vmcv_eccv2010_chapter1

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Rx An example: Image Denoising 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Reminder: the space L (Ω)
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A color image is a vector-valued function u : Ω → R3 on an open set Ω ⊂ R2 , which maps e.g. into RGB color space.
8 R
Variational Methods
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Rx Introduction 0
Variational methods
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Unifying concept: variational approach
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Variational Methods
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Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
2
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• The functional
u
2
:=
Ω
|u| dx
2
1 2
is a norm on L2 (Ω), with which it becomes a Banach space.
• The norm arises from the inner product
(u, v ) →
Ω
uv dx
if you set u 2 := (u, u). Thus, L2 (Ω) is in fact a Hilbert space. It is one of the most simple examples for an infinite dimensional Hilbert space.
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Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Images are functions
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A greyscale image is a real-valued function u : Ω → R on an open set Ω ⊂ R2 .
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Rx An example: Image Denoising 0
D. Cremers, B. Goldlucke, T. Pock ¨
A simple (but important) example: Denoising
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ECCV 2010 Tutorial Variational Methods in Computer Vision
V = Rn finitely many components xi , 1 ≤ i ≤ n (x, y ) = |x|2 =
n i=1 n i=1
V = L2 (Ω) infinitely many “components” u(x), x ∈ Ω (u, v ) = u
2 Ω
xi yi xi2
uv dx
Variational Methods in Computer Vision ECCV Tutorial, 5.9.2010
Chapter 1
Introduction
Mathematical Foundations
Daniel Cremers and Bastian Goldlucke ¨ Computer Vision Group Technical University of Munich Thomas Pock Institute for Computer Graphics and Vision Graz University of Technology
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Rx An example: Image Denoising 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
R vs. L (Ω)
n
2
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Elements Inner Product Norm
2
x 0
Definition Let Ω ⊂ Rn open. The space L2 (Ω) of square-integrable functions is defined as L2 (Ω) := u:Ω→R :
Ω
|u| dx
2
1 2
<∞ቤተ መጻሕፍቲ ባይዱ.
R 11
Variational Methods
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Variational methods
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Unifying concept: variational approach
Problem solution is the minimizer of an energy functional E, argmin E(u).
u∈V
6
R
Variational Methods
Rx 0
Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Variational methods
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Unifying concept: variational approach
• In the following, we assume functions to be in L2 (Ω), and
convergence, continuity etc. is defined with respect to the above norm.
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Variational Methods
Problem solution is the minimizer of an energy functional E, argmin E(u).
u∈V
In the variational framework, we adopt a continuous world view.
6
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Variational Methods
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Surfaces are manifolds
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vs.
Continuous
Discrete
9
R
Variational Methods
3 Convex analysis
Convex functionals Duality Conjugate functionals Legendre-Fenchel dual Subdifferential calculus Generalized variational principle
3 R
Variational Methods
2
1 2
=
|u| dx Ω
Derivatives of a functional E : V → R Gradient ´ (Frechet ) Directional ˆ (Gateaux ) Condition for minimum dE(x) = δE(x; h) = ˆ E(x ) = 0 E(x) E(x) · h dE(u) = ? δE(u; h) = ? ?
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Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨
Fundamental problems in computer vision
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ECCV 2010 Tutorial Variational Methods in Computer Vision
data / model term
Note: In Bayesian statistics, this can be interpreted as a MAP estimate for Gaussian noise.
Original
Noisy
Reconstruction
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Variational Methods
x 0
ECCV 2010 Tutorial Variational Methods in Computer Vision
3D Reconstruction
5
R
Variational Methods
Rx 0
Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Rx An example: Image Denoising 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Reminder: the space L (Ω)
Image labeling problems
Segmentation
Stereo
Optic flow
4
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Variational Methods
Rx 0
Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨
Fundamental problems in computer vision
2 Total Variation and Co-Area
The space BV(Ω) Geometric properties Co-area
3 Convex analysis
Convex functionals Duality Conjugate functionals Legendre-Fenchel dual Subdifferential calculus Generalized variational principle
Introduction An example: Image Denoising The variational principle The Euler-Lagrange equation
2 Total Variation and Co-Area
The space BV(Ω) Geometric properties Co-area
2 R
Variational Methods
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Rx 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Overview
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1 Variational Methods
The TV-L2 (ROF) model, Rudin-Osher-Fatemi 1992 For a given noisy input image f , compute argmin
u∈L2 (Ω) Ω
| u|2 dx +
1 2λ
(u − f )2 dx .
Ω
regularizer / prior
7 R
Variational Methods
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Rx Introduction 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Images are functions
1
R
Rx 0
Rx 0
D. Cremers, B. Goldlucke, T. Pock ¨ ECCV 2010 Tutorial Variational Methods in Computer Vision
Overview
x 0
1 Variational Methods
Introduction An example: Image Denoising The variational principle The Euler-Lagrange equation