International Journal of Electrical, Electronics ISSN No. (Online): 2277-2626and Computer Engineering 3(1): 48-54(2014)

Study of Curvelet and Wavelet Image Denoising by Using DifferentThreshold Estimators

Monika Shukla*, Dr. Soni Changlani** and Mr. Ayoush Johari***

*Research Scholar, Department of Electronics and Communication Engineering,LNCTS, Bhopal, (MP), India

**Professor, Department of Electronics and Communication Engineering,LNCTS, Bhopal, (MP), India

***Assistant Professor, Department of Electronics and Communication Engineering,LNCTS, Bhopal, (MP), India

(Corresponding author Monika Shukla)(Received 05 January, 2014 Accepted 22 February, 2014)

ABSTRACT: This paper describes the image denoising of Curvelet and Wavelet Image Denoising by using 4different additive noises like Gaussian noise, Speckle noise, Poisson noise and Salt & Pepper noise and also byusing 4 different threshold estimators like heursure, rigrsure, mini-maxi and squawolog for wavelet andcurvelet transform both. Our implementation offer exact reconstruction, stability against perturbation, easeof implementation and low computational complexity. From these different noises and threshold estimators,we can measure mean square errors and peak signal to noise ratio for wavelet transform and curvelettransform. In our work, we can also compare the results between wavelet and curvelet transform which one isbetter for image denoising. In our experiments, we reported here, simple threshold of the curvelet coefficientsis very competitive with “State of art” technique based on wavelet,includes threshold estimators. With thehelp of different threshold estimators, we can filtered our noisy images. Moreover, the curvelet reconstructionoffering visual sharp image and in particular, higher quality recovery of edges and of faint linear andcurvilinear features .The empirical results reported here are in encouraging agreement.

Keywords: Wavelet transform, curvelet transform, face recognition, sparse representation, feature extraction,thresholding rules, Threshold estimators and additive noises.

I. INTRODUCTION

Image denoising refers to the recovery of a digitalimage that has been contaminated by Additive whiteGaussian Noise (AWGN). AWGN is a channel modelin which the only impairment to communication is alinear addition of wideband or white noise with aconstant spectral density(expressed as watts/ Hz ofbandwidth) and a Gaussian distribution of amplitude.Although other types of noise (e.g., impulse or poissonnoise) have also been studied in the literature of imageprocessing, the term “Image Processing” is usuallydevoted to the problem associated with AWGN.Mathematically, if we use Y = X + W to denote thedegradation process(X: Clean image, Y: Noisy image,W~N., The image denoising algorithm attempts toobtain the best estimate of X from Y. The optimazationcriterion can be mean squared error(MSE) based orperceptual quality driven (Though image qualityassessment itself is a difficult problem, especially in theabsence of an original reference.

On a daily basis, hospitals are witnessing a large inflowof digital medical images and related clinical data. Themain hindrance is that an image gets often corrupted bynoise in its acquisition and transmission [1]. Imagedenoising is one of the classical problems in digitalimage processing, and has been studied for nearly half acentury due to its important role as a pre – processingstep in various electronic imaging applications. Its mainaim is to recover the best estimate of the original imagefrom its noisy versions [2]. Wavelet transform enableus to represent signals with a high degree of scarcity.This is the principle behind a non-linear wavelet basedsignal estimation technique known as waveletdenoising. In this paper we explore wavelet denoisingof images using several thresholding techniques such asSURE SHRINK, VISU SHRINK and BAYESSHRINK. Further, we use a Gaussian based model toperform combined denoising and compression fornatural images and compare the performance of wavetransform methods [3].

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In this paper, we also describe approximate of newmathematical transforms, namely as curvelet transformfor image denoising [4]. Our implementations offerexact reconstruction, stability against perturbations,ease of implementations and low computationalcomplexity. A central tool is Fourier domaincomputation of an approximate digital randomtransform. In a curvelet transform, we will use sparsityand its applications [5].In the past, researcher’s have proposed a work on

novel image denoising method which is based on DCTbasis and sparse representation [6].To achieve a good performance in these aspects, adenoising procedure should adopt to imagediscontinuities. Therefore, a comparative study onmammographic image denoising technique usingwavelet, and curvelet transform [7]. Therefore, multiresolution analysis [8] is preferred to enhance the imageoriginality. The transform domain denoising typicallyassumes that the true image can be well approximatedby a linear combination of few basis elements. That is,the image is sparsely represented in the transformdomain.Hence, by preserving the few high magnitude transformcoefficients that convey mostly the original imageproperty and discarding the rest which are mainly dueto noise, the original image can be effectively estimated[9]. The sparsity of the representation are critical forcompression of images, estimation of images and itsinverse problems. A sparse representation for imageswith geometrical structure depends on both thetransform and the original image property. In the recentyears, there has been a fair amount of research onvarious denoising methods like wavelet, curveletcontourlet and various other multi resolution analysistools. Expectation - Maximization (EM) algorithmintroduced by Figueirodo and Robert [10] for imagerestoration based on penalized livelihood formulized inwavelet domain. State-of-art Gaussian Scale Mixture(GSM) algorithms employs modelling of imagesaccording to the activity within neighbourhoods ofwavelet coefficients and attaching coefficients heavilyin inactive regions [11]. Coif man and Donoho [12]pioneered in wavelet thresholding pointed out thatwavelet algorithm exhibits visual artefacts’. Curvelettransform is a multi scale transform with strongdirectional character in which elements are highlyanisotropic at fine Scales. The developing theory ofcurvelets predict that, in recovering images which aresmooth away from edges, curvelets obtain smallerasymptotic mean square error of reconstruction thanwavelet methods [13].

II. MULTIRESOLUTION TECHNIQUESAn image can be represented at different scales bymulti resolution analysis. It preserves an imageaccording to certain levels of resolution or blurring inimages and also improves the effectiveness of anydiagnosis system [14].

A. WaveletWavelet transform is a powerful tool for signal andimage processing that had been successful used in manyscientific fields such as signal processing, imagecompression, computer graphics, pattern recognition.On contrary the traditional fourier transform, Thewavelet transform is particularly suitable for theapplications of non stationary signal which mayinstantaneous vary in time.For this reason, firstly researchers has concentrated forcontinous wavelet transform (CWT) that gives morereliable and detailed time scale representation ratherthan the classical short time fourier transform(STFT)giving a time frequency representation.The CWT technique expand a signal onto basisfunctions created by expanding shrinking and shifting asignal prototype function which named as motherwavelet, specially selected for a signal underconsiderations. This transformation decomposes thesignal into different scales with different level ofresolution. Mother wavelet has satisfy that it has a zeromean value.The CWT is computed by changing the scale of themother wavelet, shifting the scaled wavelet in time,multiplying by the signal, and integerating over alltimes. When the signal to be analysed and waveletfunctions are discredited, the CWT can be realized oncomputer and the computation can be significantlyreduced if the redundant samples removed respect tosampling theorem. This is not a true discrete wavelettransform, The fundamentals of discrete wavelettransform goes back to the sub band coding theorem.The sub-band coding encodes each part of the signalafter separating into different band of frequencies.Wavelet transform can achieve good scarcity forspatially localized details, such as edges andsingularities. For typical natural images, most of thewavelet coefficients have very small magnitudes,except for a few large ones that represent highfrequency features of the image such as edges. TheDWT (Discrete wavelet transforms) is identical to ahierarchical sub band system. In DWT, the originalimage is transformed into four pieces which is normallylabelled as A1, H1, V1 and D1 as the schematicdepicted in Fig.1.

Shukla, Changlani and Johari 50

The A1 sub-band called the approximation, can befurther decomposed into four sub-bands. The remainingbands are called detailed components. To obtain thenext level of decomposition, sub-band A1 is furtherdecomposed.

Fig. 1. DWT based Wavelet decomposition tovarious levels.

Many wavelet’s are needed to represent anedege(number depends on the lengh of the edge,not thesmoothness).In this,m-term approximation error wouldbe occur.

(││f-fm││2)2 ≈ m-1

ORIGINAL: 1% OF WAVELET COEFFS

10% OF WAVELET COEFFS

Wavelets and its GeometryThe basis function of wavelets is isotropic. They

cannot “adapt” to geometrical structure. In this weneed more refined scaling concepts.

B. CurveletCurvelets are a non-adaptive technique for multi-scaleobject representation. Being an extension of the waveletconcepts, they are becoming popular in similar fields,namely in image processing and scientific computing.Curvelet transform is a multi-scale geometric wavelettransforms, can represent edges and curves singularitiesmuch more efficiently than traditional wavelet.Curvelet combines multiscale analysis and geometricalideas to achieve the optimal rate of convergence bysimple thresholding. Multi-scale decompositioncaptures point discontinuities into linear structures.Curvelets in addition to a variable width have a variablelength and so a variable anisotropy. The length andwidth of a curvelet at fine scale due to its directionalcharacteristics is related by the parabolic scaling law:

Width (length) 2

Curvelets partition the frequency plan into dyadiccoronae that are sub partitioned into angular wedgesdisplaying the parabolic aspect ratio as shown in fig.2.Curvelets at scale 2-k, a re o f rap id decay awayf ro m a ‘ r idge ’ o f l eng th 2 - k / 2 and wid th 2 - k

and thi s r id ge i s t he e f fec t ive suppor t .

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Fig. 2. Curvelets in frequency domain.

The d iscre te t rans la t ion o f curve le tt rans form is ach ieved us ing wrapp inga lgor i thm [15] .The curve le t coe f f ic i ent s C k fo r eachsca le and ang le i s d ef ined in Fo ur ie rdo main b y

C k( r , θ) = 2-3k/4R (2-kr) A2(k/2)/2 . )

Where C k i n t h i s equa t ion represent s po la rwed ge suppor ted b y the rad ia l (R) andangular (A) windo ws . A Digital CurveletTransform can be implemented in two ways (FDCTvia USFFT and FDCT via wrapping), which differ byspatial grid used to translate curvelets at each scaleand angle [16].

III. PROPOSED WORK

In this paper, we report initial efforts at imagedenoising based on a recently introduced family oftransforms- Wavelet transform and Curvelettransform. In this paper, we compare the results fromwavelet transform and curvelet transform and we willsee which transform is better for the image denoising.Our main objective is to decrease a mean square error(MSE) and to increase a peak signal to noise ratio(PSNR) in db by adding a white noise like Gaussiannoise, Poisson noise and Speckle noise. During thisconfiguration, we will use Threshold estimator likeheursure, rigrsure, sqtwolog, and minimaxi.

We can adjust decomposition level from 1 to 5 andwe use Thresholding [17]. Thresholding is thesimplest method of image segmentation. From agreyscale image, thresholding can be used to createbinary images. Thresholding is a simple non-lineartechnique, which operates on one wavelet coefficientat a time. In its most basic form, each coefficient isthreshold by comparing against threshold. If thecoefficient is smaller than threshold, set to zero,otherwise it is kept or modified. On replacing thesmall noisy coefficients by zero and inverse wavelettransform. In both case (Soft thresholding and Hardthresholding) the coefficients that are below a certainthreshold are set to zero.

In hard thresholding,the remaining coefficientsare left unchanged. In soft thresholding, themagnitudes of the coefficients above threshold arereduced by an amount equal to the value of thethreshold. In both cases, each wavelet coefficient ismultiplied by a given shrinkage factor, which is afunction of the magnitude of the coefficient.In our thesis, we will use a curvelet transform as wellas wavelet transform for removing a additive noisewhich is present in our images.

IV. MATERIALS AND METHODS

Image from MIAS database was denoised usingwavelet and curve let transforms. Various types ofnoise like the Random noise, Gaussian noise, Salt &Pepper and speckle noise were added to this image.Denoising procedure followed here is performed bytaking wavelet/curvelet transform of the noisy imageRandom, Salt and Pepper, Poisson, Speckle andGaussian noises) and then applying hard thresholdingtechnique to eliminate noisy coefficients. Thealgorithm is as follows (Fig. 3):

Step 1: Computation of thresholdStep 2: Apply wavelet/curvelet/contourlet transformto imageStep 3: Apply computed thresholds on noisy imageStep 4: Apply inverse transform on the noisy imageto transform image from transform domain to spatialdomain.

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Fig. 3. Algorithm.

V. EXPERIMENTAL RESULTS

The Experiment was done on several natural imageslike lena, Barbara, baboon, cameraman etc. usingmultiple denoising procedures for several noises. In ourexperiment, we have considered a image of A cricketer

Mahendra Singh dhoni. In this image we have used adifferent additive noises like Gaussian noise, poissonnoise, and speckle noise with different noise levels σ =10, 15, 20, 25, 30, 35 etc. And before adding a noise,mean value is always be 0.

Table. 1: Comparison of Wavelet and Curvelet with Different Noise in PSNR.

NOISES NOISYIMAGESPSNR/db

WAVELETPSNR/db

CURVELETPSNR/db

Poisson 27.7344 27.0602 33.8397Gaussian 24.9825 26.2889 32.4896

Speckle 30.2455 27.4944 34.8447

Salt andpepper

27.0201 27.1197 27.5421

Fig. 4: Graph indicating comparative results of the PSNR values of wavelet and curvelet based thresholding forimage denoising.

010203040 NOISY IMAGE

PSNR

WAVELETPSNR

CURVELETPSNR

NOISYIMAGE

APPLYTRANSFORM

APPLYTHRESHOLDAPPLY

INVERSETRANSFORM

DENOISEDIMAGE

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Table 1. shows the comparison of wavelet and curveletwith different noises and we measures the peak signalto noise ratio(in Db) and Fig. 4 shows a graph whichindicates a comparative results of the PSNR values ofwavelet and curvelet based thresholding (soft/hard) for

image denoising and there is, we apply a different typesof threshold estimator like rigrsure, heursure, sqtwolog,mini-maxi. And different decomposition levels like 1,2, 3, 4, 5 & so on.

Table 2: Comparison of Wavelet and Curvelet with Different Noise in MSE.

NOISES NOISYIMAGESMSE

WAVELETMSE

CURVELETMSE

Poisson 109.5562 127.9571 26.8605

Gaussian 207.5685 152.8252 36.6541

Speckle 61.4507 115.7913 23.3111

Salt andPepper

129.144 126.2137 114.5175

Fig. 5: Graph indicating comparative results o the MSE values of wavelet and curvelet based thresholding for imagedenoising.

Table 2. shows the comparison of wavelet and curveletwith different noises. We measures the mean squareerror(MSE) and Fig. 5 shows a graph which indicates acomparative results of the MSE values of wavelet andcurvelet based thresholding (soft/hard) for image

denoising and there is, we apply a different types ofthreshold estimator like rigrsure, heursure, sqtwolog,mini-maxi. And different decomposition levels like 1,2, 3, 4, 5 and so on.

0100200300 NOISY IMAGE

PSNRWAVELETPSNRCURVELETPSNR

Shukla, Changlani and Johari 54

VI. CONCLUSION

The comparison of wavelet transform and curvelettransform technique is rather a new approach, and it hasa big advantage over the other techniques that it lessdistorts spectral characteristics of the image denoising.The experimental results show that the curvelettransform gives better results/performance than wavelettransform method.

ACKNOWLEDGEMENT

The Author would like to thanks The Rajiv GandhiProudyogiki Vishwavidyalaya, Bhopal, (MP). For itsgenerous support , and the Lakshmi Narain College OfTechnology and Science, Bhopal, (MP). For theirhospitality, during my academic period 2011-2013.Shewishes to thanks Dr. Soni Changlani and Mr. AyoushJohari for their help and encouragement.

REFERENCES

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