Transcript Folie 1

Regularisierung mir
Singulären Energien
Martin Burger
Institut für Numerische und Angewandte Mathematik
Westfälische Wilhelms Universität Münster
[email protected]
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Collaborations

Stan Osher, Jinjun Xu, Guy Gilboa (UCLA)

Lin He (Linz / UCLA)

Klaus Frick, Otmar Scherzer (Innsbruck)

Carola Schönlieb (Vienna)

Don Goldfarb, Wotao Yin (Columbia)
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Introduction
Classical regularization schemes for inverse
problems and image smoothing are based on
Hilbert spaces and quadratic energy functionals

Example: Tikhonov regularization for linear
operator equations

¸k
1k k !
¡
k
Au
f 2 + Lu 2
min
2
2
u
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Introduction
These energy functionals are strictly convex
and differentiable – standard tools from analysis
and computation (Newton methods etc.) can be
used
 Disadvantage: possible oversmoothing, seen
from first-order optimality condition
 Tikhonov yields
¤
¡ ¤

L Lu =
¸A (Auf )
Hence u is in the range of (L*L)-1A*
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Introduction
Classical inverse problem: integral equation of
the first kind, regularization in L2 (L = Id), A =
Z Z integral operator with kernel k
Fredholm

u=¸
k(y; x)(¡k(y; z)u(z) + f (z)) dy dz
Smoothness of regularized solution is
determined by smoothness of kernel
 For typical convolution kernels like Gaussians,
u is analytic !

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Image Smoothing
Classical image smoothing: data in L2 (A = Id),
L = gradient¡
(H1-Seminorm)

¢u + ¸u = ¸f
On a reasonable domain, standard elliptic
regularity implies

u 2 H 2 ( ) ,! C( )
Reconstruction contains no edges, blurs the
image (with Green kernel)

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Sparse Reconstructions
?
`2(Z)
Let A be an operator on
(basis representation of a Hilbert space operator, wavelet)
 Penalization by squared norm (L = Id)
 Optimality condition for components of u
¤ ¡

uk = ¸ (A ( Au + f ))
k
Decay of components determined by A*. Even if
data are generated by sparse signal (finite number
of nonzeros), reconstruction is not sparse !

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Error estimates
Error estimates for ill-posed problems can be
obtained only under
conditions
(source
¤
9w :stronger
u=A w
conditions)

cf. Groetsch, Engl-Hanke-Neubauer, Colton-Kress,
Natterer.
Engl-Kunisch-Neubauer.
Equivalent to u being minimizer of Tikhonov
functional with data
 For many inverse problems unrealistic due to
extreme smoothness assumptions

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Error estimates

Condition can be weakened¤to
9v :
u = f (A A)v
cf. Neubauer et al (algebraic), Hohage (logarithmic),
Mathe-Pereverzyev (general).

Advantage: more realistic conditions

Disadvantage: Estimates get worse with f
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Singular Energies
`2(Z)
P
Let A be the identity on
rk (uk )
 Nonlinear Penalization by
Optimality condition for components of u
0

r (uk ) + ¸uk = ¸fk
k
If rk is smooth and strictly convex, then Taylor
expansion
00 yields
¼ 00

r (fk )uk + ¸uk
k
r (fk )fk + ¸fk
k
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Singular Energies
Example becomes more interesting for singular
(nonsmooth) energy

rk (t) = jtj

Take

Then optimality condition becomes
sign (uk ) + ¸uk = ¸fk
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Singular Energies
Result is well-known soft-thresholding of
8 et al, Chambolle et al
wavelets Donoho
¡


< fk
fk +
uk =
:
0
1
¸
1
¸
fk > 1
¸
¡
fk < 1
¸
else
Yields a sparse signal
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Singular Energies
ZImage
smoothing: try nonlinear energy
r(ru)

for penalization
Optimality condition is nonlinear PDE
¡r ¢ ((rr)(ru)) + ¸u = ¸f
If r is strictly convex usual smoothing behaviour
 If r is not convex problem not well-posed
 Try singular case at the borderline

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Total Variation Methods
r(p) = jpj
Simplest choice
yields total variation method
 Total variation methods are popular in imaging
(and inverse problems), since

- they keep sharp edges
eliminate oscillations (noise)
- create new nice mathematics
-
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ROF Model

ROF model for denoising
Rudin-Osher Fatemi 89/92,
Acar-Vogel 93, Chambolle-Lions 96, Vogel 95/96,
Scherzer-Dobson 96, Chavent-Kunisch 98,
Meyer 01,…
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ROF Model

Optimality condition for ROF denoising
p + ¸u = ¸f;
2
j
j
p @u

Dual variable p enters !

Subgradient offconvex
¤ j 8 2
2 functional
TV
@J(u) = p X
v X:
¡
h
¡
i
·
g
J(u)
p; v u
J(v)
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ROF Model
Reconstruction (code by Jinjun Xu)
clean
noisy
ROF
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ROF Model
ROF model denoises cartoon images resp.
computes the cartoon of an arbitrary image

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Numerical Differentiation with TV

From Master Thesis of Markus Bachmayr, 2007
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Singular energies
Methods with singular energies offer great
potential, but still have some shortcomings

- difficult to analyze and to obtain error estimates
systematic errors (clean images not
reconstructed perfectly)
- computational challenges
- some extensions to complicated imaging tasks
are not well understood (e.g. inpainting)
-
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Singular energies
 General
¸ k problem
Au ¡ f k2 + J(u) ! min
2
u
leads to optimality
condition
¤
¤
p + ¸A Au = ¸A f;
p 2 @J(u)
First of all „dual smoothing“, subgradient p is in
the range of A*

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Singular energies
For smooth and strictly convex energies, the
subdifferential is a singleton
f 0 g

@J(u) = J (u)
Dual smoothing directly results in a primal one !
 For singular energies, subdifferentials are not
usually multivalued. The consequence is a
possibility to break the primal smoothing
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Error Estimation
First question for error estimation:
estimate difference of u (minimizer of ROF) and f
in terms of l

Estimate in the L2 norm is standard, but does
not yield information about edges

Estimate in the BV-norm too ambitious: even
arbitrarily small difference in edge location can
yield BV-norm of order one !

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Error Estimation
We need a better error measure, stronger than
L2, weaker than BV
 Possible choice: Bregman distance Bregman 67

Real distance for a strictly convex differentiable
functional – not symmetric
 Symmetric version

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Error Estimation
Bregman distances reduce to known measures
for standard energies 1
k uk 2
J
(u)
=
 Example 1:
2

Subgradient = Gradient = u
 Bregman distance becomes
1k ¡ k
DJ (u; v) = u
v 2
2

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Error Estimation
Bregman distances reduce
Z to known
Z measures
for standard energies
J (u) = u log u
u
 Example 2:

Subgradient = Gradient = log u
 Bregman distance becomes Kullback-Leibler
divergence (relative
Z Entropy) Z
u
DJ (u; v) = u log + (v ¡ u)
v

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Error Estimation
Total variation is neither symmetric nor
differentiable
 Define generalized Bregman distance for each
subgradient


Symmetric version
Kiwiel 97, Chen-Teboulle 97
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Error Estimation
For energies homogeneous of degree one, we
have


Bregman distance becomes
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Error Estimation
Bregman distance for singular energies is not a
strict distance, can be zero for
 In particular dTV is zero for contrast change

Resmerita-Scherzer 06
Bregman distance is still not negative
(convexity)
 Bregman distance can provide information
about edges

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Error Estimation
Let v be piecewise constant with white
background and color values on regions
 Then we obtain subgradients of the form

with signed distance function
and
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Error Estimation

Bregman distances given by
In the limit we obtain for being piecewise
continuous

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Error Estimation
For estimate in terms of l we need smoothness
condition on data


Optimality condition for ROF
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Error Estimation

Subtract q

Estimate for Bregman distance, mb-Osher 04
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Error Estimation
In practice we have to deal with noisy data f
(perturbation of some exact data g)


Estimate for Bregman distance
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Error Estimation

Optimal choice of the penalization parameter
i.e. of the order of the noise variance
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Error Estimation
Direct extension to deconvolution / linear
inverse¸problems
k
¡ k
j j
!

2
Au
f
2
+ u TV
min
u2BV
under standard source condition
mb-Osher 04
Extension: stronger estimates under stronger
conditions, Resmerita 05
Nonlinear inverse problems, Resmerita-Scherzer 06

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Discretization
Natural choice: primal discretization with
piecewise constant functions on grid

Problem 1: Numerical analysis (characterization
of discrete subgradients)
 Problem 2: Discrete problems are the same for
any anisotropic version of the total variation

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Discretization
In multiple dimensions, nonconvergence of the
primal discretization for the isotropic TV (p=2)
can be shown

Convergence of anisotropic TV (p=1) on
rectangular aligned grids

Fitzpatrick-Keeling 1997
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Primal-Dual Discretization
 Alternative:
perform primal-dual discretization
for optimality system (variational inequality)
with convex set
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Primal-Dual Discretization

Discretization
Discretized convex set with appropriate
elements (piecewise linear in 1D, RaviartThomas in multi-D)

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Primal / Primal-Dual Discretization
In 1 D primal, primal-dual, and dual
discretization are equivalent
 Error estimate for Bregman distance by
analogous techniques


Note that only the natural condition
is needed to show
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Primal / Primal-Dual Discretization
In multi-D similar estimates, additional work
since projection of subgradient is not discrete
subgradient.

Primal-dual discretization equivalent to
discretized dual minimization (Chambolle 03,
Kunisch-Hintermüller 04). Can be used for existence
of discrete solution, stability of p

Mb 07 ?
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Cartesian Grids
For most imaging applications Cartesian grids
are used. Primal dual discretization can be
reinterpreted as a finite difference scheme in this
setup.
 Value of image intensity corresponds to color in
a pixel of width h around the grid point.
 Raviart-Thomas elements on Cartesian grids
particularly easy. First component piecewise
linear in x, pw constant in y,z, etc.
 Leads to simple finite difference scheme with
staggered grid
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Iterative Refinement & ISS
ROF minimization has a systematic error, 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
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Iterative Refinement & ISS
Idea: add the residual („noise“) back to the
image to pronounce the features decreased to
much. Then do ROF again. Iterative procedure

Osher-mb-Goldfarb-Xu-Yin 04
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Iterative Refinement & ISS

Improves reconstructions significantly
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Iterative Refinement & ISS
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Iterative Refinement & ISS

Simple observation from optimality condition
Consequently, iterative refinement equivalent to
Bregman iteration

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Iterative Refinement & ISS
Choice of parameter l less important, can be
kept small (oversmoothing). Regularizing effect
comes from appropriate stopping.
 Quantitative stopping rules available, or
„stop when you are happy“ – S.O.
 Limit l to zero can be studied. Yields gradient
flow for the dual variable („inverse scale space“)

mb-Gilboa-Osher-Xu 06, mb-Frick-Osher-Scherzer 06
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Iterative Refinement & ISS
Non-quadratic fidelity is possible, some caution
needed for L1 fidelity

He-mb-Osher 05, mb-Frick-Osher-Scherzer 06

Error estimation in Bregman distance
mb-He-Resmerita 07
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Iterative Refinement
MRI Data Siemens Magnetom Avanto 1.5 T
Scanner He, Chang, Osher, Fang, Speier 06
 PenalizationTV + Wavelet

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Iterative Refinement
MRI Data Siemens Magnetom Avanto 1.5 T
Scanner He, Chang, Osher, Fang, Speier 06

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Iterative Refinement
MRI Data Siemens Magnetom Avanto 1.5 T
Scanner He, Chang, Osher, Fang, Speier 06

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Surface Smoothing

Smoothing of surfaces obtained as level sets

3D Ultrasound, Kretz / GE Med.
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Inverse Scale Space
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Iterative Refinement & ISS
 Application
to other regularization techniques,
e.g. wavelet thresholding is straightforward
Starting from soft shrinkage, iterated
refinement yields firm shrinkage, inverse scale
space becomes hard shrinkage

Osher-Xu 06
Bregman distance natural sparsity measure,
source condition just requires sparse signal,
number of nonzero components is smoothness
measure in error estimates

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Inpainting
Difficult to construct total variation techniques
for inpainting
 Original extensions of ROF failed to obtain
natural connectivity (see book by Chan, Shen 05)
 Inpainting region
, image f (noisy) given on


Try to minimize
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Inpainting

Optimality condition will have the form
with A being a linear operator defining the norm

In particular p = 0 in D !
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Inpainting
Different iterated approach (motivated by
Cahn-Hilliard inpainting, Bertozzi et al 05)
 Minimize in each step

First term for damping, second for fidelity (fit to
f where given, and to old iterate in the inpainting
region), third term for smoothing

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Inpainting

Continuous flow for damping parameter to zero

Fourth order flow for H-1 norm

Stationary solution (existence ?) satisfies
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Inpainting

Result: Penguins
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Download and Contact

Papers and Talks:
www.math.uni-muenster.de/u/burger

e-mail: [email protected]
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