Advection-Based Function Matching on Surfaces

A tangent vector field on a surface is the generator of a smooth family of maps from the surface to itself, known as the flow. Given a scalar function on the surface, it can be transported, or advected, by composing it with a vector field's flow. Such transport is exhibited by many physical phe...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Computer graphics forum Jg. 35; H. 5; S. 55 - 64
Hauptverfasser: Azencot, Omri, Vantzos, Orestis, Ben-Chen, Mirela
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Oxford Blackwell Publishing Ltd 01.08.2016
Schlagworte:
Advection
Analysis
and systems
Categories and Subject Descriptors (according to ACM CCS)
Computation
Computer graphics
Fields (mathematics)
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems
Image processing systems
Inverse problems
languages
Mathematical analysis
Mathematical models
Operators
Scalars
Studies
Transport
ISSN:0167-7055, 1467-8659
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:A tangent vector field on a surface is the generator of a smooth family of maps from the surface to itself, known as the flow. Given a scalar function on the surface, it can be transported, or advected, by composing it with a vector field's flow. Such transport is exhibited by many physical phenomena, e.g., in fluid dynamics. In this paper, we are interested in the inverse problem: given source and target functions, compute a vector field whose flow advects the source to the target. We propose a method for addressing this problem, by minimizing an energy given by the advection constraint together with a regularizing term for the vector field. Our approach is inspired by a similar method in computational anatomy, known as LDDMM, yet leverages the recent framework of functional vector fields for discretizing the advection and the flow as operators on scalar functions. The latter allows us to efficiently generalize LDDMM to curved surfaces, without explicitly computing the flow lines of the vector field we are optimizing for. We show two approaches for the solution: using linear advection with multiple vector fields, and using non‐linear advection with a single vector field. We additionally derive an approximated gradient of the corresponding energy, which is based on a novel vector field transport operator. Finally, we demonstrate applications of our machinery to intrinsic symmetry analysis, function interpolation and map improvement.
Bibliographie:ArticleID:CGF12963
istex:0F94F78DA4B2F20B2F739605881928AF7EE545FF
ark:/67375/WNG-0PTR3MHZ-1
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ObjectType-Article-1
ObjectType-Feature-2
content type line 23
ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.12963