Inverse Total Variation Flow

In this paper we analyze iterative regularization with the Bregman distance of the total variation seminorm. Moreover, we prove existence of a solution of the corresponding flow equation as introduced in [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with ${\rm L}^p$-norm fit-to-data terms and Bregman distance regularization terms. For the associated flow equations well-posedness is derived using recent results on metric gradient flows from [L. Ambrosio, N. Gigli, and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 2005]. In contrast to previous work the results of this paper apply for the analysis of variational denoising methods with the Bregman distance under adequate noise assumptions. Aside from the theoretical results we introduce a ...

[1]  Fuensanta Andreu Vaíllo,et al.  The total variation flow , 2003 .

[2]  Jesús Ildefonso Díaz Díaz,et al.  Some qualitative properties for the total variation flow , 2002 .

[3]  V. Caselles,et al.  Parabolic Quasilinear Equations Min-imizing Linear Growth Functionals , 2004 .

[4]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[5]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[6]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[7]  Jürgen Elstrodt,et al.  Maß-und Integrationstheorie , 1996 .

[8]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[9]  W. Rudin.  Real and complex analysis , 1968 .

[10]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[11]  G. Aubert,et al.  Modeling Very Oscillating Signals. Application to Image Processing , 2005 .

[12]  O. Scherzer,et al.  Non‐stationary iterated Tikhonov–Morozov method and third‐order differential equations for the evaluation of unbounded operators , 2000 .

[13]  M. Novaga,et al.  Explicit Solutions of the Eigenvalue Problem $div \left(\frac Du\vert Du \vert \right)=u$ in $R^2$ , 2005 .

[14]  Joachim Weickert,et al.  Relations Between Regularization and Diffusion Filtering , 2000, Journal of Mathematical Imaging and Vision.

[15]  ANTONIN CHAMBOLLE,et al.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[16]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[17]  R. Temam.  Navier-Stokes Equations , 1977 .

[18]  J. Bourgain,et al.  On the equation DIV Y = f and applications to control of phases , 2002 .

[19]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[20]  Y. Meyer,et al.  Variational methods in image processing , 2004 .

[21]  Guy Gilboa,et al.  Nonlinear Inverse Scale Space Methods for Image Restoration , 2005, VLSM.

[22]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[23]  Robert E. Megginson.  An Introduction to Banach Space Theory , 1998 .

[24]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[25]  G. Anzellotti,et al.  Pairings between measures and bounded functions and compensated compactness , 1983 .

[26]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[27]  Michael G. Crandall,et al.  GENERATION OF SEMI-GROUPS OF NONLINEAR TRANSFORMATIONS ON GENERAL BANACH SPACES, , 1971 .

[28]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .