Decomposition and modeling in the non manifold domain

Decomposition and Modeling
in the Non-Manifold Domain
Phd. Thesis; Phd student: Franco Morando
University of Genoa
Genoa (Italy)
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F. Morando: Decomposition and Modeling in the Non-Manifold Domain
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What is a Non-Manifold Object?
 Manifold object in nD:: every point has
a neighborhood homeomorphic to
either a n-ball (internal point) or to a n-
dimensional closed half-plane
(boundary point).
 Non-manifold points do not have a
neighborhood that fulfills the definition
above.
 Non-manifold objects contain some
non-manifold point.
 Singularity:: a connected set of non-
manifold points

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 A k-dimensional simplex (or k-simplex) σ in Ed
is the locus of the points in Ed
that can be
expressed as the convex combination of k+1
linearly independent points.
 Such points are called vertices of σ
 Given a k-simplex σ, any s-simplex τ
generated by a subset of s+1<= k vertices of σ
is called an s-face of σ.
Geometric Simplex

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Examples of Non-Manifold Objects

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Non-Manifold Modeling: Why?
 To represent and manipulate objects which
combine wire-frames, surfaces, and solid parts
in CAD/CAM applications.
 Complex spatial objects are described by
meshes with a non-manifoldnon-manifold and non-regularnon-regular
domain.
 Non-manifold and non-regular meshes are
generated by topology-modifying simplificationtopology-modifying simplification
algorithms.

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Non-Manifold Objects are Difficult
 The mathematics of non-manifolds is not well
understood.
 Identifying and classifying singularities is not
easy.
 Data structures must support object traversal and
editing, but traversing singularities could be
difficult.

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Related work - Non-manifold
decomposition
 Non-manifold spineNon-manifold spine [Desaulniers and Stewart, 1992]
Manifold components, efficient, supports extended Euler
operators
2D only, r-sets only
 Cutting and StitchingCutting and Stitching [Gueziec et al., 1998]
Manifold components, efficient, supports compression
2D only, r-sets only
 MatchmakerMatchmaker [Rossignac and Cardoze, 1999]
Compact and efficient
Pseudomanifold components, 2D only, r-sets only

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Outline
 Modeling with simplicial complexes
 A scalable data structure for the 2D case
 Decomposition of non-manifold meshes
 The decomposition lattice
 The standard decomposition
 The class of initial-quasi manifold complexes
 A two-level data structure for the 2D and 3D
cases
 Conclusions
 Applications

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 A finite collection Σ of simplexes such that:
 for each simplex σ ∈ Σ, all faces of σ belong to Σ
 for each pair of simplexes σ and τ, either σ∩τ = ∅ or
σ∩τ is a face of both σ and τ.
Simplicial complex

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The 2D case (triangle-segment
meshes)
Wire-edge:Wire-edge: no
triangle incident at it
Different types of edges (in red)
Triangle-edge:Triangle-edge: at least one
triangle incident at it
Connected simplicial complex
in 2D (triangle-segmenttriangle-segment mesh)

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Manifold and non-manifold edges
in 2D
 Manifold edge:Manifold edge: exactly one or two triangles
incident at it
in red 8 manifold triangle
edges
in red a non manifold edge

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Manifold and non-manifold vertices in 2D
 Manifold vertex:Manifold vertex:
– No incident triangle and one/two incident edges
OR
– No incident wire-edges and incident triangles form a
single fan
in red 6 manifold vertices in red 3 non-manifold vertices

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Decomposition
 Basic idea in this thesis:
 Cut the complex at non-manifold cells
 Duplicate cells at cuts
 Cuts remove singularities and reveal
components.
 Object disconnected into components:components:
 Each component has a regular structure
 Components are linked together at non-
manifold jointsjoints

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The nD case - Decomposition
 A uniform approach to the nD case:
 2D decompose mesh of triangles and segments
 3D decompose 2D+tetrahedralizations
 4D and above.

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Examples of Decomposition (2D)

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Examples of Decomposition (2D-3D)
2D
2D 3D

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Examples of Decomposition
This decomposition is possible only if the object on the left is
represented as a surface

F. Morando Decomposition and Modeling in the Non-Manifold Domain
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Minimal decomposition adding just one point is possible in
2D (surface mesh) and not possible in 3D (volume mesh)

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Hard to decide where to cut. Need a standard
solution.
In 3D, we can only cut through the volume

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In 2D, we can also inflate the surface
introducing another point
q
p'
p
p
r

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Requirements for decomposition
 Eliminate as many singularities as possible
 Do not cut through manifold parts
 Do not create too many parts (Compactness)
 Algorithm must be deterministic (Uniqueness)
 Contrasting requirements:
 Uniqueness
 Compactness

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Eliminate as many singularities as
possible (red edge must be fixed)
Do not cut through manifold parts
(green edge must stay)

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Compactness: Do not create too
many parts (2 parts seems
enough)
Shall we avoid this?

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Uniqueness
 In green the three possible manifold
decompositions with least number of parts
 Minimal decomposition is not unique
 In yellow maximal decomposition
 Maximal decomposition is
unique

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A formal approach to decomposition
 Consider all possible decompositions of a complex
 Reject decompositions:
 that cut through manifold faces (too much decomposed)
 that still contain singularities which could be eliminated
without cutting through manifold faces (not decomposed
enough)
 The remaining decompositions are admissible
[De Floriani, Mesmoudi, Morando, Puppo, DGCI01]

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The decomposition lattice
Decompositions form a lattice having its bottom at
the input complex and its top at the totally
exploded version of the same complex
Top
Bottom

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Stitching equivalences
 Each arc in the lattice is an operation that
identifies/splits two vertices (stitching
equivalence)

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The standard decomposition
 The standard decompositionstandard decomposition is the least upper
bound (the most decomposed complex) in the
sub-lattice of admissible decompositions that
do not cut through manifold faces
 The standard decomposition can be computed
in linear time w.r.t. the number of vertices in
the output and in t log t w.r.t. the number of
simplices t in the input.

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Standard
decomposition

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Standard
decomposition
Admissible
decompositions

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 Takes the most decomposed among
admissible decompositions (blue area)
 is Unique
 disconnects only at singular joints

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 The most compact among admissible
decompositions
 is not unique.
 there could be decompositions with the same
number of parts that are not even
topologically equivalent
 may not reveal parts correctly
Non standard decompositions…

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 More compact decompositions that are
not topologically equivalent
Non standard decompositions…
Non orientable
Orientable
Standard
Decomposition

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 The standard decomposition can be computed
in linear time w.r.t. the number of vertices in
the output and in t log t w.r.t. the number of
simplices t in the input.

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A local and recursive algorithm
 Initialize output complex ∇ at input complex Σ
 Analyze iteratively each vertex v of Σ :
 Consider the link of v. It’s a (d-1)-complex
 Decompose it
 If the link decomposition has more than one
connected component split v in ∇
 Introduce a copy of v for each connected
component in the decomposed link
(the link of a vertex v is the set of faces of cells in
the star of v, which are not incident at v)

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An example of decomposition
Link of u
Link of v

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Decomposed link of v in Σ
v splits into
two copies

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Decomposed link of u in Σ
u splits into
four
copies

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The result is the standard decomposition

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A light decomposition algorithm
 Computational complexity: O(n + t log t) where
 n is the number of vertices in the output
decomposition
 t is the number of top cells in the input complex

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 The standard decomposition may leave non
manifoldness whenever it unclear how to split.
 This reveals a new class of non-manifolds we
called initial quasi-manifoldinitial quasi-manifold (IQM).
 The connected components of the standard
decomposition are initial quasi-manifold..

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Initial quasi-manifolds
 The connected components of the standard
decomposition are initial quasi-manifoldinitial quasi-manifold (IQM)
 Some examples:
 The pinched pie considered as a set of tetrahedra
is a 3D IQM. Its boundary is not a 2D IQM since
the standard decomposition inflate it (on the right).
p'
pp

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Some examples of IQM…
 The double pinched pie considered as a
set of tetrahedra is a 3D IQM. Its
boundary is not a 2D IQM the standard
decomposition inflate it.
a
ba

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Initial quasi manifoldness is not…
 If we add a vertex in on the pinched
edge the resulting complex is not an
IQM
 The standard decomposition decompose
it into a torus hence...

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Initial quasi manifoldness is not…
 Initial quasi manifoldness depends on
the particular triangulation
 It is not a topoligical property
 It is a combinatorial property

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Initial quasi-manifolds facts…
 A complex is initial quasi-manifold if and only if
the star of each vertex is (d-1)-manifold
connected
 The class of initial quasi-manifolds is the class
of all those complexes that cannot be
decomposed without cutting through manifold
faces
NO
YES

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Initial quasi-manifolds facts…
 In 2D: initial quasi-manifold coincides with
manifold
 The standard 2-decomposition is made of
manifold components
 For D≥3 initial quasi-manifolds could be non-
manifold and possibly even non-pseudomanifold
 There are IQM tetrahedralizations where, 3
tetrahedra are incident to a triangle.
 For D=3 an IQM complex embedded in 3D
space, must be a pseudomanifold
 Thus the standard decomposition is made of
pseudomanifold components: it is easy to
navigate components through face adjacencies

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Initial quasi-manifolds
 Encoding IQMs:
 In an IQM the star of each vertex is (d-1)-manifold
connected
 This means that we can easily navigate the whole
star of each vertex through (d-1)-adjacencies
 This is important to design data structures

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A two-level data structure for 3D
meshes in E3
 Encode separately decomposition and IQM
components.
 In IQMs the star of each vertex is (d-1)-
manifold connected so we can easily encode
the whole star of each vertex through (d-1)-
adjacencies
[De Floriani, Morando, Puppo, ongoing work]

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Related work - Non-manifold data
structures
 Radial-edge and extensionsRadial-edge and extensions [Weiler, 1988; Gursoz
et al., 1990; Yamaguchi and Kimura, 1995; Lee and Lee,
2001; McCains and Hellerstein, 2001]
General and efficient
Expensive, 2D only
 Selective Geometric ComplexesSelective Geometric Complexes [Rossignac and
O’Connor, 1990]
Very general, nD
Not efficient in mesh traversal (implemented with incidence
graphs)
 Directed edgesDirected edges [Campagna et al., 1999]
Compact and efficient
2D only, regular objects only

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A data structure for triangle-
segment meshes [De Floriani, Magillo, Puppo, Sobrero,
ACM-SMA 2002]
Stores only triangles and vertices. Edges are stored
implicitly as either triangle sides (triangle-edges) or vertex
adjacencies (wire-edges).
Maintains a reduced set of relations that permit to
retrieve all relations involving triangles and vertices in
optimal time

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segment meshes[De Floriani, Magillo, Puppo, Sobrero,
ACM-SMA 2002]
 For each triangle t stores links to its three
vertices (Triangle-Vertex relationTriangle-Vertex relation)
 For each edge e of t stores two triangles adjacent
to t if e is non-manifold, one triangle otherwise
(partial Triangle-Triangle relationpartial Triangle-Triangle relation)

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segment meshes [De Floriani, Magillo, Puppo,
Sobrero, ACM-SMA 2002]
 For each vertex v maintain:For each vertex v maintain:
 All vertices adjacent to v
through wire-edges
(Vertex-Vertex relationVertex-Vertex relation)
 One triangle incident for each
edge-connected component of
triangles incident at v
(partial Vertex-Trianglepartial Vertex-Triangle
relationrelation)

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Storage cost:Storage cost:
24m + 16n + 16l + 8c + 4a bytes and 3m +2n bits, where
 n number of vertices
 m number of triangles
 l number of wire edges
 c number of edge-connected components of triangles at
non-manifold vertices
 a number of triangles incident at non-manifold edges
In the manifold case:In the manifold case:
 m ≈ 2n, l=c=a=0
 Total cost is ≈ 65n bytes, only one byte more than
indexed structure with adjacencies

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It can be traversed through edge adjacencies and it
can be traversed around a vertex both in optimal timeoptimal time
(in linear w.r.t. the output size)
Downscale well to manifold meshes.Downscale well to manifold meshes. Overhead
limited to one byte per vertexone byte per vertex w.r.t. indexed data
structure with adjacencies when applied to manifold
meshes
More compactMore compact than specialization of existing non-
manifold data structures to 2D simplicial complexes
(cons?)… it does not represent explicitly singularitiessingularities
and componentscomponents of the object

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meshes in E3
 For each non-manifold vertex split by decomposition,
maintain a table of its vertex copies in the components of
the standard decomposition.
 Vertex tables can be accessed through hashing.
 Vertex tables are sufficient to navigate among different
components.
 Vertex-based and Tetrahedron-based relations can be
retrieved in optimal time.
 Edge-based and Triangle-based relations can be
retrieved in time proportional to the size of star of their
vertices
[De Floriani, Morando, Puppo, ongoing work]

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meshes in E3
 For each component of the standard decomposition:
 Store only tetrahedra and vertices
 For each vertex maintain:
 One incident tetrahedron (partial Vertex-
Tetrahedron relation)
 For each tetrahedron maintain
 List of vertices (Tetrahedron-Vertex relation)
 List of adjacent tetrahedra (Tetrahedron-
Tetrahedron relation)

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Summary
 A decomposition of nD complexes into initial quasi-
manifold components:
 Mathematically sound and unique decomposition
 Efficient algorithm
 A data structure for direct encoding of triangle-
segment meshes:
 Efficient
 More compact than other existing structures
 Scalable to the manifold case
 A two-level data structure for 3D meshes based on
standard decomposition:
 Compact and efficient
 Represents components and non-manifold joints explicitly

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Current and future work
 Extension of data structures to arbitrary
dimensions
 Non-manifold multiresolution modeling
 Non-manifold simplification
 Retrieval-by-sketch from 3D object databases
 3D icons

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Non-manifold simplification
 Iterative local simplification of geometry
 Simplification may change topology and
dimension
 Reveals the part-based structure of the object
[De Floriani, Mo, Morando, Pupagillpo, IWVF01]

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22288
triangles
161 triangles + 14 edges
18 tri + 7 edges 4 tri + 5 edges 6 edges

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11098
triangles
166 triangles
22 tri + 2 edges 3 edges5 tri + 1 edge

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1906 triangles 10 tri +30 edges

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Non-Manifold Multi-Tessellation (NMT)
[De Floriani, Magillo, Puppo, Sobrero, ACM-SMA 2002]
 Non-manifold simplification define a Directed Acyclic
Graph (NMT-DAG)

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Selective refinement on a
NMT
 Selective refinement:Selective refinement: extract a mesh from a NMT
satisfying some application-dependent requirements
(LOD criterion + maximal size)
 Extracted meshesExtracted meshes correspond to set S of
modifications closedclosed with respect to the partial
order defined by the DAG.

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 Different shapes are merged into a single non-manifold
3D icon by overlapping NMT-DAGs
 The standard decomposition of the non-manifold 3D
icon can be the starting point to define keys for the
database
Retrieval-by-sketch from 3D
object databases

F. Morando Decomposition and
Modeling in the Non-Manifold
Domain72
Thank you!
Joint work with:
L. De Floriani, P. Magillo, E. Puppo,
M.Mesmoudi, D. Sobrero.

Decomposition and modeling in the non manifold domain

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