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