Statistical Geometry Processing Winter Semester 2011/2012
Introduction
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Research Direction
Statistical geometry processing
• Geometric datasets
• Machine learning / statistical data analysis in graphics
Goals
• Discover structure in geometric data
• Long term goal: “Shape understanding”
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Ambitious Problem
What does shape understanding mean?
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Practical Relevance
Computer graphics
• Content creation bottleneck
• 3D scanning: data quality
Geometry beyond graphics
• Science and engineering (CT, MR, [f]MRI, ET, Cryo-EM, ...)
[data sets: G. Wolf, J. Bernauer]
[data set: Institute for Cartography, Leibnitz University Hannover]
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Scientific Challenge
Philosophical question
• What is “structure”?
• Insight about our own reasoning
[data sets: G. Wolf, Dosch]
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Research Group at MPI / MMCI
Main focus: low-level problems
• Correspondence problems
• Basic low-level building block
Correspondences
• Formal modelling?
• Efficient computation?
• Analyse the structure of corresponding geometry
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Example Projects
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Correspondences
Questions
• What parts are essientially the same?
• Dense-point-to-point correspondences
Variants
• Rigid, deformable, semantic
• Global vs. partial
• Self-matching: symmetry detection
[data sets: C. Theobald, C. Stoll, M. Wacker]
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Animation Reconstruction
[data set: P. Phong, Stanford University]
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Real-Time 3D Scanners
space-time stereo
courtesy of James Davis University of California
at Santa Cruz
color-coded structured light
courtesy of Phil Fong Stanford University
high-speed structured light
courtesy of Stefan Gumhold TU Dresden
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Time-of-Flight / PMD Devices
PMD Time-of-flight camera
Minolta Laser Scanner (static)
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Animation Reconstruction
Problems
• Noisy data
• Incomplete data (acquisition holes)
• No correspondences
• Just point clouds
noise
holes
missing correspondences
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Animation Reconstruction
Remove noise, outliers
Fill-in holes (from all frames)
Dense correspondences
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Representation
t = 0 t = 1 t = 2
data
urshape S
f
f f
deformation
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Matching Algorithm
Variational model
• Maximize data fit
• Penalize deformations
• Penalize abrupt motion
• Favour smooth surfaces
Algorithm
• Numerical optimization
• Greedy assembly
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Problem: Local Optima
[Data set: Hao Li, ETH Zürich]
data reconstruction
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Global Deformable Shape Matching
[data set: Stanford 3D Scanning Repository]
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Contributions
[data sets: TOSCA]
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Global Animation Reconstruction
[Data set: Hao Li, ETH Zürich]
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Symmetry Detection
[data set: M. Wacker, HTW Dresden]
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Symmetry Detection
[data sets: IKG, University Hannover / M. Wacker, HTW Dresden]
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Symmetry Detection
[DataSet : IKG, University Hannover]
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Denoising from Symmetry
overlay of
16 parts [data set: IKG, Leibnitz University Hannover]
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2D Bitmap Images
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Rigid, Isometic, Relaxed Isometric
rigid isometric relaxed isometric
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Learning
Window Variants [data set: IKG, Leibnitz University Hannover]
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Learning
Window Variants
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Inverse Procedural Modeling [data set: G. Wolf]
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Overview
Rules from example geometry
• Example model
• Compute rules describing a class of similar models
Input
Output
[data set: G. Wolf]
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r-Similarity
• Local neighborhoods match exemplar
What are similar models?
input
output
radius r radius r
radius r
[data set: G. Wolf]
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Simple Example
input output [data set: G. Wolf]
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Shape Grammar
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Results
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Shape Variations
input
output
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Shape Variations
input
output
[data set: Dosch]
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1D Grids
input output
[data set: G. Wolf]
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2D Grids
~ 500K triangles
input output
[data set: Dosch]
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3D Scanner Data
input
output
[data set: IKG, Leibnitz University Hannover]
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Results
Topics
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Topics (tentative)
Mathematical tools
• Basic statistics
Probability theory, machine learning
Dependencies and graphical models
Maximum likelihood, maximum a posteriori, least-squares
Inverse problems and regularization
• Variational methods
• Geometry
Representations, linear models, acquisition
Differential geometry, descriptors
Operators, eigenfunctions
• Numerics
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Topics (tentative)
Geometry Processing
• Rigid shape matching
Local registration (ICP)
Global registration (feature-based)
• Linear and Riemannian Shape spaces
Surface reconstruction
Shape deformation and non-rigid shape matching
• Intrinsic geometry
Isometry invariant shape matching
General metrics, high-dimensional shape spaces
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Topics (tentative)
Geometry Processing
• Symmetry
What is symmetry? Some group theory :-)
Detection algorithms & applications
Inverse procedural modeling
• Shape Learning
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Another View
Methods:
• Variational and statistical estimation
Least-squares, function estimation
Robust estimation, sparse statistics
Regularization, inverse problems
• Differential geometry
• Group theory
Symmetry groups
• Numerics
Solving large linear and non-linear systems
Eigenvalue problems
“It is tempting, if the only tool
you have is a hammer,
to treat everything as if it were a nail.”
Abraham Maslow (1908-1970)