Efficient Deformable Shape Correspondence via Kernel Matching

We present a method to match three dimensional shapes under non-isometric deformations, topology changes and partiality. We formulate the problem as matching between a set of pair-wise and point-wise descriptors, imposing a continuity prior on the mapping, and propose a projected descent optimization procedure inspired by difference of convex functions (DC) programming. Surprisingly, in spite of the highly non-convex nature of the resulting quadratic assignment problem, our method converges to a semantically meaningful and continuous mapping in most of our experiments, and scales well. We provide preliminary theoretical analysis and several interpretations of the method.Comment: Accepted for oral presentation at 3DV 2017, including supplementary materia

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

A Low-Dimensional Representation for Robust Partial Isometric Correspondences Computation

Intrinsic isometric shape matching has become the standard approach for pose invariant correspondence estimation among deformable shapes. Most existing approaches assume global consistency, i.e., the metric structure of the whole manifold must not change significantly. While global isometric matching is well understood, only a few heuristic solutions are known for partial matching. Partial matching is particularly important for robustness to topological noise (incomplete data and contacts), which is a common problem in real-world 3D scanner data. In this paper, we introduce a new approach to partial, intrinsic isometric matching. Our method is based on the observation that isometries are fully determined by purely local information: a map of a single point and its tangent space fixes an isometry for both global and the partial maps. From this idea, we develop a new representation for partial isometric maps based on equivalence classes of correspondences between pairs of points and their tangent spaces. From this, we derive a local propagation algorithm that find such mappings efficiently. In contrast to previous heuristics based on RANSAC or expectation maximization, our method is based on a simple and sound theoretical model and fully deterministic. We apply our approach to register partial point clouds and compare it to the state-of-the-art methods, where we obtain significant improvements over global methods for real-world data and stronger guarantees than previous heuristic partial matching algorithms.Comment: 17 pages, 12 figure

3D Shape Segmentation with Projective Convolutional Networks

This paper introduces a deep architecture for segmenting 3D objects into their labeled semantic parts. Our architecture combines image-based Fully Convolutional Networks (FCNs) and surface-based Conditional Random Fields (CRFs) to yield coherent segmentations of 3D shapes. The image-based FCNs are used for efficient view-based reasoning about 3D object parts. Through a special projection layer, FCN outputs are effectively aggregated across multiple views and scales, then are projected onto the 3D object surfaces. Finally, a surface-based CRF combines the projected outputs with geometric consistency cues to yield coherent segmentations. The whole architecture (multi-view FCNs and CRF) is trained end-to-end. Our approach significantly outperforms the existing state-of-the-art methods in the currently largest segmentation benchmark (ShapeNet). Finally, we demonstrate promising segmentation results on noisy 3D shapes acquired from consumer-grade depth cameras.Comment: This is an updated version of our CVPR 2017 paper. We incorporated new experiments that demonstrate ShapePFCN performance under the case of consistent *upright* orientation and an additional input channel in our rendered images for encoding height from the ground plane (upright axis coordinate values). Performance is improved in this settin

Pattern classification using a linear associative memory

Pattern classification is a very important image processing task. A typical pattern classification algorithm can be broken into two parts; first, the pattern features are extracted and, second, these features are compared with a stored set of reference features until a match is found. In the second part, usually one of the several clustering algorithms or similarity measures is applied. In this paper, a new application of linear associative memory (LAM) to pattern classification problems is introduced. Here, the clustering algorithms or similarity measures are replaced by a LAM matrix multiplication. With a LAM, the reference features need not be separately stored. Since the second part of most classification algorithms is similar, a LAM standardizes the many clustering algorithms and also allows for a standard digital hardware implementation. Computer simulations on regular textures using a feature extraction algorithm achieved a high percentage of successful classification. In addition, this classification is independent of topological transformations

On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA