Structured Regularization of Functional Map Computations

@article{Ren2019StructuredRO,
  title={Structured Regularization of Functional Map Computations},
  author={Jing Ren and Mikhail Panine and Peter Wonka and Maks Ovsjanikov},
  journal={Computer Graphics Forum},
  year={2019},
  volume={38}
}
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

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References

SHOWING 1-10 OF 51 REFERENCES
Fully Spectral Partial Shape Matching
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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