An Efficient Parameter Selection Criterion For Image Denoising
2005 IEEE InternationalSymposium on Signal Processingand Information TechnologyAn Efficient Parameter Selection Criterion for Image DenoisingHamed Pirsiavash1, Shohreh Kasaei2, Farrokh Marvasti1h pirsiavash@ee.sharif.edu, skasaei@sharif.edu, marvasti@sharif.edu1Multimedia Research Lab., Sharif University of Technology, Tehran, IranDepartment of Computer Engineering, Sharif University of Technology, Tehran, Iran2et al. for multiple wavelet threshold selection [2, 3]. Theydefined a criterion which its minimum roughly minimizesthe mean square error (MSE), but their method works insome special conditions and as proved in [2], it works onlyfor wavelet shrinkage with orthogonal transforms. Inaddition, as they mentioned in their paper, its output has lowMSE, but it is not guaranteed to yield a good visual quality.In this paper, assuming the additive noise to have anarbitrary distribution, a novel criterion for image denoisingis introduced. The minimization of this criterion leads tonear optimum parameter set for denoising purposes. In orderto evaluate the performance of this criterion, it is applied foroptimum parameter selection in a popular image denoisingalgorithm, the wavelet thresholding.The layout of this paper is as follows: Section 2introduces the proposed criterion and its efficiency inparameter selection. In section 3, wavelet shrinkagealgorithm is described briefly. The experimental results andthe performance comparison are presented in Section 4.Finally, Section 5 concludes the paper.Abstract - The performance of most image denoising systemsdepends on some parameters which should be set carefully basedon noise distribution and its variance. As in some applicationsnoise characteristics are unknown, in this research, a criterionwhich its minimization leads to the best parameter set up isintroduced. The proposed criterion is evaluated for the waveletshrinkage image denoising algorithm using the cross validationprocedure. The criterion is tested for some different values ofthresholds, and the output leading to the minimum criterion valueis selected as the final denoised output. The resulting outputs of ourmethod and the previous threshold selection scheme for the waveletshrinkage, i.e. the median absolute difference (MAD), arecompared. The objective and subjective test results show theimproved efficiency of the proposed denoising algorithm.Keywords – image denoising, wavelet shrinkage, noise estimation,parameter selection.1. INTRODUCTIONNowadays, by improving image acquisition systems,many types of cameras are available. Some of these camerasuse very simple hardware in order to have low cost and to beembedded in other devices like mobile phones. Hence, theoutput images of these devices are noisy and poor. Inaddition, in most image processing systems, the taken imageshould be fed to some processing stages like compressionand recognition. The parasitic noise in the input image couldsuffer the other processes and make them inefficient.To overcome these shortcomings, many image denoisingalgorithms have been developed during recent years. Forinstance, Gaussian smoothing, neighborhood filtering, andwavelet shrinkage can be mentioned [1].In general, all denoising methods have some parametersand thresholds which should be adjusted to gain the bestperformance. Generally, these parameters depend on thenoise distribution and its variance. Most algorithms supposethe noise to have a white Gaussian distribution with a knownvariance. However, in practical situations, we have noinformation about the noise variance. Hence, anotherproblem rises which is the parameter and threshold selectionalgorithm. During recent years, some researchers consideredthis problem and made some solutions [2, 3, and 4]. Thegeneralized cross validation method is proposed by Jansen0-7803-9314-7/05/ 20.00 2005 IEEE2. PROPOSED CRITERION FOR PARAMETERSELECTIONIn image denoising algorithms with additive noise, theinput image is assumed to be the summation of originalimage and an additive random noise. An importantknowledge which is used in the proposed criterion is theindependency of these two signals (the original image andthe additive noise). Here, the aim of denoising algorithms isto remove the parasitic noise. In fact, the difference betweenthe input and the output of the denoising stage is theestimated noise which has been removed (see Figure 1).Therefore, the distribution of the estimated noise shouldapproach that of the additive noise.The estimated noise for image denoising with twodistinct parameter sets is shown in Figure 2. In this figure,the estimated noise is exaggerated to be shown clearly. Itcould be seen that there is a large similarity between theestimated noise and the original image, but this similarity inFigure 2(f) is less than that in Figure 2(e). It means that foran optimum image denoising algorithm, the correlationbetween the estimated noise and the output image which is872
an expectation of the original image should be minimized.This result is in agreement with the assumption ofindependency between the original image and the additivenoise. Now, the correlation for each parameter set can becomputed. Consequently, by minimizing that, the bestparameter set for the denoising algorithm can be found.Input ImageImageDenoisingDifferentiatorOutput ImageEstimatedNoiseFig. 1. Noise estimationFig. 2. Form left to right and top to bottom: (a) Original image(Lena), (b) noisy image (AWGN, sigma 15), (c) blurred denoised image, (d) denoisedimage using proper parameters, (e) estimated noise of (c), (f) estimated noise of (d). (The estimated noise is exaggerated to be seen clearly).According to our knowledge, most denoisingalgorithms assume the original image to have more energyin lower frequency components compared to the noise. Inaddition, the additive noise usually has near flat spectrum(i.e. white noise). Hence, a huge energy of noise can beremoved by removing the higher frequency components.But, we know that using these methods, the edge pointswhich have higher frequency components will be blurred.As we know, because the human visualize system is moresensitive to the edges, the blurring effect will be perceivedobviously. In this research, in order to adapt to the humanvisualize system, the criterion is altered. As can be seen inFigure 2, the edge points can be found in the estimatednoise. Therefore, the correlation between the estimatednoise and the edge map of the output image is used.Because the output image for some parameter sets hashigh amount of noise, the edge map should be extractedusing a robust edge detection method; thus, here, Cannyedge detector is used [5]. Then, the following value shouldbe minimized.C Correlation( Estimated noise , output image edge)(1)With weak denoising parameters, the estimated noiseapproaches to a zero field and makes the correlation tohave a low value. Consequently, the minimum correlation873
will be found for the weak denoising parameters. In orderto suppress this defect, the estimated noise energy is usedin the proposed criterion. The final criterion can be writtenas follows:Cl Correlation ( Estimated noise , output image edge )log( Input image energy estimated noise energy )(2)The minimum of this criterion is found for theoptimum parameters set. Some optimization methods likethe genetic algorithms can be used to achieve theminimum, yet in this paper, showing the performance ofthe proposed criterion is the final goal; consequently,finding the minimum is performed using a simple method,i.e. cross validation. This criterion is computed for someparameter sets which are predefined multiplications of theparameter set of the MAD method, described in the nextsection, and the minimum value is chosen.3. WAVELET SHRINKAGEAs the defined criterion should be evaluated andcompared with the other parameter selection methods, inthis research, a usual image denoising algorithm, i.e. thewavelet shrinkage is implemented. In this section, a briefdescription of this method is presented. Theimplementation results and details are discussed in Section4.Wavelet shrinkage is an efficient signal denoisingalgorithm introduced by Donoho et al. in [1, 6, and 7].That method is based on the idea that the original imagehas large wavelet coefficients and the noise is distributedover all coefficients. Thus, by thresholding the smallcoefficients, the image will not be damaged although alarge amount of noise energy will be removed. The hardthresholding is applied using: 0x T(3)HWT(x) x T xwhere T is a predetermined threshold value. This basicidea causes some oscillations near the edges. As a result,they proposed soft thresholding method in which smallwavelet coefficients are cancelled and the others arechanged in order not to destroy the continuity in waveletcoefficients.x T 0(4)SWT(x) x T Sign( x) ( x T )where Sign(x) denotes the signum function. Using thismethod, oscillations are suppressed [7].In wavelet thresholding methods, the selection ofthresholds for each resolution level is very importantbecause according to the other denoising algorithms,wrong selection can make the output image blurred ornoisy. Some threshold selection methods are introducedfor Gaussian noise distributions with known variances.Three commonly used methods are the universal, SURE,and MiniMax. The mathematical details can be found in[6, 7]. For instance, universal method is as follows: T σ n 2 log NN(5)where N is the number of data points and σ n isthenoise variance defined below. In most denoisingalgorithms, the median of absolute difference (MAD) isused for noise variance estimation [8]. Median( x Median( x) )MAD σn (6)0.67450.6745This estimation yields to good results for Gaussiandistributed noise. A typical wavelet shrinkage algorithm isshown in Figure 3.Input nsformSoftThresholdingThresholdSelectionOutput ImageFig. 3. Block diagram of a simple wavelet shrinkage denoising system.4. EXPERIMENTAL RESULTSIn this section, the implementation results and theperformance of the proposed algorithm when compared toother available approaches are presented. An efficientcriterion is computed for several parameter sets in adenoising algorithm and the parameters leading to theminimum of the criterion are chosen as the bestparameters for the input image and the related noisestatistics. The proposed algorithm is implemented usingMatlab package for wavelet shrinkage image denoisingprocess.As briefly discussed in Section 3, wavelet shrinkage isa powerful image denoising algorithm, and thus manyresearchers have proposed different modified versions ofthat algorithm. In this research, wavelet shrinkage isimplemented in two resolution levels. Here, Daubechieswavelet with 6 tabs is used. The initial threshold for eachsubspace is chosen independently based on the MADvariance estimation and MiniMax threshold selectionmethods. Next, the criterion is computed for 11 different874
multiplications of this initial threshold set. Themultiplications are chosen uniformly in the logarithmic 0.81.2scale in the range of 10to 10 . Finally, the minimumis taken as the best solution. The MSE of this method andthe MAD method for some standard images are listed inTable 1. The last column contains the best MSE among 11tested threshold sets. As can be seen, the obtaineddenoising system (with selected variance) approaches theminimum MSE. The resulting PSNR and the calculatedcriterion for a sample image are plotted in Figure 4. Asseen in this figure, choosing the minimum value of theproposed criterion matches the maximum PSNR that leadsto the best parameter selection for denoising purposes. Asample output result is shown in Figure 5.As seen in Table 1, for most tested images the MSEobtained from the proposed algorithm is less than that ofthe MAD method. Moreover, for some cases the obtainedMSE by our algorithm is close to the minimum availableMSE.As the MSE is not the best measurement forperformance analysis in image processing systems, theoutputs should be examined in a subjective test as well.The results of the subjective test among 20 boys and girlsare presented in Table 2. Some particular cases in whichthe MSEs of our method are high are examined in thistest. For instance, in the 21st row of Table 1, the resultingMSE is higher than that of the MAD method. The outputsof the 21st case are shown in Figure 6. It is obvious thatthe output of our method is subjectively better than theMAD output, which proves that both objective andsubjective tests should be run. In fact, because thecriterion uses the edge map, our algorithm leads to lessdefects in the edge areas and thus results in highersubjective performance; although it may have a higherMSE. Another result obtained from these implementationsis that for lower input noise variances, our methodperforms much better than the MAD method. Because forimages with a low level of noise, after denoising the edgemap can be extracted more efficiently, and thus theproposed algorithm can better calculate the minimum thatmatches the maximum of the PSNR. As another result, forimages with small size (about 256x256), our methodperforms better than the MAD and MiniMax methods,because our method is less directly dependent to thestatistics of the images. This fact motivated us to examinethis method in spatially adaptive wavelet shrinkagealgorithms introduced in [9].5. CONCLUSIONIn this paper, an efficient criterion for performanceanalysis of denoising systems is introduced. It is shownthat using a cross validation procedure, we can adjust thesystem parameters to achieve a better performance. Thiscriterion is examined for wavelet shrinkage as a commondenoising algorithm. According to the results, the obtainedsubjective tests show the superiority of the proposedalgorithm when compared to the MAD approach. Anotherimportant advantage of this method is its independency onthe noise distribution and its variance. As mentionedabove, the algorithm performs even better for images withlower noise variances.Table 1. MSE of our method in comparison with MAD and minimumavailable on510152025510152051015202551551015152051013.04MSE 518.4554.47113.3881.59192.1815.1849.9784.65MSE 4548.9082.4381.59114.0715.1835.5674.19Table 2. Subjective test results among 20 boys and girls (score 5 isassigned to the original 93.192.81ACKNOWLEDGEMENTThe authors would like to acknowledge the support ofIran Telecommunication Research Center (ITRC) for thisproject.875
REFERENCES[1][2][3][4]A. Buades, B. Coll, and J. M. Morel, “On image denoisingmethods,” CMLA Preprint (http://www.cmla.ens-cachan.fr/Cmla/),to appear in SIAM Multiscale Modeling and Simulation, 2004.M. Jansen, M. Malfait and A. bultheel, "Generalized crossvalidation for wavelet thresholding," Signal Processing, Vol. 56,pp. 33-34, Jan 1997.M. Jansen and A. Bultheel, "Multiple wavelet threshold estimationby generalized cross validation for images with correlated noise,"IEEE Tarns. on Image Processing, Vol. 8, No. 7, pp. 947-953, Jul1999.Tai-Chiu Hsung and D. Pak-Kong Lun, "Generalized crossvalidation for multiwavelet shrinkage," IEEE Signal ProcessingLetters, Vol.11, No. 6, pp. 549-552, Jun 2004.[5][6][7][8][9]Canny and John, "A Computational Approach to Edge Detection,"IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.PAMI-8, No. 6, pp. 679-698, 1986.D. Donoho and I. Johnstone, "Ideal spatial adaptation via waveletshrinkage," Biomedika, vol. 81, pp. 425–455, 1994.D. Donoho, "Denoising by soft-thresholding," IEEE Trans. onInformation Theory, Vol. 41, pp.613-627, 1995.P. J. Rousseeuw and A. M. Leroy, Robust Regression and OutlierDetection, New York: Wiley, 1987.S. Grace Chang, Bin yu and Martin Vetterli, "Spatially adaptivewavelet thresholding with context modeling for image denoising,"IEEE Trans. on Image Processing, vol. 9, No. 9, pp. 1522-1531,Sept 2000.(a)(b)Fig. 4. Results for Lena256 (with Gaussian noise, standard deviation 15) (a) PSNR vs. threshold ratio, (b) proposed criterion vs. threshold ratio.(Threshold is the multiplication of the threshold ratio and the MAD threshold, i.e. setting threshold ratio equal to one leads to the MAD method).Fig. 5. From left to right and top to bottom: (a) Original image (Barbara256), (b) noisy image (Gaussian, standard deviation 15), (c) MAD output(MSE 295), (d) output of our method (MSE 131).876
Fig. 6. From left to right and top to bottom: (a) Original image (fishing boat), (b) noisy image (AWGN, standard deviation 20), (c) MAD output(MSE 157), (d) output of our method (MSE 192).877
Matlab package for wavelet shrinkage image denoising process. As briefly discussed in Section 3, wavelet shrinkage is a powerful image denoising algorithm, and thus many researchers have proposed different modified versions of that algorithm. In this research, wavelet shrinkage is
Criterion 1: ICP in place (interim) 9 Criterion 2: Full MDT ICP in place within 7 days 11 Criterion 3: Service user involvement 12 Criterion 4: MDT input developed and reviewed 13 Criterion 5: Appropriate goals for the resident 14 Criterion 6: Care and treatment to meet goals 15 Criterion 7: Resources to provide care and treatment 16 .
criterion can be applied. Mikhailov Criterion for Delayed Systems Recently, Chen and Tsay (ref. 2) derived a modified Nyquist stability criterion which, in the context of this paper, is similar to the first formulation of the Mikhailov criterion. The criterion was shown to hold for characteristic functions L(s) for which:
CRITERION 2 CRITERION 3 CRITERION 4 CRITERION 5. Figure adapted from "The RTI Approach to Evaluating Learning Disabilities," by J. Kovaleski, A. VanDerHeyden and E. Shapiro, 2013. Time series graphs do not replace . sound professional judgment. USE this resource. MTSS Data Decision Rules Guidance.
Fig. 1 Tresca's criterion Von Mises Criterion The von Mises Criterion (1913), also known as the maximum distortion energy criterion, octahedral shear stress theory, or Maxwell-Huber-Hencky-von Mises theory, is often used to estimate the yield of ductile materials. The von Mises criterion states that failure occurs when the
The Akaike Criterion Akaike's minimum information theoretic criterion (1) is commonly used to deter- mine the appropriate number of terms in an LAR model. According to this criterion, the number of terms to be included in the model is the value that minimizes AIC(J) Nlog V 2J, (13)
selection criterion. Your teacher will provide you with the guidelines from the mini lesson to read independently, additionally, refer to the list of attributes below to prompt your thinking. Selection Criterion Attribute How this helps to achieve the selection criterion SC1 Leadership Example: 1. Positive attitude towards others 2. 3. Example: 1.
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