Edge-Preserving Image Regularization Based on Morphological Wavelets and Dyadic Trees

Despite the tremendous success of wavelet-based image regularization, we still lack a comprehensive understanding of the exact factor that controls edge preservation and a principled method to determine the wavelet decomposition structure for dimensions greater than 1. We address these issues from a machine learning perspective by using tree classifiers to underpin a new image regularizer that measures the complexity of an image based on the complexity of the dyadic-tree representations of its sublevel sets. By penalizing unbalanced dyadic trees less, the regularizer preserves sharp edges. The main contribution of this paper is the connection of concepts from structured dyadic-tree complexity measures, wavelet shrinkage, morphological wavelets, and smoothness regularization in Besov space into a single coherent image regularization framework. Using the new regularizer, we also provide a theoretical basis for the data-driven selection of an optimal dyadic wavelet decomposition structure. As a specific application example, we give a practical regularized image denoising algorithm that uses this regularizer and the optimal dyadic wavelet decomposition structure.

[1]  Pingmei Xu,et al.  Learning a wavelet tree for multichannel image denoising , 2011, 2011 18th IEEE International Conference on Image Processing.

[2]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[3]  Gábor Lugosi,et al.  Concept learning using complexity regularization , 1995, IEEE Trans. Inf. Theory.

[4]  S. Loustau.  Penalized empirical risk minimization over Besov spaces , 2009 .

[5]  Henk J. A. M. Heijmans,et al.  Nonlinear multiresolution signal decomposition schemes. I. Morphological pyramids , 2000, IEEE Trans. Image Process..

[6]  Alberto Maria Segre,et al.  Programs for Machine Learning , 1994 .

[7]  Clayton Scott.  Dyadic Decision Trees , 2004 .

[8]  Peter J. Ramadge,et al.  Morphological wavelets and the complexity of dyadic trees , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[9]  Robert E. Schapire,et al.  How boosting the margin can also boost classifier complexity , 2006, ICML.

[10]  Michael T. Orchard,et al.  On the importance of combining wavelet-based nonlinear approximation with coding strategies , 2002, IEEE Trans. Inf. Theory.

[11]  Robert D. Nowak,et al.  Minimax-optimal classification with dyadic decision trees , 2006, IEEE Transactions on Information Theory.

[12]  Peter L. Bartlett,et al.  Generalization Error of Combined Classifiers , 2002, J. Comput. Syst. Sci..

[13]  David L. Donoho,et al.  Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Da , 1993 .

[14]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[15]  H. Triebel.  Theory Of Function Spaces , 1983 .

[16]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[17]  Robert D. Nowak,et al.  Controlling the error in FMRI: Hypothesis testing or set estimation? , 2008, 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro.

[18]  Wolfgang Heardle et al..  Wavelets, approximation, and statistical applications , 2013 .

[19]  Henk J. A. M. Heijmans,et al.  Nonlinear multiresolution signal decomposition schemes. II. Morphological wavelets , 2000, IEEE Trans. Image Process..

[20]  Robert D. Nowak,et al.  Near-Minimax Optimal Classification with Dyadic Classification Trees , 2003, NIPS.

[21]  Peter J. Ramadge,et al.  Morphological wavelet transform with adaptive dyadic structures , 2010, 2010 IEEE International Conference on Image Processing.

[22]  Peyman Milanfar,et al.  Kernel Regression for Image Processing and Reconstruction , 2007, IEEE Transactions on Image Processing.

[23]  Wei-Yin Loh,et al.  Classification and regression trees , 2011, WIREs Data Mining Knowl. Discov..

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

[25]  Robert D. Nowak,et al.  Minimax Optimal Level-Set Estimation , 2007, IEEE Transactions on Image Processing.

[26]  Alessandro Foi,et al.  Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering , 2007, IEEE Transactions on Image Processing.

[27]  Robert D. Nowak,et al.  Dyadic Classification Trees via Structural Risk Minimization , 2002, NIPS.

[28]  Richard G. Baraniuk,et al.  Wavelet statistical models and Besov spaces , 1999 .

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

[30]  V. Vapnik.  Estimation of Dependences Based on Empirical Data , 2006 .

[31]  Anestis Antoniadis,et al.  Wavelet kernel penalized estimation for non-equispaced design regression , 2006, Stat. Comput..

[32]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[33]  Sreerama K. Murthy,et al.  Automatic Construction of Decision Trees from Data: A Multi-Disciplinary Survey , 1998, Data Mining and Knowledge Discovery.

[34]  Lei Zhang,et al.  Sparsity-based image denoising via dictionary learning and structural clustering , 2011, CVPR 2011.

[35]  J. Ross Quinlan,et al.  C4.5: Programs for Machine Learning , 1992 .

[36]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[37]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .