Noiseless Codelength in Wavelet Denoising
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Research Article Noiseless Codelength in Wavelet Denoising Soosan Beheshti, Azadeh Fakhrzadeh, and Sridhar Krishnan Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 Correspondence should be addressed to Soosan Beheshti, [email protected] Received 27 April 2009; Revised 14 October 2009; Accepted 4 January 2010 Academic Editor: Christoph Mecklenbr¨auker Copyright © 2010 Soosan Beheshti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose an adaptive, data-driven thresholding method based on a recently developed idea of Minimum Noiseless Description Length (MNDL). MNDL Subspace Selection (MNDL-SS) is a novel method of selecting an optimal subspace among the competing subspaces of the transformed noisy data. Here we extend the application of MNDL-SS for thresholding purposes. The approach searches for the optimum threshold for the data coefficients in an orthonormal basis. It is shown that the optimum threshold can be extracted from the noisy coefficients themselves. While the additive noise in the available data is assumed to be independent, the main challenge in MNDL thresholding is caused by the dependence of the additive noise in the sorted coefficients. The approach provides new hard and soft thresholds. Simulation results are presented for orthonormal wavelet transforms. While the method is comparable with the existing thresholding methods and in some cases outperforms them, the main advantage of the new approach is that it provides not only the optimum threshold but also an estimate of the associated mean-square error (MSE) for that threshold simultaneously.
1. Introduction We can recognize different phenomena by collecting data from them. However, defective instruments, problems with the data acquisition process, and the interference of natural factors can all degrade the data of interest. Furthermore, noise can be introduced by transmission errors or compression. Thus, denoising is often a necessary step in data processing and various approaches have been introduced for this purpose. Some of these methods, such as Wiener filters, are grouped as linear techniques. While these techniques are easy to implement, their results are not always satisfactory. Over past decades, researchers have improved the performance of denoising methods by developing nonlinear approaches such as [1–6]. Although these approaches have succeeded in providing better results, they are usually computationally exhaustive, hard to implement, or use particular assumptions either on the noisy data or on the class of the data estimator. Thresholding methods are alternative approaches to the denoising problem. The thresholding problem is first formulated in [7] where VisuShrink is introduced. This threshold is a nonadaptive universal threshold and depends
only on the number of data points and noise variance. VisuShrink i
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