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

applications of manifolds and research challenges
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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 1

Applications of Manifolds and Research Challenges

Luiz Velho IMPA

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SLIDE 2

Outline

  • Concepts
  • Illumination
  • Appearance
  • Simulation
  • Faces
  • Manifold Learning
  • Wrap-up
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SLIDE 3
  • 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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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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SLIDE 7

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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SLIDE 8

The Sphere

  • Construction [Grimm 2002]
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SLIDE 9

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

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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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SLIDE 23

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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SLIDE 28

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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SLIDE 30

Questions ?