Discretization and Approximation of Surfaces Using Varifolds

Buet, Blanche; Leonardi, Gian Paolo; Masnou, Simon

doi:10.1515/geofl-2018-0004

Cited by 15 publications

(9 citation statements)

References 15 publications

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“…These fields are called currents as generalized differential forms or measures on the Grassmannian bundle. Geometric measure theory has since been discretized to perform geometric processing tasks [Mullen et al 2007;Buet et al 2018;Mollenhoff and Cremers 2019] and medical imaging [Charon and Trouvé 2014;Vaillant and Glaunes 2005;Glaunes et al 2004;Durrleman et al 2008Durrleman et al , 2009Durrleman et al , 2011.…”

Section: Related Workmentioning

confidence: 99%

“…The present paper only focuses on representing orientable surfaces, which is only a special case in this larger context of geometric measure theory. For example, there are non-orientable and non-manifold surface representations involving measures on Grassmannian bundles, whose computational aspects have been drawing increasing attention [Buet et al 2018]. We expect that these measure theoretic ways of representing geometries will bring more elegant solutions and new computational tools to previously challenging problems.…”

Section: Applicationsmentioning

confidence: 99%

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Computing minimal surfaces with differential forms

Wang

Chern

2021

ACM Trans. Graph.

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We describe a new algorithm that solves a classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve. Existing numerical methods face challenges due to the non-convexity of the problem. Using a representation of curves and surfaces via differential forms on the ambient space, we reformulate this problem as a convex optimization. This change of variables overcomes many difficulties in previous numerical attempts and allows us to find the global minimum across all possible surface topologies. The new algorithm is based on differential forms on the ambient space and does not require handling meshes. We adopt the Alternating Direction Method of Multiplier (ADMM) to find global minimal surfaces. The resulting algorithm is simple and efficient: it boils down to an alternation between a Fast Fourier Transform (FFT) and a pointwise shrinkage operation. We also show other applications of our solver in geometry processing such as surface reconstruction.

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Section: Related Workmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

Computing minimal surfaces with differential forms

Wang

Chern

2021

ACM Trans. Graph.

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“…In addition, one can check by direct application of the area formula that for any diffeomorphism φ, the pushforward varifold φ # µ X is nothing but the varifold µ φ(X) associated to the deformed set φ(X). Lastly, we conclude this brief review by pointing out that, beyond its interest for the shape analysis problems we consider here, this representation of rectifiable sets as varifolds can be also very useful in computational geometry, for example in the estimation of discrete curvatures [33] and curvature flows [34].…”

Section: Weight Metamorphoses On Varifolds 21 Diffeomorphic Varifold ...mentioning

confidence: 99%

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Hsieh¹,

Charon²

2021

Preprint

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This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.

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“…There is a long history of interactions between geometric analysis and measure theory that goes back to the early 20th century alongside the development of convex geometry [2,23] and later on, in the 1960s, with the emergence of a whole new field known as geometric measure theory [22] from the works of Federer, Fleming and their students. Since then, ideas from geometric measure theory have found their way into applied mathematics most notably in areas such as computational geometry [16,1,10,31] and shape analysis [29,26,14,41] with applications in physics, image analysis and reconstruction or computer vision among many others.…”

Section: Introductionmentioning

confidence: 99%

On length measures of planar closed curves and the comparison of convex shapes

Charon¹,

Pierron²

2020

Preprint

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In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle S 1 that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski-Fenchel-Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski-Fenchel-Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach.

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Discretization and Approximation of Surfaces Using Varifolds

Cited by 15 publications

References 15 publications

Computing minimal surfaces with differential forms

Computing minimal surfaces with differential forms

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

On length measures of planar closed curves and the comparison of convex shapes

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