Improved Functional Mappings via Product Preservation

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, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.

Section: Related Workmentioning

confidence: 99%

Structured Regularization of Functional Map Computations

Ren

Panine

Wonka

et al. 2019

“…Other recent functional map regularizations, constraints and priors [ERGB16, VLB*17, NO17, RPWO18] are complementary to our method, as they can be applied at the coarsest level instead of the basic functional map method that we used. In [NMR*18] the authors suggested an approach to transfer high frequency functions and improve existing functional maps via product preservation. Since this step comes on top of an existing functional map it is complementary to our approach, and may serve as an additional improvement.…”

Section: Introductionmentioning

confidence: 99%

“…Since this step comes on top of an existing functional map it is complementary to our approach, and may serve as an additional improvement. Finally, the point‐to‐point reconstruction step has been addressed as a separate problem in the functional framework [RMC15, EBC17, ESBC19, NMR*18], and some of these methods provide a vertex‐to‐point‐in‐triangle map as output, which can be used for transferring smooth textures. Note, though, that the meshes still need to be very fine, in order to support non‐linear texture deformation, leading to long running times and large memory consumption.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Functional Maps between Subdivision Surfaces

Shoham

Vaxman

Ben‐Chen

2019

We propose a novel approach for computing correspondences between subdivision surfaces with different control polygons. Our main observation is that the multi‐resolution spectral basis functions that are open used for computing a functional correspondence can be compactly represented on subdivision surfaces, and therefore can be efficiently computed. Furthermore, the reconstruction of a pointwise map from a functional correspondence also greatly benefits from the subdivision structure. Leveraging these observations, we suggest a hierarchical pipeline for functional map inference, allowing us to compute correspondences between surfaces at fine subdivision levels, with hundreds of thousands of polygons, an order of magnitude faster than existing correspondence methods. We demonstrate the applicability of our results by transferring high‐resolution sculpting displacement maps and textures between subdivision models.

“…Two recent papers [N017,NMR*18] consider a related problem, namely pointwise product preservation under functional maps, in two different ways. The first uses a different data term enforcing commutativity of the functional map with the pointwise product operation.…”

Section: Introductionmentioning

confidence: 99%

“…The second represents functional maps in an extended basis containing pointwise products of the basis functions. Importantly, kernelization can be applied to these and other formulations and flavors of functional maps, complementing their work by explicitly promoting preservation of not only pointwise products but also nonlinearities applied to descriptor values [NO17,NMR*18].…”

Section: Introductionmentioning

confidence: 99%

Kernel Functional Maps

Wang

Gehre

Bronstein

et al. 2018

Functional maps provide a means of extracting correspondences between surfaces using linear‐algebraic machinery. While the functional framework suggests efficient algorithms for map computation, the basic technique does not incorporate the intuition that pointwise modifications of a descriptor function (e.g. composition of a descriptor and a nonlinearity) should be preserved under the mapping; the end result is that the basic functional maps problem can be underdetermined without regularization or additional assumptions on the map. In this paper, we show how this problem can be addressed through kernelization, in which descriptors are lifted to higher‐dimensional vectors or even infinite‐length sequences of values. The key observation is that optimization problems for functional maps only depend on inner products between descriptors rather than descriptor values themselves. These inner products can be evaluated efficiently through use of kernel functions. In addition to deriving a kernelized version of functional maps including a recent extension in terms of pointwise multiplication operators, we provide an efficient conjugate gradient algorithm for optimizing our generalized problem as well as a strategy for low‐rank estimation of kernel matrices through the Nyström approximation.