total variation and related methods ii
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Total Variation and Related Methods II. Variational Methods and their Analysis. We investigate the analysis of variational methods in imaging Most general form:. Variational Methods and their Analysis. Questions: Existence Uniqueness - PowerPoint PPT PresentationTRANSCRIPT
Martin Burger Institut für Numerische und Angewandte Mathematik
European Institute for Molecular Imaging CeNoS
Total Variation and Related Methods II
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Variational Methods and their Analysis We investigate the analysis of variational methods in imagingMost general form:
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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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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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TV Regularization Under the above assumptions we have
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Mean Value Technical simplification by eliminating mean value
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Mean Value Eliminate mean value
Hence, minimum is attained among those functions with mean value c
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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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Structure of BV0 Equivalent norm
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Poincare-InequalityProof. Assume
does not hold. Then for each natural number n there is such that
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Poincare-InequalityProof (ctd).
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Poincare-InequalityProof (ctd).
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Dual Space Property Define
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Dual Space Property
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Dual Space Property
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Dual Space Property
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Dual Space Property
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Existence Basic ingredients of an existence proof are-Sequential lower semicontinuity
- Compactness
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Existence What is the correct topology ?
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Lower SemicontinuityCompactness follows in the weak* topology.Lower semicontinuity ?
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Lower Semicontinuity
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Lower Semicontinuity
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Lower Semicontinuity First term:
analogous proof implies
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Existence Theorem: Let J be sequentially lower semicontinuous and
be compact. Then there exists a minimum of JProof.
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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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Uniqueness Since the total variation is not strictly convex and definitely will not enforce uniqueness, the data term should do
Proof: