Total-Variation -- Fast Gradient Flow and Relations to Koopman Theory

The space-discrete Total Variation (TV) flow is analyzed using several mode decomposition techniques. In the one dimensional case, we provide analytic formulations to Dynamic Mode Decomposition (DMD) and to Koopman Mode Decomposition (KMD) of the TV flow and compare the obtained modes to TV spectral decomposition. We propose a computationally efficient algorithm to evolve the one-dimensional TV flow. A significant speedup by three orders of magnitude is obtained, compared to iterative minimizations. A common theme, for both mode analysis and fast algorithm, is the significance of phase transitions during the flow, in which the subgradient changes. We explain why applying DMD directly on TV-flow measurements cannot model the flow or extract modes well. We formulate a more general method for mode decomposition that coincides with the modes of KMD. This method is based on the linear decay profile, typical to TV-flow. These concepts are demonstrated through experiments, where additional extensions to the twodimensional case are given.

[1]  Thomas Brox,et al.  A TV flow based local scale estimate and its application to texture discrimination , 2006, J. Vis. Commun. Image Represent..

[2]  Clarence W. Rowley,et al.  Evaluating the accuracy of the dynamic mode decomposition , 2016, Journal of Computational Dynamics.

[3]  Adam M. Oberman,et al.  Anisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes , 2010, 1007.1035.

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

[5]  Jack Xin,et al.  A Weighted Difference of Anisotropic and Isotropic Total Variation for Relaxed Mumford-Shah Color and Multiphase Image Segmentation , 2020, SIAM J. Imaging Sci..

[6]  Arjan Kuijper,et al.  P-Laplacian Driven Image Processing , 2007, 2007 IEEE International Conference on Image Processing.

[7]  Ido Cohen,et al.  Modes of Homogeneous Gradient Flows , 2021, SIAM J. Imaging Sci..

[8]  Guy Gilboa,et al.  A Spectral Approach to Total Variation , 2013, SSVM.

[9]  J. Nathan Kutz,et al.  Variable Projection Methods for an Optimized Dynamic Mode Decomposition , 2017, SIAM J. Appl. Dyn. Syst..

[10]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization , 2006, Journal of Mathematical Imaging and Vision.

[11]  Ido Cohen,et al.  Total-Variation Mode Decomposition , 2021, SSVM.

[12]  Jeremias Sulam,et al.  Learning to solve TV regularised problems with unrolled algorithms , 2020, NeurIPS.

[13]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[14]  Wotao Yin,et al.  Parametric Maximum Flow Algorithms for Fast Total Variation Minimization , 2009, SIAM J. Sci. Comput..

[15]  Total Variation Flow and Sign Fast Diffusion in one dimension , 2011, 1107.2153.

[16]  Stephan Didas,et al.  Relations Between Higher Order TV Regularization and Support Vector Regression , 2005, Scale-Space.

[17]  Guy Gilboa,et al.  A Total Variation Spectral Framework for Scale and Texture Analysis , 2014, SIAM J. Imaging Sci..

[18]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[19]  Alfred M. Bruckstein,et al.  Scale Space and Variational Methods in Computer Vision , 2011, Lecture Notes in Computer Science.

[20]  Michael Möller,et al.  Nonlinear Spectral Analysis via One-Homogeneous Functionals: Overview and Future Prospects , 2015, Journal of Mathematical Imaging and Vision.

[21]  Jack Xin,et al.  A Weighted Difference of Anisotropic and Isotropic Total Variation Model for Image Processing , 2015, SIAM J. Imaging Sci..

[22]  Guy Gilboa,et al.  Examining the Limitations of Dynamic Mode Decomposition through Koopman Theory Analysis , 2021 .

[23]  Alexander W. Dowling,et al.  Learning spatiotemporal dynamics in wholesale energy markets with dynamic mode decomposition , 2021 .

[24]  M. Novaga,et al.  The Total Variation Flow in RN , 2002 .

[25]  V. Caselles,et al.  Minimizing total variation flow , 2000, Differential and Integral Equations.

[26]  Michael Moeller,et al.  Nonlinear spectral geometry processing via the TV transform , 2020, ACM Trans. Graph..

[27]  Tieyong Zeng,et al.  Adaptive total variation based image segmentation with semi-proximal alternating minimization , 2021, Signal Process..

[28]  A. Chambolle,et al.  Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals , 2019, Analysis & PDE.

[29]  Guy Gilboa,et al.  Nonlinear Spectral Processing of Shapes via Zero-Homogeneous Flows , 2021, SSVM.

[30]  Thomas Brox,et al.  Equivalence Results for TV Diffusion and TV Regularisation , 2003, Scale-Space.

[31]  Michael Möller,et al.  Spectral Decompositions Using One-Homogeneous Functionals , 2016, SIAM J. Imaging Sci..

[32]  Rushikesh Kamalapurkar,et al.  Singular Dynamic Mode Decompositions , 2021, ArXiv.

[33]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[34]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[35]  Ido Cohen,et al.  Introducing the p-Laplacian spectra , 2019, Signal Process..

[36]  A. Chambolle,et al.  An introduction to Total Variation for Image Analysis , 2009 .