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<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/624efb9c6caf596192b72c4a/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624efb9c6caf596192b72c4a/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">1532 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 9, SEPTEMBER 2000</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws0">Adaptive Wavelet Thresholding for Image Denoising</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">and Compression</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">S. Grace Chang, Student Member, IEEE, Bin Yu, Senior Member, IEEE, and Martin Vetterli, Fellow, IEEE</div><div class="t m0 x5 h5 y5 ff3 fs3 fc0 sc0 ls0 ws0">Abstract—The first part of this paper proposes an adaptive,</div><div class="t m0 x1 h5 y6 ff4 fs3 fc0 sc0 ls0 ws0">data-driven threshold for image denoising via wavelet soft-thresh-</div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 ls0 ws0">olding. The threshold is derived in a Bayesian framework, and the</div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 ls0 ws0">prior used on the wavelet coefficients is the generalized Gaussian</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 ls0 ws0">distribution (GGD) widely used in image processing applications.</div><div class="t m0 x1 h5 ya ff4 fs3 fc0 sc0 ls0 ws0">The proposed threshold is simple and closed-form, and it is adap-</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 ls0 ws0">tive to each subband because it depends on data-driven estimates</div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 ls0 ws0">of the parameters. Experimental results show that the proposed</div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 ls0 ws0">method, called BayesShrink, is typically within 5% of the MSE</div><div class="t m0 x1 h5 ye ff4 fs3 fc0 sc0 ls0 ws0">of the best soft-thresholding benchmark with the image assumed</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls0 ws0">known. It also outperforms Donoho and Johnstone’s SureShrink</div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 ls0 ws0">most of the time.</div><div class="t m0 x6 h5 y11 ff4 fs3 fc0 sc0 ls0 ws0">The second part of the paper attempts to further validate</div><div class="t m0 x1 h5 y12 ff4 fs3 fc0 sc0 ls0 ws0">recent claims that lossy compression can be used for denoising.</div><div class="t m0 x1 h5 y13 ff4 fs3 fc0 sc0 ls0 ws0">The BayesShrink threshold can aid in the parameter selection</div><div class="t m0 x1 h5 y14 ff4 fs3 fc0 sc0 ls0 ws0">of a coder designed with the intention of denoising, and thus</div><div class="t m0 x1 h5 y15 ff4 fs3 fc0 sc0 ls0 ws0">achieving simultaneous denoising and compression. Specifically,</div><div class="t m0 x1 h5 y16 ff4 fs3 fc0 sc0 ls0 ws0">the zero-zone in the quantization step of compression is analogous</div><div class="t m0 x1 h5 y17 ff4 fs3 fc0 sc0 ls0 ws0">to the threshold value in the thresholding function. The remaining</div><div class="t m0 x1 h5 y18 ff4 fs3 fc0 sc0 ls0 ws0">coder design parameters are chosen based on a criterion derived</div><div class="t m0 x1 h5 y19 ff4 fs3 fc0 sc0 ls0 ws0">from Rissanen’s minimum description length (MDL) principle.</div><div class="t m0 x1 h5 y1a ff4 fs3 fc0 sc0 ls0 ws0">Experiments show that this compression method does indeed re-</div><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 ls0 ws0">move noise significantly, especially for large noise power. However,</div><div class="t m0 x1 h5 y1c ff4 fs3 fc0 sc0 ls0 ws0">it introduces quantization noise and should be used only if bitrate</div><div class="t m0 x1 h5 y1d ff4 fs3 fc0 sc0 ls0 ws0">were an additional concern to denoising.</div><div class="t m0 x6 h5 y1e ff3 fs3 fc0 sc0 ls0 ws0">Index Terms—Adaptive method, image compression, image de-</div><div class="t m0 x1 h5 y1f ff4 fs3 fc0 sc0 ls0 ws0">noising, image restoration, wavelet thresholding.</div><div class="t m0 x7 h6 y20 ff1 fs4 fc0 sc0 ls0 ws0">I. INTRODUCTION</div><div class="t m0 x1 h7 y21 ff4 fs6 fc0 sc0 ls0 ws0">A</div><div class="t m0 x8 h6 y22 ff1 fs4 fc0 sc0 ls0 ws0">N IMAGE is often corrupted by noise in its acquisition or</div><div class="t m0 x8 h6 y23 ff1 fs4 fc0 sc0 ls0 ws0">transmission. The goal of denoising is to remove the noise</div><div class="t m0 x1 h6 y24 ff1 fs4 fc0 sc0 ls0 ws0">while retaining as much as possible the important signal fea-</div><div class="t m0 x1 h6 y25 ff1 fs4 fc0 sc0 ls0 ws0">tures. Traditionally, this is achieved by linear processing such</div><div class="t m0 x1 h6 y26 ff1 fs4 fc0 sc0 ls0 ws0">as Wiener filtering. A vast literature has emerged recently on</div><div class="t m0 x9 h8 y27 ff1 fs5 fc0 sc0 ls0 ws0">Manuscript received January 22, 1998; revised April 7, 2000. This work</div><div class="t m0 x1 h8 y28 ff1 fs5 fc0 sc0 ls0 ws0">was supported in part by the NSF Graduate Fellowship and the Univer-</div><div class="t m0 x1 h8 y29 ff1 fs5 fc0 sc0 ls0 ws0">sity of California Dissertation Fellowship to S. G. Chang; ARO Grant</div><div class="t m0 x1 h8 y2a ff1 fs5 fc0 sc0 ls0 ws0">DAAH04-94-G-0232 and NSF Grant DMS-9322817 to B. Yu; and NSF Grant</div><div class="t m0 x1 h8 y2b ff1 fs5 fc0 sc0 ls0 ws0">MIP-93-213002 and Swiss NSF Grant 20-52347.97 to M. Vetterli. Part of this</div><div class="t m0 x1 h8 y2c ff1 fs5 fc0 sc0 ls0 ws0">work was presented at the IEEE International Conference on Image Processing,</div><div class="t m0 x1 h8 y2d ff1 fs5 fc0 sc0 ls0 ws0">Santa Barbara, CA, October 1997. The associate editor coordinating the</div><div class="t m0 x1 h8 y2e ff1 fs5 fc0 sc0 ls0 ws0">review of this manuscript and approving it for publication was Prof. Patrick L.</div><div class="t m0 x1 h8 y2f ff1 fs5 fc0 sc0 ls0 ws0">Combettes.</div><div class="t m0 x9 h8 y30 ff1 fs5 fc0 sc0 ls0 ws0">S. G. Chang was with the Department of Electrical Engineering and Computer</div><div class="t m0 x1 h8 y31 ff1 fs5 fc0 sc0 ls0 ws0">Sciences, University of California, Berkeley, CA 94720 USA. She is now with</div><div class="t m0 x1 h8 y32 ff1 fs5 fc0 sc0 ls0 ws0">Hewlett-Packard Company, Grenoble, France (e-mail: grchang@yahoo.com).</div><div class="t m0 x9 h8 y33 ff1 fs5 fc0 sc0 ls0 ws0">B. Yu is with the Department of Statistics, University of California, Berkeley,</div><div class="t m0 x1 h8 y34 ff1 fs5 fc0 sc0 ls0 ws0">CA 94720 USA (e-mail: binyu@stat.berkeley.edu)</div><div class="t m0 x9 h8 y35 ff1 fs5 fc0 sc0 ls0 ws0">M. Vetterli is with the Laboratory of Audiovisual Communications, Swiss</div><div class="t m0 x1 h8 y36 ff1 fs5 fc0 sc0 ls0 ws0">Federal Institute of Technology (EPFL), Lausanne, Switzerland and also with</div><div class="t m0 x1 h8 y37 ff1 fs5 fc0 sc0 ls0 ws0">the Department of Electrical Engineering and Computer Sciences, University of</div><div class="t m0 x1 h8 y38 ff1 fs5 fc0 sc0 ls0 ws0">California, Berkeley, CA 94720 USA.</div><div class="t m0 x9 h8 y39 ff1 fs5 fc0 sc0 ls0 ws0">Publisher Item Identifier S 1057-7149(00)06914-1.</div><div class="t m0 xa h6 y5 ff1 fs4 fc0 sc0 ls0 ws0">signal denoising using nonlinear techniques, in the setting of</div><div class="t m0 xa h6 y3a ff1 fs4 fc0 sc0 ls0 ws0">additive white Gaussian noise. The seminal work on signal de-</div><div class="t m0 xa h6 y3b ff1 fs4 fc0 sc0 ls0 ws0">noising via wavelet thresholding or shrinkage of Donoho and</div><div class="t m0 xa h6 y3c ff1 fs4 fc0 sc0 ls0 ws0">Johnstone ([13]–[16]) have shown that various wavelet thresh-</div><div class="t m0 xa h6 y3d ff1 fs4 fc0 sc0 ls0 ws0">olding schemes for denoising have near-optimal properties in</div><div class="t m0 xa h6 y3e ff1 fs4 fc0 sc0 ls0 ws0">the minimax sense and perform well in simulation studies of</div><div class="t m0 xa h6 y3f ff1 fs4 fc0 sc0 ls0 ws0">one-dimensional curve estimation. It has been shown to have</div><div class="t m0 xa h6 y40 ff1 fs4 fc0 sc0 ls0 ws0">better rates of convergence than linear methods for approxi-</div><div class="t m0 xa h6 y41 ff1 fs4 fc0 sc0 ls0 ws0">mating functions in Besov spaces ([13], [14]). Thresholding is</div><div class="t m0 xa h6 y42 ff1 fs4 fc0 sc0 ls0 ws0">a nonlinear technique, yet it is very simple because it operates</div><div class="t m0 xa h6 y43 ff1 fs4 fc0 sc0 ls0 ws0">on one wavelet coefficient at a time. Alternative approaches to</div><div class="t m0 xa h6 y44 ff1 fs4 fc0 sc0 ls0 ws0">nonlinear wavelet-baseddenoising can be found in, for example,</div><div class="t m0 xa h6 y45 ff1 fs4 fc0 sc0 ls0 ws0">[1], [4], [8]–[10], [12], [18], [19], [24], [27]–[29], [32], [33],</div><div class="t m0 xa h6 y46 ff1 fs4 fc0 sc0 ls0 ws0">[35], and references therein.</div><div class="t m0 xb h6 y47 ff1 fs4 fc0 sc0 ls0 ws0">On a seemingly unrelated front, lossy compression has been</div><div class="t m0 xa h6 y48 ff1 fs4 fc0 sc0 ls0 ws0">proposed for denoising in several works [6], [5], [21], [25],</div><div class="t m0 xa h6 y49 ff1 fs4 fc0 sc0 ls0 ws0">[28]. Concerns regarding the compression rate were explicitly</div><div class="t m0 xa h6 y4a ff1 fs4 fc0 sc0 ls0 ws0">addressed. This is important because any practical coder must</div><div class="t m0 xa h6 y4b ff1 fs4 fc0 sc0 ls0 ws0">assume a limited resource (such as bits) at its disposal for repre-</div><div class="t m0 xa h6 y4c ff1 fs4 fc0 sc0 ls0 ws0">senting the data. Other works [4], [12]–[16] also addressed the</div><div class="t m0 xa h6 y4d ff1 fs4 fc0 sc0 ls0 ws0">connection between compression and denoising, especially with</div><div class="t m0 xa h6 y4e ff1 fs4 fc0 sc0 ls0 ws0">nonlinear algorithms such as wavelet thresholding in a mathe-</div><div class="t m0 xa h6 y4f ff1 fs4 fc0 sc0 ls0 ws0">matical framework. However, these latter works were not con-</div><div class="t m0 xa h6 y50 ff1 fs4 fc0 sc0 ls0 ws0">cerned with quantization and bitrates: compression results from</div><div class="t m0 xa h6 y51 ff1 fs4 fc0 sc0 ls0 ws0">a reduced number of nonzero wavelet coefficients, and not from</div><div class="t m0 xa h6 y52 ff1 fs4 fc0 sc0 ls0 ws0">an explicit design of a coder.</div><div class="t m0 xb h6 y53 ff1 fs4 fc0 sc0 ls0 ws0">The intuition behind using lossy compression for denoising</div><div class="t m0 xa h6 y54 ff1 fs4 fc0 sc0 ls0 ws0">may be explained as follows. A signal typically has structural</div><div class="t m0 xa h6 y55 ff1 fs4 fc0 sc0 ls0 ws0">correlations that a good coder can exploit to yield a concise rep-</div><div class="t m0 xa h6 y56 ff1 fs4 fc0 sc0 ls0 ws0">resentation. White noise, however, does not have structural re-</div><div class="t m0 xa h6 y57 ff1 fs4 fc0 sc0 ls0 ws0">dundancies and thus is not easily compressable. Hence, a good</div><div class="t m0 xa h6 y58 ff1 fs4 fc0 sc0 ls0 ws0">compression method can provide a suitable model for distin-</div><div class="t m0 xa h6 y59 ff1 fs4 fc0 sc0 ls0 ws0">guishing between signal and noise. The discussion will be re-</div><div class="t m0 xa h6 y5a ff1 fs4 fc0 sc0 ls0 ws0">stricted to wavelet-based coders, though these insights can be</div><div class="t m0 xa h6 y5b ff1 fs4 fc0 sc0 ls0 ws0">extended to other transform-domain coders as well. A concrete</div><div class="t m0 xa h6 y5c ff1 fs4 fc0 sc0 ls0 ws0">connection between lossy compression and denoising can easily</div><div class="t m0 xa h6 y5d ff1 fs4 fc0 sc0 ls0 ws0">be seen when one examines the similarity between thresholding</div><div class="t m0 xa h6 y5e ff1 fs4 fc0 sc0 ls0 ws0">and quantization, the latter of which is a necessary step in a prac-</div><div class="t m0 xa h6 y5f ff1 fs4 fc0 sc0 ls0 ws0">tical lossy coder. That is, the quantization of wavelet coefficients</div><div class="t m0 xa h6 y60 ff2 fs4 fc0 sc0 ls0 ws0">with a zero-zone is an approximation to the thresholding func-</div><div class="t m0 xa h6 y61 ff1 fs4 fc0 sc0 ls0 ws0">tion (see Fig. 1). Thus, provided that the quantization outside</div><div class="t m0 xa h6 y62 ff1 fs4 fc0 sc0 ls0 ws0">of the zero-zone does not introduce significant distortion, it fol-</div><div class="t m0 xa h6 y63 ff1 fs4 fc0 sc0 ls0 ws0">lows that wavelet-based lossy compression achieves denoising.</div><div class="t m0 xa h6 y64 ff1 fs4 fc0 sc0 ls0 ws0">With this connection in mind, this paper is about wavelet thresh-</div><div class="t m0 xa h6 y65 ff1 fs4 fc0 sc0 ls0 ws0">olding for image denoising and also for lossy compression. The</div><div class="t m0 xa h6 y66 ff1 fs4 fc0 sc0 ls0 ws0">threshold choice aids the lossy coder to choose its zero-zone,</div><div class="t m0 xa h6 y67 ff1 fs4 fc0 sc0 ls0 ws0">and the resulting coder achieves simultaneous denoising and</div><div class="t m0 xa h6 y68 ff1 fs4 fc0 sc0 ls0 ws0">compression if such property is desired.</div><div class="t m0 xc h8 y69 ff1 fs5 fc0 sc0 ls0 ws0">1057–7149/00$10.00 © 2000 IEEE</div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624efb9c6caf596192b72c4a/bg2.jpg"><div class="t m0 xd h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">CHANG et al.: ADAPTIVE WAVELET THRESHOLDING FOR IMAGE DENOISING AND COMPRESSION 1533</div><div class="t m0 xd h8 y6a ff1 fs5 fc0 sc0 ls0 ws0">Fig. 1. Thresholding function can be approximated by quantization with a</div><div class="t m0 xd h8 y6b ff1 fs5 fc0 sc0 ls0 ws0">zero-zone.</div><div class="t m0 x9 h6 y6c ff1 fs4 fc0 sc0 ls0 ws0">The theoreticalformalization offiltering additiveiid Gaussian</div><div class="t m0 xd h6 y6d ff1 fs4 fc0 sc0 ls0 ws0">noise (of zero-mean and standard deviation</div><div class="t m0 xe h6 y6e ff1 fs4 fc0 sc0 ls0 ws0">) via thresholding</div><div class="t m0 xd h6 y6f ff1 fs4 fc0 sc0 ls0 ws0">wavelet coefficients was pioneered by Donoho and Johnstone</div><div class="t m0 xd h6 y70 ff1 fs4 fc0 sc0 ls0 ws0">[14]. A wavelet coefficient is compared to a given threshold and</div><div class="t m0 xd h6 y71 ff1 fs4 fc0 sc0 ls0 ws0">issetto zeroifits magnitude isless than thethreshold; otherwise,</div><div class="t m0 xd h6 y72 ff1 fs4 fc0 sc0 ls0 ws0">it is kept or modified (depending on the thresholding rule). The</div><div class="t m0 xd h6 y73 ff1 fs4 fc0 sc0 ls0 ws0">threshold acts as an oracle which distinguishes between the</div><div class="t m0 xd h6 y74 ff1 fs4 fc0 sc0 ls0 ws0">insignificant coefficients likely due to noise, and the significant</div><div class="t m0 xd h6 y75 ff1 fs4 fc0 sc0 ls0 ws0">coefficients consisting of important signal structures. Thresh-</div><div class="t m0 xd h6 y76 ff1 fs4 fc0 sc0 ls0 ws0">olding rules are especially effective for signals with sparse or</div><div class="t m0 xd h6 y77 ff1 fs4 fc0 sc0 ls0 ws0">near-sparserepresentationswhereonlyasmallsubsetofthecoef-</div><div class="t m0 xd h6 y78 ff1 fs4 fc0 sc0 ls0 ws0">ficients represents all or most of the signal energy. Thresholding</div><div class="t m0 xd h6 y79 ff1 fs4 fc0 sc0 ls0 ws0">essentially creates a region around zero where the coefficients</div><div class="t m0 xd h6 y7a ff1 fs4 fc0 sc0 ls0 ws0">are considered negligible.Outside of this region, the thresholded</div><div class="t m0 xd h6 y7b ff1 fs4 fc0 sc0 ls0 ws0">coefficients are kept to full precision (that is, without quanti-</div><div class="t m0 xd h6 y7c ff1 fs4 fc0 sc0 ls0 ws0">zation). Their most well-known thresholding methods include</div><div class="t m0 xd h6 y7d ff2 fs4 fc0 sc0 ls0 ws0">VisuShrink [14] and SureShrink [15]. These threshold choices</div><div class="t m0 xd h6 y7e ff1 fs4 fc0 sc0 ls0 ws0">enjoy asymptotic minimax optimalities over function spaces</div><div class="t m0 xd h6 y7f ff1 fs4 fc0 sc0 ls0 ws0">such as Besov spaces. Forimage denoising, however, VisuShrink</div><div class="t m0 xd h6 y80 ff1 fs4 fc0 sc0 ls0 ws0">is known to yield overly smoothed images. This is because its</div><div class="t m0 xd h6 y81 ff1 fs4 fc0 sc0 ls0 ws0">threshold choice,</div><div class="t m0 xf h6 y82 ff1 fs4 fc0 sc0 ls0 ws0">(called theuniversalthreshold and</div><div class="t m0 x10 h6 y83 ff1 fs4 fc0 sc0 ls0 ws0">is the noise variance), can be unwarrantedly large due to its</div><div class="t m0 xd h6 y84 ff1 fs4 fc0 sc0 ls0 ws0">dependenceonthenumberofsamples,</div><div class="t m0 x11 h6 y85 ff1 fs4 fc0 sc0 ls0 ws0">,whichismorethan</div><div class="t m0 xd h6 y86 ff1 fs4 fc0 sc0 ls0 ws0">foratypicaltestimageofsize .SureShrinkusesahybrid</div><div class="t m0 xd h6 y87 ff1 fs4 fc0 sc0 ls0 ws0">of the universal threshold and the SURE threshold, derived from</div><div class="t m0 xd h6 y88 ff1 fs4 fc0 sc0 ls0 ws0">minimizing Stein’s unbiased risk estimator [30], and has been</div><div class="t m0 xd h6 y89 ff1 fs4 fc0 sc0 ls0 ws0">showntoperformwell.SureShrinkwillbethemaincomparisonto</div><div class="t m0 xd h6 y8a ff1 fs4 fc0 sc0 ls0 ws0">the method proposed here, and, as will be seen later in this paper,</div><div class="t m0 xd h6 y8b ff1 fs4 fc0 sc0 ls0 ws0">ourproposedthresholdoftenyieldsbetterresult.</div><div class="t m0 x9 h6 y8c ff1 fs4 fc0 sc0 ls0 ws0">Since the works of Donoho and Johnstone, there has been</div><div class="t m0 xd h6 y8d ff1 fs4 fc0 sc0 ls0 ws0">much research on finding thresholds for nonparametric estima-</div><div class="t m0 xd h6 y8e ff1 fs4 fc0 sc0 ls0 ws0">tion in statistics. However, few are specifically tailored for im-</div><div class="t m0 xd h6 y8f ff1 fs4 fc0 sc0 ls0 ws0">ages. In this paper, we propose a framework and a near-op-</div><div class="t m0 xd h6 y90 ff1 fs4 fc0 sc0 ls0 ws0">timal threshold in this framework more suitable for image de-</div><div class="t m0 xd h6 y91 ff1 fs4 fc0 sc0 ls0 ws0">noising. This approach can be formally described as Bayesian,</div><div class="t m0 xd h6 y92 ff1 fs4 fc0 sc0 ls0 ws0">but this only describes our mathematical formulation, not our</div><div class="t m0 xd h6 y93 ff1 fs4 fc0 sc0 ls0 ws0">philosophy. The formulation is grounded on the empirical ob-</div><div class="t m0 xd h6 y94 ff1 fs4 fc0 sc0 ls0 ws0">servation that the wavelet coefficients in a subband of a natural</div><div class="t m0 xd h6 y95 ff1 fs4 fc0 sc0 ls0 ws0">image can be summarized adequatelybyageneralized Gaussian</div><div class="t m0 xd h6 y96 ff2 fs4 fc0 sc0 ls0 ws0">distribution (GGD). This observation is well-accepted in the</div><div class="t m0 xd h6 y97 ff1 fs4 fc0 sc0 ls0 ws0">image processing community (for example, see [20], [22], [23],</div><div class="t m0 xd h6 y98 ff1 fs4 fc0 sc0 ls0 ws0">[29], [34], [36]) and is used for state-of-the-art image coders</div><div class="t m0 xd h6 y99 ff1 fs4 fc0 sc0 ls0 ws0">in [20], [22], [36]. It follows from this observation that the av-</div><div class="t m0 xd h6 y9a ff1 fs4 fc0 sc0 ls0 ws0">erage MSE (in a subband) can be approximated by the corre-</div><div class="t m0 xd h6 y9b ff1 fs4 fc0 sc0 ls0 ws0">sponding Bayesian squared error risk with the GGD as the prior</div><div class="t m0 xd h6 y9c ff1 fs4 fc0 sc0 ls0 ws0">applied to each in an iid fashion. That is, a sum is approximated</div><div class="t m0 xd h6 y9d ff1 fs4 fc0 sc0 ls0 ws0">by an integral. We emphasize that this is an analytical approx-</div><div class="t m0 xd h6 y9e ff1 fs4 fc0 sc0 ls0 ws0">imation and our framework is broader than assuming wavelet</div><div class="t m0 x12 h6 y9f ff1 fs4 fc0 sc0 ls0 ws0">coefficients are iid draws from a GGD. The goal is to find the</div><div class="t m0 x12 h6 ya0 ff1 fs4 fc0 sc0 ls0 ws0">soft-threshold that minimizes this Bayesian risk, and we call our</div><div class="t m0 x12 h6 ya1 ff1 fs4 fc0 sc0 ls0 ws0">method BayesShrink.</div><div class="t m0 x13 h6 ya2 ff1 fs4 fc0 sc0 ls0 ws0">The proposed Bayesian risk minimization is subband-depen-</div><div class="t m0 x12 h6 ya3 ff1 fs4 fc0 sc0 ls0 ws0">dent. Given the signal being generalized Gaussian distributed</div><div class="t m0 x12 h6 ya4 ff1 fs4 fc0 sc0 ls0 ws0">and the noise being Gaussian, via numerical calculation a nearly</div><div class="t m0 x12 h6 ya5 ff1 fs4 fc0 sc0 ls0 ws0">optimal threshold for soft-thresholding is found to be</div><div class="t m0 x14 h6 ya6 ff1 fs4 fc0 sc0 ls0 ws0">(where is the noise variance and the signal vari-</div><div class="t m0 x12 h6 ya7 ff1 fs4 fc0 sc0 ls0 ws0">ance). This threshold gives a risk within 5% of the minimal risk</div><div class="t m0 x12 h6 ya8 ff1 fs4 fc0 sc0 ls0 ws0">over a broad range of parameters in the GGD family. To make</div><div class="t m0 x12 h6 ya9 ff1 fs4 fc0 sc0 ls0 ws0">this threshold data-driven, the parameters</div><div class="t m0 x15 h6 yaa ff1 fs4 fc0 sc0 ls0 ws0">and are esti-</div><div class="t m0 x12 h6 yab ff1 fs4 fc0 sc0 ls0 ws0">mated from the observed data, one set for each subband.</div><div class="t m0 x13 h6 yac ff1 fs4 fc0 sc0 ls0 ws0">To achieve simultaneous denoising and compression, the</div><div class="t m0 x12 h6 yad ff1 fs4 fc0 sc0 ls0 ws0">nonzero thresholded wavelet coefficients need to be quantized.</div><div class="t m0 x12 h6 yae ff1 fs4 fc0 sc0 ls0 ws0">Uniform quantizer and centroid reconstruction is used on the</div><div class="t m0 x12 h6 yaf ff1 fs4 fc0 sc0 ls0 ws0">GGD. The design parameters of the coder, such as the number of</div><div class="t m0 x12 h6 yb0 ff1 fs4 fc0 sc0 ls0 ws0">quantizationlevelsandbinwidths,aredecidedbasedonacriterion</div><div class="t m0 x12 h6 yb1 ff1 fs4 fc0 sc0 ls0 ws0">derived from Rissanen’s minimum description length (MDL)</div><div class="t m0 x12 h6 yb2 ff1 fs4 fc0 sc0 ls0 ws0">principle [26]. This criterion balances the tradeoff between the</div><div class="t m0 x12 h6 yb3 ff1 fs4 fc0 sc0 ls0 ws0">compression rate and distortion, and yields a nice interpretation</div><div class="t m0 x12 h6 yb4 ff1 fs4 fc0 sc0 ls0 ws0">ofoperatingatafixedslopeontherate-distortioncurve.</div><div class="t m0 x13 h6 yb5 ff1 fs4 fc0 sc0 ls0 ws0">The paper is organized as follows. In Section II, the wavelet</div><div class="t m0 x12 h6 yb6 ff1 fs4 fc0 sc0 ls0 ws0">thresholding idea is introduced. Section II-A explains the</div><div class="t m0 x12 h6 yb7 ff1 fs4 fc0 sc0 ls0 ws0">derivation of the BayesShrink threshold by minimizing a</div><div class="t m0 x12 h6 yb8 ff1 fs4 fc0 sc0 ls0 ws0">Bayesian risk with squared error. The lossy compression based</div><div class="t m0 x12 h6 yb9 ff1 fs4 fc0 sc0 ls0 ws0">on the MDL criterion is explained in Section III. Experimental</div><div class="t m0 x12 h6 yba ff1 fs4 fc0 sc0 ls0 ws0">results on several test images are shown in Section IV and</div><div class="t m0 x12 h6 ybb ff1 fs4 fc0 sc0 ls0 ws0">compared with SureShrink. To benchmark against the best</div><div class="t m0 x12 h6 ybc ff1 fs4 fc0 sc0 ls0 ws0">possible performance of a threshold estimate, the compar-</div><div class="t m0 x12 h6 ybd ff1 fs4 fc0 sc0 ls0 ws0">isons also include OracleShrink, the best soft-thresholding</div><div class="t m0 x12 h6 ybe ff1 fs4 fc0 sc0 ls0 ws0">estimate obtainable assuming the original image known, and</div><div class="t m0 x12 h6 ybf ff2 fs4 fc0 sc0 ls0 ws0">OracleThresh, the best hard-thresholding counterpart. The</div><div class="t m0 x12 h6 yc0 ff2 fs4 fc0 sc0 ls0 ws0">BayesShrink method often comes to within 5% of the MSEs</div><div class="t m0 x12 h6 yc1 ff1 fs4 fc0 sc0 ls0 ws0">of OracleShrink, and is better than SureShrink up to 8% most</div><div class="t m0 x12 h6 yc2 ff1 fs4 fc0 sc0 ls0 ws0">of the time, or is within 1% if it is worse. Furthermore, the</div><div class="t m0 x12 h6 yc3 ff2 fs4 fc0 sc0 ls0 ws0">BayesShrink threshold is very easy to compute. BayesShrink</div><div class="t m0 x12 h6 yc4 ff1 fs4 fc0 sc0 ls0 ws0">with the additional MDL-based compression, as expected,</div><div class="t m0 x12 h6 yc5 ff1 fs4 fc0 sc0 ls0 ws0">introduces quantization noise to the image. This distortion may</div><div class="t m0 x12 h6 yc6 ff1 fs4 fc0 sc0 ls0 ws0">negate the denoising achieved by thresholding, especially when</div><div class="t m0 x16 h6 yc7 ff1 fs4 fc0 sc0 ls0 ws0">is small. However, for larger values of , the MSE due to the</div><div class="t m0 x12 h6 yc8 ff1 fs4 fc0 sc0 ls0 ws0">lossy compression is still significantly lower than that of the</div><div class="t m0 x12 h6 yc9 ff1 fs4 fc0 sc0 ls0 ws0">noisy image, while fewer bits are used to code the image, thus</div><div class="t m0 x12 h6 yca ff1 fs4 fc0 sc0 ls0 ws0">achieving both denoising and compression.</div><div class="t m0 x17 h6 ycb ff1 fs4 fc0 sc0 ls0 ws0">II. W</div><div class="t m0 x18 h6 ycc ff1 fs5 fc0 sc0 ls0 ws0">AVELET THRESHOLDING AND THRESHOLD SELECTION</div><div class="t m0 x13 h6 ycd ff1 fs4 fc0 sc0 ls0 ws0">Let the signal be , where is some</div><div class="t m0 x12 h6 yce ff1 fs4 fc0 sc0 ls0 ws0">integer power of 2. It has been corrupted by additive noise and</div><div class="t m0 x12 h6 ycf ff1 fs4 fc0 sc0 ls0 ws0">one observes</div><div class="t m0 x19 h6 yd0 ff1 fs4 fc0 sc0 ls0 ws0">(1)</div><div class="t m0 x12 h6 yd1 ff1 fs4 fc0 sc0 ls0 ws0">where</div><div class="t m0 x1a h6 yd2 ff1 fs4 fc0 sc0 ls0 ws0">are independent and identically distributed ( )</div><div class="t m0 x12 h6 yd3 ff1 fs4 fc0 sc0 ls0 ws0">as normal</div><div class="t m0 x1b h6 yd4 ff1 fs4 fc0 sc0 ls0 ws0">and independent of . The goal is to</div><div class="t m0 x12 h6 yd5 ff1 fs4 fc0 sc0 ls0 ws0">remove the noise, or “denoise”</div><div class="t m0 x1c h6 yd6 ff1 fs4 fc0 sc0 ls0 ws0">, and to obtain an estimate</div><div class="t m0 x1d h6 yd7 ff1 fs4 fc0 sc0 ls0 ws0">of which minimizes the mean squared error (MSE),</div><div class="t m0 x1e h6 yd8 ff1 fs4 fc0 sc0 ls0 ws0">MSE</div><div class="t m0 x19 h6 yd9 ff1 fs4 fc0 sc0 ls0 ws0">(2)</div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624efb9c6caf596192b72c4a/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">1534 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 9, SEPTEMBER 2000</div><div class="t m0 x1f h8 yda ff1 fs5 fc0 sc0 ls0 ws0">Fig. 2. Subbands of the 2-D orthogonal wavelet transform.</div><div class="t m0 x20 h6 ydb ff1 fs4 fc0 sc0 ls0 ws0">Let , , and ; that is,</div><div class="t m0 x1 h6 ydc ff1 fs4 fc0 sc0 ls0 ws0">the boldfaced letters will denote the matrix representation of the</div><div class="t m0 x1 h6 ydd ff1 fs4 fc0 sc0 ls0 ws0">signals under consideration. Let</div><div class="t m0 xe h6 yde ff1 fs4 fc0 sc0 ls0 ws0">denote the matrix</div><div class="t m0 x1 h6 ydf ff1 fs4 fc0 sc0 ls0 ws0">of wavelet coefficients of</div><div class="t m0 x21 h6 ye0 ff1 fs4 fc0 sc0 ls0 ws0">, where is the two-dimensional</div><div class="t m0 x1 h6 ye1 ff1 fs4 fc0 sc0 ls0 ws0">dyadic orthogonal wavelet transform operator, and similarly</div><div class="t m0 x22 h6 ye2 ff1 fs4 fc0 sc0 ls0 ws0">and . The readers are referred to references such as</div><div class="t m0 x1 h6 ye3 ff1 fs4 fc0 sc0 ls0 ws0">[23], [31] for details of the two-dimensional orthogonal wavelet</div><div class="t m0 x1 h6 ye4 ff1 fs4 fc0 sc0 ls0 ws0">transform. It is convenient to label the subbands of the transform</div><div class="t m0 x1 h6 ye5 ff1 fs4 fc0 sc0 ls0 ws0">as in Fig. 2. The subbands</div><div class="t m0 x23 h6 ye6 ff1 fs4 fc0 sc0 ls0 ws0">, , ,</div><div class="t m0 x1 h6 ye7 ff1 fs4 fc0 sc0 ls0 ws0">are called the details, where is the scale, with being the</div><div class="t m0 x1 h6 ye8 ff1 fs4 fc0 sc0 ls0 ws0">largest (or coarsest) scale in the decomposition, and a subband</div><div class="t m0 x1 h6 ye9 ff1 fs4 fc0 sc0 ls0 ws0">at scale</div><div class="t m0 x24 h6 yea ff1 fs4 fc0 sc0 ls0 ws0">has size . The subband is the low</div><div class="t m0 x1 h6 yeb ff2 fs4 fc0 sc0 ls0 ws0">resolution residual, and</div><div class="t m0 x25 h6 yec ff1 fs4 fc0 sc0 ls0 ws0">is typically chosen large enough such</div><div class="t m0 x1 h6 yed ff1 fs4 fc0 sc0 ls0 ws0">that</div><div class="t m0 x26 h6 yee ff1 fs4 fc0 sc0 ls0 ws0">and . Note that since the transform</div><div class="t m0 x1 h6 yef ff1 fs4 fc0 sc0 ls0 ws0">is orthogonal,</div><div class="t m0 x27 h6 yf0 ff1 fs4 fc0 sc0 ls0 ws0">are also iid .</div><div class="t m0 x20 h6 yf1 ff1 fs4 fc0 sc0 ls0 ws0">The wavelet-thresholding denoising method filters each co-</div><div class="t m0 x1 h6 yf2 ff1 fs4 fc0 sc0 ls0 ws0">efficient</div><div class="t m0 x28 h6 yf3 ff1 fs4 fc0 sc0 ls0 ws0">from the detail subbands with a threshold function</div><div class="t m0 x1 h6 yf4 ff1 fs4 fc0 sc0 ls0 ws0">(to be explained shortly) to obtain</div><div class="t m0 x29 h6 yf5 ff1 fs4 fc0 sc0 ls0 ws0">. The denoised estimate is</div><div class="t m0 x1 h6 yf6 ff1 fs4 fc0 sc0 ls0 ws0">then</div><div class="t m0 x26 h6 yf7 ff1 fs4 fc0 sc0 ls0 ws0">, where is the inverse wavelet transform</div><div class="t m0 x1 h6 yf8 ff1 fs4 fc0 sc0 ls0 ws0">operator.</div><div class="t m0 x20 h6 yf9 ff1 fs4 fc0 sc0 ls0 ws0">There are two thresholding methods frequently used. The</div><div class="t m0 x1 h6 yfa ff2 fs4 fc0 sc0 ls0 ws0">soft-threshold function (also called the shrinkage function)</div><div class="t m0 x2a h6 yfb ff1 fs4 fc0 sc0 ls0 ws0">sgn (3)</div><div class="t m0 x1 h6 yfc ff1 fs4 fc0 sc0 ls0 ws0">takes the argument and shrinks it toward zero by the threshold</div><div class="t m0 x2b h6 yfd ff1 fs4 fc0 sc0 ls0 ws0">. The other popular alternative is the hard-threshold function</div><div class="t m0 x2c h6 yfe ff1 fs4 fc0 sc0 ls0 ws0">(4)</div><div class="t m0 x1 h6 yff ff1 fs4 fc0 sc0 ls0 ws0">which keeps the input if it is larger than the threshold</div><div class="t m0 x2d h6 y100 ff1 fs4 fc0 sc0 ls0 ws0">; oth-</div><div class="t m0 x1 h6 y101 ff1 fs4 fc0 sc0 ls0 ws0">erwise, it is set to zero. The wavelet thresholding procedure re-</div><div class="t m0 x1 h6 y102 ff1 fs4 fc0 sc0 ls0 ws0">moves noise by thresholding only the wavelet coefficients of the</div><div class="t m0 x1 h6 y103 ff1 fs4 fc0 sc0 ls0 ws0">detail subbands, while keeping the low resolution coefficients</div><div class="t m0 x1 h6 y104 ff1 fs4 fc0 sc0 ls0 ws0">unaltered.</div><div class="t m0 x20 h6 y105 ff1 fs4 fc0 sc0 ls0 ws0">The soft-thresholding rule is chosen over hard-thresholding</div><div class="t m0 x1 h6 y106 ff1 fs4 fc0 sc0 ls0 ws0">for several reasons. First, soft-thresholding has been shown to</div><div class="t m0 x1 h6 y107 ff1 fs4 fc0 sc0 ls0 ws0">achieve near-optimal minimax rate over a large range of Besov</div><div class="t m0 x1 h6 y108 ff1 fs4 fc0 sc0 ls0 ws0">spaces [12], [14]. Second, for the generalized Gaussian prior</div><div class="t m0 x1 h6 y109 ff1 fs4 fc0 sc0 ls0 ws0">assumed in this work, the optimal soft-thresholding estimator</div><div class="t m0 x1 h6 y10a ff1 fs4 fc0 sc0 ls0 ws0">yields a smaller risk than the optimal hard-thresholding esti-</div><div class="t m0 x1 h6 y10b ff1 fs4 fc0 sc0 ls0 ws0">mator (to be shown later in this section). Lastly, in practice, the</div><div class="t m0 x1 h6 y10c ff1 fs4 fc0 sc0 ls0 ws0">soft-thresholding method yields more visually pleasant images</div><div class="t m0 x1 h6 y10d ff1 fs4 fc0 sc0 ls0 ws0">over hard-thresholding because the latter is discontinuous and</div><div class="t m0 x1 h6 y10e ff1 fs4 fc0 sc0 ls0 ws0">yields abrupt artifacts in the recovered images, especially when</div><div class="t m0 xa h6 y10f ff1 fs4 fc0 sc0 ls0 ws0">the noise energy is significant. In what follows, soft-thresh-</div><div class="t m0 xa h6 y110 ff1 fs4 fc0 sc0 ls0 ws0">olding will be the primary focus.</div><div class="t m0 xb h6 y111 ff1 fs4 fc0 sc0 ls0 ws0">While the idea of thresholding is simple and effective,</div><div class="t m0 xa h6 y112 ff1 fs4 fc0 sc0 ls0 ws0">finding a good threshold is not an easy task. For one-dimen-</div><div class="t m0 xa h6 y113 ff1 fs4 fc0 sc0 ls0 ws0">sional (1-D) deterministic signal of length</div><div class="t m0 x2e h6 y114 ff1 fs4 fc0 sc0 ls0 ws0">, Donoho and</div><div class="t m0 xa h6 y115 ff1 fs4 fc0 sc0 ls0 ws0">Johnstone [14] proposed for VisuShrink the universal threshold,</div><div class="t m0 x2f h6 y116 ff1 fs4 fc0 sc0 ls0 ws0">, which results in an estimate asymptotically</div><div class="t m0 xa h6 y117 ff1 fs4 fc0 sc0 ls0 ws0">optimal in the minimax sense (minimizing the maximum</div><div class="t m0 xa h6 y118 ff1 fs4 fc0 sc0 ls0 ws0">error over all possible</div><div class="t m0 x30 h6 y119 ff1 fs4 fc0 sc0 ls0 ws0">-sample signals). One other notable</div><div class="t m0 xa h6 y11a ff1 fs4 fc0 sc0 ls0 ws0">threshold is the SURE threshold [15], derived from minimizing</div><div class="t m0 xa h6 y11b ff1 fs4 fc0 sc0 ls0 ws0">Stein’s unbiased risk estimate [30] when soft-thresholding</div><div class="t m0 xa h6 y11c ff1 fs4 fc0 sc0 ls0 ws0">is used. The SureShrink method is a hybrid of the universal</div><div class="t m0 xa h6 y11d ff1 fs4 fc0 sc0 ls0 ws0">and the SURE threshold, with the choice being dependent</div><div class="t m0 xa h6 y11e ff1 fs4 fc0 sc0 ls0 ws0">on the energy of the particular subband [15]. The SURE</div><div class="t m0 xa h6 y11f ff1 fs4 fc0 sc0 ls0 ws0">threshold is data-driven, does not depend on</div><div class="t m0 x31 h6 y120 ff1 fs4 fc0 sc0 ls0 ws0">explicitly,</div><div class="t m0 xa h6 y121 ff1 fs4 fc0 sc0 ls0 ws0">and SureShrink estimates it in a subband-adaptive manner.</div><div class="t m0 xa h6 y122 ff1 fs4 fc0 sc0 ls0 ws0">Moreover, SureShrink has yielded good image denoising</div><div class="t m0 xa h6 y123 ff1 fs4 fc0 sc0 ls0 ws0">performance and comes close to the true minimum MSE of the</div><div class="t m0 xa h6 y124 ff1 fs4 fc0 sc0 ls0 ws0">optimal soft-threshold estimator (cf. [4], [12]), and thus will be</div><div class="t m0 xa h6 y125 ff1 fs4 fc0 sc0 ls0 ws0">the main comparison to our proposed method.</div><div class="t m0 xb h6 y126 ff1 fs4 fc0 sc0 ls0 ws0">In the statistical Bayesian literature, many works have con-</div><div class="t m0 xa h6 y127 ff1 fs4 fc0 sc0 ls0 ws0">centrated on deriving the best threshold (or shrinkage factor)</div><div class="t m0 xa h6 y128 ff1 fs4 fc0 sc0 ls0 ws0">based on priors such as the Laplacian and a mixture of Gaus-</div><div class="t m0 xa h6 y129 ff1 fs4 fc0 sc0 ls0 ws0">sians (cf. [1], [8], [9], [18], [24], [27], [29], [32], [35]). With an</div><div class="t m0 xa h6 y12a ff1 fs4 fc0 sc0 ls0 ws0">integral approximation to the pixel-wise MSE distortion mea-</div><div class="t m0 xa h6 y12b ff1 fs4 fc0 sc0 ls0 ws0">sure as discussed earlier, the formulation here is also Bayesian</div><div class="t m0 xa h6 y12c ff1 fs4 fc0 sc0 ls0 ws0">for finding the best soft-thresholding rule under the general-</div><div class="t m0 xa h6 y12d ff1 fs4 fc0 sc0 ls0 ws0">ized Gaussian prior. A related work is [27] where the hard-</div><div class="t m0 xa h6 y12e ff1 fs4 fc0 sc0 ls0 ws0">thresholding rule is investigated for signals with Laplacian and</div><div class="t m0 xa h6 y12f ff1 fs4 fc0 sc0 ls0 ws0">Gaussian distributions.</div><div class="t m0 xb h6 y130 ff1 fs4 fc0 sc0 ls0 ws0">The GGD has been used in many subband or wavelet-based</div><div class="t m0 xa h6 y131 ff1 fs4 fc0 sc0 ls0 ws0">image processing applications [2], [20], [22], [23], [29], [34],</div><div class="t m0 xa h6 y132 ff1 fs4 fc0 sc0 ls0 ws0">[36]. In [29], it was observed that a GGD with the shape param-</div><div class="t m0 xa h6 y133 ff1 fs4 fc0 sc0 ls0 ws0">eter</div><div class="t m0 x32 h6 y134 ff1 fs4 fc0 sc0 ls0 ws0">ranging from 0.5 to 1 [see (1)] can adequately describe the</div><div class="t m0 xa h6 y135 ff1 fs4 fc0 sc0 ls0 ws0">wavelet coefficients of a large set of natural images. Our expe-</div><div class="t m0 xa h6 y136 ff1 fs4 fc0 sc0 ls0 ws0">rience with images supports the same conclusion. Fig. 3 shows</div><div class="t m0 xa h6 y137 ff1 fs4 fc0 sc0 ls0 ws0">the histogram of the wavelet coefficients of the images shown in</div><div class="t m0 xa h6 y138 ff1 fs4 fc0 sc0 ls0 ws0">Fig. 9, against the generalized Gaussian curve, with the param-</div><div class="t m0 xa h6 y139 ff1 fs4 fc0 sc0 ls0 ws0">eters labeled (the estimation of the parameters will be explained</div><div class="t m0 xa h6 y13a ff1 fs4 fc0 sc0 ls0 ws0">later in the text.) A heuristic can be set forward to explain why</div><div class="t m0 xa h6 y13b ff1 fs4 fc0 sc0 ls0 ws0">there are a large number of “small” coefficients but relatively</div><div class="t m0 xa h6 y13c ff1 fs4 fc0 sc0 ls0 ws0">few “large” coefficients as the GGD suggests: the small ones</div><div class="t m0 xa h6 y13d ff1 fs4 fc0 sc0 ls0 ws0">correspond to smooth regions in a natural image and the large</div><div class="t m0 xa h6 y13e ff1 fs4 fc0 sc0 ls0 ws0">ones to edges or textures.</div><div class="t m0 xa h9 y13f ff2 fs4 fc0 sc0 ls0 ws0">A. Adaptive Threshold for BayesShrink</div><div class="t m0 xb h6 y140 ff1 fs4 fc0 sc0 ls0 ws0">The GGD, following [20], is</div><div class="t m0 x33 h6 y141 ff1 fs4 fc0 sc0 ls0 ws0">(5)</div><div class="t m0 x34 h6 y142 ff1 fs4 fc0 sc0 ls0 ws0">, , , where</div><div class="t m0 xa h6 y143 ff1 fs4 fc0 sc0 ls0 ws0">and</div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div><div class="d m1"></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>