Seamless surface mappings

Aigerman, Noam; Poranne, Roi; Lipman, Yaron

doi:10.1145/2766921

Cited by 82 publications

(51 citation statements)

References 40 publications

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“…which implies d ρ f = 0. The equality involving ker ∆ (1) ρ follows from a similar argument. The following decomposition results follow from standard Hodge-theoretic arguments.…”

Section: Twisted Hodge Theory and Synchronizabilitymentioning

confidence: 82%

The Geometry of Synchronization Problems and Learning Group Actions

Gao

Brodzki

Mukherjee

2019

Discrete Comput Geom

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We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph Γ with a flat principal G-bundle over Γ , thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of Γ into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.

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“…which implies d ρ f = 0. The equality involving ker ∆ (1) ρ follows from a similar argument. The following decomposition results follow from standard Hodge-theoretic arguments.…”

Section: Twisted Hodge Theory and Synchronizabilitymentioning

confidence: 82%

The Geometry of Synchronization Problems and Learning Group Actions

Gao

Brodzki

Mukherjee

2019

Discrete Comput Geom

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“…A drawback is that both these approaches (as opposed to the original invisible seams method [RNLL10]) also imposes limitations on the texture data (as sampled in the texel values). Though not directly useful for texture mapping purposes, the concept of seamless uv assignment can be generalized to generic 2D affine transformations [APL15].…”

Section: Perfecting Traditional Texture Mappingmentioning

confidence: 99%

Rethinking Texture Mapping

Yuksel

Lefèbvre

Tarini

2019

Computer Graphics Forum

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The intrinsic problems of texture mapping, regarding its difficulties in content creation and the visual artifacts it causes in rendering, are well‐known, but often considered unavoidable. In this state of the art report, we discuss various radically different ways to rethink texture mapping that have been proposed over the decades, each offering different advantages and trade‐offs. We provide a brief description of each alternative texturing method along with an evaluation of its strengths and weaknesses in terms of applicability, usability, filtering quality, performance, and potential implementation related challenges.

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“…The algorithm uses sequential convex programming, where at each iteration a convex program is solved followed by an update of the frames which enables gradual exploration of the nonconvex space. A similar approach was used in [APL14, APL15] for computing shape correspondences. The advantage of these methods is that they can generate arbitrary maps including nonsmooth maps, and have the potential for a larger feasible space.…”

Section: Previous Workmentioning

confidence: 99%

A Subspace Method for Fast Locally Injective Harmonic Mapping

Hefetz

Chien²,

Weber

2019

Computer Graphics Forum

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We present a fast algorithm for low‐distortion locally injective harmonic mappings of genus 0 triangle meshes with and without cone singularities. The algorithm consists of two portions, a linear subspace analysis and construction, and a nonlinear non‐convex optimization for determination of a mapping within the reduced subspace. The subspace is the space of solutions to the Harmonic Global Parametrization (HGP) linear system [BCW17], and only vertex positions near cones are utilized, decoupling the variable count from the mesh density. A key insight shows how to construct the linear subspace at a cost comparable to that of a linear solve, extracting a very small set of elements from the inverse of the matrix without explicitly calculating it. With a variable count on the order of the number of cones, a tangential alternating projection method [HCW17] and a subsequent Newton optimization [CW17] are used to quickly find a low‐distortion locally injective mapping. This mapping determination is typically much faster than the subspace construction. Experiments demonstrating its speed and efficacy are shown, and we find it to be an order of magnitude faster than HGP and other alternatives.

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Seamless surface mappings

Cited by 82 publications

References 40 publications

The Geometry of Synchronization Problems and Learning Group Actions

The Geometry of Synchronization Problems and Learning Group Actions

Rethinking Texture Mapping

A Subspace Method for Fast Locally Injective Harmonic Mapping

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