In many real-world settings of the shape recognition problem, the data are degraded by acquisition imperfections and noise, resulting in the need to find partial
similarity of objects. Such cases are common, for example, in face recognition, where the facial surface may be partially occluded by hair. In other applications, such as shape retrieval, correct semantic similarity of two objects is based on partial similarity – for example, a centaur is partially similar to a human because they share the human-like upper body.
In rigid shape analysis, modifications of the popular iterative closest point (ICP) algorithm are able to deal with partial shape alignment by rejecting points with bad correspondences (e.g., by thresholding the product of local normal vectors). However, it is impossible to guarantee how large and regular the resulting corresponding parts will be.
Bronstein et al.  formulated non-rigid partial similarity as a multi-criterion optimization problem, in which one tries to find the corresponding parts in two shapes by simultaneously maximizing significance and similarity criteria (in , metric distortion [14, 19, 5] was used as a criterion of similarity, and part area as significance). The problem requires the knowledge of correspondence between the shapes, and in the absence of a given correspondence, can be solved by alternating between weighted correspondence finding and maximization of part area. In , a different significance criterion based on statistical occurrence of local shape descriptors was used.
One of the drawbacks of the above method is its tendency in some cases to find a large number of disconnected components, which have the same area as a larger single component. The same authors addressed this problem using a Mumford-Shah [22, 10]-like regularization for rigid  and non-rigid  shapes.
Recent works on local shape descriptors (see, e.g., descriptors [16, 33, 24, 11, 18, 21, 32, 29, 27, 9]) have led to the adoption of bags of features  approach popular in image analysis for the description of 3D shapes [21, 23, 30]. Bags of features allows to some extent finding partial similarity, if the overlap between the parts is sufficiently large.
In this paper, we present an approach for correspondence-less partial matching of non-rigid 3D shapes. Our work is inspired by the recent work on partial matching of images . The main idea of this approach, adopted here, is to find similar parts by comparing part-wise distributions of local descriptors. This removes the need of correspondence knowledge and greatly simplifies the problem.
The rest of the paper is organized is as follows. In Section 2, we review the mathematical background of diffusion geometry, which is used for the construction of local descriptors. Section 3 deals with the partial matching problem and Section 4 addresses its discretization. Section 5 presents experimental results. Finally, Section 6 concludes the paper.
Diffusion geometry. Diffusion geometry is an umbrella term referring to geometric analysis of diffusion or random walk processes. We models a shape as a compact two-dimensional Riemannian manifold . In it simplest setting, a diffusion process on
is described by the partial differential equation
called the heat equation, where denotes the positive-semidefinite Laplace-Beltrami operator associated with the Riemannian metric of . The heat equation describes the propagation of heat on the surface and its solution is the heat distribution at a point in time . The initial condition of the equation is some initial heat distribution ; if has a boundary, appropriate boundary conditions must be added.
The solution of (1) corresponding to a point initial condition , is called the heat kernel and represents the amount of heat transferred from to in time due to the diffusion process. The value of the heat kernel
can also be interpreted as the transition probability density of a random walk of lengthfrom the point to the point .
Using spectral decomposition, the heat kernel can be represented as
Here, and(without loss of generality, we assume to be sorted in increasing order starting with ). Since the Laplace-Beltrami operator is an intrinsic geometric quantity, i.e., it can be expressed solely in terms of the metric of , its eigenfunctions and eigenvalues as well as the heat kernel are invariant under isometric transformations (bending) of the shape.
Heat kernel signatures. By setting , the heat kernel expresses the probability density of remaining at a point after time . The value , sometimes referred to as the auto-diffusivity function, is related to the Gaussian curvature through
This relation coincides with the well-known fact that heat tends to diffuse slower at points with positive curvature, and faster at points with negative curvature. Under mild technical conditions, the set is fully informative in the sense that it allows to reconstruct the Riemannian metric of the manifold .
Sun et al.  proposed constructing point-wise descriptors referred to as heat kernel signatures (HKS) by taking the values of the discrete auto-diffusivity function at point at multiple times, , where are some fixed time values and is chosen so that . Such a descriptor is a vector of dimensionality at each point. Since the heat kernel is an intrinsic quantity, the HKS is invariant to isometric transformations of the shape.
A scale-invariant version of the HKS descriptor (SI-HKS) was proposed in 
. First, the heat kernel is sampled logarithmically in time. Next, the logarithm and a derivative with respect to time of the heat kernel values are taken to undo the multiplicative constant. Finally, taking the magnitude of the Fourier transform allows to undo the scaling of the time variable.
Bags of features. Ovsjanikov et al.  and Toldo et al.  proposed constructing global shape descriptors from local descriptors using the bag of features paradigm . In this approach, a fixed “geometric vocabulary”is computed by means of an off-line clustering of the descriptor space. Next, each point descriptor is represented in the vocabulary using vector quantization. The bag of features global shape descriptor is then computed as the histogram of quantized descriptors over the entire shape.
3 Partial matching
In what follows, we assume to be given two shapes and with corresponding point-wise descriptor fields and defined on them (here we adopt HKS descriptors, though their quantized variants or any other intrinsic point-wise descriptors can be used as well). Assuming that is a part of an unknown shape that is intrinsically similar to , we aim at finding a part having the same area of such that the integral shape descriptors computed on and coincide as closely as possible. In order to prevent the parts from being fragmented and irregular, we penalize for their boundary length. The entire problem can be expressed as minimization of the following energy functional
under the constraint , where denotes area and . The first term of the functional constitutes the data term while the second one is the regularity term whose influence is controlled by the parameter .
Discretization of the above minimization problem with a crisp set results in combinatorial complexity. To circumvent this difficulty, in [2, 3] it was proposed to relax the problem by replacing the crisp part by a fuzzy membership function on , replacing the functional by a generalization of the Mumford-Shah functional  to surfaces. Here, we adopt this relaxation as well as the approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli . This yields the problem of the form
with the data term
and the Ambrosio-Tortorelli regularity term
where is the so-called phase field indicating the discontinuities of , and is a parameter.
The first term of above imposes piece-wise smoothness of the fuzzy part governed by the parameter . By setting a sufficiently large , the parts become approximately piece-wise constant as desired in the original crisp formulation (4). The second term of is analogous to the boundary length term in (4) and converges to the latter as .
We minimize (5) using alternating minimization comprising the following two iteratively repeated steps:
Step 1: fix and solve for
Step 2: fix the part and solve for
4 Discretization and numerical aspects
We represent the surface as triangular mesh with faces constructed upon the samples and denote by the corresponding area elements at each vertex. denote the diagonal matrix created out of . The membership function is sampled at each vertex and represented as the vector . Similarly, the phase field is represented as the vector .
Descriptors. The computation of the discrete heat kernel requires computing discrete eigenvalues and eigenfunctions of the discrete Laplace-Beltrami operator. The latter can be computed directly using the finite elements method (FEM) , of by discretization of the Laplace operator on the mesh followed by its eigendecomposition. Here, we adopt the second approach according to which the discrete Laplace-Beltrami operator is expressed in the following generic form,
where is a scalar function defined on the mesh, are weights, and are normalization coefficients. In matrix notation, (10) can be written as , where is an vector and .
The discrete eigenfunctions and eigenvalues are found by solving the generalized eigendecomposition  , where is a diagonal matrix of eigenvalues and
is the matrix of the corresponding eigenvectors.
Different choices of have been studied, depending on which continuous properties of the Laplace-Beltrami operator one wishes to preserve [13, 31]. For triangular meshes, a popular choice adopted in this paper is the cotangent weight scheme [25, 20], in which
where and are the two angles opposite to the edge between vertices and in the two triangles sharing the edge.
Data term. Denoting by the matrix of point-wise descriptors on (stored in columns), we have
This yields the following discretization of the data term (6)
Gradient norm. We start by deriving the discretization of a single term in some triangle of the mesh. Let us denote by and the vertices of the triangle and let be the matrix whose columns are the vectors forming the triangle, and by its area. Let also be the sparse matrix with at indices and , and at and . is constructed in such a way to give the differences of values of on the vertices of the triangle with respect to the values at the central vertex, . The gradient of the function is constant on the triangle and can be expressed in these terms by .
In order to introduce the weighting by , let be an sparse matrix with the elements for every vertex belonging to the triangle and zero otherwise. In this notation, is a per-triangle field whose elements are the average values of on each of the mesh triangles. We use the Kroenecker product of with to define the matrix formed by replicating twice each of the rows of . This yields
where is the matrix containing stacked as rows, and .
Discretized alternating minimization. We plug in the results obtained so far into the two steps of the alternating minimization problem (8)–(9). For fixed , the discretized minimization problem (8) w.r.t. can be written as
Let us now fix . In a triangle , we denote and let . Using this notation, we obtain the following discretization of the integrals in the regularization term (7)
In order to test our approach, we performed several partial matching experiments on data from the SHREC 2010 benchmark [8, 7] and the TOSCA dataset .111Both datasets are available online at http://tosca.cs.technion.ac.il The datasets contained high-resolution (10K-50K vertices) triangular meshes of human and animal shapes with simulated transformations, with known groundtruth correspondence between the transformed shapes. In our experiments, all the shapes were downsampled to approximately 2500 vertices. Parts were cut by taking a geodesic circle of random radius around a random center point.
For each part, the normalized HKS descriptor was calculated at each vertex belonging to the part. To avoid boundary effects (see Figure 1), descriptors close to the boundary were ignored when calculating in (4). The distance from the boundary was selected in accordance to the time scales of the HKS. We used ten linearly spread samples in range for the descriptors and the according distance taken from edge was set to . Two to three iterations of the alternating minimization procedure were used, exhibiting fast convergence (Figure 3). After three iterations the member function typically ceased changing significantly. The phase map assumed the values close to in places of low gradient of the membership function , and less than in high gradient areas (Figure 3). The importance of the regularization step is is evident observing the change in in Figure 3. Figure 3 shows the influence of the parameter , controlling the impact of the regularization. For too small values of , two equally weighted matches are obtained due to symmetry (left). The phenomenon decreases with the increase of the influence of the regularization penalty. However, increasing more causes incorrect matching (second and third columns from the right) due to low data term influence. Increasing it even more starts smoothing the result (rightmost column) until eventually making the membership function uniform over the entire shape. The resulting membership function was thresholded in such a way that the outcome area will be as close as possible to the query area. Figures 6–6 show examples of matching results after thresholding. Notice that in Figure 6 the matching result sometimes contain the symmetric counter part of the result due to invariance of the HKS descriptor to intrinsic symmetry. (in this figure, the threshold was adjusted to the value of when the membership functions weights are split between two symmetric parts as in Figure 3). The method is robust to shape deformations and geometric and topological noise as depicted in Figure 6. Note that the figures show part-to-whole shape matching, but because of the low scale HKS descriptors the same procedure works for matching to other parts as well.
Table 1 summarizes quantitative evaluation that was performed on a subset of the SHREC database, for which groundtruth correspondence and its bilaterally symmetric counterpart were available. This subset included a male, a dog and a horse shape classes with different geometric, topological and noise deformations ( shapes in total). The query set was generated by selecting a part from a deformed shape ( queries in each deformation category) and matched to the null shape with parameters and thresholds as described above.
|Isometry + Shotnoise||1000||75% (85%)|
|Isometry + Noise||1000||71% (82%)|
|Isometry + Microholes||1000||68% (82%)|
|Isometry + Holes||1000||66% (76%)|
Complexity. The code was implemented in Matalb with some parts written in C with MEX interface. The quadratic programming problem (8) in Step 1 was solved using QPC222 available online at http://sigpromu.org/quadprog implementation of a dual active set method. The experiments were run on 2.3GHz Intel Core2 Quad CPU, 2GB RAM in Win7 32bit environment. The running time (including re-meshing and descriptor calculation) per part was sec.
We presented a framework for finding partial similarity between shapes which does not rely on explicit correspondence. The method is based on regularized matching of region-wise local descriptors, and can be efficiently implemented. Experimental results show that our approach performs well in challenging matching scenarios, such as the presence of geometric and topological noise. In the future work, we will extend the method to the setting of two partially-similar full shapes, in which two similar parts have to be found in each shape, and then consider a multi-part matching (puzzle) scenario.
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