Martin Burger - Ricam · Martin Burger 52 Linz, 2011 4D TV Model Analysis relies on superlinear...
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Martin Burger Institute for Computational and Applied Mathematics (4D) Variational Models Preserving Sharp Edges
Transcript of Martin Burger - Ricam · Martin Burger 52 Linz, 2011 4D TV Model Analysis relies on superlinear...
Martin Burger Institute for Computational and
Applied Mathematics
(4D) Variational Models Preserving
Sharp Edges
Martin Burger
Mathematical Imaging Workgroup @WWU
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Raman Shift (cm-1)
DNAAkrosomFlagellumGlass
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Some Philosophy
„No matter what question, L1 is the answer“
Stanley O.
Regularization in data assimilation is at the same state it was 10
years ago in biomedical imaging
The understanding and methods we gained in medical imaging
can hopefully be useful in geosciences and data assimilation
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Biomedical Imaging: 2000 vs 2010
Modality State of the art 2000 State of the art 2010
Full CT Filtered Backprojection Exact Reconstruction
PET/SPECT Filtered Backprojection /EM EM-TV / Dynamic Sparse
PET-CT - EM-AnatomicalTV
Acousto-Opt. - Wavelet Sparse / TV
EEG/MEG LORETA Sparsity / Bayesian
ECG-BSPM Least Norm L1 of normal derivative
Microscopy None, linear Filter Poisson-TV / Shearlet-L1
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Based on joint work with
Martin Benning, Michael Möller, Felix Lucka, Jahn Müller
(Münster)
Stanley Osher (UCLA)
Christoph Brune (Münster / UCLA / Vancouver)
Fabian Lenz (Münster), Silvia Comelli (Milano/Münster)
Eldad Haber (Vancouver)
Mohammad Dawood, Klaus Schäfers (NucMed/EIMI Münster)
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SFB
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Regularization of Inverse Problems
We want to solve
Forward operator between Banach spaces
with finite dimensional approximation (sampling, averaging)
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Dynamic Biomedical Imaging 7
Saarbrücken, 9.7.10
Maximum Likelihood / Bayes
Reconstruct maximum-likelihood estimate
Model of posterior probability (Bayes)
Yields regularized variational problem for finite m
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Minimization of penalized log-likelihood
General variational approach
Combines nonlocal part (including K ) with local regularization
functional
Gaussian noise (note: covariance hidden in output norm)
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Example Gauss:
Additive noise, i.i.d. on each pixel, mean zero, variance s
Minimization of negative posterior log-likelihood yields
Asymptotic variational model
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Optimality
Existence and uniqueness by variational methods
General case: optimality condition
is a nonlinear integro-differential equation / inclusion
(integral operator K, differential operator in J )
Gauss:
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Robustness
Due to noisy data robustness of
with respect to errors in f is important
Problem is robust for large a, but data are only reproduced for
small a
Convergence of solutions as f converges or as a to zero in
weak* topology
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Structure of Solutions
Analysis by convex optimization techniques, duality
Structure of subgradients important
Possible solution satisfy source condition
Allows to gain information about regularity (e.g. of edges)
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Structure of Solutions
Optimality condition for
Structure of u determined completely by properties of uB and K*
For smoothing operators K, singularity not present in uB cannot
be detected
Model error goes into K resp. K* and directly modifies u
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4D VAR
Given time dynamics starting from unknown initial value
Variational Problem to estimate initial state for further
prediction
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4D VAR = 3D Variational Problem
Elimination of further states from dynamics
Effective Variational Problem for initial value in 3D
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Example: Linear Advection
Minimize quadratic fidelity + TV of initial value subject to
Upwind discretization
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4D VAR for Linear Advection
Gibbs phenomenon as usual
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4D VAR for Linear Advection
Full observations (black), noisy(blue), 40 noisy samples (red)
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4D VAR for Linear Advection
Different noise variances
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Analysis of Model Error
Optimality
Exact Operator for linear advection is almost unitary
Hence
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Beyond Gaussian Priors
Again: optimality condition for MAP estimate
If J is strictly convex and smooth, subdifferential is a singleton
containing only the gradient of J, which can be inverted to
obtain a similar relation. Again operator determines structure
Only chance to obtain full robustness: multivalued
subdifferential. Singular regularization
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Singular Regularization
Construct J such that the subdifferential at points you want to
be robust is large
Example: l1 sparsity
Zeros are robust
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TV-Methods: Structural Prior (Cartooning)
Penalization of total Variation
Formal
Exact
ROF-Model for denoising g : minimize total variation subject to
Rudin-Osher-Fatemi 89,92
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Why TV-Methods ?
Cartooning
Linear Filter TV-Method
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ROF Model
clean noisy ROF
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H2O15 PET – Left Ventricular Time Frame
EM EM-Gauss EM-TV
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Dynamic Biomedical Imaging 27
Saarbrücken, 9.7.10
H2O15 PET – Right Ventricular Time Frame
EM EM-Gauss EM-TV
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4D VAR for Linear Advection
Gibbs phenomenon as usual
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4D VAR for Linear Advection
Full observations (black), noisy(blue), 40 noisy samples (red)
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4D VAR TV for Linear Advection
Comparison for full observations
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4D VAR TV for Linear Advection
Comparison for observed samples
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4D VAR TV for Linear Advection
Comparison for observed samples with noise
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Analysis of Model Error
Variational problem as before, add
Optimality condition
As before
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Analysis of Model Error
Structures are robust: apply T in region where
If we find s solving Poisson equation
with then
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Numerical Solution: Splitting or ALM
Operator Splitting into standard problem (dependent on code)
and simple denoising-type problem
Example: Peaceman Rachford-Splitting for
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Bayes and Uncertainty
Natural prior probabilities for singular regularizations can be
constructed even in a Gaussian framework
Interpret J(u) as a random variable with variance s2
Prior probability density
MAP estimate minimizes
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Bayes and Uncertainty
Equivalence to original form via constraint regularization
For appropriate choice of a and g, minimization of
and
is equivalent to subject to
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Uncertainty Quantification
Sampling with standard MCMC schemes difficult
Novel Gibbs sampler by
F.Lucka based on analytical
integration of posterior
distribution function in 1D
Theoretical Insight:
MSc Thesis Silvia Comelli
CM Estimate for TV prior
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Uncertainty Quantification II
Error estimates in dependence on the noise,
using source conditions
Error estimates need appropriate distance
measure,generalized Bregman-distance
mb-Osher 04, Resmerita 05, mb-Resmerita-He 07, Benning-mb 09
Estimates for Bayesian distributions in Bregman transport
distances (w. H.Pikkarainen) = 2 Wasserstein distance in the
Gaussian case
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Uncertainty Quantification III
Idea: construct linear functionals from nonlinear eigenvectors
We have
For TV-denoising (also for linear advection example),
Estimate of maximal error for mean value on balls
For l1-sparsity estimate of error in single components
Benning PhD 11, Benning-mb 11
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ROF minimization loses contrast, total variation of the
reconstruction is smaller than total variation of clean image.
Image features left in residual f-u
g, clean f, noisy u, ROF f-u
mb-Gilboa-Osher-Xu 06
Loss of Contrast
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Loss of Contrast = Systematic Bias of TV
Becomes more severe in ill-posed problems with operator K
Not just simple vision effect to be corrected, but loss of
information
Simple idea for Least-Squares:
add back the noise to amplify = Augmented Lagrangian
Osher-mb-Goldfarb-Xu-Yin 2005
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Bregman Iteration
Can be shown to be equivalent to Bregman iteration
Immediate generalization to convex fidelities and regularizers
Generalization to Gauss-Newton type Methods for nonlinear K:
use linearization of K around last iterate ul
Bachmayr-mb 2009
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Bregman Iteration
Properties like iterative regularization method
Regularizing effect from appropriate termination of the iteration
Better performance for oversmoothing single steps, i.e.
regularization parameter a very large
Limit: Inverse Scale Space Method
mb-Gilboa.Osher-Xu 2006
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Why does Inverse Scale Space work ?
Singular value decomposition in fully quadratic case
Eigenfunctions:
yields
Convergence faster in small frequencies (large eigenvalues)
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Why does Inverse Scale Space work ?
Convex one-homogeneous regularization J (TV, l1, …)
Eigenfunctions:
yields
Again large frequencies appear later. Not at all for small t !
Eigenvalues in TV indeed related to jump measures
PhD-Thesis Benning, 2011
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Why does Inverse Scale Space work ?
Multiple frequencies not simple for nonlinear case
However, various theoretical and computational results
confirming exact scale decomposition
PhD-Thesis Benning, 2011 / mb-Frick-Scherzer-Osher 2007
Complete characterization of inverse scale space for discrete
l1-functionals, yields jump dynamics in time, adaptive basis
pursuit method with guaranteed convergence
mb-Möller-Benning-Osher, 2011
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Saarbrücken, 9.7.10
18F-FDG
PET
EM, 20 min EM-TV, 5s
EM, 5s BREG, 5s
Jahn Müller, 2011
Data from Nuclear
Medicine
Department, UKM
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STED
Microscopy
Christoph Brune,
2009
Data from MPI for
Biophys. Chem.
Göttingen
(K.Willig,
A.Schönle,
Hell)
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4D Reconstruction
4D imaging of transport with penalization of large
velocities:
Minimize
subject to
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Analysis of Motion Model
Functional related to Benamou-Brenier formulation of optimal
transport . Analysis different from optimal transport, since
usually no initial and final densities are given (more related to
mean-field games, Lasry-Lions 07)
Existence by transformation to
- A-priori estimate for w in L2. Weak compactness
-A-priori estimates for u in Lp(0,T;BV) and for time derivative in
Lq(0,T;W-1,s)
- Adaptation of Aubin-Lions gives strong compactness of u in
Lr(0,T; Lr), and thus of the square-root in L2r(0,T; L2r)
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4D TV Model
Analysis relies on superlinear growth of F, although F=Identity
seems a very reasonable choice
Choosing F equal to the identity would imply we seek a minimal
L1 norm of the vector of total variations. Favours sparsity, i.e.
solutions with very large total variation at some time step
allowed if small else. This does not correspond to a smooth
motion model, hence superlinear choices preferable
Some indications of this effect in numerical results
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Numerical solution
Complicated 4D variational problem combining various integral
and differential operators + nonlinearity. Convexity achieved by
formulation in momentum variable m = u V
Efficient GPU implementation by Christoph Brune on CUDA
with specially designed algorithms. All subproblems solvable
by FFT or shrinkage
Realized by introducing new variables and inexact Uzawa
Augmented Lagrangian approach
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Augmented Lagrangian
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Inexact Uzawa Augmented Lagrangian
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Update of Primal Variables
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Results: Deblurring, Synthetic Data
Exact solution Blurred Data
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Results: Deblurring, Synthetic Data
Exact solution Reconstruction
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Results: Cardiac 18F-FDG PET (Eulerian)
PET Reconstruction
(Data)
Registration to Diastole
Registration to Systole
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Info
http://imaging.uni-muenster.de
http://www.cells-in-motion.de
http://www.herzforscher.de
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