Efficient Shape Matching using Vector Extrapolation

Rodolà, Emanuele; Harada, Tatsuya; Kuniyoshi, Yasuo; Cremers, Daniel

doi:10.5244/c.27.91

Cited by 8 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other methods further relax the problem of mapping between surface points to mapping between soft distributions [SNB*12, MCSK*17] or mapping between surface functions expressed in a spectral basis [OBCS*12, RMC15, GBKS18, MRR*19]. The functional settings is attractive, because maps can be represented in a linear and low‐dimensional manner, however no guaranteed conversion to strict homeomorphisms is known.…”

Section: Related Workmentioning

confidence: 99%

Surface Maps via Adaptive Triangulations

Schmidt

Pieper

Kobbelt

2023

Computer Graphics Forum

View full text Add to dashboard Cite

We present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus‐0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete‐continuous optimization algorithm performs both energy‐based remeshing as well as global second‐order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre‐computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface‐to‐surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay‐based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi‐resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state‐of‐the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.

show abstract

Section: Related Workmentioning

confidence: 99%

Surface Maps via Adaptive Triangulations

Schmidt

Pieper

Kobbelt

2023

Computer Graphics Forum

View full text Add to dashboard Cite

show abstract

“…How to measure the matching cost between the two feature sets reliably and efficiently is a well-known key issue. It is usually an assignment problem, which can be solved by Hungarian, Dynamic Programming, Game theory [11] or some more sophisticated, probability-based techniques [12].…”

Section: Multi-view Matchingmentioning

confidence: 99%

Neural shape codes for 3D model retrieval

Bai

Liu

et al. 2015

Pattern Recognition Letters

View full text Add to dashboard Cite

“…Even though originating from distinct motivations, the first two methods share a convenient interpretation as partitioning problems in the space of potential assignments. In Section 3.3 we provide a different view on the problem, as presented in [23,22], by using the language of regression analysis.…”

Section: Minimum Distortion Correspondence Via Elastic Net Regularizamentioning

confidence: 99%

“…A detailed explanation of our approach on the computation of the unique minimumdistance projection Π α in an efficient manner is given in [23]. Also note that, for practical purposes, we adopt a more efficient alternative to the standard projected gradient descent (3.5), namely its acceleration via vector extrapolation techniques [22].…”

Section: Spectral Matchingmentioning

confidence: 99%

Relaxations for Minimizing Metric Distortion and Elastic Energies for 3D Shape Matching

Cremers¹,

Rodolà²,

Windheuser³

2013

Actes du CIRM

Self Cite

View full text Add to dashboard Cite

We present two methods for non-rigid shape matching. Both methods formulate shape matching as an energy minimization problem, where the energy measures distortion of the metric defined on the shapes in one case, or directly describes the physical deformation relating the two shapes in the other case. The first method considers a parametrized relaxation of the widely adopted quadratic assignment problem (QAP) formulation for minimum distortion correspondence between deformable shapes. In order to control the accuracy/sparsity trade-off a weighting parameter is introduced to combine two existing relaxations, namely spectral and game-theoretic. This leads to an approach for deformable shape matching with controllable sparsity. The second method focuses on computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points it matches infinitesimal surface patches while preserving the geometric structures. In this spirit, matchings are considered as diffeomorphisms between the objects' surfaces which are by definition geometrically consistent. Based on the observation that such diffeomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes, this leads to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, the approach results in a binary linear program whose relaxed version can be solved efficiently in a globally optimal manner.

show abstract

Efficient Shape Matching using Vector Extrapolation

Cited by 8 publications

References 22 publications

Surface Maps via Adaptive Triangulations

Surface Maps via Adaptive Triangulations

Neural shape codes for 3D model retrieval

Relaxations for Minimizing Metric Distortion and Elastic Energies for 3D Shape Matching

Contact Info

Product

Resources

About