Statistical Geometry Processing Winter Semester 2011/2012

Introduction

2

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”

3

Ambitious Problem

What does shape understanding mean?

4

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]

6

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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18

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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

36

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]

40

2D Grids

~ 500K triangles

input output

[data set: Dosch]

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3D Scanner Data

input

output

[data set: IKG, Leibnitz University Hannover]

42

Results

Topics

44

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

45

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

46

Topics (tentative)

Geometry Processing

• Symmetry

What is symmetry? Some group theory :-)

Detection algorithms & applications

Inverse procedural modeling

• Shape Learning

47

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)