Error Estimation in TV Imaging Martin Burger Institute for Computational and Applied Mathematics...

Error Estimation in TV Imaging

Martin Burger

Institute for Computational and Applied MathematicsEuropean Institute for Molecular Imaging (EIMI)

Center for Nonlinear Science (CeNoS)

Westfälische Wilhelms-Universität Münster

4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

Stan Osher (UCLA)mb-Osher, Inverse Problems 04

Elena Resmerita, Lin He (Linz)mb-Resmerita-He, Computing 07

Joint Work with

¸2kAu ¡ f k2 +

12kLuk2 ! min

u


Total variation methods are one of the most popular techniques in modern imaging

Basic idea is to model image, resp. their main structure (cartoon) as functions of bounded variation

Reconstructions seek images of as small total variation as possible

TV Imaging

¸2kAu ¡ f k2 +

12kLuk2 ! min

u


Total variation is a convex, but not differentiable and not strictly convex functional

„ “

Banach space BV consisting of all L1 functions of bounded variation

TV Imaging

¸2kAu ¡ f k2 +

12kLuk2 ! min

u

jujB V =

Zjr uj dx

jujB V = supg2C 1

0 ;kgk1 · 1

Zu(r ¢g) dx


ROF model

Rudin-Osher Fatemi 89/92, Chambolle-Lions 96, Scherzer-Dobson 96, Meyer 01,…

TV flow

Caselles et al 99-06, Feng-Prohl 03, ..

Denoising Models

u(t = 0) = f

¸2

Z(u ¡ f )2 +jujT V ! min

u2B V

@tu = r ¢(r ujr uj

) 2 ¡ @jujT V


Optimality condition for ROF denoising

Dual variable p enters in ROF and TV flow – related to mean curvature of edges for total variation

Subdifferential of convex functional

ROF Model

@J (u) = fp2 X ¤ j 8v 2 X :

J (u) +hp;v ¡ ui · J (v)g

p+¸u = ¸f ; p2 @jujT V


ROF Model

Reconstruction (code by Jinjun Xu)

clean noisy ROF


ROF model denoises cartoon images resp. computes the cartoon of an arbitrary image, natural spatial multi-scale decomposition by varying

ROF Model


First question for error estimation: estimate difference of u (minimizer of ROF) and f in terms of

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 !

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

Error Measure

D J (u;v) = J (u) - J (v) - hJ 0(v);u - vi

dJ (u;v) = DJ (u;v) +DJ (v;u) = hJ 0(u) - J 0(v);u - vi


Bregman distances reduce to known measures for standard energies

Example 1:

Subgradient = Gradient = u

Bregman distance becomes

Bregman Distance

J (u) =12kuk2

DJ (u;v) =12ku ¡ vk2


Example 2:

Subgradient = Gradient = log u

Bregman distance becomes Kullback-Leibler divergence (relative Entropy)

Bregman Distance

DJ (u;v) =

Zulog

uv

+Z

(v¡ u)

J (u) =

Zulogu -

Zu


Total variation is neither symmetric nor differentiable Define generalized Bregman distance for each subgradient

Symmetric version

Kiwiel 97, Chen-Teboulle 97

Bregman Distance

DpJ (u;v) = J (u) - J (v) - hp;u - vi

p2 @J (v)

dJ (u;v) = hpu - pv;u - vi


For energies homogeneous of degree one, we have

Bregman distance becomes

Bregman Distance

J (v) = hp;vi ; p2 @J (v)

DpJ (u;v) = J (u) - hp;vi ; p2 @J (v)


Bregman distance for total variation 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

Bregman Distance

dT V (u;f (u)) = 0

u 6= v


For estimate in terms of we need smoothness condition on data

Optimality condition for ROF

Error Estimation

q2 @jf jT V \ L2( )

p+¸u = ¸f ; p2 @jujT V

p - q+¸(u - f ) = - q


Apply to u – v

Estimate for Bregman distance, mb-Osher 04

Error Estimation

dT V (u;f ) = hp - q;u - f i ·kqk2

4 = O(¸¡ 1)

h (u - f ) +p - q;u - f i = hq;f ¡ ui

·kqk2

4+ ¸ku ¡ f k2


In practice we have to deal with noisy data f (perturbation of some exact data g)

Analogous estimate for Bregman distance

Optimal choice of the parameter

i.e. of the order of the noise variance

Error Estimation

dT V (u;f ) = hp - q;u - f i ·kqk2

4+

¸2kf - gk2

¸¡ 1 » kg - f k

q2 @jgjT V \ L2( )


Analogous estimate for TV flow mb-Resmerita-He 07

Regularization parameter is stopping time T of the flow T ~ -1

Note: all estimates multivalued ! Hold for any subgradient satisfying

Error Estimation

q2 @jgjT V \ L2( )


Let g be piecewise constant with white background and color values ci on regions i

Then we obtain subgradients of the form

with signed distance function di and s.t.

Interpretation

q² = r ¢(Ã²(di )r di )

0 · Ã² · 1; Ã²(0) = 1

supp (Ã²) ½(¡ ²;²)


chosen smaller than distance between two region boundaries

Note: on the region boundary (di = 0)

subgradient equals mean curvature of edge

Interpretation

q² = r ¢(Ã²(di )r di )

= Ã²(0)¢ di +(Ã²)0(0)jr di j2 = ¢ di


Bregman distances given by

If we only take the sup over those g with

and let tend to zero we obtain

Interpretation

Dq²

T V (u;g) = supkqk1 · 1

Zur ¢(q - Ã²(di )r di )

q= r di on @ i

liminf²

Dq²

T V (u;g) ¸ jujT V ( nS

@ i )


Multivalued error estimates imply quantitative estimate for total variation of u away from the discontinuity set of g

Other geometric estimates possible by different choice of subgradients, different limits

Interpretation


Direct extension to deconvolution / linear inverse problems: A linear operator

under standard source condition

mb-Osher 04

Nonlinear problems Resmerita-Scherzer 06, Hofmann-Kaltenbacher-Pöschl-Scherzer 07

Extensions

¸2kAu ¡ f k2 +jujT V ! min

u2B V

q= A¤w 2 @jgjT V


Stronger estimates under stronger conditions Resmerita 05

Numerical analysis for appropriate discretizations (correct discretization of subgradient crucial) mb 07

Extensions


Extension to other fitting functionals (relative entropy, log-likelihood functionals for different noise models)

Extension to anisotropic TV (Interpretation of subgradients)

Extension to geometric problems (segmentation by Chan-Vese, Mumford-Shah): use exact relaxation in BV with bound constraints Chan-Esedoglu-Nikolova 04

Future Tasks


Papers and talks at

www.math.uni-muenster.de/u/burger

Email

[email protected]

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