesics, and a divergence function. The generalized Pythagorean theorem and dual projections theorem are derived therefrom.We construct alpha-geometry, extending this convex analysis. In this review, geometry of a manifold of proba- bility distributions is then given, and a plenty of applications are touched upon. Appendix presents an easily understable introduction to differential geometry and its duality. Jun ZHANG Department of Psychology, University of Michigan, USA Information Geometry: Duality, Convexity and Divergences In this talk, I explore the mathematical relationships between duality in in- formation geometry, convex analysis, and divergence functions. First, from the fundamental inequality of a convex function, a family of divergence measures can be constructed, which specializes to the familiar Bregman divergence, Jen- son difference, beta-divergence, and alpha-divergence, etc. Second, the mixture parameter turns out to correspond to the alpha ¡-¿ -alpha duality in informa- tion geometry (which I call “referential duality”, since it is related to the choice of a reference point for computing divergence). Third, convex conjugate oper- ation induces another kind of duality in information geometry, namely, that of biorthogonal coordinates and their transformation (which I call “representa- tional duality”, since it is related to the expression of geometric quantities, such as metric, affine connection, curvature, etc of the underlying manifold). Under this analysis, what is traditionally called “+1/-1 duality” and “e/m duality” in information geometry reflect two very different meanings of duality that are nevertheless intimately interwined for dually flat spaces. Hiroshi MATSUZOE Department of Computer Science and Engineering Graduate School of Engineer- ing, Nagoya Institute of Technology, NITECH, Japan Computational Geometry from the Viewpoint of Affine Differential Geometry Incidence relations (configurations of vertexes, edges, etc.) are important in computational geometry. Incidence relations are invariant under the group of affine transformations. On the other hand, affine differential geometry is to study hypersurfaces in an affine space that are invariant under the group of
- 19. Abstracts of the LIX Fall Colloquium 2008 7 affine transformation. Therefore affine differential geometry gives a new sight in computational geometry. From the viewpoint of affine differential geometry, algorithms of geometric transformation and dual transformation are discussed. The Euclidean distance function is generalized by a divergence function in affine differential geometry. A divergence function is an asymmetric distance-like func- tion on a manifold, and it is an important object in information geometry. For divergence functions, the upper envelope type theorems on statistical mani- folds are given. Voronoi diagrams determined from divergence functions are also discussed. Richard NOCK CEREGMIA, University of Antilles-Guyane, France The Intrinsic Geometries of Learning In a seminal paper, Amari (1998) proved that learning can be made more effi- cient when one uses the intrinsic Riemanian structure of the algorithms’ spaces of parameters to point the gradient towards better solutions. In this paper, we show that many learning algorithms, including various boosting algorithms for linear separators, the most popular top-down decision-tree induction algorithms, and some on-line learning algorithms, are spawns of a generalization of Amari’s natural gradient to some particular non-Riemanian spaces. These algorithms exploit an intrinsic dual geometric structure of the space of parameters in rela- tionship with particular integral losses that are to be minimized. We unite some of them, such as AdaBoost, additive regression with the square loss, the logistic loss, the top-down induction performed in CART and C4.5, as a single algorithm on which we show general convergence to the optimum and explicit convergence rates under very weak assumptions. As a consequence, many of the classifica- tion calibrated surrogates of Bartlett et al. (2006) admit efficient minimization algorithms. Frédéric BARBARESCO Thales Air Systems, France Applications of Information Geometry to Radar Signal Processing Main issue of High Resolution Doppler Imagery is related to robust statistical estimation of Toeplitz Hermitian positive definite covariance matrices of sensor data time series (e.g. in Doppler Echography, in Underwater acoustic, in Elec- tromagnetic Radar, in Pulsed Lidar). We consider this problem jointly in the framework of Riemannian symmetric spaces and the framework of Information Geometry. Both approaches lead to the same metric, that has been initially con- sidered in other mathematical domains (study of Bruhat-Tits complete metric Space Upper-half Siegel Space in Symplectic Geometry). Based on Frechet- Karcher barycenter definition geodesics in Bruhat-Tits space, we address prob- lem of N Covariance matrices Mean estimation. Our main contribution lies in the development of this theory for Complex Autoregressive models (maximum
- 20. 8 F. Nielsen entropy solution of Doppler Spectral Analysis). Specific Blocks structure of the Toeplitz Hermitian covariance matrix is used to define an iterative parallel algorithm for Siegel metric computation. Based on Affine Information Geom- etry theory, we introduce for Complex Autoregressive Model, Khler metric on reflection coefficients based on Khler potential function given by Doppler signal Entropy. The metric is closely related to Khler-Einstein manifold and complex Monge-Ampere Equation. Finally, we study geodesics in space of Khler poten- tials and action of Calabi Khler-Ricci Geometric Flows for this Complex Au- toregressive Metric. We conclude with different results obtained on real Doppler Radar Data in HF X bands : X-band radar monitoring of wake vortex turbu- lences, detection for Coastal X-band HF Surface Wave Radars. Frank NIELSEN LIX, Ecole Polytechnique, Paris, France Sony Computer Science Laboratories Inc., Tokyo, Japan Computational Geometry in Dually Flat Spaces: Theory, Applications and Per- spectives Computational information geometry emerged from the fruitful interactions of geometric computing with information geometry. In this talk, we survey the re- cent results obtained in that direction by first describing generalizations of core algorithms of computational geometry and machine learning to broad and ver- satile classes of distortion measures. Namely, we introduce the generic classes of Bregman, Csiszar and Burbea-Rao parametric divergences and explain their relationships and properties with respect to algorithmic design. We then present few applications of these meta-algorithms to the field of statistics and data anal- ysis and conclude with perspectives. Tetsuo ASANO School of Information Science, Japan Advanced Institute of Science and Tech- nology, JAIST, Japan Constant-Working-Space Algorithms for Image Processing This talk surveys recent progress in constant-working-space algorithms for prob- lems related to image processing. An extreme case is when an input image is given as read-only memory in which reading an array element is allowed but writing any value at any array element is prohibited, and also the number of working storage cells available for algorithms is at most some constant. This chapter shows how a number of important fundamental problems can be solved in such a highly constrained situation. Stéphane MALLAT Ecole Polytechnique, Centre de Mathmatiques Appliques, CMAP, France
- 21. Abstracts of the LIX Fall Colloquium 2008 9 Sparse Geometric Super-Resolution What is the maximum signal resolution that can be recovered from partial noisy or degraded data? This inverse problem is a central issue, from medical to satel- lite imaging, from geophysical seismic to HDTV visualization of Internet videos. Increasing an image resolution is possible by taking advantage of “geometric regularities”, whatever it means. Super-resolution can indeed be achieved for signals having a sparse representation which is “incoherent” relatively to the measurement system. For images and videos, it requires to construct sparse rep- resentations in redundant dictionaries of waveforms, which are adapted to geo- metric image structures. Signal recovery in redundant dictionaries is discussed, and applications are shown in dictionaries of bandlets for image super-resolution. Martin VETTERLI School of Computer and Communication Sciences, EPFL, Switzerland Sparse Sampling: Variations on a Theme by Shannon Sampling is not only a beautiful topic in harmonic analysis, with an interesting history, but also a subject with high practical impact, at the heart of signal pro- cessing and communications and their applications. The question is very simple: when is there a one-to-one relationship between a continuous-time function and adequately acquired samples of this function? A cornerstone result is of course Shannon’s sampling theorem, which gives a sufficient condition for reconstruct- ing the projection of a signal onto the subspace of bandlimited functions, and this by taking inner products with a sinc function and its shifts. Many variations of this basic framework exist, and they are all related to a subspace structure of the classes of objects that can be sampled. Recently, this framework has been extended to classes of non-bandlimited sparse signals, which do not have a sub- space structure. Perfect reconstruction is possible based on a suitable projection measurement. This gives a sharp result on the sampling and reconstruction of sparse continuous-time signals, namely that 2K measurements are necessary and sufficient to perfectly reconstruct a K-sparse continuous-time signal. In accor- dance with the principle of parsimony, we call this sampling at Occam’s rate. We first review this result and show that it relies on structured Vandermonde mea- surement matrices, of which the Fourier matrix is a particular case. It also uses a separation into location and value estimation, the first being non-linear, while the second is linear. Because of this structure, fast, O(K3 ) methods exist, and are related to classic algorithms used in spectral estimation and error correction coding. We then generalize these results to a number of cases where sparsity is present, including piecewise polynomial signals, as well as to broad classes of sampling or measurement kernels, including Gaussians and splines. Of course, real cases always involve noise, and thus, retrieval of sparse signals in noise is considered. That is, is there a stable recovery mechanism, and robust practical algorithms to achieve it. Lower bounds by Cramer-Rao are given, which can also be used to derive uncertainty relations with respect to position and value of sparse signal estimation. Then, a concrete estimation method is given using
- 22. 10 F. Nielsen an iterative algorithm due to Cadzow, and is shown to perform close to opti- mal over a wide range of signal to noise ratios. This indicates the robustness of such methods, as well as their practicality. Next, we consider the connection to compressed sensing and compressive sampling, a recent approach involving random measurement matrices, a discrete set up, and retrieval based on convex optimization. These methods have the advantage of unstructured measurement matrices (actually, typically random ones) and therefore a certain universality, at the cost of some redundancy. We compare the two approaches, highlighting differences, similarities, and respective advantages. Finally, we move to appli- cations of these results, which cover wideband communications, noise removal, and superresolution imaging, to name a few. We conclude by indicating that sampling is alive and well, with new perspectives and many interesting recent results and developments. Joint work with Thierry Blu (CUHK), Lionel Coulot, Ali Hormati (EPFL), Pier-Luigi Dragotti (ICL) and Pina Marziliano (NTU). Michel BARLAUD I3S CNRS, University of Nice-Sophia-Antipolis, Polytech’Nice Institut Uni- versitaire de France, France Image Retrieval via Kullback Divergence of Patches of Wavelets Coefficients in the k-NN Framework This talk presents a framework to define an objective measure of the similar- ity (or dissimilarity) between two images for image processing. The problem is twofold: 1) define a set of features that capture the information contained in the image relevant for the given task and 2) define a similarity measure in this feature space. In this paper, we propose a feature space as well as a statistical measure on this space. Our feature space is based on a global description of the image in a multiscale transformed domain. After decomposition into a Laplacian pyra- mid, the coefficients are arranged in intrascale/ interscale/interchannel patches which reflect the dependencies of neighboring coefficients in presence of spe- cific structures or textures. At each scale, the probability density function (pdf) of these patches is used as a description of the relevant information. Because of the sparsity of the multiscale transform, the most significant patches, called Sparse Multiscale Patches (SMP), describe efficiently these pdfs. We propose a statistical measure (the Kullback-Leibler divergence) based on the comparison of these probability density function. Interestingly, this measure is estimated via the nonparametric, k-th nearest neighbor framework without explicitly build- ing the pdfs. This framework is applied to a query-by-example image retrieval method. Experiments on two publicly available databases showed the potential of our SMP approach for this task. In particular, it performed comparably to a SIFT-based retrieval method and two versions of a fuzzy segmentation-based method (the UFM and CLUE methods), and it exhibited some robustness to different geometric and radiometric deformations of the images.
- 23. Abstracts of the LIX Fall Colloquium 2008 11 Francis BACH INRIA/ENS, France Machine learning and kernel methods for computer vision Kernel methods are a new theoretical and algorithmic framework for machine learning. By representing data through well defined dot-products, referred to as kernels, they allow to use classical linear supervised machine learning algorithms to non linear settings and to non vectorial data. A major issue when applying these methods to image processing or computer vision is the choice of the kernel. I will present recent advances in the design of kernels for images that take into account the natural structure of images. Sylvain LAZARD VEGAS, INRIA LORIA Nancy, France 3D Visibility and Lines in Space Computing visibility information in a 3D environment is crucial to many applica- tions such as computer graphics, vision and robotics. Typical visibility problems include computing the view from a given point, determining whether two objects partially see each other, and computing the umbra and penumbra cast by a light source. In a given scene, two points are visible if the segment joining them does not properly intersect any obstacle in the scene. The study of visibility is thus intimately related to the study of the set of free line segments in a scene. In this talk, I will review some recent combinatorial and algorithmic results related to non-occluded segments tangent to up to four objects in three dimensional scenes. Suresh VENKATASUBRAMANIAN School of Computing, University of Utah, USA Non-standard Geometries and Data Analysis Traditional data mining starts with the mapping from entities to points in a Euclidean space. The search for patterns and structure is then framed as a geo- metric search in this space. Concepts like principal component analysis, regres- sion, clustering, and centrality estimation have natural geometric formulations, and we now understand a great deal about manipulating such (typically high dimensional) spaces. For many domains of interest however, the most natural space to embed data in is not Euclidean. Data might lie on curved manifolds, or even inhabit spaces endowed with different distance structures than lp spaces. How does one do data analysis in such domains? In this talk, I’ll discuss two specific domains of interest that pose challenges for traditional data mining and geometric methods. One space consists of collections of distributions, and the other is the space of shapes. In both cases, I’ll present ongoing work that at- tempts to interpret and understand clustering in such spaces, driven by different applications.
- 24. 12 F. Nielsen Markus GROSS Department of Computer Science, Institute of Scientific Computing, Swiss Fed- eral Institute of Technology Zurich, ETHZ, Switzerland 3D Video: A Fusion of Graphics and Vision In recent years 3-dimensional video has received a significant attention both in research and in industry. Applications range from special effects in feature films to the analysis of sports events. 3D video is concerned with the computation of virtual camera positions and fly-throughs of a scene given multiple, conven- tional 2D video streams. The high-quality synthesis of such view-independent video representations poses a variety of technical challenges including acquisi- tion, reconstruction, processing, compression, and rendering. In this talk I will outline the research in this area carried out at ETH over the past years. I will discuss various concepts for passive and active acquisition of 3D video using combinations of multiple cameras and projectors. Furthermore, I will address topics related to the representation and processing of the massive amount data arising from such multiple video streams. I will highlight the underlying techni- cal concepts and algorithms that draw upon knowledge both from graphics and from vision. Finally I will demonstrate some commercial applications targeting at virtual replays for sports broadcasts.
- 25. From Segmented Images to Good Quality Meshes Using Delaunay Refinement Jean-Daniel Boissonnat1 , Jean-Philippe Pons2 , and Mariette Yvinec1 1 INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France Jean-Daniel.Boissonnat@sophia.inria.fr 2 CSTB, 290 route des Lucioles, BP 209, 06904 Sophia-Antipolis Cedex, France Jean-Philippe.Pons@cstb.fr Abstract. This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the so-called Delaunay refinement technique allows to mesh surfaces and volumes bounded by surfaces, with theoretical guarantees on the quality of the approximation, from a geometrical and a topologi- cal point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field. We show how this general paradigm can be adapted to pro- duce anisotropic meshes, i.e. meshes elongated along prescribed direc- tions. Lastly, we discuss extensions to higher dimensions, and especially to space-time for producing time-varying 3D models. This is also of inter- est when input images are transformed into data points in some higher dimensional space as is common practice in machine learning. 1 Introduction Motivation. The ubiquity of digital imaging in scientific research and in indus- try calls for automated tools to extract high-level information from raster rep- resentations (2D, 3D, or higher-dimensional rectilinearly-sampled scalar/vector fields), the latter often being not directly suitable for analysis and interpreta- tion. Notably, the computerized creation of geometric models from digital images plays a crucial role in many medical imaging applications. A precondition for extracting geometry from images is usually to partition image pixels (voxels) into multiple regions of interest. This task, known as im- age segmentation, is a central long-standing problem in image processing and computer vision. Doing a review of this area is out of the scope of this paper. Let us only mention that it is a highly ill-posed problem due to various per- turbing factors such as noise, occlusions, missing parts, cluttered data, etc. The interested reader may refer to e.g. [1] for a specific survey on segmentation of medical images. This paper focuses on a step posterior to image segmentation: the automatic generation of discrete geometric representations from segmented images, such F. Nielsen (Ed.): ETVC 2008, LNCS 5416, pp. 13–37, 2009. c Springer-Verlag Berlin Heidelberg 2009
- 26. 14 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec as surface meshes representing boundaries between different regions of inter- est, or volume meshes of their interior. This step is determinant in numer- ous applications. For instance, in medicine, an increasing number of numerical simulations of physical or physiological processes call for geometric models of anatomical structures: electroencephalography (EEG) and magnetoencephalog- raphy (MEG), image-guided neurosurgery, electromagnetic modeling, blood flow simulation, etc. However, this topic has attracted less interest than image segmentation so far. As a result, reliable fully-automated tools for the unstructured discretization of segmented images, and in particular of medical datasets, are still lacking. So that simplistic or low-quality geometric models are still of wide use in some ap- plications. For example, in electromagnetic modeling, such as specific absorption rate studies, finite element methods (FEM) on unstructured grids conforming to anatomical structures would be desirable; but due to the difficulty of produc- ing such models, most numerical simulations so far have been conducted using finite difference methods on rectilinear grids, although the poor definition of tis- sue boundaries (stair-casing effect) strongly limits their accuracy. Similarly, in the EEG/MEG source localization problem using the boundary element method (BEM), simplistic head models consisting of a few nested tissue layers remain more popular than realistic models featuring multiple junctions. The generation of geometric models from segmented images presents many challenges. The output must fulfill many requirements in terms of geometric ac- curacy and topological correctness, smoothness, number, type, size and shape of mesh elements, in order to obtain acceptable results and make useful predictions, avoid instabilities in the simulations, or reduce the overall processing time. No- tably, the conditioning of stiffness matrices in FEM directly depends on the sizes and shapes of the elements. Another example is image-guided neurosurgery, for which real-time constraints impose strong limitations on the complexity of the geometric brain model being dynamically registered onto the patient anatomy. Grid-based methods. Commonly used techniques do not meet the aforemen- tioned specifications. The most popular technique for producing surface meshes from raster data is undoubtedly the marching cubes algorithm, introduced by Lorensen and Cline [2]. Given a scalar field sampled on a rectilinear grid, the marching cubes algorithm efficiently generates a triangular mesh of an isosurface by tessellating each cubic cell of the domain according to a case table constructed off-line. Unfortunately, this technique, as well as its many subsequent variants, typ- ically produces unnecessarily large meshes (at least one triangle per boundary voxel) of very low quality (lots of skinny triangles). This may be acceptable for visualization purposes, but not for further numerical simulations. In order to ob- tain suitable representations, the resulting meshes often have to be regularized, optimized and decimated, while simultaneously controlling the approximation accuracy and preserving some topological properties, such as the absence of self- intersections. Sometimes, good tetrahedral meshes of the domains bounded by
- 27. From Segmented Images to Good Quality Meshes 15 the marching cubes surfaces also have to be generated. Most of the time, these tasks are overconstrained. Recently, the interest in grid-based techniques has been renewed by a few methods with theoretical guarantees. Plantiga and Vegter [3] propose an algo- rithm to mesh implicit surfaces with guaranteed topology, based on an adaptive octree subdivision controlled by interval arithmetic. But in its current form, this algorithm is relevant to closed-form expressions, not to sampled data. The recent algorithm of Labelle and Shewchuck [4] fills an isosurface with a uniformly sized tetrahedral mesh whose dihedral angles are bounded between 10.7◦ and 164.8◦ . The algorithm is very fast, numerically robust, and easy to implement because, like the marching cubes algorithm, it generates tetrahedra from a small set of precomputed stencils. Moreover, if the isosurface is a smooth 2-manifold with bounded curvature, and the tetrahedra are sufficiently small, then the boundary of the mesh is guaranteed to be a geometrically and topo- logically accurate approximation of the isosurface. However, this algorithm lacks flexibility: notably, it is limited to uniform surface meshes, and isotropic surface and volume meshes. Delaunay-based methods. This paper surveys Delaunay-based meshing tech- niques for curved objects. It is recognized as one of the most powerful techniques for generating surface and volume meshes with theoretical guarantees on the quality of the approximation, from a geometrical and topological point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field. It also allows to mesh sev- eral domains simultaneously. Recent extensions show that this general paradigm can be adapted to produce anisotropic meshes, i.e. meshes elongated along pre- scribed directions, as well as meshes in higher dimensions. In this paper, we show how Delaunay-based meshing can be applied in medi- cal imaging and in computer vision to the extraction of meshes from segmented images, with all the desired specifications. The rest of the paper is organized as follows. We first introduce the notion of restricted Delaunay triangulation in Section 2. We then show how to mesh surfaces (Section 3) and volumes bounded by surfaces (Section 4) using the so-called Delaunay refinement tech- nique. Anisotropic meshes are discussed in Section 5. Lastly, we tackle extensions of Delaunay refinement to higher dimensions (Section 6), and especially to space- time for producing time-varying 3D models. This is also of interest when input images are transformed into data points in some higher dimensional space as is common practice in machine learning. 2 Restricted Delaunay Triangulations In this section, we recall the definitions of Voronoi diagrams and Delaunay tri- angutions, and their generalization known as power (or Laguerre) diagrams and weighted Delaunay (or regular) triangulations. We then introduce the concept of restricted Delaunay triangulation which is central in this paper.
- 28. 16 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec 2.1 Voronoi Diagrams and Delaunay Triangulations Voronoi diagrams are versatile structures which encode proximity relationships between objects. They are particularly relevant to perform nearest neighbor search and motion planning (e.g. in robotics), and to model growth processes (e.g. crystal growth in materials science). Delaunay triangulations, which are geometrically dual to Voronoi diagrams, are a classical tool in the field of mesh generation and mesh processing due to their optimality properties. In the sequel, we call k-simplex the convex hull of k + 1 affinely independent points. For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle and a 3-simplex is a tetrahedron. Let E = {p1, . . . , pn} be set of points in Rd , called sites. Note that in this paper, we are mainly interested in d = 3, except in Section 6, where the case d 3 is studied. The Voronoi region, or Voronoi cell, denoted by V (pi), associated to a point pi ∈ E is the region formed by points that are closer to pi than to all other sites in E: V (pi) = {x ∈ Rd : ∀j, x − pi ≤ x − pj}. V (pi) is the intersection of n − 1 half-spaces bounded by the bisector planes of segments [pipj], j = i. V (pi) is therefore a convex polyhedron, possibly un- bounded. The Voronoi diagram of E, denoted by Vor(E), is the subdivision of space induced by the Voronoi cells V (p1), . . . , V (pn). See Fig. 1 for a two-dimensional example of a Voronoi diagram. In two di- mensions, the edges shared by two Voronoi cells are called Voronoi edges and the points shared by three Voronoi cells are called Voronoi vertices. Similarly, in three dimensions, we term Voronoi facets, edges and vertices the geometric objects shared by respectively two, three and four Voronoi cells, respectively. The Voronoi diagram is the collection of all these k-dimensional objects, with 0 ≤ k ≤ d, which we call Voronoi faces. In particular, note that Voronoi cells V (pi) correspond to d-dimensional Voronoi faces. To simplify the presentation and without real loss of generality, we will assume in the sequel that E does not contain any subset of d + 2 points that lie on a same hypersphere. We say that the points of E are then in general position. The Delaunay triangulation of E, noted Del(E), is the geometric dual of Vor(E), and can be described as an embedding of the nerve1 of Vor(E). The nerve of Vor(E) is the abstract simplicial complex that contains a simplex σ = (pi0 , . . . , pik ) iff V (pi0 ) ∩ . . . ∩ V (pik ) = ∅. Specifically, if k + 1 Voronoi cells have a non-empty intersection, this intersection constitutes a (d−k)-dimensional face f of Vor(E). The convex hull of the associated k + 1 sites constitutes a k- dimensional simplex in the Delaunay triangulation and this simplex is the dual of face f. In 3D, the dual of a Delaunay tetrahedron is the Voronoi vertex that coincides with the circumcenter of the tetrahedron, the dual of a Delaunay facet is a Voronoi edge, the dual of a Delaunay edge is a Voronoi facet, and the dual of a Delaunay vertex pi is the Voronoi cell V (pi). See Fig. 1. 1 The notion of nerve of a covering is a basic concept in algebraic topology [5].
- 29. From Segmented Images to Good Quality Meshes 17 Fig. 1. The voronoi diagram of a set of points (left). Its dual Delaunay triangulation (right). The Voronoi vertex v that is the dual of a d-dimensional simplex σ of Del(E) is the circumcenter of σ and, since v is closer to the vertices of σ than to all other points of E, the interior of the ball centered at v that circumscribes σ does not contain any point of E. We say that such a ball is empty. This property turns out to characterize Delaunay triangulations. Hence, Del(E) can be equivalently defined as the unique (under the general position assumption) triangulation of E such that each simplex in the triangulation can be circumscribed by an empty ball. 2.2 Power Diagrams Weighted Delaunay Triangulations In this section, we introduce an extension of Voronoi diagrams that will be useful in the sequel. Point sites p1, . . . , pn are replaced by hyperspheres Σ = {σ1, . . . , σn} and the Euclidean distance from a point x to a point site pi is replaced by the power distance to hypersphere σi, i.e. the quantity σi(x) = x−ci2 −r2 i if ci and ri denote the center and radius of σi. One can then define the power cell of site σi as V (σi) = {x ∈ Rd : ∀j, σi(x) ≤ σj(x)}. Like Voronoi cells, power cells are convex polyhedra. The subdivision of space induced by the power cells V (σ1), . . . , V (σn), constitutes the power diagram V (σ) of Σ. As in the case of Voronoi diagrams, we define the geometric dual of the power diagram V (σ) as an embedding of the nerve of V (σ), where the dual of a face f = i=1,...k V (σi) is the convex hull of the centers c1, . . . ck. If the spheres Σ are in general position, the geometric dual of the power diagram is a triangulation. This triangulation is called the weighted Delaunay (or regular)
- 30. 18 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec triangulation of Σ. Note that because some spheres of Σ may have an empty power cell, the set of vertices in the weighted Delaunay triangulation is only a subset of the centers of the σi. It is a remarkable fact that weighted Delaunay triangulations can be computed almost as fast as non weighted Delaunay triangulations. An efficient implemen- tation of both types of triangulations can be found in the Cgal library [6,7]. It is robust to degenerate configurations and floating-point errors through the use of exact geometric predicates. 2.3 Restricted Delaunay Triangulations We introduce the concept of restricted Delaunay triangulation which, as the con- cept of Delaunay triangulation, is related to the notion of nerve. Given a subset Ω ⊂ Rd and a set E of points, we call Delaunay triangulation of E restricted to Ω, and note Del|Ω(E), the subcomplex of Del(E) composed of the Delaunay simplices whose dual Voronoi faces intersect Ω. We refer to Fig. 2 to illustrate this concept in 2D. Fig. 2 (left) shows a Delaunay triangulation restricted to a curve C, which is composed of the Delaunay edges whose dual Voronoi edges intersect C. Fig. 2 (right) shows the Delaunay triangulation of the same set of points restricted to the region R bounded by the curve C. The restricted triangu- lation is composed of the Delaunay triangles whose circumcenters are contained in R. For an illustration in R3 , consider a region O bounded by a surface S and a sample E, the Delaunay triangulation restricted to S, Del|S(E), is composed of the Delaunay facets in Del(E) whose dual Voronoi edges intersect S while the Delaunay triangulation restricted to O, Del|O(E), is made of those tetrahedra in Del(E) whose circumcenters belong to O. The attentive reader may have noticed that in both cases of Figure 2, the restricted Delaunay triangulation forms a good approximation of the object. Actually, this is a general property of the restricted Delaunay triangulation. It can be shown that, under some assumptions, and especially if E is a sufficiently dense sample of a smooth surface S, Del|S(E) is a good approximation of S, both in a topological and in a geometric sense. Specifically, Del|S(E) is a triangulated surface that is isotopic to S; the isotopy moves the points by a quantity that becomes arbitrarily small when the density increases; in addition, normals of S of can be consistently approximated from Del|S(E). Before stating precise results, we define what “sufficiently dense” means. The definition is based on the notion of medial axis. In the rest of the paper, S will denote a closed smooth surface of R3 . Definition 1 (Medial axis). The medial axis of a surface S is the closure of the set of points with at least two closest points on S. Definition 2 (lfs). The local feature size at a point x on a surface S, noted lfs(x), is the distance from x to the medial axis of S. We write lfs(S) = infx∈S lfs(x). It can be shown that lfs(x) does not exceed the reach of S at x, denoted by rch(x). The reach at x is defined as the radius of the largest open ball tangent
- 31. From Segmented Images to Good Quality Meshes 19 Fig. 2. The Voronoi diagram (in red) and the Delaunay triangulation (in blue) of a sample of red points on a planar closed curve C (in black). On the left: the edges of the Voronoi diagram and of the Delaunay triangulation that are restricted to the curve are in bold lines. On the right: the triangles belonging to the Delaunay triangulation of the sample restricted to the domain bounded by C are in blue. Fig. 3. The medial axis of a planar curve (only the portion inside the domain bounded by the curve is shown). The thin curves are parallel to the boundary of the domain. to S at x whose interior does not contain any point of S. Plainly, rch(x) cannot exceed the smallest radius of curvature at x and can be strictly less at points where the thickness of the object bounded by S is small. As shown by Federer [8], the local feature size of a smooth surface object is bounded away from 0.2 The following notion of ε-sample has been proposed by Amenta and Bern in their seminal paper on surface reconstruction [9]. 2 In fact, Federer proved the stronger result that the local feature size is bounded away from 0 as soon as S belongs to the class C1,1 of surfaces that admit a normal at each point and whose normal field is Lipschitz. This class is larger than the class of C2 surfaces and includes surfaces whose curvature may be discontinuous at some points. An example of a surface that is C1,1 but not C2 is the offset of a cube.
- 32. 20 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec Fig. 4. A surface Delaunay ball whose center is a candidate for being inserted in E Definition 3 (ε-sample). Let ε 1 and S be a smooth surface. We say that a finite point set E ⊂ S is an ε-sample of S if any point x of S is at distance at most ε lfs(x) from a point of E. The notion of ε-sample is not very handy since it requires that any point of the surface is close to a sample point. A more convenient notion of sample, called loose ε-sample, only requires a finite set of points of S to be close to the sample set [10]. More precisely, consider the Voronoi edges of Vor(E) that intersect S. We require that each such intersection point is close to the sample set. By definition, these Voronoi edges are dual to the facets of Del|S(E). An intersection point of such an edge with S is thus the center of a so-called surface Delaunay ball, i.e. a ball circumscribing a facet of Del|S(E) and centered on the surface S (see Fig. 4). Definition 4 (Loose ε-sample). Let ε 1 be a constant and S be a smooth surface. A point set E ⊂ S is a loose ε-sample of S if Del|S(E) has a vertex on each connected component of S and if, in addition, any surface Delaunay ball B(cf , rf ) circumscribing a facet f of Del|S(E) is such that rf εlfs(cf ). The following theorem states that, for sufficiently dense samples, Del|S(E) is a good approximation of S. Theorem 1. If E is a loose ε-sample of a smooth compact surface S, with ε 0.12, then the restriction of the orthogonal projection πS : R3 M(S) → S, induces an isotopy that maps Del|S(E) to S. The isotopy does not move the points of Del|S(E) by more than O(ε2 ). The angle between the normal to a facet f of Del|S(E) and the normals to S at the vertices of f is O(ε). Weaker variants of this theorem have been proved by Amenta and Bern [9] and Boissonnat and Oudot [10]. Cohen-Steiner and Morvan have further shown that one can estimate the tensor of curvatures from Del|S(E) [11].
- 33. From Segmented Images to Good Quality Meshes 21 3 Surface Sampling and Meshing In this section, we show how the concept of restricted Delaunay triangulation can be used to mesh smooth surfaces. The algorithm is proven to terminate and to construct good-quality meshes, while offering bounds on the accuracy of the original boundary approximation and on the size of the output mesh. The refinement process is controlled by highly customizable quality and size criteria on triangular facets. A notable feature of this algorithm is that the surface needs only to be known through an oracle that, given a line segment, detects whether the segment intersects the surface and, in the affirmative, returns an intersection point. This makes the algorithm useful in a wide variety of contexts and for a large class of surfaces. The paradigm of Delaunay refinement has been first proposed by Ruppert for meshing planar domains [12]. The meshing algorithm presented in this section is due to Chew [13,14]. 3.1 Delaunay Refinement for Meshing Surfaces Let S be a surface of R3 . If we know a loose ε-sample E of S, with ε 0.12, then, according to Theorem 1, the restricted Delaunay triangulation Del|S(E) is a good approximation of S. In this section, we present an algorithm that can construct such a sample and the associated restricted Delaunay triangulation. We restrict the presentation to the case of smooth, compact and closed surfaces. Hence, lfs(S) = infx∈S lfs(x) 0. The algorithm is greedy. It inserts points one by one and maintains the current set E, the Delaunay triangulation Del(E) and its restriction Del|S(E) to S. Let ψ be a function defined over S such that ∀x ∈ S, 0 ψinf ≤ ψ(x) ≤ εlfs(x). where ψmin = infx∈S ψ(x). Function ψ will control the sampling density and is called the sizing field. The shape quality of the mesh facets is controlled through their radius-edge ratio, where the radius-edge ratio of a facet is the ratio between the circumradius of the facet and the length of its shortest edge. We define a bad facet as a facet f of Del|S(E) that: – either has a too big surface Delaunay ball Bf = B(cf , rf ), meaning that rf ψ(cf ), – or is badly shaped, meaning that its radius-edge ratio ρ is such that ρ β for a constant β ≥ 1. Bad facets will be removed from the mesh by inserting the centers of their surface Delaunay balls, The algorithm is initialized with a (usually small) set of points E0 ⊂ S. Three points per connected component of S are sufficient. Then the algorithm maintains, in addition to Del(E) and Del|S(E), a list of bad facets and, as long as there remain bad facets, applies the following procedure
- 34. 22 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec refine facet(f) 1. insert in E the center cf of a surface Delaunay ball circumscribing f, 2. update Del(E), Del|S(E) and the list of bad facets An easy recurrence proves that the distance between any two points inserted in the sample is at least ψinf 0. Since S is compact, the algorithm terminates after a finite number of steps. It can be shown that the number of inserted points is O S dx ψ2(x) . Upon termination, any facet f of Del|S(E) has a circumscribing surface Delau- nay ball Bf of center cf and radius rf ψ(cf ). To be able to apply Theorem 1, we need to take ψ ≤ 0.12 lfs and to ensure that Del|S(E) has at least one vertex on each connected component of S. This can be done by taking in E0 three points per component of S that are sufficiently close. We sum up the results in the following theorem. Theorem 2. Given a compact smooth and closed surface S, and a positive Lip- schitz function ψ ≤ ε lfs on S, one can compute a loose ε-sample E of S, of size O S dx ψ2(x) . If ε ≤ 0.12, the restricted Delaunay triangulation is Del|S(E) is a triangulated surface isotopic and close to S. 3.2 Implementation Note that the surface is only queried through an oracle that, given a line segment f∗ (to be the edge of Vor(E) dual to a facet f of Del|S(E)), determines whether f∗ intersects S and, in the affirmative, returns an intersection point and the value of ψ at this point. Still, deciding whether a line segment intersects the surface may be a costly operation. However, a close examination of the proof of correctness of the algo- rithm shows that Theorems 1 and 2 still hold if we replace the previous oracle by a weaker one that checks if a given line segment s intersects S an odd number of times and, in the affirmative, computes an intersection point. Consider the case where S is an implicit surface f(x) = 0, e.g. an isosurface defined by interpola- tion in a 3D image. To know if s intersects S an odd number of times, we just have to evaluate the sign of f at the two endpoints of the segment. It is only in the case where the two signs are different that we will compute an intersection point (usually by binary search). This results in a dramatic reduction of the computing time. Although the algorithm is quite simple, it is not easy in general to know lfs or even to bound lfs from below, which is required by the oracle. In practice, good results have been obtained using the following simple heuristics. We redefine bad facets to control the distance cf − c f between the center cf of the surface De- launay ball circumscribing a facet f of Del|S(E) and the center c f of the smallest ball circumscribing f. This strategy nicely adapts the mesh density to the lo- cal curvature of S. The local feature size lfs(x) depends also on the thickness of S
- 35. From Segmented Images to Good Quality Meshes 23 Fig. 5. Meshing an isosurface in a 3D image of the brain at x, which is a global parameter and therefore difficult to estimate. However, if the sample is too sparse with respect to the object thickness, the restricted Delaunay triangulation is likely to be non manifold and/or to have boundaries. The algorithm can check on the fly that Del|S(E) is a triangulated surface with no boundary by checking that each edge in the restricted triangulation is incident to two facets, and that the link of each vertex (i.e. the boundary of the union of the facets incident to the vertex) is a simple polygon. The issue of estimating lfs can also be circumvented by using a multiscale approach that has been first proposed in the context of manifold reconstruc- tion [15,16]. We slightly modify the algorithm so as to insert at each step the candidate point that is furthest from the current sample. This will guarantee that the sample remains roughly uniform through the process. If we let the algo- rithm insert points, the topology of the triangulated surface maintained by the algorithm may well change. Consider, for instance, the case of an isosurface in a noisy image, say the brain in Fig. 5. Depending on the sampling density, the topology of the surface may be a topological sphere (which the brain is indeed) or a sphere with additional handles due to noise. Accordingly, the algorithm will produce intermediate meshes of different topologies approximating surfaces of various lfs. Since the changes of topology can be detected by computing at each step the Betti numbers of the current triangulated surface, we can output the various surfaces and the user can decide what is the best one. The surface meshing algorithm is available in the open source library Cgal [7]. Fig. 5 shows a result on a medical image. A thorough discussion of the implementation of the algorithm and other experimental results can be found in [14].
- 36. 24 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec 4 Meshing Volumes with Curved Boundaries Let O be an object of R3 bounded by a surface S. The meshing algorithm of the previous section constructs the 3D Delaunay triangulation of the sample E and extracts from Del(E) the restricted Delaunay triangulation Del|S(E). Hence, the algorithm constructs a 3D triangulation T of O as well as a polyhedral surface approximating S. However, since the algorithm does not insert points inside O, the aspect ratio of the tetrahedra of T cannot be controlled. If further computations are to be performed, it is then mandatory to improve the shape of the tetrahedra by sampling also the interior of O. We present in this section a modification of the Delaunay-based surface mesher of the previous section due to Oudot et al. [17]. This algorithm samples the inte- rior and the boundary of the object at the same time so as to obtain a Delaunay refinement volume mesher. Delaunay refinement removes all badly shaped tetra- hedra except the so-called slivers. A special postprocessing is required to remove those slivers. 4.1 3D Mesh Refinement Algorithm The algorithm is still a greedy algorithm that builds a sample E while main- taining the Delaunay triangulations Del(E) and its restrictions Del|O(E) and Del|S(E) to the object O and its bounding surface S. The sampling density is controlled by a function ψ(x) defined over O called the sizing field. Using constant α, β and γ, we define two types of bad elements. As above, a facet f of Del|S(E) is considered as bad if – either it has a too big surface Delaunay ball Bf = B(cf , rf ), i.e. rf αψ(cf ), – or it is badly shaped, meaning that its radius-edge ratio ρf is such that ρf β. A tetrahedron t of Del|O is considered as bad if – either its circumradius rt is too big, i.e. rt ψ(ct) – or it is badly shaped, meaning that its radius-edge ratio ρt is such that ρt γ. The radius-edge ratio ρt of a tetrahedron t is the ratio between the circumradius and the length of the shortest edge. The algorithm uses two basic procedures, refine facet(f), which has been defined in Section 3 and the following procedure refine tet(t). refine tet(t) 1. insert in E the center ct of the ball circumscribing t 2. update Del(E), Del|S(E), Del|O(E) and the lists of bad elements. The algorithm is initialized as the surface meshing algorithm. Then it applies the following refinement rules in order, Rule 2 being applied only when Rule 1 can no longer be applied.
- 37. From Segmented Images to Good Quality Meshes 25 Rule 1. If Del|S(E) contains a facet f which has a vertex in O S or is bad, refine facet(f) Rule 2. If there is a bad tetrahedron t ∈ Del|O(E) 1. compute the center ct of the circumscribing ball 2. if ct is included in the surface Delaunay ball of some facet f ∈ Del|S(E), refine facet(f) 3. else refine tet(t). It is proved in [17] that, for appropriate choices of parameters α, β and γ, the algorithm terminates. Upon termination, Del|S(E) = Del|S(E ∩S) and DelO(E) is a 3D-triangulation isotopic to O. 4.2 Sliver Removal While Delaunay refinement techniques can be proven to generate tetrahedra with a good radius-edge ratio, they may create flat tetrahedra of a special type called slivers. A sliver is a tetrahedron whose four vertices lie close to a plane and whose projection to that plane is a quadrilateral with no short edge. Slivers have a good radius-edge ratio but a poor radius-radius ratio (ratio between the circumradius and the radius of the largest contained sphere). Unfortunately, the latter measure typically influences the numerical conditioning of finite element methods. Slivers occur for example if one computes the Delaunay triangulation of points on a regular grid (slightly pertubed to avoid degeneracies). Each square in the grid can be circumscribed by an empty ball and is therefore a sliver of the triangulation. Fig. 6. A sliver Two techniques are known to remove slivers from volume meshes. One consists of a post-processing step called sliver exudation [18]. This step does not include any new vertex in the mesh, nor does it move any of them. Each vertex is assigned a weight and the Delaunay triangulation is turned into a weighted Delaunay triangulation. The weights are carefully chosen so that no vertex disappear from the mesh, nor any change occurs in the boundary facets (i. e. the facets of Del|S(E)). Within these constraints, the weight of each vertex is optimized in turn to maximize the minimum dihedral angles of the tetrahedra incident to that vertex. Although the guaranteed theoretical bound on dihedral angles is known
- 38. 26 J.-D. Boissonnat, J.-P. Pons, and M. Yvinec Fig. 7. The new point to be inserted is taken from the grey disk centered at the circumcenter of the bad element τ but not in the black annulus to prevent the creation of slivers to be miserably low, this algorithm is quite efficient in practice at removing slivers. Another technique, due to Li [19], avoids the appearance of small slivers in the mesh by relaxing the choice of the refinement points of a bad element (tetra- hedron or boundary facet). The new points are no longer inserted at the cir- cumcenters of Delaunay balls or surface Delaunay balls but in small picking regions around those circumcenters. Within such a picking region, we further avoid inserting points that would create slivers. (see Fig. 7). 4.3 Implementation We present two results in Fig. 8 on both uniform and non uniform sizing fields. The uniform model is an approximation of an isosurface in a 3D medical image. The initial mesh of the surface had 33,012 vertices while the second step of the algorithm added 53,762 new vertices in the interior of the object and 2,471 new vertices on its boundary. The total CPU time was 20s on a Pentium IV (1.7 GHz). A thorough discussion of the implementation of the algorithm and other experimental results can be found in [17,20]. The algorithm will be soon available in the open source library Cgal [7]. 4.4 Meshing of Multi-label Datasets The above method seamlessly extends to the case of non-binary partitions, so that it can be applied to the generation of high quality geometric models with multiple junctions from multi-label datasets, frequently encountered in medical applications.
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- 40. with a full orchestra. I mean soon to publish six preludes and fugues, two of which you have already seen; this is the sort of life I like to lead, but not that of an intendant. How vexatious it is, that at the close of such well-spent days we cannot all assemble together to enjoy each other’s society![18] I enclose my translation of “Alexander’s Feast;” you must read it aloud to the family in the evening, and in various passages where the rhymes are rugged or deficient, if you will let me have your amendments I shall be grateful. One stipulation, however, I must make, that Ramler, or rather, I should say, the English text, should not be sacrificed. Apropos, since then I have once more mounted Pegasus, and translated Lord Byron’s poem, the first strophe of which, by Theremin, is incomprehensible, and the second false. I find, however, that my lines halt a little; perhaps, some evening, you may discover something better. Schlafloser Augensonne, heller Stern! Der du mit thränenvollem Schein, unendlich fern, Das Dunkel nicht erhellst, nur besser zeigst, O wie du ganz des Glücks Erinn’rung gleichst! So funkelt längst vergangner Freuden Licht, Es scheint, doch wärmt sein matter Schimmer nicht, Der wache Gram erspäht die Nachtgestalt, Hell, aber fern, klar—aber ach! wie kalt! The poem is very sentimental, and I think I should have set it to music repeatedly in G sharp minor or B major, (but, at all events, with no end of sharps,) had it not occurred to me that the music of Löwe pleases you and Fanny; so this prevents my doing so, and there is an end of it, and of my letter also. Adieu, love me as ever.—Your Felix.
- 41. To Carl Klingemann, London. Düsseldorf, December 16th, 1834. ... So now in these lines you have read my whole life and occupations since I came here; for that I am well and happy, and often think of you, is included in them, and that I am also diligent and working hard at many things, is the natural result. I really believe that Jean Paul, whom I am at this moment reading with intense delight, has also some influence in the matter, for he invariably infects me for at least half a year with his strange peculiarities. I have been reading ‘Fixlein’ again; but my greatest pleasure in doing so, is the remembrance of the time when I first became acquainted with it, by your reading it aloud to me beside my sick-bed, when it did me so much good. I also began ‘Siebenkäs’ again, for the first time for some years, and have read from the close of the prologue to the end of the first part, and am quite enchanted with this noble work. The prologue itself is a masterpiece such as no one else could write, and so it is with the whole book, the friends, and the school-inspector, and Lenette. It revives my love for my country, and makes me feel proud of being a German, although in these days they all abuse each other. Yet such people do sometimes rise to the surface, and I do believe that no country can boast of such a sterling fellow as this.
- 42. To Rebecca Dirichlet, Berlin. Düsseldorf, December 23rd, 1834. Dear Rebecca, Why should we not, like established correspondents, exchange repeated letters on any particular subject about which we differ? I on my part will represent a methodical correspondent, and must absolutely resume the question of révolution. This is chiefly for Fanny’s benefit, but are not you identical? Can you not therefore discuss the subject together, and answer me together, if you choose? And have I not pondered and brooded much over this theme since I got your letter, which now prompts me to write? You must, however, answer me in due form, till not one jot or tittle more remains to be said in favour of révolution. Observe, I think that there is a vast distinction between reformation or reforming, and revolution, etc. Reformation is that which I desire to see in all things, in life and in art, in politics and in street pavement, and Heaven knows in what else besides. Reformation is entirely negative against abuses, and only removes what obstructs the path; but a revolution, by means of which all that was formerly good (and really good) is no longer to continue, is to me the most intolerable of all things, and is, in fact, only a fashion. Therefore, I would not for a moment listen to Fanny, when she said that Lafont’s playing could inspire no further interest since the revolution effected by Paganini; for if his playing ever had the power to interest me, it would still do so, even if in the meantime I had heard the Angel Gabriel on the violin. It is just this, however, that those Frenchmen I alluded to can form no conception of; that what is good, however old, remains always new, even although the present must differ from the past, because it emanates from other and dissimilar men. Inwardly they are only ordinary men like the former, and have only outwardly learned that something new must come, so they strive to accomplish this, and if they are even moderately applauded or flattered, they instantly declare that they have effected a révolution du goût. This is why I behave so badly when they do me the honour (as you call it) to rank me among the leaders of this movement, when I well know that, for thorough self-cultivation, the whole of a man’s life is required (and often does not suffice); and also because no Frenchman, and no newspaper, knows or ever can know what the future is to
- 43. give or to bring; and, in order to guide the movements of others, we must first be in motion ourselves, while such reflections cause us to look back on the past, not forward. Progress is made by work alone, and not by talking, which those people do not believe. But, for Heaven’s sake, don’t suppose that I wish to disown either reformation or progress, for I hope one day myself to effect a reform in music; and this, as you may see, is because I am simply a musician, and I wish to be nothing more. Now answer me, I beg, and preach to me again. To-day I have completed and transcribed an entire chorus for “St. Paul.” I may as well at once reply here to a letter I received this morning, dictated by my father to Fanny, and to which my mother added a postscript. First of all, I thank you for writing, and then, dear Father, I would entreat of you not to withhold from me your advice, as you say, for it is always clear gain to me; and if I cannot rectify the old faults, I can at least avoid committing new ones. The non-appearance of St. Paul at the stoning of Stephen is certainly a blemish, and I could easily alter the passage in itself; but I could find absolutely no mode of introducing him at that time, and no words for him to utter in accordance with the Scriptural narrative; therefore it seemed to me more expedient to follow the Bible account, and to make Stephen appear alone. I think, however, that your other censure is obviated by the music; for the recitative of Stephen, though the words are long, will not occupy more than two or three minutes, or—including all the choruses—till his death, about a quarter of an hour; whereas subsequently, at and after the conversion, the music becomes more and more diffuse, though the words are fewer.
- 44. To Pastor Bauer, Beszig. Düsseldorf, January 12th, 1835. [About a proposal as to some words for sacred music.] ... What I do not understand is the purport—musical, dramatic, or oratorical, or whatever you choose to call it—that you have in view. What you mention on the subject—the time before John, and then John himself, till the appearance of Christ—is to my mind equally conveyed in the word ‘Advent,’ or the birth of Christ. You are aware, however, that the music must represent one particular moment, or a succession of moments; and how you intend this to be done you do not say. Actual church music,—that is, music during the Evangelical Church service, which could be introduced properly while the service was being celebrated,—seems to me impossible; and this, not merely because I cannot at all see into which part of the public worship this music can be introduced, but because I cannot discover that any such part exists. Perhaps you have something to say which may enlighten me on the subject.... But even without any reference to the Prussian Liturgy, which at once cuts off everything of the kind, and will neither remain as it is nor go further, I do not see how it is to be managed that music in our Church should form an integral part of public worship, and not become a mere concert, conducive more or less to piety. This was the case with Bach’s “Passion;” it was sung in church as an independent piece of music, for edification. As for actual church music, or, if you like to call it so, music for public worship, I know none but the old Italian compositions for the Papal Chapel, where, however, the music is a mere accompaniment, subordinate to the sacred functions, co-operating with the wax candles and the incense, etc. If it be this style of church music that you really mean, then, as I said, I cannot discover the connecting link which would render it possible to employ it. For an oratorio, one principal subject must be adopted, or the progressive history of particular persons, otherwise the object would not be sufficiently defined; for if all is to be only contemplative with reference to the coming of Christ, then this theme has already been more grandly and beautifully treated in Handel’s “Messiah,” where he begins with Isaiah, and, taking the Birth as a central point, closes with the Resurrection.
- 45. When you however say “our poor Church,” I must tell you what is very strange; I have found, to my astonishment, that the Catholics, who have had music in their churches for several centuries, and sing a musical Mass every Sunday if possible, in their principal churches, do not to this day possess one which can be considered even tolerably good, or in fact which is not actually distasteful and operatic. This is the case from Pergolese and Durante, who introduce the most laughable little trills into their “Gloria,” down to the opera finales of the present day. Were I a Catholic, I would set to work at a Mass this very evening; and whatever it might turn out, it would at all events be the only Mass written with a constant remembrance of its sacred purpose. But for the present I don’t mean to do this; perhaps at some future day, when I am older.
- 46. To Herr Conrad Schleinitz, Leipzig. Düsseldorf, January 26th, 1835. Sir, Pray receive my thanks for your kind letter, and the friendly disposition which it evinces towards myself. You may well imagine that it would be a source of infinite pleasure to me, to find in your city the extensive sphere of action you describe, as my sole wish is to advance the cause of music on that path which I consider the right one; I would therefore gladly comply with a summons which furnished me with the means of doing so. I should not like, however, by such acceptance to injure any one, and I do not wish, by assuming this office, to be the cause of supplanting my predecessor. In the first place, I consider this to be wrong; and, moreover, great harm ensues to music from such contentions. Before, then, giving a decided answer to your proposal, I must beg you to solve some doubts,—namely, at whose disposal is the appointment you describe? with whom should I be in connection— with a society, or individuals, or a Board? and should I by my acceptance injure any other musician? I hope you will answer this last question with perfect candour, imagining yourself in my place; for, as I previously said, I have no wish to deprive any one either directly or indirectly of his situation. Further, it is not quite clear to me from your letter, how the direction of an academy for singing can be combined with my six months’ summer vacation; for you must be well aware how indispensable continual supervision is to such an institution, and that anything which can be accomplished in one half-year, may be easily forgotten in the next; or is there another director for the purpose of undertaking the duties instead of me? Finally, I must also confess that in a pecuniary point of view, I do not wish to accept any position that would be less profitable than my present one; but as you mention a benefit concert, no doubt this is a matter that might be satisfactorily arranged, and we should have no difficulty in coming to an agreement on this point. I have been quite candid with you, and hope, in any event, you will not take it amiss; be so good as to oblige me by sending an answer as soon as possible, and to believe that I shall ever be grateful to you for your kind letter, as well as for the honour you have done me.
- 47. To Capellmeister Spohr, Cassel. Düsseldorf, March 8th, 1835. Respected Capellmeister, I thank you much for your friendly communication. The intelligence from Vienna was most interesting to me; I had heard nothing of it. It strongly revived my feeling as to the utter impossibility of my ever composing anything with a view to competing for a prize. I should never be able to make even a beginning; and if I were obliged to undergo an examination as a musician, I am convinced that I should be at once sent back, for I should not have done half as well as I could. The thoughts of a prize, or an award, would distract my thoughts; and yet I cannot rise so superior to this feeling as entirely to forget it. But if you find that you are in a mood for such a thing, you should not fail to compose a symphony by that time, and to send it, for I know no man living who could dispute the prize with you (this is the second reason), and then we should get another symphony of yours (first reason). With regard to the members of the Judicial Committee in Vienna, I have my own thoughts, which, however, are not very legitimate, but, on the contrary, somewhat rebellious. Were I one of the judges, not a single member of the Comité should obtain a prize, if they competed for one. You wish me to write to you on the subject of my works, and I cordially thank you for asking about them. I began an oratorio about a year ago, which I expect to finish next month, the subject of which is St. Paul. Some friends have compiled the words for me from the Bible, and I think that both the subject and the compilation are well adapted to music, and very solemn,—if the music only prove as good as I wish; at all events I have enjoyed the most intense delight, while engaged in writing it. I also composed, some time since, a new overture to the “Lovely Melusina,” and have another in my head at this moment. How gladly would I write an opera; but far and near I can find no libretto and no poet. Those who have the genius of poetry cannot bear music, or know nothing of the theatre; others are neither acquainted with poetry nor with mankind, only with the boards, and lamps, and side scenes, and canvas. So I never succeed in finding the opera which I have so eagerly, yet vainly striven to procure. Each day I regret this more, but I hope at last to meet with the man I wish for this purpose. I have also written a
- 48. good deal of instrumental music of late, chiefly for the piano, but others besides; perhaps you will permit me to send you some of these as soon as I have an opportunity to do so. I am, with the highest esteem and consideration, your devoted Felix Mendelssohn Bartholdy.
- 49. To Felix Mendelssohn Bartholdy, from his Father. [19] Berlin, March 10th, 1835. This is the third letter I have written to you this week, and if this goes on, reading my letters will become a standing article in the distribution of the budget of your time; but you must blame yourself for this, as you spoil me by your praise. I at once pass to the musical portion of your last letter. Your aphorism, that every room in which Sebastian Bach is sung is transformed into a church, I consider peculiarly appropriate; and when I once heard the last movement of the piece in question, it made a similar impression on myself; but I own I cannot overcome my dislike to figured chorales in general, because I cannot understand the fundamental idea on which they are based, especially where the contending parts are maintained in an equal balance of power. For example, in the first chorus of the “Passion,”—where the chorale forms only a more important and consistent part of the basis; or where, as in the above-mentioned movement of the cantata (if I remember it rightly, having only heard it once), the chorale represents the principal building, and the individual parts only the decorations,—I can better comprehend the purpose and the conception; but not so certainly where the figure, in a certain manner, carries out variations on the theme. No liberties ought ever assuredly to be taken with a chorale. Its highest purpose is, that the congregation should sing it in all its purity to the accompaniment of the organ; all else seems to me idle and inappropriate for a church. At Fanny’s last morning’s music the motett of Bach, “Gottes Zeit ist die allerbeste Zeit,” and your “Ave Maria,” were sung by select voices. A long passage in the middle of the latter, as well as the end also, appeared to me too learned and intricate to accord with the simple piety, and certainly genuine catholic spirit, which pervades the rest of the music. Rebecca remarked that there was some confusion in the execution of those very passages which I considered too intricate; but this only proves that I am an ignoramus, but not that the conclusion is not too abstrusely modulated. With regard to Bach, the composition in question seems to me worthy of the
- 50. highest admiration. It is long since I have been so struck, or surprised by anything, as by the Introduction, which Fanny played most beautifully; and I could not help thinking of Bach’s solitary position, of his isolated condition with regard to his associates and his contemporaries, of his pure, mild, and vast power, and the transparency of its depths. The particular pieces which at the time were for ever engraved on my memory, were “Bestelle dein Haus,” and “Es ist der alte Bund.” I cared less for the bass air, or the alt solos. What first, through his “Passion,” seemed quite clear to me—that Bach is the musical type of Protestantism—becomes either negatively or positively more apparent to me every time that I hear a new piece of his; and thus it was recently with a Mass that I heard in the Academy, and which I consider most decidedly anti-Catholic; and, consequently, even all its great beauties seemed as unable to reconcile the inward contradiction, as if I were to hear a Protestant clergyman performing Mass in a Protestant Church. Moreover, I felt more strongly than ever what a great merit it was on Zelter’s part to restore Bach to the Germans; for, between Forkel’s day and his, very little was ever said about Bach, and even then principally with regard to his “wohltemperirte Clavier.” He was the first person on whom the light of Bach clearly dawned, through the acquisition of his other works, with which, as a collector of music, he became acquainted, and, as a genuine artist, imparted this knowledge to others. His musical performances on Fridays were indeed a proof that no work begun in earnest, and followed up with quiet perseverance, can fail ultimately to command success. At all events, it is an undoubted fact, that without Zelter, your own musical tendencies would have been of a totally different nature. Your intention to restore Handel in his original form, has led me to some reflections on his later style of instrumentation. A question is not unfrequently raised as to whether Handel, if he wrote in our day, would make use of all the existing musical facilities in composing his oratorios,— which, in fact, only means whether the wonted artistic form to which we give the name of Handel, would assume the same shape now that it did a hundred years ago; and the answer to this presents itself at once. The question, however, ought to be put in a different form,—not whether Handel would compose his oratorios now as he did a century since, but rather, whether he would compose any oratorios whatever; hardly—if they must be written in the style of those of the present day.
- 51. From my saying this to you, you may gather with what eager anticipations and confidence I look forward to your oratorio, which will, I trust, solve the problem of combining ancient conceptions with modern appliances; otherwise the result would be as great a failure as that of the painters of the nineteenth century, who only make themselves ridiculous by attempting to revive the religious elements of the fifteenth, with its long arms and legs, and topsy-turvy perspective. These new resources seem to me, like everything else in the world, to have been developed just at the right time, in order to animate the inner impulses which were daily becoming more feeble. On the heights of religious feeling, on which Bach, Handel, and their contemporaries stood, they required no numerous orchestras for their oratorios; and I can remember perfectly in my earliest years, the “Messiah,” “Judas,” and “Alexander’s Feast” being given exactly as Handel wrote them, without even an organ, and yet to the delight and edification of every one. But how is this to be managed nowadays, when vacuity of thought and noise in music are gradually being developed in inverse relation to each other? The orchestra, however, is now established, and is likely long to maintain its present form without any essential modification. Riches are only a fault when we do not know how to spend them. How, then, is the wealth of the orchestra to be applied? What guidance can the poet give for this, and to what regions? or is music to be entirely severed from poetry, and work its own independent way? I do not believe it can accomplish the latter, at least, only to a very limited extent, and not available for the world at large; to effect the former, an object must be found for music as well as for painting, which, by its fervour, its universal sufficiency and perspicuity, may supply the place of the pious emotions of former days. It seems to me that both the oratorios of Haydn were, in their sphere, also very remarkable phenomena. The poems of both are weak, regarded as poetry; but they have replaced the old positive and almost metaphysical religious impulses, by those which nature, as a visible emanation from the Godhead, in her universality, and her thousandfold individualities, instils into every susceptible heart. Hence the profound depth, but also the cheerful efficiency, and certainly genuine religious influence, of these two works, which hitherto stand alone; hence the combined effect of the playful and detached passages, with the most noble and sincere feelings of gratitude produced by the whole; hence is it also, that I individually could as little endure to lose in the “Creation” and in the “Seasons” the crowing of the cock, the singing of the lark, the lowing of
- 52. the cattle, and the rustic glee of the peasants, as I could in nature herself; in other words, the “Creation” and the “Seasons” are founded on nature and the visible service of God,—and are no new materials for music to be found there? The publication of Goethe’s “Correspondence with a Child” I consider a most provoking and pernicious abuse of the press, through which, more and more rapidly, all illusions will be destroyed, without which life is only death. You, I trust, will never lose your illusions, and ever preserve your filial attachment to your father.
- 53. To his Father. Düsseldorf, March 23rd, 1835. Dear Father, I have still to thank you for your last letter and my “Ave.” I often cannot understand how it is possible to have so acute a judgment with regard to music, without being yourself technically musical; and if I could express, what I assuredly feel, with as much clearness and intuitive perception as you do, as soon as you enter on the subject, I never would make another obscure speech all my life long. I thank you a thousand times for this, and also for your opinion of Bach. I ought to feel rather provoked that after only one very imperfect hearing of my composition, you at once discovered what after long familiarity on my part, I have only just found out; but then again it pleases me to see your definite sense of music, for the deficiencies in the middle movement and at the end consist of such minute faults, which might have been remedied by a very few notes (I mean struck out), that neither I, nor any other musician would have been aware of them, without repeatedly hearing the piece, because we in fact seek the cause much deeper. They injure the simplicity of the harmony, which at the beginning pleases me; and though it is my opinion that these faults would be less perceptible if properly executed, that is, with a numerous choir, still some traces of them will always remain. Another time I shall endeavour to do better. I should like you, however, to hear the Bach again, because there is a part of it which you care less for, but which pleases me best of all. I allude to the alto and bass airs; only the chorale must be given by a number of alto voices, and the bass very well sung. However fine the airs “Bestelle dein Haus” and “Es ist der alte Bund” may be, still there is something very sublime and profound in the plan of the ensuing movements, in the mode in which the alto begins, the bass then interposing with freshness and spirit, and continuing the same words, while the chorale comes in as a third, the bass closing exultantly, but the chorale not till long afterwards, dying away softly and solemnly. There is one peculiarity of this music,—its date must be placed either very early or very late, for it entirely differs from his usual style of writing in middle age; the first choral movements and the final chorus being of a kind that I should never have attributed to Sebastian Bach, but to some other composer of his
- 54. day; while no other man in the world could have written a single bar of the middle movements. My Mother does not judge Hiller rightly, for, in spite of his pleasures and honours in Paris, and the neglect he met with in Frankfort, he writes to me that he envies me my position here on the Rhine, even with all its drawbacks; and as, no doubt, a similar one may still be met with in Germany, I do not give up the hope of prevailing on him to forsake the Parisian atmosphere of pleasures and honours, and return to his studio. Now farewell, dear Father. I beg you soon let me hear from you again.—Your Felix.
- 55. To his Father. Düsseldorf, April 3rd, 1835. Dear Father, I am delighted to hear that you are satisfied with the programme of the Cologne Musical Festival. I shall not be able to play the organ for “Solomon,” as it must stand in the background of the orchestra and accompany almost every piece, the choruses and other performers here being accustomed to constant beating of time. I must therefore transcribe the whole of the organ part in the manner in which I think it ought to be played, and the cathedral organist there, Weber, will play it; I am told he is a sound musician and first-rate player. This is all so far well, and only gives me the great labour of transcribing, as I wish to have the performance as perfect as possible. I have had a good deal of trouble too with the “Morgengesang,”[20] as there is much in it that requires alteration, owing to the impossibility of executing it as written, with the means we have here. In doing so, however, it again caused me extreme pleasure, especially the stars, the moon, the elements, and the whole of the admirable finale. At the words “und schlich in dieser Nacht,” etc., it becomes so romantic and poetical, that each time I hear it I feel more touched and charmed; it therefore gratifies me to be of any use to so noble a man. The Comité were very much surprised when I maintained that it was a fine composition, and scarcely would consent to have it, but at that moment they were in a mood to be persuaded to anything. I would also have insisted on their giving an overture of Bach’s, if I had not dreaded too strong a counter-revolution. There is to be nothing of mine; therefore (from gratitude, I presume) they persist that my “admirable likeness” shall appear and be published by Whitsunday, a project from which I gallantly defend myself, refusing either to sit or stand for the purpose, having a particular objection to such pretensions. You must be well aware that your presence at the festival would not only be no gêne to me, but on the contrary, would cause me first to feel true joy and delight in my success. Allow me to take this opportunity to say to you, that the approbation and enjoyment of the public, to which I am certainly very sensible, only causes me real satisfaction when I can write to tell you of it, because I know it rejoices you, and one word of praise from you is more
- 56. truly precious to me, and makes me happier, than all the publics in the world applauding me in concert; and thus to see you among the audience, would be the dearest of all rewards to me for my labours. My oratorio[21] is to be performed in Frankfort in November, so Schelble writes to me; and much as I should like you to hear it soon, still I should prefer your hearing it first next year, at the Musical Festival. Before decidedly accepting the proposal, I have stipulated to wait till after the performance at Frankfort, that I may judge whether it be suitable for the festival; but should this prove to be the case, as I hope and wish it may, it will have a much finer effect there, and besides it is the festival that you like, and Whitsunday instead of November; and above all, I shall then know whether it pleases you or not, on which point I feel by no means sure. I cannot close this letter without speaking of the heavenly weather that delights us here. Light balmy air and sunshine, and a profusion of green, and larks! To-day I rode through the forest, and stopped for at least a quarter of an hour to listen to the birds, who in the deep solitude were fluttering about incessantly and warbling.—Your Felix.
- 57. To Herr Conrad Schleinitz, Leipzig. Düsseldorf, April 16th, 1835. Sir, I thank you cordially for your last letter, and for the friendly interest which you take in me, and in my coming to Leipzig. As I perceive by the Herr Stadtrath Porsche’s letter, as well as by that of the Superintendent of the concerts, that my going there does not interfere with any other person, one great difficulty is thus obviated. But another has now arisen, as the letter of the Superintendent contains different views with regard to the situation from yours. The direction of twenty concerts and extra concerts is named as among the duties, but a benefit concert (about which you wrote to me) is not mentioned. I have consequently said in my reply what I formerly wrote to you, that in order to induce me to consent to the exchange, I wish to see the same pecuniary advantages secured to me that I enjoy here. If a benefit concert, as you say, would bring from 200 to 300 dollars, this sum would certainly be a considerable increase to my salary; but I must say that I never made such a proposal, and indeed would not have accepted it, had it been made to me. It would be a different thing if the association chose to give an additional concert, and to devote a share of the profits towards the increase of my established salary. During my musical career, I have always resolved never to give a concert for myself (for my own benefit). You probably are aware that, personally, pecuniary considerations would be of less importance to me, were it not that my parents (and I think rightly) exact from me that I should follow my art as a profession, and gain my livelihood by means of it. I, however, reserved the power of declining certain things which, in reference to my favoured position in this respect, I will never do; for example, giving concerts or lessons. But I quite acknowledge the propriety of what my parents insist on so strongly, that in all other relations I shall gladly consider myself as a musician who lives by his profession. Thus, before giving up my present situation, I must ascertain that one equally advantageous is secured to me. I do not consider that what I require is at all presumptuous, as it has been offered to me here, and on this account I trust that a similar course may be pursued in Leipzig. An association was at that time formed here, who entrusted to me the duty of conducting the Vocal
- 58. Association, concerts, etc., and made up my salary partly in common with the Vocal Association, and partly by the profits of the concerts. Whether anything of this kind be possible with you, or whether it could be equalized by an additional concert, or whether the execution of particular duties is to be imposed on me, I cannot of course pretend to decide. I only wish that, in one way or another, a definite position should be assured to me, like the one I enjoy here; and if your idea about the benefit concert could be modified and carried out, there would then be a good hope for me that the affair might turn out according to my wish. If you can induce the directors to fulfil the wishes I have expressed, you will exceedingly oblige me, for you know how welcome a residence and active employment in your city would be to me. In any event, continue your friendly feelings towards me, and accept my thanks for them.
- 59. To the Herr Regierungs-Secretair Hixte, Cologne. Düsseldorf, May 18th, 1835. Sir, I thank you much for the kind letter you have gratified me by addressing to me. The idea which you communicate in it is very flattering for me, and yet I confess that I feel a certain degree of dislike to do what you propose, and for a long time past I have entertained this feeling. It is now so very much the fashion for obscure or commonplace people to have their likeness given to the public, in order to become more known, and for young beginners to do so at first starting in life, that I have always had a dread of doing so too soon. I do not wish that my likeness should be taken, until I have accomplished something to render me more worthy, according to my idea, of such an honour. This, however, not being yet the case, I beg to defer such a compliment till I am more deserving of it; but receive my best thanks for the friendly good-nature with which you made me this offer.[22]—I am, etc., Felix Mendelssohn Bartholdy.
- 60. To his Family. Leipzig, October 6th, 1835. For a week past I have been seeking for a leisure hour to answer, and to thank you for the charming letters I have received from you; but the London days, with their distractions, were not worse than the time has been since Fanny left this till now. At length, after the successful result of the first concert, I have at last a certain degree of rest. The day after I accompanied the Hensels to Delitsch, Chopin came; he intended only to remain one day, so we spent this entirely together in music. I cannot deny, dear Fanny, that I have lately found that you by no means do him justice in your judgment of his talents; perhaps he was not in a humour for playing when you heard him, which may not unfrequently be the case with him. But his playing has enchanted me afresh, and I am persuaded that if you, and my Father also, had heard some of his better pieces, as he played them to me, you would say the same. There is something thoroughly original in his pianoforte playing, and at the same time so masterly, that he may be called a most perfect virtuoso; and as every style of perfection is welcome and acceptable, that day was most agreeable to me, although so entirely different from the previous ones with you,—the Hensels. It was so pleasant for me to be once more with a thorough musician, and not with those half virtuosos and half classics, who would gladly combine les honneurs de la vertu et les plaisirs du vice, but with one who has his perfect and well-defined phase; and however far asunder we may be in our different spheres, still I can get on famously with such a person; but not with those half-and-half people. Sunday evening was really very remarkable when Chopin made me play over my oratorio to him, while curious Leipzigers stole into the room to see him, and when between the first and second part he dashed into his new Études and a new concerto, to the amazement of the Leipzigers, and then I resumed my “St. Paul;” it was just as if a Cherokee and a Kaffir had met to converse. He has also such a lovely new notturno, a considerable part of which I learnt by ear for the purpose of playing it for Paul’s amusement. So we got on most pleasantly together; and he promised faithfully to return in the course of the winter, when I intend to compose a new symphony, and to perform it in honour of him. We vowed
- 61. these things in the presence of three witnesses, and we shall see whether we both adhere to our word. My collection of Handel’s works arrived before Chopin’s departure, and were a source of quite childish delight to him; they really are so beautiful that I am charmed with them; thirty-two great folios, bound in thick green leather, in the regular nice English fashion, and on the back, in big gold letters, the title and contents of each volume; and in the first volume, besides, there are the following words, “To Director F. M. B., from the Committee of the Cologne Musical Festival, 1835.” The books were accompanied by a very civil letter, with the signatures of all the Committee, and on taking up one of the volumes at random it happened to be “Samson,” and just at the very beginning I found a grand aria for Samson which is quite unknown, because Herr von Mosel struck it out, and which yields in beauty to none of Handel’s; so you see what pleasure is in store for me in all the thirty-two volumes. You may imagine my delight. Before setting off on his journey Moscheles came to see me, and during the first half-hour he played over my second book of “songs without words” to my extreme pleasure. He is not the least changed, only somewhat older in appearance, but otherwise as fresh and in as good spirits as ever, and playing quite splendidly; another kind of perfect virtuoso and master combined. The rehearsals of the first subscription gradually drew near, and the day before yesterday my Leipzig music-directorship commenced. I cannot tell you how much I am satisfied with this beginning, and with the whole aspect of my position here. It is a quiet, regular, official business. That the Institute has been established for fifty-six years is very perceptible, and moreover, the people seem most friendly and well-disposed towards me and my music. The orchestra is very good, and thoroughly musical; and I think that six months hence it will be much improved, for the sympathy and attention with which these people receive my suggestions, and instantly adopt them, were really touching in both the rehearsals we have hitherto had; there was as great a difference as if another orchestra had been playing. There are still some deficiencies in the orchestra, but these will be supplied by degrees; and I look forward to a succession of pleasant evenings and good performances. I wish you had heard the introduction to my “Meeresstille” (for the concert began with that); there was such profound silence in the hall and in the orchestra, that the most delicate notes could be distinctly heard, and they played the adagio from first to last in the most masterly manner; the allegro not quite so well; for being accustomed to a slower tempo, they rather dragged; but at the end, where the slow time 4/4 ff begins, they went
- 62. capitally; the violins attacking it with a degree of vehemence that quite startled me and delighted the publicus. The following pieces, an air in E major of Weber, a violin concerto by Spohr, and the introduction to “Ali Baba” did not go so well; the one rehearsal was not sufficient, and they were often unsteady; but, on the other hand, Beethoven’s B flat symphony, which formed the second part, was splendidly given, so that the Leipzigers shouted with delight at the close of each movement. I never in any orchestra saw such zeal and excitement; they listened like—popinjays, Zelter would say. After the concert I received, and offered in turn, a mass of congratulations: first the orchestra, then the Thomas School collegians (who are capital fellows, and go to college, and are dismissed so punctually that I have promised them an order); then came Moscheles, with a Court suite of dilettanti, then two editors of musical papers, and so on. Moscheles’ concert is on Friday, and I am to play his piece for two pianos[23] with him, and he is to play my new pianoforte-concerto. My “Hebrides” have also contrived to creep into the concert. This afternoon Moscheles, Clara Wieck, and I, play Sebastian Bach’s triple concerto in D minor. How amiable Moscheles is towards myself, how cordially he is interested in my situation here, how it delights me that he is so satisfied with it, how he plays my rondo in E flat to my great admiration, and far better than I originally conceived it, and how we dine together every forenoon in his hotel, and every evening drink tea and have music in mine,—all this you can imagine for yourself, for you know him,—especially you, dear Father. These are pleasant days; and if I have not much leisure to work, I mean to make up for it hereafter, and shall derive as much benefit from it then as now. My first concert caused me no perturbation, dear Mother, but to my shame I confess, that I never felt so embarrassed at the moment of appearing as on that occasion; I believe it arose from our long correspondence and treaty on the subject, and I had never before seen a concert of the kind. The locality and the lights confused me. Now farewell all. May you be well and happy, and pray write to me very often.—Your Felix.
- 63. To Pastor Julius Schubring, Dessau. Leipzig, December 6th, 1835. Dear Schubring, You have no doubt heard of the heavy stroke that has fallen on my happy life and those dear to me.[24] It is the greatest misfortune that could have befallen me, and a trial that I must either strive to bear up against, or must utterly sink under. I say this to myself after the lapse of three weeks, without the acute anguish of the first days, but I now feel it even more deeply; a new life must now begin for me, or all must be at an end,—the old life is now severed. For our consolation and example, our Mother bears her loss with the most wonderful composure and firmness; she comforts herself with her children and grandchildren, and thus strives to hide the chasm that never can be filled up. My Brother and Sisters do what they can to fulfil their duties better than ever, the more difficult they have become. I was ten days in Berlin, that by my presence my Mother should at least be surrounded by her whole family; but I need scarcely tell you what these days were; you know it well, and no doubt you thought of me in that dark hour. God granted to my Father the prayer that he had often uttered; his end was as peaceful and quiet, and as sudden and unexpected as he desired. On Wednesday, the 18th, he was surrounded by all his family, went to bed late the same evening, complained a little early on Thursday, and at half-past eleven his life was ended. The physicians can give his malady no name. It seems that my grandfather Moses died in a similar manner,—so my uncle told us,—at the same age, without sickness, and in a calm and cheerful frame of mind. I do not know whether you are aware that more especially for some years past, my Father was so good to me, so thoroughly my friend, that I was devoted to him with my whole soul, and during my long absence I scarcely ever passed an hour without thinking of him; but as you knew him in his own home with us, in all his kindliness, you can well realize my state of mind. The only thing that now remains is to do one’s duty, and this I strive to accomplish with all my strength, for he would wish it to be so if he were still present, and I shall never cease to endeavour to gain his approval as I formerly did, though I can no longer enjoy it. When I delayed answering your letter, I little thought that I should have to answer it thus; let me thank you for it now, and
- 64. for all your kindness. One passage for “St. Paul” was excellent, “der Du der rechte Vater bist.” I have a chorus in my head for it which I intend shortly to write down. I shall now work with double zeal at the completion of “St. Paul” for my Father urged me to it in the very last letter he wrote to me, and he looked forward very impatiently to the completion of my work. I feel as if I must exert all my energies to finish it, and make it as good as possible, and then think that he takes an interest in it. If any good passages occur to you, pray send them to me, for you know the intention of the whole. To-day, for the first time, I have begun once more to work at it, and intend now to do so daily. When it is concluded, what is to come next, God will direct. Farewell, dear Schubring, bear me in your thoughts.—Your Felix Mendelssohn Bartholdy.
- 65. To Pastor Bauer, Beszig. Leipzig, December 9th, 1835. I received your kind letter here, on the very day when the christening in your family was to take place, on my return from Berlin, where I had gone in the hope of alleviating my Mother’s grief, immediately after the loss of my Father. So I received the intelligence of your happiness, on again crossing the threshold of my empty room, when I felt for the first time in my inmost being, what it is to suffer the most painful and bitter anguish. Indeed the wish which of all others every night recurred to my mind, was that I might not survive my loss, because I so entirely clung to my Father, or rather still cling to him, that I do not know how I can now pass my life, for not only have I to deplore the loss of a father (a sorrow which of all others from my childhood I always thought the most acute), but also that of my best and most perfect friend during the last few years, and my instructor in art and in life. It seemed to me so strange, reading your letter, which breathed only joy and satisfaction, calling on me to rejoice with you on your future prospects, at the moment when I felt that my past was lost and gone for ever; but I thank you for wishing me, though so distant, to become your guest at the christening; and though my name may make a graver impression now than you probably thought, I trust that impression will only be a grave, and not a painful one, to you and your wife; and when, in later years, you tell your child of those whom you invited to his baptism, do not omit my name from your guests, but say to him that one of them on that day recommenced his life afresh,—though in another sense, with new purposes and wishes, and with new prayers to God. My Mother is well, and bears her sorrow with such composure and dignity that we can all only wonder and admire, and ascribe it to her love for her children, and her wish for their happiness. As for myself, when I tell you that I strive to do my duty and thus to win my Father’s approval now as I always formerly did, and devote to the completion of “St. Paul,” in which he took such pleasure, all the energies of my mind, to make it as good as I possibly can; when I say that I force myself to the performance of my duties here, not to pass quite unprofitably these first days of sorrow, when to be
- 66. perfectly idle is most consonant to one’s feelings; that, lastly, the people here are most kind and sympathizing, and endeavour to make life as little painful to me as they can,—you know the aspect of my inner and outer life at this moment. Farewell.
- 67. To Ferdinand Hiller. Leipzig, January 24th, 1836. My dear Ferdinand, I now send you my promised report of the performance of your D minor overture, which took place last Thursday evening. It was well executed by the orchestra; we had studied it repeatedly and carefully, and a great many of the passages sounded so well as to exceed my expectations. The most beautiful of all was the first passage in A minor, piano, given by wind instruments, followed by the melody,—which had an admirable effect; and also at the beginning of the free fantasia, the forte in G minor, and then the piano, (your favourite passage,) likewise the trombones and wind instruments, piano, at the end in D major. The Finale, too, exceeded my expectations in the orchestra. But, trusting to our good understanding, I could not resist striking out, after the first rehearsal, the staccato double- basses in the melody in A major, and each time the passage recurred in F and D major, replacing them by sustained notes; you can’t think how confused the effect was, and therefore I hope you will not take this liberty amiss. I am convinced you would have done the same; it did not sound as you would have liked. I have something else, too, on my conscience that I must tell you. The Overture neither excited myself nor the musicians during its performance as I could have wished; it left us rather cold. This would have been of little consequence, but it was remarkable that all the musicians to whom I spoke said the same. The first theme and all the beginning, the melodies in A minor and A major, particularly delighted them; and up to that point they had all felt enthusiastic, but then their sympathy gradually subsided; till, when the close came, they had quite forgotten the striking impression of the theme, and no longer felt any interest in the music. This seems to me important, for I think it is connected with the difference which we have so repeatedly discussed together, and the want of interest with which you at all times regard your art, being now at length become perceptible to others. I would not say this to you, were it not that I am perfectly convinced of this being a point which must be left to each individual, as neither nature nor talents, even of the highest order, can remedy it; a man’s own will alone can do so.
- 68. Nothing is more repugnant to me than casting blame on the nature or genius of any one; it only renders him irritable and bewildered, and does no good. No man can add one inch to his stature: in such a case all striving and toiling is vain, therefore it is best to be silent. Providence is answerable for this defect in his nature. But if it be the case, as it is with this work of yours, that precisely those very themes, and all that requires talent or genius (call it as you will), is excellent and beautiful and touching, but the development not so good,—then, I think, silence should not be observed; then, I think, blame can never be unwise, for this is the point where great progress can be made by the composer himself in his works; and as I believe that a man with fine capabilities has the absolute duty imposed on him of becoming something really superior, so I think that blame must be attributed to him, if he does not develope himself according to the means with which he is endowed. And I maintain that it is the same with a musical composition. Do not tell me that it is so, and therefore it must remain so. I know well that no musician can alter the thoughts and talents which Heaven has bestowed on him; but I also know that when Providence grants him superior ones, he must also develope them properly. Do not declare, either, that we were all mistaken, and that the execution was as much in fault as the composition. I do not believe it. I do believe that your talents are such that you are inferior to no musician, but I scarcely know one piece of yours that is systematically carried out. The two overtures are certainly your best pieces, but the more distinctly you express your thoughts, the more perceptible are the defects, and in my opinion you must rectify them. Do not ask me how, for that you know best yourself. After all, it is only the affair of a walk, or a moment,—in short, of a thought. If you laugh at me for this long lecture, perhaps you may be quite right; but certainly not so if you are displeased, or bear me a grudge for it; though indeed it is very stupid in me even to suggest such a possibility. But how many musicians are there who would permit another to address them thus? And though you must see in every expression of mine how much I love and revere your genius, still I have told you that you are not absolute perfection, and this musicians usually take highly amiss. But you will not: you know my sincere interest in you too well.
- 69. To Fanny Hensel, Berlin. Leipzig, January 30th, 1836. Dear Fanny, To-day at length I can reply to your charming letters, and lecture you severely for saying in your first letter that it was long since you had been able to please me by your music, and asking me how this was. I totally deny this to be the fact, and assure you that all you compose pleases me. If two or three things in succession did not satisfy me as entirely as others of yours, I think the ground lay no deeper than this, that you have written less than in former days, when one or two songs that did not exactly suit my taste were so rapidly composed, and replaced so quickly by others, that neither of us considered much why it was that they were less attractive; we only laughed together about them, and there was an end of it. I may quote here “Die Schönheit nicht, O Mädchen,” and many others in the “prima maniera of our master” which we heartily abused. Then came beautiful songs in their turn, and so it is at present, only they cannot follow each other in such quick succession, because you must often now have other things to occupy your thoughts besides composing pretty songs, and that is a great blessing. But if you suppose that your more recent compositions seem to me inferior to your earlier ones, you are most entirely and totally mistaken, for I know no song of yours better than the English one in G minor, or the close of the “Liederkreis,” and many others of later date; besides, you are aware that formerly there were entire books of your composition that were less acceptable to me than others, because my nature always was to be a screech-owl, and to belong to the savage tribe of brothers. But you know well how much I love all your productions, and some are especially dear to my heart; so I trust that you will write to me forthwith that you have done me injustice, by considering me a man devoid of taste, and that you will never again do so. And then, neither in this letter nor in your former one do you say one word about “St. Paul” or “Melusina,” as one colleague should write to another,—that is, remarks on fifths, rhythm, and motion of the parts, on conceptions, counterpoint, et cætera animalia. You ought to have done so, however, and should do so still, for you know the value I attach to this; and
- 70. as “St. Paul” is shortly to be sent to the publisher, a few strictures from you would come just at the right moment. I write to you to-day solely in the hope of soon receiving an answer from you, for I am very weary and exhausted from yesterday’s concert, where, in addition to conducting three times, I was obliged to play Mozart’s D minor concerto. In the first movement I made a cadenza, which succeeded famously, and caused a tremendous sensation among the Leipzigers. I must write down the end of it for you. You remember the theme, of course? Towards the close of the cadence, arpeggios come in pianissimo in D minor, thus—
- 71. Then again G minor arpeggios; then Then arpeggios, and
- 72. etc., to the close in D minor. Our second violin player, an old musician, said to me afterwards, when he met me in the passage, that he had heard it played in the same Hall by Mozart himself, but since that day he had heard no one introduce such good cadenzas as I did yesterday, which gave me very great pleasure. Do you know Handel’s “Coronation Anthem”? It is most singular. The beginning is one of the finest which not only Handel, but any man, ever composed; and all the remainder, after the first short movement, horridly dry and commonplace. The performers could not master it, but are certainly far too busy to grieve much about that. Many persons here consider “Melusina” to be my best overture; at all events, it is the most deeply felt; but as to the fabulous nonsense of the musical papers, about red coral and green sea monsters, and magic palaces, and deep seas, this is stupid stuff, and fills me with amazement. But now I take my leave of water for some time to come, and must see how things are going on elsewhere.[25] I received to-day a letter from Düsseldorf, with the
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