Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Total Variation and related Methods: Error Estimation

Martin Burger Total Variation 2 Cetraro, September 2008 Error Estimation Start with the quadratic case D generalizes gradient Optimality

Martin Burger Total Variation 3 Cetraro, September 2008 Error Estimation Estimate 1: Two Solutions of Variational Problems Difference Scalar product with

Martin Burger Total Variation 4 Cetraro, September 2008 Error Estimation Use Young‘s inequality

Martin Burger Total Variation 5 Cetraro, September 2008 Error Estimation Estimate 2: Asymptotic for exact data Need regularity for : Source condition

Martin Burger Total Variation 6 Cetraro, September 2008 Error Estimation Source Condition Equivalent to existence of saddle point for

Martin Burger Total Variation 7 Cetraro, September 2008 Error Estimation

Martin Burger Total Variation 8 Cetraro, September 2008 Error Estimation Estimate 3: Asymptotic for noisy data

Martin Burger Total Variation 9 Cetraro, September 2008 Error Estimation Similar estimation as above yields

Martin Burger Total Variation 10 Cetraro, September 2008 Error Estimation Nonlinear Variational Method Optimality condition

Martin Burger Total Variation 11 Cetraro, September 2008 Error Estimation Stability Estimate between two solutions ´ Same procedure as before: take difference and use duality product with

Martin Burger Total Variation 12 Cetraro, September 2008 Error Estimation Error measure: symmetric Bregman distance ´ Note: symmetric Bregman distance is sum of non-symmetric ones

Martin Burger Total Variation 13 Cetraro, September 2008 Bregman distance R smooth and strictly convex in some H-space Same for symmetric Bregman distance

Martin Burger Total Variation 14 Cetraro, September 2008 Error Estimation R nonsmooth: Bregman distance multivalued and depends on the choice of the subgradient Note: error estimate possible for any appropriate subgradient

Martin Burger Total Variation 15 Cetraro, September 2008 Error Estimation R not strictly convex: Bregman distance is not a strict distance, possibly

Martin Burger Total Variation 16 Cetraro, September 2008 Error Estimation Bregman distance example

Martin Burger Total Variation 17 Cetraro, September 2008 Error Estimation Sparsity measure

Martin Burger Total Variation 18 Cetraro, September 2008 Error Estimation Total Variation Contrast change

Martin Burger Total Variation 19 Cetraro, September 2008 Error Estimation Contrast Change

Martin Burger Total Variation 20 Cetraro, September 2008 Error Estimation Estimate 2: Asymptotic for exact data

Martin Burger Total Variation 21 Cetraro, September 2008 Error Estimation Asymptotic

Martin Burger Total Variation 22 Cetraro, September 2008 Error Estimation Source condition

Martin Burger Total Variation 23 Cetraro, September 2008 Error Estimation Error estimate in Bregman distance Analogous in the noisy case

Martin Burger Total Variation 24 Cetraro, September 2008 Error Estimation Multivalued estimate Note: error estimate holds for any Open interpretation for total variation and

Martin Burger Total Variation 25 Cetraro, September 2008 Error Estimation TV Subgradients and edges

Martin Burger Total Variation 26 Cetraro, September 2008 Error Estimation TV subgradients

Martin Burger Total Variation 27 Cetraro, September 2008 Error Estimation

Martin Burger Total Variation 28 Cetraro, September 2008 Error Estimation

Martin Burger Total Variation 29 Cetraro, September 2008 Error Estimation

Martin Burger Total Variation 30 Cetraro, September 2008 Error Estimation Mean Curvature Source condition means smoothness of edge sets !!

Martin Burger Total Variation 31 Cetraro, September 2008 Error Estimation Bregman distance

Martin Burger Total Variation 32 Cetraro, September 2008 Error Estimation Second term