Higher-order Projected Power Iterations for Scalable Multi-Matching

The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take geometric consistency between points into account. Computationally, the multi-matching problem is difficult. It can be phrased as simultaneously solving multiple (NP-hard) quadratic assignment problems (QAPs) that are coupled via cycle-consistency constraints. The main limitations of existing multi-matching methods are that they either ignore geometric consistency and thus have limited robustness, or they are restricted to small-scale problems due to their (relatively) high computational cost. We address these shortcomings by introducing a Higher-order Projected Power Iteration method, which is (i) efficient and scales to tens of thousands of points, (ii) straightforward to implement, (iii) able to incorporate geometric consistency, (iv) guarantees cycle-consistent multi-matchings, and (iv) comes with theoretical convergence guarantees. Experimentally we show that our approach is superior to existing methods

Addressing Computational Bottlenecks in Higher-Order Graph Matching with Tensor Kronecker Product Structure

Graph matching, also known as network alignment, is the problem of finding a correspondence between the vertices of two separate graphs with strong applications in image correspondence and functional inference in protein networks. One class of successful techniques is based on tensor Kronecker products and tensor eigenvectors. A challenge with these techniques are memory and computational demands that are quadratic (or worse) in terms of problem size. In this manuscript we present and apply a theory of tensor Kronecker products to tensor based graph alignment algorithms to reduce their runtime complexity from quadratic to linear with no appreciable loss of quality. In terms of theory, we show that many matrix Kronecker product identities generalize to straightforward tensor counterparts, which is rare in tensor literature. Improved computation codes for two existing algorithms that utilize this new theory achieve a minimum 10 fold runtime improvement.Comment: 14 pages, 2 pages Supplemental, 5 figure

Mesh-based variational autoencoders for localized deformation component analysis

Spatially localized deformation components are very useful for shape analysis and synthesis in 3D geometry processing. Several methods have recently been developed, with an aim to extract intuitive and interpretable deformation components. However, these techniques suffer from fundamental limitations especially for meshes with noise or large-scale nonlinear deformations, and may not always be able to identify important deformation components. In this paper we propose a novel mesh-based variational autoencoder architecture that is able to cope with meshes with irregular connectivity and nonlinear deformations. To help localize deformations, we introduce sparse regularization along with spectral graph convolutional operations. Through modifying the regularization formulation and allowing dynamic change of sparsity ranges, we improve the visual quality and reconstruction ability. Our system also provides a nonlinear approach to reconstruction of meshes using the extracted basis, which is more effective than the current linear combination approach. We further develop a neural shape editing method, achieving shape editing and deformation component extraction in a unified framework and ensuring plausibility of the edited shapes. Extensive experiments show that our method outperforms state-of-the-art methods in both qualitative and quantitative evaluations. We also demonstrate the effectiveness of our method for neural shape editing

Higher-order Projected Power Iterations for Scalable Multi-Matching

The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take geometric consistency between points into account. Computationally, the multi-matching problem is difficult. It can be phrased as simultaneously solving multiple (NP-hard) quadratic assignment problems (QAPs) that are coupled via cycle-consistency constraints. The main limitations of existing multi-matching methods are that they either ignore geometric consistency and thus have limited robustness, or they are restricted to small-scale problems due to their (relatively) high computational cost. We address these shortcomings by introducing a Higher-order Projected Power Iteration method, which is (i) efficient and scales to tens of thousands of points, (ii) straightforward to implement, (iii) able to incorporate geometric consistency, and (iv) guarantees cycle-consistent multi-matchings. Experimentally we show that our approach is superior to existing methods