Total Generalized Variation

The novel concept of total generalized variation of a function $u$ is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of $u$. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.

[1]  Karl Kunisch,et al.  Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem , 2004, SIAM J. Appl. Math..

[2]  Charles Warlow.  Utrecht , 2008, Practical Neurology.

[3]  Matthew MacDonald,et al.  Shapes and Geometries , 1987 .

[4]  Jean-François Aujol,et al.  Some First-Order Algorithms for Total Variation Based Image Restoration , 2009, Journal of Mathematical Imaging and Vision.

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

[6]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[7]  Los Angeles,et al.  Dual Methods for Total Variation-Based Image , 2001 .

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

[9]  Joachim Weickert,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Properties of Higher Order Nonlinear Diffusion Filtering Properties of Higher Order Nonlinear Diffusion Filtering , 2022 .

[10]  V. Sharafutdinov.  Integral Geometry of Tensor Fields , 1994 .

[11]  Antonin Chambolle,et al.  The Discontinuity Set of Solutions of the TV Denoising Problem and Some Extensions , 2007, Multiscale Model. Simul..

[12]  Roger Temam,et al.  Mathematical Problems in Plasticity , 1985 .

[13]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[14]  Mila Nikolova,et al.  Local Strong Homogeneity of a Regularized Estimator , 2000, SIAM J. Appl. Math..

[15]  O. Scherzer,et al.  Characterization of minimizers of convex regularization functionals , 2006 .

[16]  Tony F. Chan,et al.  High-Order Total Variation-Based Image Restoration , 2000, SIAM J. Sci. Comput..

[17]  R. Bishop,et al.  Tensor Analysis on Manifolds , 1980 .

[18]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

[19]  G. Steidl,et al.  Variational Methods with Higher–Order Derivatives in Image Processing , 2007 .

[20]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[21]  W. Ring.  Structural Properties of Solutions to Total Variation Regularization Problems , 2000 .

[22]  Irene Fonseca,et al.  A Higher Order Model for Image Restoration: The One-Dimensional Case , 2007, SIAM J. Math. Anal..

[23]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[24]  Tony F. Chan,et al.  A fourth order dual method for staircase reduction in texture extraction and image restoration problems , 2010, 2010 IEEE International Conference on Image Processing.

[25]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .