Structured Regularization of Functional Map Computations

  title={Structured Regularization of Functional Map Computations},
  author={Jing Ren and Mikhail Panine and Peter Wonka and Maks Ovsjanikov},
  journal={Computer Graphics Forum},
We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel… Expand

Topics from this paper

Discrete Optimization for Shape Matching
A novel discrete solver is proposed for optimizing functional map‐based energies, including descriptor preservation and promoting structural properties such as area‐preservation, bijectivity and Laplacian commutativity among others, and achieves the state‐of‐the‐art accuracy on the SHREC'19 benchmark. Expand
Fast Sinkhorn Filters: Using Matrix Scaling for Non-Rigid Shape Correspondence with Functional Maps-Supplementary Material
In this supplementary material, we provide additional theoretical insights and qualitative examples that could not be fitted into the main paper due of lack of space. Specifically, we organize theExpand
DPFM: Deep Partial Functional Maps
This paper proposes the first learning method aimed directly at partial non-rigid shape correspondence, which uses the functional map framework, can be trained in a supervised or unsupervised manner, and learns descriptors directly from the data, thus both improving robustness and accuracy in challenging cases. Expand
Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence
This work presents a novel learning-based approach for computing correspondences between non-rigid 3D shapes that can learn from less training data than existing supervised approaches and generalizes significantly better than current descriptor-based learning methods. Expand
A functional skeleton transfer
This paper proposes a functional approach for skeleton transfer that uses limited information and does not require a complete match between the geometries, and suggests a novel representation for the skeleton properties, namely the functional regressor, which is compact and invariant to different discretizations and poses. Expand
Robust Shape Collection Matching and Correspondence from Shape Differences
This work proposes a method to automatically match two shape collections with a similar shape space structure, e.g. two characters in similar poses, and compute the inter‐maps between the collections, which compare favorably with automatic state‐of‐the‐art methods for non‐isometric shape correspondence. Expand
Isometric Multi-Shape Matching
A novel optimisation formulation for isometric multi-shape matching that obtains multi-matchings that are by construction provably cycle-consistent and provides a convergence and complexity analysis. Expand
Recent advances in shape correspondence
This survey covers the period from 2011, their stopping point, to 2019, inclusive, to present the recent updates on correspondence computation between surfaces or point clouds embedded in 3D. Expand
A Dual Iterative Refinement Method for Non-rigid Shape Matching
In this work, a simple and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes that is not only efficient but also robust to render accurate results within a few iterations. Expand


Fully Spectral Partial Shape Matching
An efficient procedure for calculating partial dense intrinsic correspondence between deformable shapes performed entirely in the spectral domain is proposed and a variant of the JAD problem with an appropriately modified coupling term allows to construct quasi‐harmonic bases localized on the latent corresponding parts. Expand
Coupled Functional Maps
An unbiased formulation of the shape matching problem is considered, in which both sides solve simultaneously for a low-distortion map relating the two given shapes and its inverse, resulting in an especially compact and efficient optimization problem. Expand
Adjoint Map Representation for Shape Analysis and Matching
It is shown that the adjoint operators can be used within the cycle‐consistency framework to encode and reveal the presence or lack of consistency between distortions in a collection, in a way that is complementary to the previously used purely map‐based consistency measures. Expand
Kernel Functional Maps
This paper presents a kernelized version of functional maps including a recent extension in terms of pointwise multiplication operators, and provides an efficient conjugate gradient algorithm for optimizing the authors' generalized problem as well as a strategy for low‐rank estimation of kernel matrices through the Nyström approximation. Expand
Functional maps
A novel representation of maps between pairs of shapes that allows for efficient inference and manipulation and supports certain algebraic operations such as map sum, difference and composition, and enables a number of applications, such as function or annotation transfer without establishing point-to-point correspondences. Expand
Point-wise Map Recovery and Refinement from Functional Correspondence
This paper analyzes the general problem of point-wise map recovery from arbitrary functional maps and devise an efficient recovery process based on a simple probabilistic model that achieves remarkable accuracy improvements in very challenging cases. Expand
Deblurring and Denoising of Maps between Shapes
This work names this problem map deblurring and proposes a robust method, based on a smoothness assumption, for its solution, which is suitable for non‐isometric shapes, is robust to mesh tessellation and accurately recovers vertex‐to‐point, or precise, maps. Expand
Topological Function Optimization for Continuous Shape Matching
The method is based on using the previously‐proposed persistence diagrams associated with real‐valued functions, and on the analysis of the derivatives of these diagrams with respect to changes in the function values allows for continuous optimization techniques to modify a given function, while optimizing an energy based purely on the values in the persistence diagrams. Expand
Partial Functional Correspondence
P perturbation analysis is used to show how removal of shape parts changes the Laplace–Beltrami eigenfunctions, and exploit it as a prior on the spectral representation of the correspondence. Expand
Interactive Curve Constrained Functional Maps
A user interface implementing an interactive process for constructing shape correspondence, allowing the user to update the functional map at interactive rates by introducing feature curve correspondences, and proposing an efficient numerical method to optimize the map with immediate feedback. Expand