Minimum-Distortion Isometric Shape Correspondence Using EM Algorithm

Abstract

We present a purely isometric method that establishes 3D correspondence between two (nearly) isometric shapes. Our method evenly samples high-curvature vertices from the given mesh representations, and then seeks an injective mapping from one vertex set to the other that minimizes the isometric distortion. We formulate the problem of shape correspondence as combinatorial optimization over the domain of all possible mappings, which then reduces in a probabilistic setting to a log-likelihood maximization problem that we solve via the Expectation-Maximization (EM) algorithm. The EM algorithm is initialized in the spectral domain by transforming the sampled vertices via classical Multidimensional Scaling (MDS). Minimization of the isometric distortion, and hence maximization of the log-likelihood function, is then achieved in the original 3D euclidean space, for each iteration of the EM algorithm, in two steps: by first using bipartite perfect matching, and then a greedy optimization algorithm. The optimal mapping obtained at convergence can be one-to-one or many-to-one upon choice. We demonstrate the performance of our method on various isometric (or nearly isometric) pairs of shapes for some of which the ground-truth correspondence is available.

Registration of 3D Point Clouds and Meshes: A Survey from Rigid to Nonrigid

“…When there is no ground truth correspondences, evaluation of techniques using dataset types 1 and 2 are usually based on visual assessment by connecting spare correspondences with lines [9]. Given ground truth correspondences in dataset types 3 and 4, additional metrics can be defined.…”

“…This can be visualized per vertex, assuming there is a one-to-one mapping. [9] sums these isometric distortions all over the surface to obtain one single number. Note that isometric distortion is meaningful only if the datasets are undergoing a (near-)isometric deformation.…”

“…[5] extends the latter to work on 1D skeletons for spacetime surface registration. There are techniques that formulate correspondences as a minimization of overall geodesic distortion [9]. Isometry can also be formulated as a transformation model (Section 3.5).…”

“…Non-Rigid Registration: [27] extends the above method to non-rigid registration by regularizing the displacement fields using coherence. Non-Rigid Correspondence: [9] formulates the problem of shape correspondences as a combinatorial optimization, solved using EM approach, with the goal of minimizing the overall isometric distortion of correspondences all over the surface.…”

“…Yet many useful techniques have been developed. These include techniques that target articulation, like [5] that can handle large gaps in 4D sequences, reduce high dimensionality of deformations through automatic construction of consensus skeletons and [6] that can register 4D surfaces and produce an urshape template; [7] that supports facial capture and animation of avatar heads in real-time; and [8], [9] that can handle near-to approximateisometric deformation. All these have changed the landscape of digital geometry processing, migrating the focuses to dynamic scenes and motions.…”