Weighted Triangulations for Geometry Processing

Goes, Fernando de; Memari, Pooran; Mullen, Patrick; Desbrun, Mathieu

doi:10.1145/2602143

Cited by 31 publications

(32 citation statements)

References 66 publications

(113 reference statements)

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“…The transportation cost from continuous to pointwise measures, for instance, can be computed either via multiscale algorithms [Mérigot 2011;Schwartzburg et al 2014] or through Newton iterations on Euclidean spaces [de Goes et al 2012;Zhao et al 2013]. More recently, this Newton-based approach for optimal transportation was extended to discrete surfaces [de Goes et al 2014]. Transportation distances between point clouds and line segments also were approximated in 2D based on a triangulation tiling of the plane and greedy point-tosegment clustering [de Goes et al 2011].…”

Section: Related Workmentioning

confidence: 99%

Convolutional wasserstein distances

Goes²,

et al. 2015

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Figure 1: Shape interpolation from a cow to a duck to a torus via convolutional Wasserstein barycenters on a 100×100×100 grid, using the method at the beginning of §7. AbstractThis paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

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Section: Related Workmentioning

confidence: 99%

Convolutional wasserstein distances

Goes²,

et al. 2015

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“…Zeng et al [ZGLG12] showed that the converse also holds for discrete metrics, and formulated the problem of discrete metric reconstruction from the Laplacian. It was shown later by [dGMMD14] that this problem boils down to minimizing the conformal energy.…”

Section: Related Workmentioning

confidence: 99%

“…De Goes et al [dGMMD14] derived a closedform expression of (7), which turns out to be the classical conformal energy…”

Section: Metric-from-laplacianmentioning

confidence: 99%

“…Note that our shape-from-Laplacian problem is different from the metric-from-Laplacian problems considered by [ZGLG12] and [dGMMD14] in the sense that we are additionally looking for an embedding that realizes the discrete metric. Secondly, unlike [ZGLG12,dGMMD14], we allow for arbitrary (not necessarily bijective) correspondence between X and Y .…”

Section: Shape-from-operatormentioning

confidence: 99%

“…Secondly, unlike [ZGLG12,dGMMD14], we allow for arbitrary (not necessarily bijective) correspondence between X and Y .…”

Section: Shape-from-operatormentioning

confidence: 99%

See 2 more Smart Citations

Shape‐from‐Operator: Recovering Shapes from Intrinsic Operators

Boscaini

Eynard

Kourounis

et al. 2015

Computer Graphics Forum

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Figure 1: Examples of three different shape-from-operator problems considered in the paper. Left: shape analogy synthesis as shape-from-difference operator problem (shape X is synthesized such that the intrinsic difference operator between C, X is as close as possible to the difference between A, B). Center: style transfer as shape-from-Laplacian problem. The Laplacian of the leftmost shape (fat man) captures the style, while the initial thin man shape (middle) captures the pose. Right: we deform the human shape (bottom leftmost) such that its Laplacian is diagonalized by the first 10 eigenfunctions of the alien Laplacian (top row). The result (bottom rightmost) is an 'intrinsic hybrid' between the two shapes. AbstractWe formulate the problem of shape-from-operator (SfO), recovering an embedding of a mesh from intrinsic operators defined through the discrete metric (edge lengths). Particularly interesting instances of our SfO problem include: shape-from-Laplacian, allowing to transfer style between shapes; shape-from-difference operator, used to synthesize shape analogies; and shape-from-eigenvectors, allowing to generate 'intrinsic averages' of shape collections. Numerically, we approach the SfO problem by splitting it into two optimization sub-problems: metric-from-operator (reconstruction of the discrete metric from the intrinsic operator) and embedding-from-metric (finding a shape embedding that would realize a given metric, a setting of the multidimensional scaling problem). We study numerical properties of our problem, exemplify it on several applications, and discuss its imitations.

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Variational Wasserstein Clustering

Liang

Zhang

et al. 2018

Computer Vision – ECCV 2018

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We propose a new clustering method based on optimal transportation. We discuss the connection between optimal transportation and k-means clustering, solve optimal transportation with the variational principle, and investigate the use of power diagrams as transportation plans for aggregating arbitrary domains into a fixed number of clusters. We drive cluster centroids through the target domain while maintaining the minimum clustering energy by adjusting the power diagram. Thus, we simultaneously pursue clustering and the Wasserstein distance between the centroids and the target domain, resulting in a measure-preserving mapping. We demonstrate the use of our method in domain adaptation, remeshing, and learning representations on synthetic and real data.

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Weighted Triangulations for Geometry Processing

Cited by 31 publications

References 66 publications

Convolutional wasserstein distances

Convolutional wasserstein distances

Shape‐from‐Operator: Recovering Shapes from Intrinsic Operators

Variational Wasserstein Clustering

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