error estimation in tv imaging
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Error Estimation in TV Imaging. Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster. Joint Work with. Stan Osher (UCLA) - PowerPoint PPT PresentationTRANSCRIPT
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 + 1
2kLuk2 ! minu
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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 + 1
2kLuk2 ! minu
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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 + 1
2kLuk2 ! minu
jujB V =Z
jr uj dx
jujB V = supg2C 1
0 ;kgk1 · 1
Zu(r ¢g) dx
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
ROF Model Reconstruction (code by Jinjun Xu) clean noisy ROF
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
ROF model denoises cartoon images resp. computes the cartoon of an arbitrary image, natural spatial multi-scale decomposition by varying
ROF Model
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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 ?
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
Bregman distances reduce to known measures for standard energies
Example 1:
Subgradient = Gradient = uBregman distance becomes
Bregman Distance
J (u) = 12kuk2
DJ (u;v) = 12ku ¡ vk2
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
Example 2:
Subgradient = Gradient = log uBregman distance becomes Kullback-Leibler divergence (relative Entropy)
Bregman Distance
DJ (u;v) =Z
uloguv +
Z(v¡ u)
J (u) =Z
ulogu -Z
u
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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)
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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( )
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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( )
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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 ¢(ò(di )r di )
0 · ò · 1; ò(0) = 1supp (ò) ½(¡ ²;²)
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
chosen smaller than distance between two region boundaries
Note: on the region boundary (di = 0)
subgradient equals mean curvature of edge
Interpretation
q² = r ¢(ò(di )r di )= ò(0)¢ di +(ò)0(0)jr di j2 = ¢ di
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
Bregman distances given by
If we only take the sup over those g with
and let tend to zero we obtain
Interpretation
Dq²T V (u;g) = sup
kqk1 · 1
Zur ¢(q - ò(di )r di )
q= r di on @ i
liminf²
Dq²T V (u;g) ¸ jujT V ( nS @ i )
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
Stronger estimates under stronger conditions Resmerita 05
Numerical analysis for appropriate discretizations (correct discretization of subgradient crucial) mb 07
Extensions
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
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
4.6.2007 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07
Papers and talks at
www.math.uni-muenster.de/u/burger
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