Spectral Processing of Tangential Vector Fields

Brandt, Christopher; Scandolo, Leonardo; Eisemann, Elmar; Hildebrandt, Klaus

doi:10.1111/cgf.12942

Cited by 16 publications

(29 citation statements)

References 80 publications

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“…Song et al [SLMR14] defined a saliency measure on surfaces that combined spectral and spatial information and takes advantage of the global nature of information embedded in the low‐frequency eigenfunction. Spectral methods have also been used for tangential vector and n‐vector field processing on surfaces [ABCCO13, AOCBC15,BSEH17,BSEH18].…”

Section: Related Workmentioning

confidence: 99%

Fast Approximation of Laplace‐Beltrami Eigenproblems

Nasikun

Brandt

Hildebrandt

2018

Computer Graphics Forum

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The spectrum and eigenfunctions of the Laplace‐Beltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a large‐scale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the Laplace‐Beltrami operator. Our experiments indicate that the resulting spectra well‐approximate reference spectra, which are computed with state‐of‐the‐art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.

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Section: Related Workmentioning

confidence: 99%

Fast Approximation of Laplace‐Beltrami Eigenproblems

Nasikun

Brandt

Hildebrandt

2018

Computer Graphics Forum

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“…The topological and geometric properties of vector fields and differential operators on spaces of vector fields are well studied, and discretizations of classical operators on triangle meshes have been established [Desbrun et al 2005;Polthier and Preuß 2003;Tong et al 2003;Wardetzky 2006]. In [Brandt et al 2017;Fisher et al 2007] quadratic fairness energies have been used for variational vector field design. In both papers, the benefits of using biharmonic energies over harmonic energies for vector field design and modeling are emphasized.…”

Section: Related Workmentioning

confidence: 99%

“…Recently, this approach has been generalized to fluid simulation on surfaces, where the eigenmodes of the Hodge-Laplace operator are used [Liu et al 2015]. In a parallel development, the eigenmodes of the Hodge-Laplace operator have been used for spectral processing of tangential vector fields on surfaces [Brandt et al 2017]. In a different recent development, eigenfields of the Hodge-Laplace operator have been used for functional representation of linear operators on spaces of tangential vector fields [Azencot et al 2015].…”

Section: Related Workmentioning

confidence: 99%

“…with l e i j being the length of the common edge between triangles T i and T j . This harmonic energy is a natural extension to n-vector fields of the harmonic energy for face-based, piecewise constant, tangential vector fields, see [Brandt et al 2017]. A full derivation of the energy, and the weights w i j in particular, can be found in the appendix.…”

Section: Background: N-vector Fieldsmentioning

confidence: 99%

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Modeling n -Symmetry Vector Fields using Higher-Order Energies

et al. 2018

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We introduce a variational approach for modeling n-symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing real-time responses. The approach is based on novel biharmonic and m-harmonic energies for n-fields on surface meshes and the integration of hard constraints to the resulting optimization problems. Real-time computation rates are achieved by a model reduction approach employing a Fourier-like n-vector field decomposition, which associates frequencies and modes to n-vector fields on surfaces. To demonstrate the benefits of the proposed n-field modeling approach, we use it for controlling stroke directions in line-art drawings of surfaces and for the modeling of anisotropic BRDFs which define the reflection behavior of surfaces.

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“…In the recent work [Brandt et al 2016] a discrete Hodge-Laplace operator and a spectral decomposition of the space of piecewise constant vector fields compatible with the discrete Hodge decomposition have been introduced.…”

Section: Discrete Harmonic Vector Fieldsmentioning

confidence: 99%

Directional field synthesis, design, and processing

Vaxman

Campen

Diamanti

et al. 2017

ACM SIGGRAPH 2017 Courses

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SummaryDirection fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector-set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This course provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges. PrerequisitesThe audience should have some prior experience with triangle mesh representation of geometric models, and a working knowledge of vector calculus, linear algebra, and general computer graphics fundamentals. Some familiarity with the basics of differential geometry and numerical optimization are helpful, but not required. Intended AudienceThe course targets researchers and developers who seek to understand the concepts and technologies used in direction field and vector field synthesis, learn about the most recent developments, and discern how this powerful tool, which has had impact in a variety of research and application areas, might benefit their area of work. Participants will get a broad overview, and obtain the knowledge on how to choose the proper combination of techniques for many relevant tasks. SourcesThese notes are largely based on the following state-of-the-art report by the lecturers. It has been extended to include updates on the most recent developments. • The course was subsequently given at SIGGRAPH Asia 2016, including demos and real-time coding sessions. The entire course, including the notes, the presentation slides, and the demos, is provided in the following open-source GitHub repository: https://github.com/avaxman/DirectionalFieldSynthesis Further ReadingBeing a relatively young and developing topic, no textbooks covering the various aspects of directional field synthesis in the context of computer graphics and geometry processing are available. The notes of a recent course on vector field processing offer another perspective on parts of the topic, with a focus on the discrete differential geometry aspects:• F. Her current interests are in geometry processing and modeling, specifically on vector field design, surface parametrizations, and inter-surface mappings. David Bommes RWTH Aachen University, GermanyDavid Bommes is an assistant professor in the Computer Science ...

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Spectral Processing of Tangential Vector Fields

Cited by 16 publications

References 80 publications

Fast Approximation of Laplace‐Beltrami Eigenproblems

Fast Approximation of Laplace‐Beltrami Eigenproblems

Modeling n -Symmetry Vector Fields using Higher-Order Energies

Directional field synthesis, design, and processing

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