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Martin Burger Institut für Numerische und Angewandte Mathematik
European Institute for Molecular Imaging CeNoS
Total Variation and Related Methods II
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Cetraro, September 2008Martin Burger
Variational Methods and their Analysis We investigate the analysis of variational methods in imagingMost general form:
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Cetraro, September 2008Martin Burger
Variational Methods and their Analysis Questions:- Existence- Uniqueness- Optimality conditions for solutions (-> numerical methods)- Structural properties of solutions- Asymptotic behaviour with respect to
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Cetraro, September 2008Martin Burger
Variational Methods and their Analysis Two simplifying assumptions:
-Noise is Gaussian (variance can be incorporated into )
- A is linear ´
Y Hilbert space
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Cetraro, September 2008Martin Burger
TV Regularization Under the above assumptions we have
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Cetraro, September 2008Martin Burger
Mean Value Technical simplification by eliminating mean value
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Cetraro, September 2008Martin Burger
Mean Value Eliminate mean value
Hence, minimum is attained among those functions with mean value c
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Cetraro, September 2008Martin Burger
Mean Value We can minimize a-priori over the mean value and restrict the image to mean value zeroW.r.o.g.
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Cetraro, September 2008Martin Burger
Structure of BV0 Equivalent norm
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Cetraro, September 2008Martin Burger
Poincare-InequalityProof. Assume
does not hold. Then for each natural number n there is such that
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Cetraro, September 2008Martin Burger
Poincare-InequalityProof (ctd).
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Cetraro, September 2008Martin Burger
Poincare-InequalityProof (ctd).
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Cetraro, September 2008Martin Burger
Dual Space Property Define
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Cetraro, September 2008Martin Burger
Dual Space Property
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Cetraro, September 2008Martin Burger
Dual Space Property
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Cetraro, September 2008Martin Burger
Dual Space Property
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Cetraro, September 2008Martin Burger
Dual Space Property
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Cetraro, September 2008Martin Burger
Existence Basic ingredients of an existence proof are-Sequential lower semicontinuity
- Compactness
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Cetraro, September 2008Martin Burger
Existence What is the correct topology ?
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Cetraro, September 2008Martin Burger
Lower SemicontinuityCompactness follows in the weak* topology.Lower semicontinuity ?
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Cetraro, September 2008Martin Burger
Lower Semicontinuity
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Cetraro, September 2008Martin Burger
Lower Semicontinuity
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Cetraro, September 2008Martin Burger
Lower Semicontinuity First term:
analogous proof implies
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Cetraro, September 2008Martin Burger
Existence Theorem: Let J be sequentially lower semicontinuous and
be compact. Then there exists a minimum of JProof.
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Cetraro, September 2008Martin Burger
Existence Proof (ctd). Due to compactness, there exists a subsequence, again denoted by such that
By lower semicontinuity
Hence, u is a minimizer
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Cetraro, September 2008Martin Burger
Uniqueness Since the total variation is not strictly convex and definitely will not enforce uniqueness, the data term should do
Proof:
