SLIDE 1
Applications of Manifolds and Research Challenges
Luiz Velho IMPA
Outline
- Concepts
- Illumination
- Appearance
- Simulation
- Faces
- Manifold Learning
- Wrap-up
Applications of Manifolds and Research Challenges Luiz Velho - - PowerPoint PPT Presentation
Applications of Manifolds and Research Challenges Luiz Velho - - PowerPoint PPT Presentation
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 -
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SLIDE 3
- Two Points of
View
- Functions on surfaces
- Functions defining surfaces
M
Manifolds & Parametrization
SLIDE 4
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
SLIDE 6
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
x
x
Σ
x
Σ
…
SLIDE 12
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
SLIDE 13
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:
SLIDE 21
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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