Spectral Generalized Multi-dimensional Scaling

We present a novel approach for solving the correspondence problem between a given pair of input shapes with non‐rigid, nearly isometric pose difference. Our method alternates between calculating a deformation field and a sparse correspondence. The deformation field is constructed with a low rank Fourier basis which allows for a compact representation. Furthermore, we restrict the deformation fields to be divergence‐free which makes our morphings volume preserving. This can be used to extract a correspondence between the inputs by deforming one of them along the deformation field using a second order Runge‐Kutta method and resulting in an alignment of the inputs. The advantages of using our basis are that there is no need to discretize the embedding space and the deformation is volume preserving. The optimization of the deformation field is done efficiently using only a subsampling of the orginal shapes but the correspondence can be extracted for any mesh resolution with close to linear increase in runtime. We show 3D correspondence results on several known data sets and examples of natural intermediate shape sequences that appear as a by‐product of our method.

Section: Related Workmentioning

confidence: 99%

Divergence‐Free Shape Correspondence by Deformation

Eisenberger

Lähner

Cremers

2019

“…Functional Maps. Our approach fits within the functional map framework, which was originally introduced in [OBCS*12] for solving non‐rigid shape matching problems, and extended significantly in follow‐up works, including [KBB*13, ADK16, KBBV15, RCB*17, EBC17, BDK17] among many others (see also [OCB*17] for an overview). The key observation in these techniques is that it is often easier to estimate correspondences between real‐valued functions, rather than points on the shapes.…”

Section: Related Workmentioning

confidence: 99%

Structured Regularization of Functional Map Computations

Ren

Panine

Wonka

et al. 2019

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.

“…Multiple follow‐up works aimed mainly at improving the stability and applicability of the framework; see [COC*17] for a survey. Notable extensions of functional maps include incorporation of sparsity‐based regularization [PBB*13], optimization on the manifold of orthogonal matrices [KBB*13, KGB16], design [ADK16] and tuning [COC14, LRR*17] of descriptors, correspondence between shape collections [HWG14, KGB16], partial correspondence [LRB*16,RCB*17], incorporation of descriptors through operators rather than least‐squares [NO17], use of adjoint operators to incorporate information about the reverse map [ERGB16, HO17], and regularization based on conserving products in addition to linear combinations [NMR*18]. Functional maps also have been combined with deep learning to learn how to extract dense correspondences [LRR*17].…”

Section: Related Workmentioning

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.