Earth mover's distances on discrete surfaces

Solomon, Justin; Rustamov, Raif M.; Guibas, Leonidas J.; Butscher, Adrian

doi:10.1145/2601097.2601175

Cited by 117 publications

(99 citation statements)

References 54 publications

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“…Each scheme has its advantages and disadvantages. The p ‐Laplacian scheme is computationally expensive but it can be adapted for approximating optimal transportation problems [EG99, Amb03] which are currently a subject of intensive interest in geometry processing and computer graphics [SRGB14]. The

L_{p}

‐distance fields approximate the signed distance function.…”

Section: Discussionmentioning

confidence: 99%

“…Efficient computation of distance maps and their approximations and generalizations is an important component of many recent studies including those on object and action recognition [Zuc13, WBR14], surface reconstruction [CT11], computational mechanics [FST11], computational photography [CSCRP10], medical imaging [RC13], computer graphics [CWW13, CHK13, YWH13], architectural geometry [PHD*10] and computational fluid dynamics [Tuc11, RS13]. In particular, in the computer graphics field, ‘shape aware’ substitutes and approximations of the exact distance functions are becoming increasingly popular [RLF09, LRF10, PBDSH13, SRGB14]. See also references therein.…”

Section: Introductionmentioning

confidence: 99%

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On Variational and PDE‐Based Distance Function Approximations

Belyaev

Fayolle

2015

Computer Graphics Forum

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In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson-like equations and generalized double-layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE-based distance function approximations.

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L_{p}

‐distance fields approximate the signed distance function.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Variational and PDE‐Based Distance Function Approximations

Belyaev

Fayolle

2015

Computer Graphics Forum

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show abstract

“…A number recent techniques have also considered a more general problem of finding soft (or functional) correspondences between shapes, rather than point‐to‐point ones [SNB*12, OBCS*12, SRGB14]. These methods are based on the idea that in many cases, it is both more efficient and less error‐prone to establish correspondences between probability distributions, or more generally real‐valued functions on the shapes, rather than between pairs of points.…”

Section: Related Workmentioning

confidence: 99%

Stable Region Correspondences Between Non‐Isometric Shapes

Ganapathi-Subramanian

Thibert

Ovsjanikov

et al. 2016

Computer Graphics Forum

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We consider the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar. When the shapes are quite different, a point‐to‐point map is not always appropriate, so our focus in this paper is a method to build a set of correspondences between shape regions or parts. The proposed approach exploits a variety of feature functions on the shapes and makes use of the key observation that points in matching parts have similar ranks in the sorting of the corresponding feature values. Our algorithm proceeds in two steps. We first build an affinity matrix between points on the two shapes, based on feature rank similarity over many feature functions. We then define a notion of stability of a pair of regions, with respect to this affinity matrix, obtained as a fixed point of a nonlinear operator. Our method yields a family of corresponding maximally stable regions between the two shapes that can be used to define shape parts. We observe that this is an instance of the biclustering problem and that it is related to solving a constrained maximal eigenvalue problem. We provide an algorithm to solve this problem that mimics the power method. We show the robustness of its output to noisy input features as well its convergence properties. The obtained part correspondences are shown to be almost perfect matches in the isometric case, and also semantically appropriate even in non‐isometric cases. We provide numerous examples and applications of this technique, for example to sharpening correspondences in traditional shape matching algorithms.

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“…Recent applications include astronomy [9, 18, 19], biomedical sciences [3, 25–27, 77, 81, 82, 88, 89], colour transfer [14, 17, 49, 62, 63], computer vision and graphics [7, 44, 60, 65, 68, 74, 75], imaging [36, 40, 64], information theory [78], machine learning [1, 15, 20, 34, 37, 48, 76], operational research [69] and signal processing [54, 58]. …”

Section: Introductionmentioning

confidence: 99%

A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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Transport based distances, such as the Wasserstein distance and earth mover'sdistance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the TLp distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled non-positive one-dimensional signals and two-dimensional images, and color transfer.

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Earth mover's distances on discrete surfaces

Cited by 117 publications

References 54 publications

On Variational and PDE‐Based Distance Function Approximations

On Variational and PDE‐Based Distance Function Approximations

Stable Region Correspondences Between Non‐Isometric Shapes

A Transportation $$L^p$$ Distance for Signal Analysis

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