# Bayesian Inference of Bijective Non-Rigid Shape Correspondence

###### Abstract

Many algorithms for the computation of correspondences between deformable shapes rely on some variant of nearest neighbor matching in a descriptor space. Such are, for example, various point-wise correspondence recovery algorithms used as a post-processing stage in the functional correspondence framework. In this paper, we show that such frequently used techniques in practice suffer from lack of accuracy and result in poor surjectivity. We propose an alternative recovery technique guaranteeing a bijective correspondence and producing significantly higher accuracy. We derive the proposed method from a statistical framework of Bayesian inference and demonstrate its performance on several challenging deformable 3D shape matching.

## 1 Introduction

In geometry processing, computer graphics, and vision, estimating correspondence between 3D shapes affected by different transformations is one of the fundamental problems with a wide spectrum of applications ranging from texture mapping to animation [17]. These problems are becoming increasingly important due to the emergence of affordable 3D sensing technology. Of particular interest is the setting in which the objects are allowed to deform non-rigidly.

### 1.1 Related works

A traditional approach to correspondence problems is finding a point-wise matching between (a subset of) the points on two or more shapes. Minimum-distortion methods establish the matching by minimizing some structure distortion, which can include similarity of local features [30, 11, 6, 44], geodesic [27, 10, 12] or diffusion distances [14], or a combination thereof [40]. Windheuser et al. [43] used the thin shell elastic energy of triangles, while Zeng et al. [45] used higher-order structures. Typically, the computational complexity of such methods is high, and there have been several attempts to alleviate the computational complexity using hierarchical [36] or subsampling [39] methods. Several approaches formulate the correspondence problem as quadratic assignment and employ different relaxations thereof [41, 23, 33, 2, 12, 18]. Algorithms in this category typically produce guaranteed bijective correspondences between a sparse set of points, or a dense correspondence suffering from poor surjectivity.

Embedding methods try to exploit some assumption on the correspondence (e.g. approximate isometry) in order to parametrize the correspondence problem with a few degrees of freedom. Elad and Kimmel [16] used multi-dimensional scaling to embed the geodesic metric of the matched shapes into a low-dimensional Euclidean space, where alignment of the resulting “canonical forms” is then performed by simple rigid matching (ICP) [13, 8]. The works of [25, 37] used the eigenfunctions of the Laplace-Beltrami operator as embedding coordinates and performed matching in the eigenspace. Lipman et al. [24, 19, 20] used conformal embeddings into disks and spheres to parametrize correspondences between homeomorphic surfaces as Möbius transformations. By using locally injective flattenings, [4] achieve guaranteed bijective matching. However, the majority of the matching procedures performed in the embedding space often produces noisy correspondences at fine scales, and suffers from poor surjectivity.

As opposed to point-wise correspondence methods, soft correspondence approaches assign a point on one shape to more than one point on the other.
Several methods formulated soft correspondence as a mass-transportation problem [26, 38].
Ovsjanikov *et al.* [29] introduced the functional correspondence framework, modeling the correspondence as a linear operator between spaces of functions on two shapes, which has an efficient representation in the Laplacian eigenbases.
This approach was extended in several follow-up works [31, 22, 1] .
A point-wise map is typically recovered from a low-rank approximation of the functional correspondence by a matching procedure in the representation basis, which also suffers from poor surjectivity.

### 1.2 Main contributions

As the main contribution of this paper we see the formulation of the intrinsic map *denoising* problem: Given a set of point-wise correspondences between two shapes coming from any correspondence algorithm (for example, using one of the recovery algorithms outlined in Section 2), we consider them as a noisy realization of a latent bijective correspondence. We estimate this bijection using an intrinsic equivalent of the standard minimum mean squared error (MMSE) or minimum mean absolute error (MMAE) Bayesian estimators. To the best of our knowledge, despite their simplicity, these tools have not been previously used for deformable shape analysis.

We show that the considered family of Bayesian estimators leads to a linear assignment problem (LAP) guaranteeing bijective correspondence between the shapes. Despite the common wisdom, we demonstrate that the problem is efficiently solvable for relatively densely sampled shapes by means of the well-established auction algorithm [7] and a simple multi-scale approach.

Finally, we present a significant amount of empirical evidence that the proposed denoising procedure consistently improves the quality of the input correspondence coming from different algorithms.

## 2 Pointwise map recovery

We start by briefly overviewing several recent techniques used for the computation of pointwise correspondences between non-rigid shapes. We focus on approaches relying on the functional map formalism merely because these techniques produce state-of-the-art results, emphasizing that the proposed algorithm can accept any point-wise correspondence as the input.

We model shapes as connected two-dimensional Riemannian manifolds (possibly with boundary) endowed with the standard measure induced by the volume form. Shape is equipped with the symmetric Laplace-Beltrami operator , generalizing the notion of Laplacian to manifolds. The manifold Laplacian yields an eigen-decomposition for , with eigenvalues and eigenfunctions forming an orthonormal basis of . Due to the latter property, any function can be represented via the (manifold) Fourier series expansion

(1) |

where we use the standard manifold inner product .

Consider two manifolds and , and let be a bijective mapping between them. In [29] it was proposed to consider an operator , mapping functions on to functions on via the composition . This simple change in paradigm remarkably allows to identify maps between manifolds as linear operators (named functional maps) between Hilbert spaces. Because is a linear operator, it admits a matrix representation with respect to a choice of bases and on and , respectively. Assuming the bases to be orthogonal, the matrix with the elements provides a representation of . In particular, by choosing the delta functions supported on the shape vertices as basis functions, one obtains a permutation as a matrix representation for the functional map.

A more compact way to represent in matrix form is obtained by taking the Laplacian eigenfunctions , of the respective manifolds as the choice for a basis. In case is a (near) isometry, the equality holds (approximately) for all , leading to the matrix representation being diagonally dominant, i.e., .

With this choice, Ovsjanikov et al. [29] proposed to truncate the matrix after the first coefficients as a low-pass approximation of the functional map (typical values for are in the range ). This is especially convenient for correspondence problems, where one is required to solve for . At the same time, in analogy to classical Fourier analysis, the truncation has a blurring effect on the correspondence. As a result, recovering the original bijection from the spectral coefficients leads to a non-trivial inverse problem.

Assume shapes and have points each, and let the matrices contain the first eigenvectors of the respective Laplacians. For the sake of simplicity we assume and to be area-weighted, allowing us to consider the standard dot product in all equations. The expression for can now be compactly written as

(2) |

Note that the matrix is now a rank- approximation of . The pointwise map recovery problem [32], which is highly underdetermined, consists in finding a permutation satisfying (2). The following techniques have been proposed for this purpose.

#### Linear assignment problem (LAP).

If we assume , the terms on either side of (2) have the same rank and the relation can be straightforwardly inverted to yield . Since in the truncated setting we have , the best possible solution in the sense can be obtained by looking for a permutation minimizing . This leads to the equivalent linear assignment problem:

(3) | ||||

(4) |

The equality between the two expressions comes from the observation that for permutation matrices . Minimizing with respect to , and recalling that , yields the equivalence.

The problem above admits an intuitive interpretation. Denoting by the indicator vector having the value 1 in the th position and 0 otherwise, we see that each column of contains the spectral coefficients of delta functions for and . Hence, the image via of all indicator functions on is given by the columns of . Problem (3) seeks for a permutation minimizing the distance between columns of and columns of in a sense.

#### Nearest neighbors.

In [29] the authors proposed to recover a pointwise correspondence between and by solving the nearest-neighbor problem

(5) | ||||

(6) |

This can be seen as a simplified version of the LAP where the bi-stochasticity constraints (4) are relaxed, including all (binary) column-stochastic matrices in the feasible set. A global solution to (5) can be obtained in an efficient manner by solving for each column of separately: It is sufficient to seek for the nearest column of with respect to each column of .

#### Iterative closest point (ICP).

In [29] it was additionally proposed to solve for the (not necessarily bijective) according to the nearest-neighbor approach (5), followed by a refinement of via the orthogonal Procrustes problem:

(7) | ||||

(8) |

The - and - steps are alternated until convergence. In analogy to classical Iterative Closest Point (ICP) refinement [13, 8] operating in , this can be seen as a rigid alignment between point sets (columns of) and in .

#### Coherent point drift (CPD).

The orthogonal refinement of (5), (7) assumes the underlying map to be area-preserving [29], and is therefore bound to fail in case the two shapes are non-isometric. Rodolà et al. [32] proposed to consider the non-rigid counterpart, for a given :

(9) | ||||

(10) |

where denotes the Kullback-Leibler divergence between probability distributions, is a low-pass operator promoting smooth velocity vectors, and controls the regularity of the assignment. Problem (9) can be seen as a Tikhonov regularization of the displacement field relating the two sets of spectral coefficients, with a measure of proximity given by the KL divergence between the two. The problem is then solved via expectation-maximization by the coherent point drift algorithm [28].

## 3 Bayesian map estimation

We describe a Bayesian formulation of bijective map estimation that views the given correspondence as a realization of a random process adding noise to a latent ideal correspondence. We denote by the latent bijective correspondence between the shapes. Let denote a random point on drawn from a uniform distribution, in the sense that for every measurable set , . Given , we denote by the conditional random variable a point on with a Gaussian distribution with some variance centered at that accounts for the uncertainty in the map. The Gaussian distribution is interpreted in the sense that for every measurable set ,

Using Bayes’ theorem, we can express the probability density of the conditional random variable as

Given some (possibly noisy and not necessarily bijective) correspondence , we consider as a realization of for every . Our goal is to estimate the bijection or its inverse from these data.

Let us fix some . A Bayesian estimator of given the observations can be expressed as

In the Euclidean case, the above Bayesian estimator coincides with the minimum mean absolute error (MMAE) for and the minimum mean squared error (MMSE) for ; in both cases, it has a closed-form solution as the geometric median and the centroid, respectively. The more general case discussed here can be thought of as the intrinsic counterpart of the median and the centroid.

We estimate the whole inverse map by minimizing

over all bijections . Note that due to the additional constraint that has to be a bijection, the estimation cannot be done for each point independently. We also observe that iterating the process several times consistently improves the estimated map accuracy.

Finally, we note that when is area-preserving or, more generally, scales the metric uniformly, the estimator (LABEL:eq:BayesianEst) can be equivalently rewritten in terms of as

(12) |

It is worthwhile mentioning that while being natural, the assumption of uniform prior distribution of on (embodied in the use of the standard area measure in the above integral) can be replaced by other measures emphasizing regions where errors are less tolerable. Also, non-Gaussian noise models may be more suitable for data coming from a specific correspondence algorithm. We defer these interesting questions to future study.

#### Discretization.

We consider the discretization of (12). We assume the shape to be discretized at points with the corresponding discrete area elements and pairwise geodesic distance matrix . Similarly, the shape is discretized as the same number of points, and its pairwise distance matrix is denoted by .

The bijective correspondence is represented by the permutation matrix sought by minimizing

(13) |

where is an matrix with the elements

and is an matrix with . Note that (13) is a linear assignment problem (LAP). For directly solving the LAP with the specific structure of the score matrix given by , we found the auction algorithm [7] to perform the best in practice. Its average runtime complexity is , with storage complexity if a full score matrix is used. On regular hardware, this translates to several seconds for , which quickly grows to seconds for and almost minutes for , taking tens of gigabytes of memory. We therefore conclude that directly solving the full LAP is practical for , and in the following section propose a multi-scale scheme that can scale to much larger numbers of points.

Another computational bottleneck stems from the computation of pairwise geodesic distances. For example, using fast marching [21] the computation requires computations and storage. While the computations can be thoroughly parallelized and executed on a GPU, reducing the complexity by orders of magnitude [42], the storage of a full distance matrix is still prohibitive for . However, since geodesic distance maps are almost everywhere smooth with constant gradient, their approximation in a truncated harmonic basic is optimal in the sense [3]. Instead of storing an matrix , we store the representation coefficient matrix

where contains the first eigenfunctions of the Laplace-Beltrami operator on . In order to ”decompress” the -th row of used in the computation of the LAP score, the corresponding column of is multiplied from the left by .

It is also worthwhile mentioning that while the geodesic metric is a natural candidate to compute intrinsic distances on a manifold, the proposed estimator can work with other choices. For example, diffusion distances [14] or other approximations of the geodesic distances [15] are likely to work equally well while being better amenable both for faster computation and more compact storage.

#### Multiscale solution.

In order to reduce the computation and storage complexity associated with the direct solution of the LAP for large values of , we adopt a multi-scale strategy. Both shapes are discretized in a hierarchical fashion using farthest point sampling, while the distances and the harmonics are calculated at the finest scale and sub-sampled.

First, a full LAP (13) is solved at a coarse scale. The produced correspondence is interpolated to the next scale and is used as the input correspondence to the LAP. While numerous interpolation techniques exist, we found that simple nearest neighbour interpolation produces satisfactory results. At the finer scale, the space of possible bijections is restricted to the points falling into a fixed radius around each (note that has to be larger than the coarse sampling radius). This is equivalent to assigning infinite score to the prohibited permutations. For a sufficiently small , this strategy results in sparse score matrices, with density significantly lower than .

## 4 Experiments

We start by evaluating the influence of the parameters and on the quality of the Bayesian estimator (LABEL:eq:BayesianEst). We initialize with noisy correspondences coming from a nearest neighbour (5) result and evaluate on two datasets with different global scales, see Figure 3. The optimal choice of amounts to approximately 6% of the target shapes area for both choices of but the quality is shown to be stable in a vicinity.

We test our method recover bijections from on two types of initialization, namely functional maps of low rank and sparse correspondences.

We conduct quantitative experiments on the FAUST dataset [9] (7K vertices) and on downsampled versions of the SCAPE [5] and KIDS [35] datasets (1K vertices). As quantitative quality criteria we evaluate the geodesic errors, the run times and the lack of surjectivity of the different methods. We further show that our approach can directly tackle shapes having more then 10K vertices.

### 4.1 Recovery from a functional map

In this set of experiments the low rank approximation is given in terms of a functional map of different ranks in the harmonic basis. Comparisons are done against nearest neighbors (NN) (5), bijective NN (3), ICP (7) and CPD (9).

#### Approximation of the groundtruth.

Here we construct the low rank functional map using the known groundtruth correspondences between the shapes. This is supposed to be the ideal input for all the competing methods. As the input to our method we use the matchings found by nearest neighbors and its bijective version. We show quantitative comparisons on 71 pairs from the SCAPE dataset (near isometric, 1K vertices) and 100 pairs from the FAUST dataset (including inter-class pairs, 7K vertices). In Figures 4 and 5 we compare the accuracy, in Figure 6 the lack of surjectivity is analyzed. We only show the performance of a single application of the Bayesian estimator yet adumbrate experiments with multiple iterations in the following sections. Even after one iteration, our method outperforms the state of the art method (9) as well in accuracy as in run time (Table 1). Even on shapes having more then 10K vertices just one iteration of the Bayesian estimator gives very good results, as can be seen in Figure 7. Memory consumption and run times however limit the direct applicability of the single-scale Bayesian estimator.

#### Using a functional map coming from an optimization process.

We follow the approach from [31] to construct a realistic functional map matching, which typically requires region features as input. These features were detected using the consensus-segmentation method proposed in [34], and the resulting regions were matched by intersection w.r.t. the ground-truth. Both of these two methods were executed with the same parameters as in their publicly available implementation. In Figure 8 this initialization is evaluated on the SCAPE dataset.

Nearest neighbors | 0.04 | 0.06 | 1.35 | 2.88 |
---|---|---|---|---|

Bijective NN | 2.79 | 2.30 | 463.66 | 253.03 |

ICP | 0.14 | 0.24 | 12.72 | 30.08 |

CPD | 4.79 | 4.67 | 1745.06 | 2085.65 |

NN + Bayesian | 1.75 | 1.28 | 382.86 | 244.10 |

Bij. NN + Bayesian | 4.06 | 3.44 | 746.00 | 440.94 |

### 4.2 Recovery from a sparse correspondence

In this set of experiments the low rank approximation is given in terms of sparse correspondences between high resolution shapes. This type of input can for instance be obtained by minimizing energies under -constraints, such as [33], or appears in multiresolution settings. We make use of groundtruth correspondences of a few points and interpolate the matching with the technique described in the caption of Figure 9. Figure 10 illustrates how iterations of the Bayesian estimator improve the matching.

## 5 Conclusion

We considered the problem of bijective correspondence recovery by means of denoising a given set of matches coming from any of the existing algorithms (including those not guaranteeing bijection, or producing sparse correspondences). Viewing the denosing as a Bayesian estimation problem, we formulated the intrinsic equivalent of the mean and median filters frequently employed in signal processing, with the additional constraint of bijectivity embodied through an LAP.

We find surprising the fact that such a simple idea demonstrates a consistent improvment in the correspondence quality in all experiments we have conducted. We believe that tools from estimation theory that have been heavily used in other domains of science and engineering might be very useful in shape analysis, and invite the community to further explore this direction.

Of special interest are the choice of the loss function in the posterior expectation (which in this paper was restricted to the absolute and squared distance), the prior distribution of (which we assumed uniform), and the noise distribution (which was assumed Gaussian). Alternative estimators making use of Bayesian statistics, such as maximum a posteriori (MAP) estimators, should also be explored.

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