Applications of Manifolds and Research Challenges Luiz Velho - PowerPoint PPT Presentation

Feb 07, 2023 •109 likes •412 views

Applications of Manifolds and Research Challenges Luiz Velho IMPA Outline Concepts Illumination Appearance Simulation Faces Manifold Learning Wrap-up Manifolds & Parametrization Two Points of View -

Applications of Manifolds and Research Challenges Luiz Velho IMPA
Outline • Concepts • Illumination • Appearance • Simulation • Faces • Manifold Learning • Wrap-up
Manifolds & Parametrization • Two Points of View - Functions on surfaces - Functions defining surfaces M
Desirable Properties • Minimal Distortion - Angle - Area • Smoothness - Differentiability - Continuity
Graphical Objects • Shape U O = (U, f) - Topology ( domain ) Abstract Manifold • - Geometry ( function ) Embedding • Texture • Attributes f Color - Functions ( co-domain )
G.O. Manifold Setting • Canonical Surfaces - Fixed Shape (defined apriori ) - Variable Functions (complex) ex: Sphere • • Arbitrary Surfaces - Complex Shape - Computation on Surfaces (attributes) - Building / Transforming (shape) ex: Triangle Meshes •
Applications • Illumination - Canonical Manifold + Functions • Appearance and Simulation - Pseudo-Manifold + Attributes • Faces - Manifold + Geometric Deformation • Surface Reconstruction - Pseudo-Manifold / Topology Estimation
The Sphere • Construction [Grimm 2002]
Illumination • Functions on the Sphere - Light Fields / BRDFs • Applications - Capture / Synthesis
Illumination Maps • Environment Maps - Area Sampling • Light Maps - Stratification
Material Properties • Spatially Varying BRDFs Σ x x Σ Σ … x
Spherical Panoramas • Panoramic Cameras - Processing • Multi-Camera Assembly - Stitching / Blending Cam 5 Cam 3 Cam 2 Cam 4 Cam 1 Cam 0 Cam 0 Cam 1 Cam 2 Cam 3 Cam 4 Cam 5
Omnidirectional Images • Processing Large Spherical Imagery - Example: Sharpening Obs: Metric Aware Operators
Polygonal Surfaces • Building from Images • Projective Map
Surface Properties • Texture Atlas - Albedo - Normal Field - etc...
Painting • Editing Ops - Color - Normals
Solving Equations on Manifolds • Global Structure - Surface Points - Local Neighborhoods
Simulation • Metric Aware Operators
Fluids • Vector Fields on Surfaces
Biological Processes • Reaction Diffusion • Examples:
Texture Synthesis • Stationary / Quasi Stationary
Faces • Geometry + Appearance [ G. Borshukov et al SIGGRAPH 2003 ]
Facial Expressions • Deformations
Manifold Learning • Estimate from Data Samples - Topology - Geometry
Surfaces • Point Sets
N-Dimensional Case • ex: Facial Expressions
Multiresolution • Manifold T -Spline (Gu, et al)
Adaptation • Hierarchical Surface Reconstruction
Challenges • Representation - Simple / Emcompassing • Operators - Efficient / Accurate • Multi-Resolution - Hierarchical Atlas / Dynamic Setting • API - Intuitive / General
Questions ?

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