Graph Signal Denoising via Trilateral Filter on Graph Spectral Domain

This paper presents a graph signal denoising method with the trilateral filter defined in the graph spectral domain. The original trilateral filter (TF) is a data-dependent filter that is widely used as an edge-preserving smoothing method for image processing. However, because of the data-dependency, one cannot provide its frequency domain representation. To overcome this problem, we establish the graph spectral domain representation of the data-dependent filter, i.e., a spectral graph TF (SGTF). This representation enables us to design an effective graph signal denoising filter with a Tikhonov regularization. Moreover, for the proposed graph denoising filter, we provide a parameter optimization technique to search for a regularization parameter that approximately minimizes the mean squared error w.r.t. the unknown graph signal of interest. Comprehensive experimental results validate our graph signal processing-based approach for images and graph signals.

[1]  Alexei A. Efros,et al.  Fast bilateral filtering for the display of high-dynamic-range images , 2002 .

[2]  Masashi Sugiyama,et al.  Subspace Information Criterion for Model Selection , 2001, Neural Computation.

[3]  Abderrahim Elmoataz,et al.  Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing , 2008, IEEE Transactions on Image Processing.

[4]  Mithun Uliyar,et al.  Fast Non-Local algorithm for image denoising , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[5]  Dimitri Van De Ville,et al.  Tight Wavelet Frames on Multislice Graphs , 2013, IEEE Transactions on Signal Processing.

[6]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[7]  Peyman Milanfar,et al.  A Tour of Modern Image Filtering: New Insights and Methods, Both Practical and Theoretical , 2013, IEEE Signal Processing Magazine.

[8]  Richard Szeliski,et al.  Computer Vision - Algorithms and Applications , 2011, Texts in Computer Science.

[9]  Nannan Yu,et al.  LLSURE: Local Linear SURE-Based Edge-Preserving Image Filtering , 2013, IEEE Transactions on Image Processing.

[10]  Pascal Frossard,et al.  The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains , 2012, IEEE Signal Processing Magazine.

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

[12]  C. L. Mallows.  Some comments on C_p , 1973 .

[13]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[14]  Sunil K. Narang,et al.  Perfect Reconstruction Two-Channel Wavelet Filter Banks for Graph Structured Data , 2011, IEEE Transactions on Signal Processing.

[15]  Daniel Cohen-Or,et al.  Bilateral mesh denoising , 2003 .

[16]  Antonio Ortega,et al.  A graph-based joint bilateral approach for depth enhancement , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[17]  Pascal Frossard,et al.  Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Data Domains , 2012, ArXiv.

[18]  Yuichi Tanaka,et al.  M-Channel Oversampled Graph Filter Banks , 2014, IEEE Trans. Signal Process..

[19]  Sunil K. Narang,et al.  Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs , 2012, IEEE Transactions on Signal Processing.

[20]  Jean-Michel Morel,et al.  A non-local algorithm for image denoising , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[21]  Camille Couprie,et al.  Dual constrained TV-based regularization , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[22]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

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

[24]  Jack Tumblin,et al.  The Trilateral Filter for High Contrast Images and Meshes , 2003, Rendering Techniques.

[25]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[26]  Pierre Vandergheynst,et al.  Wavelets on Graphs via Spectral Graph Theory , 2009, ArXiv.

[27]  Thierry Blu,et al.  The SURE-LET Approach to Image Denoising , 2007, IEEE Transactions on Image Processing.

[28]  Masashi Sugiyama,et al.  A unified method for optimizing linear image restoration filters , 2002, Signal Process..

[29]  Chandra Sekhar Seelamantula,et al.  Sure-fast bilateral filters , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[30]  Guillermo Sapiro,et al.  Non-local sparse models for image restoration , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[31]  Yuichi Tanaka,et al.  Trilateral filter on graph spectral domain , 2014, 2014 IEEE International Conference on Image Processing (ICIP).

[32]  Nikolas P. Galatsanos,et al.  Regularized reconstruction to reduce blocking artifacts of block discrete cosine transform compressed images , 1993, IEEE Trans. Circuits Syst. Video Technol..

[33]  Roberto Manduchi,et al.  Bilateral filtering for gray and color images , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[34]  Sunil K. Narang,et al.  Bilateral filter: Graph spectral interpretation and extensions , 2013, 2013 IEEE International Conference on Image Processing.

[35]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

[36]  Rafael C. Gonzales,et al.  Digital Image Processing -3/E. , 2012 .

[37]  Jian Sun,et al.  Guided Image Filtering , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[38]  Jack Tumblin,et al.  The trilateral filter for high contrast images and meshes , 2003, Rendering Techniques.

[39]  I. Yamada,et al.  Total generalized variation for graph signals , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[40]  Albert C. S. Chung,et al.  Trilateral filtering for biomedical images , 2004, 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821).