Comparison Of Various Thresholding Techniques Of Image .
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013Comparison of Various Thresholding Techniques ofImage DenoisingShivani Mupparaju 1, B Naga Venkata Satya Durga JahnaviDepartment of ECE, VNR VJIET, Hyderabad, A.P, India2Abstractreconstruction retaining back the essential signalcharacteristics and giving less noise image.2. Objectives and Tools Employed2.1. Objective of the projectThe main objective of this paper is study variousthresholding techniques such as Sure Shrink, VisuShrink and Bayes Shrink and determine the best one forimage denoising.IJERTDenoising using wavelets attempts to remove the noisepresent in the signal while signal characteristics arepreserved, regardless of its frequency content. It can behandled using three steps: a linear forward wavelettransform, nonlinear thresholding step and a linearinverse wavelet transform. Wavelet denoising is a lotdifferent from smoothing; smoothing is used to removethe high frequencies and retains the lower frequencies.Wavelet shrinkage is a non-linear process and it isused to distinguish from entire linear denoisingtechnique. Wavelet shrinkage depends on the choice ofa thresholding parameter and the choice of how thethreshold is determined, and the efficacy of denoisingvarious techniques can be used for choosing denoisingparameters and so far there is no “best” universalthreshold determination technique. So, Variousdenoising techniques such as Sure Shrink, Bayes Shrinkand Visu Shrink determines the best one for imagedenoising.1. IntroductionDuring acquisition and transmission, image denoisingcan used to remove the additive noise while keeping theimportant signal features. In the recent years waveletthresholding and threshold selection for signal denoising has gain more interest because wavelet gives anappropriate basis for separating noisy signal from theimage signal. The motivation is the wavelet transformis good at compacting energy, the small coefficient aremore likely due to noise and large coefficient due toimportant signal features . These small coefficients canbe thresholded without affecting the main features ofthe image.A simple non-linear technique called 'thresholding'which operates on one wavelet coefficient at a time.Each coefficient is thresholded by comparing againstthreshold. And if the coefficient is smaller than thethreshold, set it to zero; otherwise it is modified.Replacing the small coefficients by zero and applyinginverse wavelet transform on the result may lead toIJERTV2IS90812www.ijert.org2.2. Tools UsedSoftware: MATLAB3. Types of Noise3.1. Gaussian NoiseBecause of its mathematical tractability in both spatialand frequency domains, Gaussian (also called normal)noise models are used frequently in practice. In fact,this tractability is so convenient that it often results inGaussian models being used in situations in which theyare marginally applicable at best.The Probability density function of Gaussianrandom variable z, is given by:(1.1)where z represents intensity,is the mean(average) value of z , and σ is its standard deviation.The standard deviation squaredof z., is called variance3294
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 20133.2 Rayleigh Noise3.5. Uniform NoiseThe probability density function of Rayleigh Noise isgiven byThe Probability density function is given byP(z) (z-a) 0P(z)for z a if a z b 0otherwise(1.11)for z aThe mean of this density function is given by(1.2)The mean and variance of this density are given by(1.12)And its variance by(1.3)(1.13)And4. DenoisingIn many cases, additive noise is evenly distributed overthe frequency domain (i.e., white noise), whereas animage contains mostly low frequency information. Thenoise is a characteristic at high frequencies and itseffects can be reduced using low-pass filter. So afrequency filter or with a spatial filter can also be used.Often a spatial filter is preferably used, as it iscomputationally less expensive than a frequency filter.Denoising can be done in various domains and by usingvarious methods(1.4)3.3. Erlang (Gamma) NoiseThe probability density function of Erlang noise isgiven byP(z) for z 0 0for z 0(1.5)IJERTWhere the parameters are such that a 0, b is a positiveinteger. The mean and variance of this density aregiven by(1.6)And5. Thresholding(1.7)3.4. Exponential NoiseThe Probability density function of exponential noise isgiven byP(z) for z 0 0for z 0(1.8)Where a 0. The mean and variance of this densityfunction are(1.9)And(1.10)IJERTV2IS90812a) Spatial domainb) Frequency domainc) Wavelet domain andd) Curvelet domain5.1 Motivation for Wavelet thresholdingThe plot of wavelet coefficients suggests that smallcoefficients are decreased due to noise, whilecoefficients with a large absolute value carry moresignal information. Replacing noisy coefficients (smallcoefficients below a certain threshold value) by zeroand an inverse wavelet transform may lead to areconstruction that has lesser noise. Stated moreprecisely, we are motivated to this thresholding ideabased on the following assumptions:i. The deco-relating property of a wavelet transformcreates a sparse signal: most coefficients, which arezero or close to zero when they are left untouched.ii. Noise is spread out equally along all coefficients.iii. The noise level is not too high so that we candistinguish the signal wavelet coefficients from thenoisy ones.As it turns out, this method is indeed effective andthresholding is a simple and efficient method for noisereduction. Further, inserting zeros creates more sparsityin the wavelet domain and here we see a link betweenwavelet de-noising and compression.www.ijert.org3295
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 20135.2 Hard and soft thresholding:Hard and soft thresholding with threshold are definedas follows The hard thresholding operator is defined asD(U,λ) Ufor all U λ 0otherwise(2.1)The soft thresholding operator can be defined asD(U,λ) sgn(U)max(0,U -λ)(2.2)5.3 Threshold determinationThreshold determination is an important in imagedenoising. A small threshold gives a result close to theinput, but the result can still have the noise component.Whereas a large threshold, produces a signal with alarge number of zero coefficients. This results insmooth noiseless signal.The effect of smoothness, however, destroys detailsand in image processing may cause blur and artifacts.To investigate the effect of threshold selection, weperformed wavelet denoising using hard and softthresholds on four signals popular in wavelet literature:Blocks and Doppler.The setup is as follows:1. The original signals have length 2048.2. We step through the thresholds from 0 to 5 withsteps of 0.2 and at each step denoise the four noisysignals by both hard and soft thresholding with thatthreshold.3. For each threshold, the MSE of the denoised signal iscalculated.4. Repeat the above steps for different orthogonalbases, namely, Haar, Daubechies 2,4 and 8.Fig:Table 1IJERTV2IS908125.4 Comparison with Universal threshold:The threshold λUNIV 2ln N (N being the signallength, σ2being the noise variance) is well known inwavelet literature as the Universal threshold. It is theoptimum threshold and minimizes the cost function ofthe difference between the function and the softthreshold version of the same in the L2.In our case,N 2048, σ 1, therefore theoretically,2ln(2048)(1) 3.905As seen from the table, the best empirical thresholds forboth hard and soft thresholding are much lower thanthis value, independent of the wavelet used. It thereforeseems that the universal threshold is not useful todetermine a threshold. However, it is useful for obtain astarting value when nothing is known of the signalcondition. One can surmise that the universal thresholdmay give a better estimate for the soft threshold if thenumbers of samples are larger.IJERTHard threshold is a “keep or kill” procedure and ismore intuitively appealing. The transfer function of thesame is shown. The alternative, soft thresholding(whose transfer function is shown), shrinks coefficientsabove the threshold in absolute value. While hardthresholding may seem goodl, the continuity of softthresholding has some advantages. Moreover, hardthresholding does not even work with some algorithmssuch as the GCV procedure. At times, pure noisecoefficients may pass the hard threshold and appear asannoying ’blips’ in the output. Soft thresholdingshrinks these false structures.The results are tabulated in the Table 1 and represents Blocksand Doppler of both hard and soft for different filters.www.ijert.org5.5 Image Denoising using Thresholding:An image is often corrupted by noise in its acquisitionor transmission. The underlying concept of denoising inimages is similar to the 1D case. The goal is to removethe noise while retaining the important signal featuresas much as possible.The noisy image is represented as a two-dimensionalmatrix {xij},i ,j 1, .N. The noisy version of the imageis modeled asyij xij nijij 1, .N.(2.3)Where {nij} are iid as N (0,σ 2). We can use the sameprinciples of thresholding and shrinkage to achievedenoising as in 1-D signals. The problem again boilsdown to finding an optimal threshold such that themean squared error between the signal and its estimateis minimized.The wavelet decomposition of an image is done asfollows: In the first level of decomposition, the imageis split into 4 sub bands, namely the HH, HL, LH andLL sub bands. The HH sub band gives the diagonaldetails of the image the HL sub band gives thehorizontal features while the LH sub band representsthe vertical structures. The LL sub band is the lowresolution residual consisting of low frequencycomponents and it is this sub band, which is furthersplit at higher levels of decomposition. The different3296
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013methods for denoising we investigate differ only in theselection of the threshold. The basic procedure remainsthe same:i.ii.iii.Calculate the DWT of the image.Threshold the wavelet coefficients.Compute the IDWT to get the denoisedestimate.Moreover, it is also found to yield visually morepleasing images. Hard thresholding introduces artifactsin the recovered images.The three thresholding techniques- Visu Shrink, SureShrink and Bayes Shrink and investigate theirperformance for denoising various standard images5.5.1 VisuShrink: Visu Shrink was introduced byDonoho . It can be defined as σ 2log I where σ is thenoise variance and I is the number of pixels in theimage.The maximum of any I values can be given byN(0,σ2) with the probability approaching 1 as numberof pixels in the given image increases. Therefore if ithas high probability, a pure noise signal is calculated asbeing identically zero.Sure Shrink uses a Hybrid scheme. The idea behind thishybrid scheme is that the losses while using anuniversal threshold,tend to be largerthan SURE for dense situations, but much smaller forsparse cases .So the threshold is set to tFd in densesituations and to tS in sparse situations. Thus the(4.12)Whereand(2.5)IJERTHowever, for denoising images, Visu shrink is foundto yield an overly smoothed estimate as seen. This isbecause the universal threshold (UT) is derived underthe constraint that with high probability the estimateshould be at least as smooth as the signal. The universalthreshold is high for large values of I, killing manysignal coefficients along with the noise. Thus, thethreshold doesnt perform well at discontinuities in thesignal.5.5.3 Threshold Selection in Sparse Cases :Thedrawback of SURE in situations of extreme sparsity ofthe wavelet coefficients. In such cases the noisecontributed to the SURE profile by the manycoordinates at which the signal is zero swamps theinformation contributed to the SURE profile by the fewcoordinates where the signal is nonzero. Consequently,5.5.2 Sure Shrink: Let μ (μi : i 1, . d) be alength-d vector, and let x {xi} (with xi distributed asN(¹i,1)) be multivariate normal observations with meanvector μ. Let be a fixed estimate based on theobservations x. SURE can be defined as Stein’sunbiased Risk Estimator). It is a special method forestimating the loss in an unbiased fashion where isthe soft threshold estimator. Stein’s result to get anunbiased estimate of the risk is applied. For anobserved vector x is the set of noisy waveletcoefficients in a sub band we find out the threshold tsthatminimizesSURE(t,x),i.e.(2.4)The above optimization problem is computationallystraightforward. Without loss of generality, we canreorder x in order of increasing. Then on intervalsof t that lie between two values of, SURE (t) isstrictly increasing. Therefore the minimum value of ts isone of the data valuesthere are only d values andthe threshold can be obtained.IJERTV2IS90812www.ijert.orgη being the thresholding operator.estimator in the hybrid method works as follows5.5.4 SURE applied to image denoising: The waveletdecomposition of the noisy image is obtained. TheSURE threshold is determined for each sub band usingthe above equations. We choose between this thresholdand the universal threshold using the equation .Theexpressionsandin the equations, given for σ 1have to suitably modified according to the noisevariance σ and the variance of the coefficients in thesub band. The results obtained for the image ’lina’(512*512pixels) using Sure Shrink are shown inresults. The Db4' wavelet was used with 4 levels ofdecomposition. Clearly, the results are much better thanVisu Shrink. The sharp features of the image areretained and the MSE is considerably lower. This isbecause Sure Shrink is sub band adaptive- a separatethreshold is computed for each detail sub band.5.5.5 Bayes Shrink: In Bayes Shrink we determine thethreshold for each subband assuming a GeneralizedGaussian Distribution (GGD) . The GGD is given byGG x, ( x) C ( x , )exp [ ( x , ) x ] x , 0(2.6)3297
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013Where5.5.6 Parameter Estimation to determine theThreshold:The GGD parameters, σx and β, need to be estimated tocompute TB (σx) . The noise variance σ2 is estimatedfrom the subband HH1 by the robust median estimator ,12 (3 (1 ) x , X 1 andC x , (2.7) . x , 2 1 (2.8) ˆ 0.6745Yij subbandHH1 and (t ) e u u t 1du(2.9)0The parameter σx is the standard deviation and β is theshape parameter It has been observed that with a shapeparameter β ranging from 0.5 to 1, we can describe thedistribution of coefficients in a sub band for a large setof natural images. Assuming such a distribution for thewavelet coefficients, we empirically estimate β and σxfor each sub band and try to find the threshold T whichminimizes the Bayesian Risk, i.e., the expected value ofthe mean square error. Y2 X2 22Where Y2as zero-mean,Where X̂ T(Y ); Y/X N(x,σ2) and X G Gx,β.The optimal threshold T* is then given byT*(σx,β ) arg min Γ(T)It is a function of the parameters σx and β. Since there isno closed form solution for T*, numerical calculation isused to find its value.It is observed that the threshold value set byTB ( x ) 2 x(2.10)is very close to T*. The estimated threshold TB σ2 σxis not only nearly optimal but also has an intuitiveappeal. The normalized threshold, TB/σ. is inverselyproportional to σx, the standard deviation of X, andproportional to σx, the noise standard deviation. Whenσ/σx 1, the signal is much stronger than the noise, Tb/σis chosen to be small in order to preserve most of thesignal and remove some of the noise; when σ σx 1,the noise dominates and the normalized threshold ischosen to be large to remove the noise which hasexacted the signal. Thus, this threshold choice adapts toboth the signal and the noise characteristics as reflectedin the parameters σ and σx.(2.11)The parameter β does not explicitly enter into theexpression of TB (σx). Therefore it suffices to estimatedirectly the signal standard deviation σx. Theobservation gives Y X V, with X and V asindependent of each other, henceIJERT T E ( Xˆ X )2 Ex E y x Xˆ Xmedian Yij (2.12)is the variance of Y. Since Y is modeled Y2 can be found empirically by1 n ˆY2 Yi ,2jn i , j 1(2.13)(4.20)Where nXn is the size of the subband underconsideration. ˆTˆB ( ˆ x ) ˆ X2Thus(2.14)Where ˆ x max ˆY2 ˆ 2 ,0 (2.15)22In the case that ˆ ˆ Y , ˆ x is taken to be zero, i.e,TˆB ( ˆ x ) is , or, in practice, TˆB ( ˆ x ) max Yij and all coefficients are set to zero.To summarize, Bayes Shrink performsthresholding, sub band dependent threshold, ˆTˆB ( ˆ x ) ˆ Xsoft2(2.16)The reconstruction using Bayes Shrink is smoother andmore visually appealing than the one obtained usingIJERTV2IS90812www.ijert.org3298
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013Figure 2: Noisy ImageSure Shrink. This not only validates the approximationof the wavelet Coefficients to the GGD but alsojustifies the approximation to the threshold to a valueindependent of β.5.5.7 Image Reconstruction:Image reconstruction can be carried out by thefollowing procedure:(i) Lets sample the data by a factor of two on all thefour sub bands at the coarsest scale(ii) Filter the sub bands in each dimension.(iii) Sum the four filtered sub bands to reach the lowlow sub band at the next finer scale.This above process can be repeated until the image isfully reconstructed.òr20F(r,f )drdf [s r2 /(2m0 )](2.17) ò exp(-l z j -zi )l -1 J1 (lr2 )J 0 (lri )dl. 0IJERT6. ResultsFigure 3: Visu ShrinkFigure 1: Original ImageIJERTV2IS90812www.ijert.org3299
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013Figure 5: Bayes ShrinkFigure 6: SNR of Bayes shrinkIJERTFigure 4: SNR of Visu shrinkFigure 7: Sure ShrinkFigure 8: SNR of SURE shrink7. ConclusionWe have seen that wavelet thresholding is an effectivemethod of denoising noisy signals.Then weIJERTV2IS90812www.ijert.org3300
International Journal of Engineering Research & Technology (IJERT)ISSN: 2278-0181Vol. 2 Issue 9, September - 2013investigated many soft and hard thresholding schemesusing Visu Shrink, Sure Shrink and Bayes Shrink fordenoising images. It was found that sub band adaptivethresholding performs better than a universalthresholding. Among these, Bayes Shrink gave the bestresults. This validates the assumption that theGeneralized Gaussian Distribution (GGD) is a verygood model for the wavelet coefficient distribution in asub band.International Journal of Advanced Computer Scienceand Applications,Vol. 2, No.3, March 2011An important point to be noted is that although SureShrink performed worse than Bayes Shrink, it adaptswell to sharp discontinuities in the signal. This was notevident in the natural images that were used for testingpurpose specially while comparing the performance ofthese algorithms on artificial images withdiscontinuities (such as medical images).8. 2IJERTImage denoising using thresholding by waveletsprovides an extension for research into fast and ainting and image feature extraction. Wavelettransform can also be used in the analysis and synthesisof multi-scale models of stochastic processes. Theconcept of wavelet transform finds an importantapplication in speech coding, communications, radar,sonar, denoising, edge detection and feature detectionDigital Image Processing, 3rd edition, by RafaelC.Gonzalez, Richard E.Woods ,Pearson PublicationsDigital Image Processing using MATLAB, 2ndedition, by Rafael C.Gonzalez,Martin Vetterli , S Grace Chang, Bin Yu. Adaptivewavelet thresholding for image denoising andcompression.IEEE Transactions on ImageProcessing, 9(9):1532–1546, Sep 2000.David L Donoho. De-noising by soft thresholding.IEEE Transactions on Information Theory, 41(3):613–627, May 1995.Iain M.Johnstone David L Donoho. Adapting tosmoothness via wavelet shrinkage. Journal of theStatistical Association, 90(432):1200–1224, Dec1995.David L Donoho. Ideal spatial adaptation by waveletshrinkage . Biometrika, 81(3):425–455,August 1994.Maarten Jansen. Noise Reduction by WaveletThresholding ,volume 161. Springer Verlag, UnitedStates of America, 1 edition, 2001.Carl Taswell. The what, how and why of waveletshrinkage denoising. Computing in Science andEngineering , pages 12–19, May/June 2000.Sachin D Ruikar and Dharmpal D Doye . Waveletbased image denoising technique . (IJACSA)www.ijert.org3301
characteristics and giving less noise image. 2. Objectives and Tools Employed. 2.1. Objective of the project . The main objective of this paper is study various thresholding techniques such as Sure Shrink, Visu Shrink and Bayes Shrink and determine the best one for image denoising. 2.2. Tools Used . Software: MATLAB . 3. Types of Noise . 3.1.Cited by: 1Publish Year: 2013Author: Shivani Mupparaju, B Naga Venkata Satya Durga Jahnavi
Cloud Detection Techniques Clouds are bright, cold, often non-uniform Spectral – Brightness values thresholding – Spectral ratios and differences thresholding Texture (spatial coherence) Continuity (temporal coherence) Compositing (assuming that at least one day is clear during the compositing period, e.g. a week or a month)
Keywords: Medical Image Denoising, Multiscale Transforms, Shrinkage Thresholding. 1. Introduction Medical imaging has become new research focus area and is playing a significant role in diagnosing diseases. There are many imaging modalities for different applications. All these modalities will introduce some amount of noise like Gaussian,
implemented on medical images. The quality of segmented image is measured by statistical parameters: Jaccard Similarity Coefficient, Peak Signal to Noise Ratio (PSNR). KEYWORDS Thresholding, Niblack, Sauvola, PSNR, Jaccard 1. INTRODUCTION Image segmentation is a fundamental process in many image, video, and computer vision applications.
AND ITS APPLICATION TO WAVELET ANALYSIS Harrison H. Zhou* and J. T. Gene Hwang** Cornell University May 1, 2003 Abstract. Many statistical practices involve selecting a model (a reduced model from the full model) and then use it to do estimation with possible thresholding. Is it possible to do so and still come up with an estimator always .
Regularization in Tomographic Reconstruction Using Thresholding Estimators Jérôme Kalifa*, Andrew Laine, and Peter D. Esser Abstract— In tomographicmedical devicessuch as single photon emission computed tomography or positron emission tomography cameras, image reconstruction is an unstable inverse problem, due to the presence of additive noise.
accuracy in the case of more realistic inhomogeneous and irregular clinical lesions, using clinical or simulated data [1, 2], in particular when using fixed thresholding methods, which are highly dependent on the image type [3]. The use of advanced PET-AS beyond thresholding was recommended to reduce dosimetry errors, especially in the case of het-
Implementation of Multi level thresholding based Ant Colony Optimization Algorithm for Edge Detection of Images By Spoorthy Kanajal Chandrakanth The Supervisory Committee certifies that this disquisition complies with North Dakota State University's regulations and meets the accepted standards for the degree of MASTER OF SCIENCE
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