variational methods and partial differential...
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Daniel Cremers
Computer Science & Mathematics
TU Munich
Variational Methods and
Partial Differential Equations
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2 Daniel Cremers Variational Methods and Partial Differential Equations
Spatially Dense Reconstruction
infinite-dimensional optimization
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3 Daniel Cremers Variational Methods and Partial Differential Equations
Which path is the fastest?
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4 Daniel Cremers Variational Methods and Partial Differential Equations
Bernoulli & The Brachistochrone
Johann Bernoulli (1667-1748)
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Image segmentation:
Optimization in Computer Vision
Kass et al. ’88, Mumford, Shah ’89, Caselles et al. ‘95,
Kichenassamy et al. ‘95, Paragios, Deriche ’99,
Chan, Vese ‘01, Tsai et al. ‘01, …
Multiview stereo reconstruction:
Faugeras, Keriven ’98, Duan et al. ‘04, Yezzi, Soatto ‘03,
Labatut et al. ’07, Kolev et al. IJCV ‘09…
Optical flow estimation:
Horn, Schunck ‘81, Nagel, Enkelmann ‘86, Black, Anandan ‘93,
Alvarez et al. ‘99, Brox et al. ‘04, Zach et al. ‘07, Sun et al. ‘08,
Wedel et al. ’09, Werlberger et al. ‘10, …
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Compute solutions via energy minimization
Optimization and Variational Methods
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Inverse Problems: Denoising
input image denoised
Rudin, Osher, Fatemi, Physica D ‘92
For compute:
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Variational Methods & PDEs
Extremality principle as a necessary condition:
But: How to define the derivative with respect to a function?
For consider the functional
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The Gateaux Derivative
Derivative in “direction” :
Taylor expansion:
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The Gateaux Derivative
Integration by parts:
Necessary optimality condition:
Euler-Lagrange equation
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Leonhard Euler
(1703-1783)
Joseph-Louis Lagrange
(1736 – 1813)
Variational Methods
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12 Daniel Cremers Variational Methods and Partial Differential Equations
Gradient Descent
Iteratively walk “down-hill”:
direction of
steepest descent
For the energy
we obtain:
diffusion equation
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Diffusion
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Edge-preserving Denoising
Slightly modify the regularization (Rudin, Osher, Fatemi ‘92):
Gradient descent:
nonlinear diffusion
Perona, Malik ‘90, Rudin et al. ’92, Weickert ‘98
total variation
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Linear vs. Nonlinear Diffusion
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Variational Deblurring
original image deblurred image blurred image
Lions, Osher, Rudin, ‘92
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Variational Super-Resolution
Schoenemann, Cremers, IEEE TIP ’12
One of several input images Super-resolution estimate
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Variational Optical Flow
Input video Optical flow field
Horn & Schunck ‘81, Zach et al. DAGM ’07, Wedel et al. ICCV ’09
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*
* 60 fps @ 640x480
Input video Optical flow field
Variational Optical Flow
Horn & Schunck ‘81, Zach et al. DAGM ’07, Wedel et al. ICCV ’09
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one of two input images depth reconstruction
Variational Stereo Reconstruction
Pock , Schoenemann, Graber, Bischof, Cremers ECCV ’08
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Variational Scene Flow
Wedel & Cremers, “Scene Flow”, Springer 2011
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2 label segmentation 10 label segmentation
Variational Image Segmentation
Mumford, Shah ‘89, Chambolle, Cremers, Pock ’12:
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Prior Knowledge for Segmentation
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Cremers, Osher, Soatto, IJCV ’06
Statistical Interpretation
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Cremers, Osher, Soatto, IJCV ’06
Statistical Interpretation
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Dynamical Shape Priors
Cremers, IEEE PAMI ’06
Statistically synthesized embedding functions
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Cremers, IEEE PAMI ’06
Dynamical Shape Priors
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Dynamical Shape Priors
Cremers, IEEE PAMI ’06
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Cremers, IEEE PAMI ’06
Dynamical Shape Priors
Dynamical shape prior
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30 Daniel Cremers Variational Methods and Partial Differential Equations
Dynamical Shape Priors
Cremers, IEEE PAMI ’06
Dynamical prior of shape and translation
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Variational Multiview Reconstruction
Kolev et al., IJCV ‘09
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Overview
variational methods
multiview reconstruction
scene flow
image segmentation
optical flow
statistical shape priors
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