On the effective measure of dimension in total variation minimization

Total variation (TV) is a widely used technique in many signal and image processing applications. One of the famous TV based algorithms is TV denoising that performs well with piecewise constant images. The same prior has been used also in the context of compressed sensing for recovering a signal from a small number of measurements. Recently, it has been shown that the number of measurements needed for such a recovery is proportional to the size of the edges in the sampled image and not the number of connected components in the image. In this work we show that this is not a coincidence and that the number of connected components in a piecewise constant image cannot serve alone as a measure for the complexity of the image. Our result is not limited only to images but holds also for higher dimensional signals. We believe that the results in this work provide a better insight into the TV prior.

[1]  M. Davies,et al.  Greedy-like algorithms for the cosparse analysis model , 2012, 1207.2456.

[2]  Emmanuel J. Candès,et al.  Modern statistical estimation via oracle inequalities , 2006, Acta Numerica.

[3]  M. Rudelson,et al.  Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[4]  Raja Giryes,et al.  On the Effective Measure of Dimension in the Analysis Cosparse Model , 2014, IEEE Transactions on Information Theory.

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

[6]  M. Lustig,et al.  Compressed Sensing MRI , 2008, IEEE Signal Processing Magazine.

[7]  A. Bruckstein,et al.  On Over-Parameterized Model Based TV-Denoising , 2007, 2007 International Symposium on Signals, Circuits and Systems.

[8]  Deanna Needell,et al.  Stable Image Reconstruction Using Total Variation Minimization , 2012, SIAM J. Imaging Sci..

[9]  T. Chan,et al.  Edge-preserving and scale-dependent properties of total variation regularization , 2003 .

[10]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

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

[12]  Michael Elad,et al.  The Cosparse Analysis Model and Algorithms , 2011, ArXiv.

[13]  Deanna Needell,et al.  Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization , 2012, IEEE Transactions on Image Processing.

[14]  Yonina C. Eldar,et al.  Uniqueness conditions for low-rank matrix recovery , 2011, Optical Engineering + Applications.

[15]  Emmanuel J. Candès,et al.  Matrix Completion With Noise , 2009, Proceedings of the IEEE.

[16]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[17]  Raja Giryes,et al.  Sampling in the Analysis Transform Domain , 2014, ArXiv.

[18]  Michael Elad,et al.  Sparsity Based Methods for Overparametrized Variational Problems , 2014, SIAM J. Imaging Sci..