Functional maps representation on product manifolds

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices

QuAnt: Quantum Annealing with Learnt Couplings

Modern quantum annealers can find high-quality solutions to combinatorialoptimisation objectives given as quadratic unconstrained binary optimisation(QUBO) problems. Unfortunately, obtaining suitable QUBO forms in computervision remains challenging and currently requires problem-specific analyticalderivations. Moreover, such explicit formulations impose tangible constraintson solution encodings. In stark contrast to prior work, this paper proposes tolearn QUBO forms from data through gradient backpropagation instead of derivingthem. As a result, the solution encodings can be chosen flexibly and compactly.Furthermore, our methodology is general and virtually independent of thespecifics of the target problem type. We demonstrate the advantages of learntQUBOs on the diverse problem types of graph matching, 2D point cloud alignmentand 3D rotation estimation. Our results are competitive with the previousquantum state of the art while requiring much fewer logical and physicalqubits, enabling our method to scale to larger problems. The code and the newdataset will be open-sourced.<br

Q-{M}atch: {I}terative Shape Matching via Quantum Annealing

Finding shape correspondences can be formulated as an NP-hard quadratic assignment problem (QAP) that becomes infeasible for shapes with high sampling density. A promising research direction is to tackle such quadratic optimization problems over binary variables with quantum annealing, which allows for some problems a more efficient search in the solution space. Unfortunately, enforcing the linear equality constraints in QAPs via a penalty significantly limits the success probability of such methods on currently available quantum hardware. To address this limitation, this paper proposes Q-Match, i.e., a new iterative quantum method for QAPs inspired by the alpha-expansion algorithm, which allows solving problems of an order of magnitude larger than current quantum methods. It implicitly enforces the QAP constraints by updating the current estimates in a cyclic fashion. Further, Q-Match can be applied iteratively, on a subset of well-chosen correspondences, allowing us to scale to real-world problems. Using the latest quantum annealer, the D-Wave Advantage, we evaluate the proposed method on a subset of QAPLIB as well as on isometric shape matching problems from the FAUST dataset

Geometric deep learning

The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas

SIZER: A Dataset and Model for Parsing 3D Clothing and Learning Size Sensitive 3D Clothing

While models of 3D clothing learned from real data exist, no method can predict clothing deformation as a function of garment size. In this paper, we introduce SizerNet to predict 3D clothing conditioned on human body shape and garment size parameters, and ParserNet to infer garment meshes and shape under clothing with personal details in a single pass from an input mesh. SizerNet allows to estimate and visualize the dressing effect of a garment in various sizes, and ParserNet allows to edit clothing of an input mesh directly, removing the need for scan segmentation, which is a challenging problem in itself. To learn these models, we introduce the SIZER dataset of clothing size variation which includes $100$ different subjects wearing casual clothing items in various sizes, totaling to approximately 2000 scans. This dataset includes the scans, registrations to the SMPL model, scans segmented in clothing parts, garment category and size labels. Our experiments show better parsing accuracy and size prediction than baseline methods trained on SIZER. The code, model and dataset will be released for research purposes.Comment: European Conference on Computer Vision 202

Shape Correspondence with Isometric and Non-Isometric Deformations

The registration of surfaces with non-rigid deformation, especially non-isometric deformations, is a challenging problem. When applying such techniques to real scans, the problem is compounded by topological and geometric inconsistencies between shapes. In this paper, we capture a benchmark dataset of scanned 3D shapes undergoing various controlled deformations (articulating, bending, stretching and topologically changing), along with ground truth correspondences. With the aid of this tiered benchmark of increasingly challenging real scans, we explore this problem and investigate how robust current state-of- the-art methods perform in different challenging registration and correspondence scenarios. We discover that changes in topology is a challenging problem for some methods and that machine learning-based approaches prove to be more capable of handling non-isometric deformations on shapes that are moderately similar to the training set