Joint3DShapeMatching

Joint 3D Shape Matching
December 2015
Abstract
In this paper we extend the global optimization-based
approach of jointly matching a set of images to jointly
matching a set of 3D meshes. The estimated correspon-
dences simultaneously maximize pairwise feature affini-
ties and cycle consistency across multiple models. We
show that the low-rank matrix recovery problem can be
efficiently applied to the 3D meshes as well. The fast
alternating minimization algorithm helps to handle real
world practical problems with thousands of features. Ex-
perimental results show that, unlike the state-of-the-art
algorithm which rely on semi-definite programming, our
algorithm provides an order of magnitude speed-up along
with competitive performance. Along with the joint shape
matching we propose an approach to apply a distortion
term in pairwise matching, which helps in successfully
matching the reflexive sub-parts of two models distinc-
tively. In the end, we demonstrate the applicability of
the algorithm to match a set of 3D meshes of the SCAPE
benchmark database.
1 Introduction
Finding feature correspondences between two meshes is a
new problem in computer vision with various applications
such as structure from motion, model registration, shape
analysis, to name a few. While previous efforts were
mostly focused on matching a pair of images, many tasks
require to find correspondences across multiple meshes in
fact. A typical example is nonrigid structure from motion
[4], where one can hardly reconstruct a nonrigid shape
from two frames. Furthermore, recent work has shown
that leveraging multi-way information can dramatically
improve matching results compared to pairwise match-
ing. The most important constraint for joint matching
is the cycle consistency, i.e., the composition of matches
along a loop of meshes should be identity, as illustrated
in Figure 1. Given pairwise matches, one can possibly
identify true or false matches by checking all cycles in
the mesh collection. But there are many difficulties for
this approach. For example, the input pairwise matches
are often very noisy Figure 1. An illustration of consis-
tent multi-mesh matching. With many false matches and
missing matches, and the features detected from different
meshes may only have a partial overlap even if the same
feature detector is applied. Therefore, it is likely that very
few consistent cycles can be found. Moreover, how to
sample cycles is not straightforward due to the huge num-
ber of possibilities. Recent work on joint matching has
shown that, if all feature correspondences within multi-
ple meshes are denoted by a large binary matrix, the cy-
cle consistency can be translated into the fact that such
a matrix should be positive semi-definite and low-rank .
Based on this observation, convex optimization-based al-
gorithms were proposed, which achieved the state-of-the-
art performances with theoretical guarantees . But these
algorithms rely on semi-definite programming (SDP) [9],
which is not computationally efficient to handle image
matching problems in practice.
In this paper, we make use of MatchALS [16], a novel
algorithm for multi image matching, and use it for 3D
model meshes. The inputs to our algorithm are orig-
inal similarities between feature descriptors which we
used Heat Kernel Signature as. The outputs are fea-
ture correspondences between all pairs of images. Un-
like many previous methods starting from quantized pair-
wise matches , we postpone the decision until we opti-
mize for both pairwise affinities and multi-mesh consis-
tency. Instead of using Semi Definite Programming (SDP)
relaxation, we formulate the problem as a low-rank ma-
1

Figure 1: Illustration of Consistent Multi-Image Match-
ing
trix recovery problem and employ the nuclear-norm re-
laxation for rank minimization. We show that the posi-
tive semi-definiteness of a desired solution can be spon-
taneously fulfilled. Moreover, we derive a fast alternat-
ing minimization algorithm to globally solve the problem
in the low-dimensional variable space . Besides validat-
ing our method on both simulated and real benchmark
data sets, we also demonstrate the applicability of the pro-
posed method combined with Re-weighted Random Walk
(RRWM) [7] graph matching to match meshes with dif-
ferent models. We introduce a random walk view on the
problem and propose a robust graph matching algorithm
against outliers and deformation. Matching between two
graphs is formulated as node selection on an association
graph whose nodes represent candidate correspondences
between the two graphs. The solution is obtained by sim-
ulating random walks with re-weighting jumps enforcing
the matching constraints on the association graph. This
algorithm achieves noise-robust graph matching by iter-
atively updating and exploiting the confidences of candi-
date correspondences. In a practical sense, this RRWM
Distortion term is of particular importance since the real-
world matching problem is made difficult by the presence
of noise and outliers. Extensive and comparative experi-
ments demonstrate that it outperforms the state-of-the-art
graph matching algorithms especially in the presence of
outliers and deformation.
2 Preliminaries and notation
Suppose we have n models and pi features from each
model i. The objective is to find feature correspondences
between all pairs of models. Before introducing the pro-
posed method, we first give a brief introduction to pair-
wise matching techniques and the definition of cycle con-
sistency.
2.1 Pairwise matching
To match an image pair (i,j), one can compute similarities
for all pairs of feature points from two images and store
them in a matrix Sij ∈ Rpi×pj
. We represent the feature
correspondences for mesh pair (i,j) by a partial permuta-
tion matrix Xij ∈ 0, 1pi×pj
, which satisfies the doubly
stochastic constraints:
0 ≤ Xij1 ≤ 1, 0 ≤ XT
ij1 ≤ 1 (1)
To find Xij , we can maximize the inner product be-
tween Xij and Sij subject to the constraints in (1) result-
ing in a linear assignment problem, which has been well
studied and can be efficiently solved by the Hungarian al-
gorithm. In image matching, spatial rigidity is usually
preferred, i.e., the relative location between two features
in an mesh should be similar to that between their cor-
respondences in the other mesh. This problem is well
known as graph matching and formulated as a quadratic
assignment problem (QAP). While QAP is NP-hard [2],
many efficient algorithms have been proposed to solve it
approximately, e.g. Those solvers basically relax the bi-
nary constraint on the permutation matrix, solve the opti-
mization, and output the confidence of a candidate match
being correct. We refer readers to the related literature
for details. Here we aim to emphasize that the outputs
of graph matching solvers are basically optimized affinity
scores of candidate matches, which consider both feature
similarity and spatial rigidity. We will use these scores
(saved in Sij) as our input in some cases.
2.2 Cycle Consistency
For any three models (i,j,z), cycle consistency can be de-
scribed as
Xij = XizXzj, (2)
and can be extended to the case with more than three
models.The cycle consistency can be described more con-
cisely by introducing a virtual universe that is defined as
the set of unique features that appear in the image collec-
tion [13]. Each point in the universe may be observed by
2

several images and the corresponding image points should
be matched. In this way, consistent matching should sat-
isfy Xij = AiAT
j , where Ai ∈ {0, 1}
pi×k
denotes the
map from Image i to the universe, k is the number of
points in the universe, and k ≥ pi for all i.
Suppose the correspondences for all m =
n
i=1 pi
features in the image collection is denoted by X ∈
{0, 1}
m×m
and all Ais are concatenated as rows in a ma-
trix A ∈ {0, 1}
m×k
. Then one can write X as
X =





X11 X12 . . . X1n
X21 X22 . . . X2n
...
...
...
...
Xn1 Xn2 . . . Xnn





, (3)
X = AAT
(4)
From (4), it is clear to see that desired X should be both
positive semi-definite and low rank, which effectively im-
pose cycle consistency without checking all cycles of pair-
wise matches.
X 0, rank(X) ≤ k (5)
3 Joint Matching via Rank Mini-
mization
Given the affinity scores {Sij|1 ≤ i, j ≤ n} for all mod-
els, we aim to find globally consistent matches X. Affin-
ity scores can be computed from either feature similarities
or graph matching solvers according to specific scenarios,
For our experiments we use Heat Kernel Signature (HKS)
feature descriptor to define the affinity score matrices.
3.1 Cost Function
To formulate the cost function, we treat the problem as
a low rank matrix recovery problem. The inner product
between Xij and Sij is maximized for all i, j as multiple
linear assignment problems. Alongside, to enforce cycle
consistency, the rank of X is minimized. We make fol-
lowing two relaxations on X to optimize the solution:
• X is treated as a real matrix X ∈ [0, 1]
m×m
in-
stead of a binary matrix. Experimental results show
that the solution values can be stably quantized by a
threshold of 0.5.
• Rank of X is replaced by nuclear norm ||X||∗ (sum
of singular values), which is a tight convex relaxation
proven to be very effective in various low-rank prob-
lems such as matrix completion [6].
The estimated X should be sparse since at most one
value in each row of Xij can be nonzero. To induce spar-
sity, we minimize the sum of values in X. Combining all
three terms, we obtain the following cost function:
f(X) = −
n
i=1
n
j=1
Sij, Xij + α 1, X + λ||X||∗,
= − S − α1, X + λ||X||∗, (6)
where ., . denotes the inner product, S ∈ Rm×m
is a
collection of Sijs, α is is the weight of sparsity, λ controls
the weight of the nuclear norm.
Besides the doubly stochastic constraints in (1), addi-
tional constraints shall be imposed on X after relaxation:
Xii = Ipi
, 1 ≤ i ≤ n, (7)
Xij = XT
ji, 1 ≤ i, j ≤ n, i = j, (8)
0 ≤ X ≤ 1, (9)
Finally, we obtain the following optimization problem:
min
X
W, X + λ||X||∗,
s.t.X ∈ C, (10)
where W = α1 − S and C denotes a set of matrices
satisfying the constraints given in (1), (7), (8), and (9).
4 Fast Alternating Minimization
The state-of-the-art methods to solve the convex nuclear
norm minimizatiion in (10) are proximal method [14] or
ADMM [3] based on iterative single value thresholding
[5]. However, singular value decomposition (SVD) needs
to be performed in each iteration, which is extremely ex-
pensive even for a mediumsized problem.
3

Similar low-rank optimization was solved in [16] for
2D images more efficiently via a change of variables
X = ABT
, where A, B ∈ Rm×k
are new variables with
a smaller dimension k < m.
We use the following lowrank factorization-based for-
mulation implemented for 2D images in [16] and extend
it to 3D mesh based models:
min
A,B
W, ABT
+ λ||ABT
||∗,
s.t.ABT
∈ C, (11)
Now rewriting the nuclear norm minimization problem
as SDP [15],
||X||∗ = min
A,B:ABT =X
1
2
||A||2
F + ||B||2
F (12)
Finally the following formulation is obtained:
min
A,B
W, ABT
+
λ
2
||A||2
F +
λ
2
||B||2
F ,
s.t.ABT
∈ C, (13)
The selection of matrix dimension k is critical to the
success of change of variables, while it directly affects
the computational complexity.
4.1 Algorithm
We extend the following algorithm described in [16] for
2D Images to 3D models. The algorithm applies ADMM
to solve (13). The augmented Lagrangian of (13) is as
follows:
£µ(X, A, B, Y ) = W, X +
λ
2
||A||2
F +
λ
2
||B||2
F
+ Y X − ABT
+
µ
2
||X − ABT
||2
F , (14)
where Y is the dual variable and µ is a parameter control-
ling the step size in optimization. The ADMM alternately
updates each primal variable by minimizing £µ and up-
dates the dual variable via gradient ascent while fixing all
other variables.
Result: Globally consistent matches X
randomly initialize A and B, Y = 0 ;
W = α1 − S ;
while not converged do
A ← X + 1
µ Y B BT
B + λ
µ I
τ
;
B ← X + 1
µ Y A AT
A + λ
µ I
τ
;
X ← Pc ABT
− 1
µ (W + Y ) ;
Y ← Y k
+ µ X − ABT
;
end
quantize X with a threshold equal to 0.5;
Algorithm 1: Matching via Alternating Least Squares
(MatchALS)
5 RRWM
Introduction to problem: Graph matching is an essen-
tial problem in computer vision and machine learning. In
this paper, we implemented a random walk view on the
problem and use a robust graph matching algorithm [7]
against outliers and deformation. Matching between two
graphs is formulated as node selection on an association
graph whose nodes represent candidate correspondences
between the two graphs. The solution is obtained by sim-
ulating random walks with re-weighting jumps enforcing
the matching constraints on the association graph. Our
algorithm achieves noise-robust graph matching by iter-
atively updating and exploiting the confidences of candi-
date correspondences. In a practical sense, our work is of
particular importance since the real-world matching prob-
lem is made difficult by the presence of noise and outliers.
Extensive and comparative experiments demonstrate that
it outperforms the state-of-the-art graph matching algo-
rithms especially in the presence of outliers and deforma-
tion.
Recent resurgence of combinatorial optimization ap-
proaches to feature matching has changed the situation
and firmly settled graph matching formulations based on
Integer Quadratic Programming (IQP), which is a gener-
alization of the classical graph matching problems. IQP
explicitly takes into consideration both unary and pair-
wise terms reflecting the compatibilities in local appear-
ance as well as the pair-wise geometric relationships be-
tween the matching features. Since IQP is known to be
4

NP-hard, approximate solutions are required. This pro-
vides a novel interpretation of graph matching in a ran-
dom walk view and relates it to the IQP formulation. In-
troducing an association graph constructed with nodes as
candidate correspondences and edges as pair-wise com-
patibilities between candidate correspondences, we show
that the search for correspondences between the given
two graphs can be cast as a node ranking [11] and selec-
tion problem in the association graph. For this ranking,
we introduce an affinity-preserving random walk and de-
rive a ranking based on its quasi-stationary distribution,
and prove its equivalence to the spectral relaxation for
the IQP formulation. Then, in this random walk view,
we adopt the personalization strategy of Web ranking al-
gorithms [8] and propose the re-weighted random walk
algorithm by re-weighting jumps for the graph matching
constraints. It achieves noise-robust graph matching by si-
multaneously updating and exploiting the confidences of
candidate correspondences. In a practical sense, our work
is of particular importance since the real-world matching
problem is made difficult by the presence of deformation
and outliers.
5.1 Problem Formulation
The objective of graph matching is to determine the cor-
rect correspondences between two attributed graphs Gp =
(Vp, Ep, Ap) and Gq = (Vq, Eq, Aq) where V represents
a set of nodes, E, edges, and A, attributes. Each node
vP
i ∈ V P
or edge eP
i j ∈ EP
has an associated attribute
vector aP
i ∈ AP
or aP
i j ∈ AP
. In feature correspondence
problems, a node attribute aP
i usually describes a local ap-
pearance of feature i in model P, and an edge attribute
aP
i j represents the geometric relationship between fea-
tures i and j in the image P. For each pair of edges eP
i j ∈
EP
and eQ
a b ∈ EQ
, there is an affinity or compatibility
Wia; jb = f aP
i , aP
j , aP
i j, aQ
a , aQ
b , aQ
a b that measures
the mutual consistency of attributes between the pairs
of candidate correspondences vP
i , vQ
a and vP
j , vQ
b .
Thus, using a matrix form W, a non-diagonal element
Wia; jb contains a pair-wise affinity between two corre-
spondences vP
i , vQ
a and vP
j , vQ
b , and a diagonal term
Wia; ia represents a uanry affinity of a correspondence
vP
i , vQ
a . Representing the correspondence with an as-
signment or permutation matrix X ∈ {0, 1}
nP
nQ
is com-
mon, such that Xia = 1 implies that node vP
i corresponds
to node vQ
a , e.g., feature i in the image P is matched to
feature a in the image Q, and Xia = 0 otherwise. In
this paper, we denote x ∈ {0, 1}
nP
nQ
as a column-wise
vectorized replica of X. The graph matching problem
can be formulated as an integer quadratic program (IQP),
that is, finding the indicator vector x∗
that maximizes the
quadratic score function as follows.
x∗
= argmax(xT
Wx) (15)
s.t.x ∈ {0, 1}
nP
nQ
, ∀i
nq
a=1
xiq ≤ 1, ∀a
np
i=1
xia ≤ 1
where the two-way constraints refer to the one-to-one
matching from GP
to GQ
. In general, no efficient al-
gorithm exists that can guarantee the optimality bounds
since the IQP is NP-hard, thus approximate solutions are
required.
Figure 2: Association Graphs for Graph Matching by
Random Walks
5.2 Algorithnm
In the previous affinity-preserving random walks, the
matching constraints are ignored and not reflected in the
random walk process. Inducing the matching constraints
only as a post-processing discretization step like leads
to a weak local optimum. How then can we reflect the
two-way matching constraints in the affinity-preserving
random walk? We adopts the personalization approach
widely used in Web ranking methods, which strength-
ens the effects of reliable nodes in random walks. This
is achieved by adopting a jump or teleport in the ran-
dom walk: the random walker moves by traversing an
edge with probability alpha(α) or by performing a jump
5

to some constrained nodes with probability 1 - α. α rep-
resents the bias between the two possible actions, i.e., fol-
lowing an edge or jumping. To address the lack of person-
alization or users focus, Web ranking algorithms adopted
this approach in topic-sensitive or query dependent vari-
ants. In our formulation, adopting the personalized jump,
the probability distribution is updated using the following
equation:
x(n+1)T
x
(n+1)
abs = α x(n)T
x
(n)
abs P + (1 − α) rT
(16)
where a re-weighting jump vector r is added to the
affinity-preserving random walk of. In this approach, we
use the jumps for generating a biased random walk to the
matching constraints.
6 Experimental Results
We evaluate the performance of the proposed method us-
ing SCAPE benchmark data [1]. We use the blended in-
trisic maps described in [10] to evaluate the performance
of MatchALS for 3D meshes with RRWM Distortion term
added and compare the matching errors for different mod-
els. The size of universe is fixed as 100 points and in each
mesh a random sample of the points are observed with a
probability denoted by ρo. The number of meshes is de-
noted by n, which is generic in our code but for this sim-
ulation purposes it was kept to 4. Then, the ground-truth
pairwise matches are established, and random corruptions
are simulated by removing some true matches and adding
some false matches to achieve an error rate of ρe. Fi-
nally, the corrupted permutation matrix is fed into our Al-
gorithm as the input affinity scores.
Figure 3 shows the results we obtained before we ap-
plied the RRWM distortion term. We observed 1-to-many
feature correspondences prior to RRWM application, and
this can bee seen by the same color of Figure 3 meshes at
the limbs which give false matches sometimes due to the
inherent geometrical symmetry of the object.
Next we demonstrate the results obtained from run-
ning the MatchALS for 3D Meshes in Figure 4 using the
RRWM Distortion term, and the results can be clearly
seen to weed out the false positive matches in Figure 4.
Given the weight matrix W, the reweight factor α,
and the inflation factor β ;
Prevent conflicting walks by setting Wia; jb = 0for
all conflicting match pairs ;
Set the maximum degree
dmax = maxia jb Wia;jb ;
Initialize the transition matrix P = W/dmax, the
starting probability x as uniform ;
while x not converged do
( Affinity-preserving random walking by edges ) ;
xT
= xT
P ;
( Reweighting with two-way constraints ) ;
yT
= exp (βx/maxx) ;
while y not converged do
normalize across rows by
yai = yai/
I
i=1 yai;
normalize across columns by
yai = yai/
A
q=1 yai;
end
y = y/ yai ;
( Affinity-preserving random walking with
reweighted jumps) ;
xT
= αxT
+ (1 − α) yT
;
x = x/ xai ;
end
Discretize x by the matching constraints;
Algorithm 2: Reweighted Random Walk Graph Match-
ing
Preciously colored the same, the limbs are now of dif-
ferent colors, which signifies that RRWM allows one-to-
many matches that appear due to inherent symmetry of
models, to be filtered out in a very precise manner.
Figure 5 next shows the 1-to-1 correspondences we
have achieved between one pair of meshes we matched
and the run time for the simulation was 4.53 seconds
Next we demonstrate the matching errors between the
4 meshes we obtained. For each pair of possible match-
ing the evaluation was performed by calculating the av-
erage Geodesic Error. For the SCAPE Data-set [12] pro-
vides the ground truth vertex-to-vertex correspondence.
To evaluate performance of our algorithm we map every
vertex that has a ground truth correspondence. A geodesic
distance between the predicted correspondence and the
6

(a) Mesh1 matched to Mesh3 (b) Mesh2 matched to Mesh3 (c) Mesh3 matched to Mesh4
Figure 4: Results of MatchALS combined with RRWM
Figure 3: Results of MatchALS performed on 4 Meshes
from SCAPE Benchmark Dataset
true correspondence is recorded. Then the error per sam-
ple is averaged to calculate the Average Geodesic Error
of a mapping between a pair of meshes. Table 1 shows
the evaluation results of the pairwise matching of the 4
meshes we used to demonstrate our results.
Mesh1 Mesh2 Mesh3 Mesh4
Mesh1 0 1.904% 2.637% 13.909%
Mesh2 1.904% 0 6.824% 18.363%
Mesh3 2.637% 6.824% 0 8.236%
Mesh4 13.909% 18.636% 8.236% 0
Table 1: Average Geodesic Error for matching a pair of
Meshes using MatchALS and RRWM
Figure 5: Illustration of 1 to 1 Mapping of feature corre-
spondences in 2 Meshes
7 Conclusion
In this paper, we extended the joint matching algorithm
MatchALS for 2D images to 3D mesh matching. We
use pairwise feature similarities, the heat kernel signature
feature descriptor, as inputs and obtain accurate matches
by an efﬁcient algorithm (MatchALS) that globally op-
timizes for both feature afﬁnities and cycle consistency
of matches. The experiments not only validate the effec-
tiveness of the proposed method but also demonstrate that
joint matching is a promising approach to matching a col-
lection of meshes obtained from crowd sourced 3D model
collections MatchALS is a powerful algorithm that can be
7

extended to 3-D mesh models and the performance gen-
erated by the algorithm is almost the same as the state-
of-the art algorithm but with exponential orders of incre-
ment in speed, as we ourselves observed in our simula-
tion. Coupling the MatchALS with Re-weighted Random
Walk Matching is something that has not been done be-
fore and our novel implementation of stitching the two
algorithms has given very fruitful and promise-bearing re-
sults applicable to real world 3-D model matching appli-
cations .The results reveal that the matching accuracy in
the challenging situations largely depends on the effective
exploitation of the matching constraints.
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