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 Combinatorial Embedding Networks for Deep Graph Matching

Graph matching refers to finding node correspondence between graphs, such that the corresponding node and edge's affinity can be maximized. In addition with its NP-completeness nature, another important challenge is effective modeling of the node-wise and structure-wise affinity across graphs and the resulting objective, to guide the matching procedure effectively finding the true matching against noises. To this end, this paper devises an end-to-end differentiable deep network pipeline to learn the affinity for graph matching. It involves a supervised permutation loss regarding with node correspondence to capture the combinatorial nature for graph matching. Meanwhile deep graph embedding models are adopted to parameterize both intra-graph and cross-graph affinity functions, instead of the traditional shallow and simple parametric forms e.g. a Gaussian kernel. The embedding can also effectively capture the higher-order structure beyond second-order edges. The permutation loss model is agnostic to the number of nodes, and the embedding model is shared among nodes such that the network allows for varying numbers of nodes in graphs for training and inference. Moreover, our network is class-agnostic with some generalization capability across different categories. All these features are welcomed for real-world applications. Experiments show its superiority against state-of-the-art graph matching learning methods.Comment: ICCV2019 oral. Code available at https://github.com/Thinklab-SJTU/PCA-G

An Approach to the Category of Net Computations

We introduce the notion of strongly concatenable process as a refinement of concatenable processes [DMM89] which can be expressed axiomatically via a functor $Q[-]$ from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net $N$, the strongly concatenable processes of $N$ are isomorphic to the arrows of $Q[N]$. In addition, we identify a coreflection right adjoint to $Q[-]$ and characterize its replete image, thus yielding an axiomatization of the category of net computations

On the Category of Petri Net Computations

We introduce the notion of strongly concatenable process as a refinement of concatenable processes [DMM89] which can be expressed axiomatically via a functor $Q[-]$ from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net $N$, the strongly concatenable processes of $N$ are isomorphic to the arrows of $Q[-]$. In addition, we identify a coreflection right adjoint to $Q[-]$ and characterize its replete image, thus yielding an axiomatization of the category of net computations