Graph Signal Denoising via Trilateral Filter on Graph Spectral Domain
(IEEE Trans. on Signal and Information Processing over Networks)
1Tokyo University of Agriculture and Technology
2Tokyo Institute of Technology
Abstract
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.
Sample code
SGTF toolbox is available here.
Note: Our code requires the graph signal processing tool box (GSPBox). To run our code, you must download the tool box and place it into our tool box.
Summary of SGTF
The trilateral filter (TF) [1] can be defined by using an adjacency matrix
where
A normalized graph Laplacian matrix is defined as
where
In this paper, we adjust SGTF to denoise graph signals
where
where
where
Experimental Results
※graphBior [5]
※OSGFB: oversampled graph filter bank [6]
※SGWT: spectral graph wavelet transform [7]
References
[1] P. Choudhury and J. Tumblin, “The trilateral filter for high contrast
images and meshes,” Eurographics Symp. Rendering, pp. 1-11, 2003.
[2] D. Shuman, S. Narang, P. Frossard, A. Ortega, and P. Vandergheynst,
“Signal processing on graphs: Extending high-dimensional data analysis
to networks and other irregular data domains,” IEEE Signal Processing
Magazine, vol. 30, issue. 3, pp. 83-98, May, 2013.
[3] C. Stein, “Estimation of the mean of a multivariate normal distribution,” Ann. Statist., vol. 9, pp. 1135-1151, 1981.
[4] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color
images,” in Proc. IEEE Int. Conf. Computer Vision (ICCV), pp. 839-
846, Bombay, India, Jan., 1998.
[5] S. K. Narang and A. Ortega, “Compact support biorthogonal wavelet
filter banks for arbitrary undetected graphs,” IEEE Trans. on Signal
Processing, vol. 61, pp. 4672-4685, 2013.
[6] Y. Tanaka and A. Sakiyama, “M-channel oversampled graph filter
banks,” IEEE Trans. Signal Processing, vol. 62, no. 13, pp. 3578–3590,
Jul, 2014.
[7] D. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on graphs
via spectral graph theory,” Applied and Computational Harmonic Anal-
ysis, vol. 30, no. 2, pp. 129-150, 2011. |
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