Generalized Metric Energies for Continuous Shape Deformation

Esturo, Janick Martinez; Rössl, Christian; Theisel, Holger

doi:10.1007/978-3-642-54382-1_8

Cited by 10 publications

(8 citation statements)

References 28 publications

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“…More advanced presentations are given by Petersen [37], and do Carmo [15]. Approximate Killing fields have been used in shape deformation [33], shape space exploration [28], planar as-Killing-as-possible vector fields [43], on curved surfaces [6], and for designing approximate Killing fields on meshes [3,4]. Approximate Killing fields have also been used for the decomposition of 2-tensors in a compact region of 2D Euclidean space [14], and for non-rigid image registration [13].…”

Section: Related Workmentioning

confidence: 99%

Time-Dependent Flow seen through Approximate Observer Killing Fields

Hadwiger

Mlejnek

Theußl

et al. 2019

IEEE Trans. Visual. Comput. Graphics

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Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

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Section: Related Workmentioning

confidence: 99%

Time-Dependent Flow seen through Approximate Observer Killing Fields

Hadwiger

Mlejnek

Theußl

et al. 2019

IEEE Trans. Visual. Comput. Graphics

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“…Non-linear properties of a deformation, such as area/volume preservation, or isometry, are represented as properties of the displacement field. Recently, [MERT14] created a generalized formulation that includes these two cases. Hence, deformations that are approximately volume preserving can be generated by successively deforming a shape by small displacement fields, that are divergence free [vFTS06].…”

Section: Deformationmentioning

confidence: 99%

“…In a similar spirit, deformations which are as isometric as possible can be generated using approximate Killing vector fields [SBCBG11a]. Recently, [MERT14] created a generalized formulation that includes these two cases.…”

Section: Deformationmentioning

confidence: 99%

Directional Field Synthesis, Design, and Processing

Vaxman

Campen

Diamanti

et al. 2016

Computer Graphics Forum

111

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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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“…In contrast, our technique is applicable to such scans by constraining transformations to be conformal directly in three‐dimensional space. Recently, Paille et al [PP12] and Martinez et al [MRT13] proposed the formulations of three‐dimensional conformal energies for volumetric parametrization and shape deformation, respectively. Paille et al [PP12] extended the Cauchy‐Riemann equations to 3D.…”

Section: Introductionmentioning

confidence: 99%

“…Paille et al [PP12] extended the Cauchy‐Riemann equations to 3D. Martinez et al [MRT13] formulated their deformation framework as continuous shape deformation, which is the generalization of as‐killing‐as possible deformation [SBCBG11].…”

Section: Introductionmentioning

confidence: 99%

As‐Conformal‐As‐Possible Surface Registration

Yoshiyasu

Yoshida

et al. 2014

Computer Graphics Forum

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We present a non-rigid surface registration technique that can align surfaces with sizes and shapes that are different from each other, while avoiding mesh distortions during deformation. The registration is constrained locally as conformal as possible such that the angles of triangle meshes are preserved, yet local scales are allowed to change. Based on our conformal registration technique, we devise an automatic registration and interactive registration technique, which can reduce user interventions during template fitting. We demonstrate the versatility of our technique on a wide range of surfaces.

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Generalized Metric Energies for Continuous Shape Deformation

Cited by 10 publications

References 28 publications

Time-Dependent Flow seen through Approximate Observer Killing Fields

Time-Dependent Flow seen through Approximate Observer Killing Fields

Directional Field Synthesis, Design, and Processing

As‐Conformal‐As‐Possible Surface Registration

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