ZoomOut

Abstract

We present a simple and efficient method for refining maps or correspondences by iterative upsampling in the spectral domain that can be implemented in a few lines of code. Our main observation is that high quality maps can be obtained even if the input correspondences are noisy or are encoded by a small number of coefficients in a spectral basis. We show how this approach can be used in conjunction with existing initialization techniques across a range of application scenarios, including symmetry detection, map refinement across complete shapes, non-rigid partial shape matching and function transfer. In each application we demonstrate an improvement with respect to both the quality of the results and the computational speed compared to the best competing methods, with up to two orders of magnitude speed-up in some applications. We also demonstrate that our method is both robust to noisy input and is scalable with respect to shape complexity. Finally, we present a theoretical justification for our approach, shedding light on structural properties of functional maps.155:2 • Melzi. et al the same or better quality at a fraction of the cost compared to the current top performing methods.(2) We demonstrate how higher-frequency information can be extracted from low-frequency spectral map representations. (3) We introduce a novel variational optimization problem and develop a theoretical justification of our method, shedding light on structural properties of functional maps. RELATED WORKShape matching is a very well-studied area of computer graphics. Below we review the methods most closely related to ours, concentrating on spectral techniques for finding correspondences between non-rigid shapes. We refer the interested readers to recent surveys including [Biasotti et al. 2016;Tam et al. 2013;Van Kaick et al. 2011] for a more in-depth treatment of the area.Point-based Spectral Methods. Early spectral methods for shape correspondence were based on directly optimizing pointwise maps between spectral shape embeddings based on either adjacency or Laplacian matrices of graphs and triangle meshes [Jain and Zhang 2006;Jain et al. 2007;Mateus et al. 2008;Ovsjanikov et al. 2010;Scott and Longuet-Higgins 1991;Umeyama 1988]. Such approaches suffer from the requirement of a good initialization, and rely on restricting assumptions about the type of transformation relating the shapes. An initialization algorithm with optimality guarantees, although limited to few tens of points, was introduced in [Maron et al. 2016] and later extended to deal with intrinsic symmetries in [Dym and Lipman 2017]. Spectral quantities (namely, sequences of Laplacian eigenfunctions) have also been used to define pointwise descriptors, and employed within variants of the quadratic assignment problem in Kimmel 2010, 2011]. These approaches have been recently generalized by spectral generalized multidimensional scaling [Aflalo et al. 2016], which explicitly formulates minimumdistortion shape correspondence in the spectral domain.

CCuantuMM: Cycle-Consistent Quantum-Hybrid Matching of Multiple Shapes

“…Our experiments show that CCuantuMM significantly outperforms several variants of the previous quantumhybrid method Q-Match [42]. It is even competitive with several non-learning-based classical state-of-the-art shape methods [22,36] and can match more shapes than them. In a broader sense, this paper demonstrates the very high potential of applying (currently available and future) quantum hardware in computer vision.…”

“…Matching shape pairs is a classical problem in geometry processing [36]. When more than two shapes of the same class exist, stronger geometric cues can be leveraged to improve results by matching all of them simultaneously.…”

“…HiPPI [2] is a computationally efficient method that takes geometric relations into account while generalising permutation synchronization but is still limited in resolution. Instead of looking at permutations directly, ZoomOut [36] reduces the dimensionality of the problem by projecting it onto the spectral decomposition. This idea has been extended to take cycle consistency within the spectral space into account [25], which does not guarantee a point-wise consistent matching.…”

Non‐Isometric Shape Matching via Functional Maps on Landmark‐Adapted Bases

“…For each shape of the SHREC'19 dataset [ MMR*19b ], we compute the matrix with entries , where designates the i ‐th basis vector. We use 7 landmarks, 10 Dirichlet‐Steklov eigenfunctions, leading to a Dirichlet‐Steklov block of size 70 × 70, and 120 Dirichlet Laplacian eigenfunctions.…”

WTFM Layer: An Effective Map Extractor for Unsupervised Shape Correspondence

“…Therefore, many studies try to formulate powerful constraints such as preservation of geometric quantities (e.g., descriptors) combined with commutativity to optimize the structural properties of the functional maps [OBCS*12]. Later, follow‐up research have been extended to partial shapes [RCB*17, LRB*16], refined pointwise maps [RMC17], direction‐preserving maps [RPWO18], and iteratively spectral upsampling maps [MRR*19], as well as maps combined with the matrix scaling schemes from computational optimal transport [PRM*21] and with extrinsic shape alignment [ELC20]. Despite great successes achieved by these axiomatic methods, their performances still heavily depend on the quality of the inputting handcrafted descriptor and most of them make restrictive assumptions about the discretization, topology, or deformation of the considered shapes.…”