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