1 University of Wisconsin – Madison Electrical Computer Engineering ECE533 Digital Image Processing Image Denoising Using Wavelet Thresholding Techniques Yang Yang2 Abstract Wavelet transforms enable us to represent signals with a high degree of scarcity. Wavelet thresholding is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. The aim of this project was to study various thresholding techniques such as SureShrink, VisuShrink and BayeShrink and determine the best one for image denoising.3TABLE OF CONTENTS 1 INTRODUCTION.......................................................... 4 2 THRESHOLDING......................................................... 4 2.1 INTRODUCTION............................................................ 4 2.2 HARD AND SOFT THRESHOLDING................................. 5 2.3 THRESHOLD SELECTION .............................................. 6 2.4 COMPARISON WITH UNIVERSAL THRESHOLD .............. 7 3 IMAGE DENOISING USING THRESHOLDING .... 8 3.1 VISUSHRINK................................................................ 9 3.2 SURESHRINK ............................................................... 9 3.3 BAYESSHRINK ........................................................... 10 4 EXPERIMENTAL RESULTS .................................... 10 5 CONCLUSIONS.......................................................... 24 REFERENCES................................................................ 2541 Introduction In many applications, image denoising is used to produce good estimates of the original image from noisy observations. The restored image should contain less noise than the observations while still keep sharp transitions (i.e. edges). Wavelet transform, due to its excellent localization property, has rapidly become an indispensable signal and image processing tool for a variety of applications, including compression and denoising [1, 2, 3]. Wavelet denoising attempts to remove the noise present in the signal while preserving the signal characteristics, regardless of its frequency content. It involves three steps: a linear forward wavelet transform, nonlinear thresholding step and a linear inverse wavelet transform. Wavelet thresholding (first proposed by Donoho [1, 2, 3]) is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. It removes noise by killing coefficients that are insignificant relative to some threshold, and turns out to be simple and effective, depends heavily on the choice of a thresholding parameter and the choice of this threshold determines, to a great extent the efficacy of denoising. Researchers have developed various techniques for choosing denoising parameters and so far there is no “best” universal threshold determination technique. The aim of this project was to study various thresholding techniques such as SureShrink[1], VisuShrink[3] and BayesShrink[5] and determine the best one for image denoising. 2 Thresholding 2.1 Introduction The plot of wavelet coefficients in Fig 1 suggests that small coefficients are dominated by noise, while coefficients with a large absolute value carry more signal information than noise. Replacing noisy coefficients (small coefficients below a certain threshold value)5by zero and an inverse wavelet transform may lead to a reconstruction that has lesser noise. Fig 1: A noisy signal in time domain and wavelet domain. Note the scarcity of coefficients. 2.2 Hard and soft thresholding Hard and soft thresholding with threshold ¸ are defined as follows: The hard thresholding operator is defined as: D(U, λ) = U for all |U|> λ = 0 otherwise The soft thresholding operator on the other hand is defined as: D(U, λ) = sgn(U)max(0, |U| - λ ) Fig2: Hard Thresholding Fig3: Soft Thresholding Hard threshold is a “keep or kill” procedure and is more intuitively appealing. The transfer function of the same is shown in Fig 2. The alternative, soft thresholding (whose transfer function is shown in Fig3), shrinks coefficients above the threshold in absolute value. While at first sight hard thresholding may seem to be natural, the 0 500 1000 1500 2000 2500−10−505101520250 500 1000 1500 2000 2500−6−4−2024686continuity of soft thresholding has some advantages. It makes algorithms mathematically more tractable [3]. Moreover, hard thresholding does not even work with some algorithms such as the GCV procedure [4]. Sometimes, pure noise coefficients may pass the hard threshold and appear as annoying ’blips’ in the output. Soft thesholding shrinks these false structures. 2.3 Threshold selection As one may observe, threshold selection is an important question when denoising. A small threshold may yield a result close to the input, but the result may still be noisy. A large threshold on the other hand, produces a signal with a large number of zero coefficients. This leads to a smooth signal. Paying too much attention to smoothness, however, destroys details and in image processing may cause blur and artifacts. The setup is as follows: 1. The original signals have length 2048. 2. We step through the thresholds from 0 to 5 with steps of 0.2 and at each step denoise the four noisy signals by both hard and soft thresholding with that threshold. 3. For each threshold, the MSE of the denoised signal is calculated. 4. Repeat the above steps for different orthogonal bases, namely, Haar, Daubechies. The results are tabulated in the table 1 Table 1: Best thresholds, empirically found with different denoising schemes, in terms of MSE72.4 Comparison with Universal threshold The threshold¸σλNUNIVln2= (N being the signal length, 2σ being the noise variance) is well known in wavelet literature as the Universal threshold. It is the optimal threshold in the asymptotic sense and minimizes the cost function of the difference between the function and the soft thresholded version of the same in the L2 norm sense (i.e. it minimizes2OrigThreshYYE − ). In our case, N=2048, σ= 1, therefore theoretically, ()905.3)1(2048ln2 ==UNIVλ. As seen from the table, the best empirical thresholds for both hard and soft thresholding are much lower than this value, independent of the wavelet used. It therefore seems that the universal threshold is not useful to determine a threshold. However, it is useful for
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