A New Image Denoising Method Based on Wave Atoms and Cycle Spinning

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1 6 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY A New Iage Denoising Method Based on Wave Atos and Cycle Spinning Wei-qiang Zhang College of Matheatics and coputational Science, Shenzhen University, Shenzhen 586, China. Eail: wqzhang@szu.edu.cn Yi-ei Song School of Science, Xidian University, Xi an 77, China. Eail: songyiei5@6.co Ji-qiang Feng Institute of Intelligent Coputing Science, Shenzhen University, Shenzhen 586, China. Eail: athlove@6.co Abstract A new ethod for iage denoising was presented,which colligated the strong point of wave atos transfor and Cycle Spinning. Due to lack of translation invariance of wave atos transfor,iage denoising by coefficient thresholding would lead to Pseudo-Gibbs phenoena. Cycle Spinning was eployed to avoid the artifacts. Experiental results show that the ethod can reove noisy and reain edges, while Pseudo-Gibbs phenoena are controlled efficiently, and can get better visual effect and PSNR gains copared with the ethods like siplex wave atos or wavelet denoising using Cycle Spinning.And in heavy background noise, this advantage is significant. Index Ters iage processing, denoising, wavelet transfors, wave atos, translation invariance, Cycle Spinning I. INTRODUCTION Wavelet theory is widely used in signal processing, but the traditional wavelet transforation showed soe liitations in the processing of two-diensional iage[,]. The iage processing ethod cobined partial differential equations and wavelet theory can better retain the iage edge inforation [, ]. In the past two years, Deanet and Ying proposed a variant of wavelet packet-wave atos[5,6]. Wave atos transforation is a new type of two-diensional ulti-scale transforation, and still eets the parabolic proportional scaling relation and anisotropic characteristics of curve wave. In the wave atos, the oscillation function or director texture is sparser than that in the wavelet, Gabor atos or curve wave [6]. Wave atos applies to any local direction of the ode and can sparsely spread in the anisotropy ode in the axis direction. Copared with the curve wave, wave atos can not only capture the vibration ode, but can characterize the pattern through the oscillation. Although wave atos transforation can sparsely show the two-diensional iage, due to its lack of translation invariance, the artificial visual distortion will be Proect supported by National nature Science Foundation of China (No.6787,68, 9). Corresponding author: Ji-qiang Feng, athlove@6.co. introduced at the sae tie of being applied to iage denoising; especially for the part of iage edge, the Pseudo-Gibbs phenoenon is particularly obvious. The Cycle Spinning technology [7] proposed by Coifan and Donoho well avoided this visual distortion. Cobined with the effective representation of the wave atos on the oscillation texture, Cycle Spinning technology was introduced to iprove the wave atos hard threshold denoising, and a denoising algorith based on the wave atos transforation was proposed by this paper. The experiental results showed that, copared with traditional denoising ethod, the algorith better iproved the visual effect of iage denoising and obtained a higher PSNR gain, especially had a better effect on the iages with rich details and texture. In the strong noise level, this advantage was ore apparent. II. WAVE ATOMS We write wave atos as φ ( x),with subscript = ( n,, ) = (,,, n, n),,,, n, n Z, index a point ( x, w ) in phase space, as x = nw, = π, C ax C i=, where C, C > are two positive constants. x and w are the centers of φ ( x) in spatial and frequency doain respectively. Definition. The eleents of a frae of wave packets φ ( x) are called wave atos when { } φ M ( ) ( x C + x x ) M () M φ ( w) CM (+ w w ) () M + CM (+ w+ w ) for all M >. Definition only presents a qualitative description for wave atos with spatial frequency location restriction. In practice, Deanet uses the strategy of frequency localization given by Villeose to construct wave atos fro tensor products of adequately chosen D wave packets. i ACADEMY PUBLISHER doi:./sw.9..6-

2 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY 7 The trick consists in exhibiting adequate syetric paris of copactly supported bups in frequency, given by the forula iw/ iα ψ( w) = e e g( ε( w π ( + ))) ] iα + e g( ε+ ( w+ π ( + ))) [π where ε ( ) = and α = ( + ). The function g is an appropriate real-valued, C bup function, copactly supported on an interval of length π, and chosen such that ψ ( w ) =. Let g supported on [ 7 π /6,5 π /6],and such that for w π /, g( π / w) + g( π / + w) = and g( w π /) = g( π / + w). Then the translates { ψ ( x n )} for an orthonoral basis of L ( R ).This construction provides a unifor, or Gabor, tiling of the frequency axis. We need to introduce the subscript to index scale, and write our basis functions as /, ( ) ψ ( ) ( n x = ψ x n = ψ x n) Then the resulting basis of wavelet packets ψ, ( x ) for an orthonoral basis of L ( R ). We ephasize here that these constructed basis functions have a good property, naely the uniforly bounded location in both tie and frequency, which is the ost iportant difference with wavelet packets fro a standard ulti-resolution analysis and plays a key role in designing wave atos. For all f ( x) L ( R), the coefficients can be seen as a deciated convolution at scale, Cn,, = ψ( x n) f( x) dx = ψ( x n) f( x) By Plancherel, i nw ψ Cn,, = e ( w) u( w) dw π In two diension, let us abbreviate = ( n,, ), where = (, ) and n = ( n, n). H be Hilbert Transfor. We define an orthonoral basis + φ ( x, x) = ψ ( x n) ψ ( x n) A dual orthonoral basis can be defined fro the Hilbert-transfored wavelet packets, φ ( x, x ) = Hψ ( x n ) Hψ ( x n ) We + + φ () + φ φ () φ denote φ = and φ =, () () φ = φ, φ for the wave atos frae in two then{ } { } diension, and satisfy () () ϕ, f + ϕ, f = f The coefficients of two-diension wave atos transfor n can be obtained as follows: () () WA( f ) = f, ϕ + f, ϕ Figure and figure show the space-frequency doain fors of one-diensional wave atos at increasingly scales. Figure. The frequency bands divide of one-diensional wave atoes (a) =,= ACADEMY PUBLISHER

3 8 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY (b) =,= (c) =5,=8 Figure. One-diensional wave atos in space-frequency doain at increasingly scales. Figure shows two-diensional wave atos at increasingly scales. The upper panels represent wave atos in the spatial doain and the nether panels show wave atos in the frequency doain. Figure. Two-diensional wave atos at increasingly scales. ACADEMY PUBLISHER

4 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY 9 III. ALGORITHM DESCRIPTION A. Wave Atos Hard Threshold Denoising The basic idea of hard threshold denoising based on wave atos transforation is consistent with the wavelet based denoising ethod. Assue the noisy iage u can be expressed as u = u + η, where u is the noise clean iage, η is the Gaussian noise of zero ean and variance σ ; the purpose of iage denoising is to recover a clear iage fro u. Soft threshold function because of its continuity akes the edge of the iage denoising too vague, and too uch detail is lost. However, hard threshold ethod can better retain the local characteristics of iage edge, so this paper uses the ethod of wave atos hard threshold denoising. B. Cycle Spinning In the threshold denoising process, if the transforation is lack of translation invariance, pseudo-gibbs phenoenon will be produced in the iage discontinuous point neighborhood area (edges and textures), leading to iage distortion; this distortion is closely related to the location of iage discontinuous points. For exaple, for Haar wavelet, the pseudo-gibbs phenoenon will not be produced in the discontinuous point neighborhood area of n, but obvious pseudo-gibbs phenoenon will occur in the discontinuous point neighborhood in other location (such as n ). A ethod to prevent this phenoenon is to change the location of iage discontinuous points via iage translation, conduct threshold denoising on the translated iage and then reversely translated the denoised iage to avoid the pseudo-gibbs phenoenon. If, however, the iage to be analyzed contains a plurality of discontinuous points, the optial translation of a certain discontinuous point ay result in pseudo-gibbs phenoenon in the neighborhood area of another discontinuous point. So it is difficult to find a translation aount that can satisfy the requireents of all discontinuous points. To inhibit the pseudo-gibbs phenoenon occurred due to the lack of translation invariance in threshold denoising process, Coifan and Donoho proposed Cycle Spinning technology, that is, to carry out cycle spinning-threshold denoising-reverse cycle spinning. As the threshold denoising on the iage after each translation will ake the occurrence of pseudo-gibbs phenoenon in different places; therefore, single translation is not used, but a different denoising result s ) i, will be obtained fro each translation in iage rows and coluns, and the denoising result s ) inhibiting pseudo-gibbs phenoenon by linear average on all the denoising results, that is: ) s = S ( T ( Λ[ T( S ( x))]), i, i, i, K K ) ) s s = i, KK i = = K, K is the axiu translation aount in the row and colun direction, S is the cycle spinning operator,. the subscript is the translation aount in the i, row and colun directions, TT, is the transforation operator and its inverse operator respectively, and Λ is the threshold operator. C. Cycle Spinning Based Wave Atos Denoising Algorith Although the hard threshold can well preserve the iage details, the processed iage will have the vision distortions such as ringing, pseudo-gibbs phenoenon; to inhibit pseudo-gibbs phenoenon in the process of hard threshold denoising, the wave atos based iage denoising new algorith is proposed by cobining with Cycle Spinning technology; the specific algorith steps are as follows: ) Conduct cycle spinning on the noisy iage u by the use of cycle spinning operator S, and obtain iage Su ( ); ) Conduct wave atos transforation T on the iage Su ( ) after cycle spinning and get the transforation coefficient TS( u ); ) Process these coefficients by hard threshold operator Λh and obtain the transforation coefficient Λ ( TS h ( u )) after denoising; ) Carry out inverse wave atos transforation on the wave atos coefficient Λ ( TS h ( u )) after hard threshold process and get the denoised iaget ( Λ h ( TS( u))) ; 5) The restored iage u S % = T ( Λh ( TS( u))) can be obtained by conducting reverse cycle spinning on the denoised iage T ( Λ h ( TS( u))), where S represents the reverse cycle spinning operator, and the final denoising results can be derived by averaging on all results. IV. EXPERIMENTAL RESULTS AND ANALYSIS In order to verify the correctness and validity of the proposed algorith, select soe iages with the size of about 5 5 and the white Gaussian noise with the ean of zero for experients, such as the seisic profile with rich texture inforation, fingerprint iage and Lena figure (figure ) with rich edge details, and Barbara figure (figure ), etc. Select 8 as the axiu translation aount in the iage row and colun direction. In the experient, the coparison of denoising effects has been ade of the wavelet hard threshold denoising (WT), cycle spinning wavelet hard threshold denoising (WT+ CS), wave atos hard threshold denoising (WA) and the proposed ethod in this paper (WA + CS). ACADEMY PUBLISHER

5 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY superior to traditional wavelet. TABLE I. THE COMPARISON ON THE PSNRS OF THE DENOISED IMAGES WITH DIFFERENT NOISE VARIANCES(DB) (a) WT (b) WT+CS Varian -ces W T Lena WT+ W CS A WA+ CS W T Barbara WT+ W CS A WA+ CS (c) WA (d) WA+CS Figure. The coparison of denoising effects of Lena( σ =., PSNR=9.99dB). (a) WT (c) WA (b) WT+CS (d) WA+CS Figure 5. The coparison of denoising effects of Barbara( σ =.5, PSNR=6.5dB). Fro the visual effects, wave atos denoising ethod can better retain the edge (the bri of Lena figure) and the texture inforation (hair of Lena figure and the pants stripes of Barbara figure). It indicates wave atos can well retain the curved edge contour of the iage, and is superior to the other two ethods in ters of PSNR gain and characterization texture. The texture details in the figures such as the hair and pants stripes, figure (c) and figure (d) are uch clearer than figure (a) and figure (b). In the edge area such as the hat bri, the effect of wave atos ethod is better than that of the wavelet ethod. Moreover, the visual effect of the wave atos denoising by the use of Cycle Spinning is significantly better than the traditional wave atos denoising, and the peak signal to noise ratio has been iproved by ore than db. Figure (d) has the highest peak signal to noise ratio of the iage by the use of Cycle Spinning wave atos denoising, which has ore effectively inhibited the pseudo-gibbs phenoenon caused by the lack of translation invariance in the process of threshold denoising, and significantly iproved the visual quality of the iage. The wave atos anisotropy and its efficient representation of oscillation texture deterine that the wave atos is To illustrate the algorith proposed by this paper is significantly better than other ethods in the obective perforance, coparison has been ade on the PSNRs of the denoised iages with different noise variances (see Table ). Whether the traditional wave atos threshold denoising algorith or the Cycle Spinning based wave atos threshold denoising algorith, the effect is better than the traditional wavelet threshold denoising, and even better than the cycle spinning wavelet denoising. Furtherore, in the strong noise level, this advantage is ore obvious. The coparison of the CPU tie consued by the denoising of iages with different noise variances has been ade, and the results showed the wavelet ethod consued less tie than the wave atos ethod. After the use of Cycle Spinning technology, the speeds of the algoriths are slower and the new algorith is the ost tie consuing. This is due to the coplexity of the wave atos transforation coputing; the repeated translation invariance will also increase the workload, so the cost of the new algorith coputing is great. V. CONCLUSIONS Wave atos are an eerging new direction ulti -scale transforation used for iage processing and nuerical analysis, with the oscillation cycle and support size satisfying the parabolic scaling relation. Its notable feature is the ulti-scale and anisotropy, which can sparsely spread the sooth oscillation function (such as texture). In the processing of traditional iage, pseudo-gibbs phenoenon often arises fro the lack of translation invariance and thus leading to iage distortion, Cycle Spinning is an effective way to eliinate this distortion. On the basis of literature [6], this paper proposed an iage denoising algorith on the basis of wave atos and Cycle Spinning. At the sae tie of effective reoval of the iage noise, this algorith better retained the iage edge and texture, and could effectively inhibit the pseudo-gibbs phenoenon, aking the denoised iage look ore realistic and natural, with better visual effects. The texture sharpness and contrast of the processed iage were superior to the traditional wavelet threshold, cycle spinning wavelet threshold and wave atos threshold ethods. For iages with rich texture inforation, particularly the texture ACADEMY PUBLISHER 5

6 JOURNAL OF SOFTWARE, VOL. 9, NO., JANUARY iages, it has ideal denoising effect, and this advantage is ore obvious in the context of strong noise. Due to the costly new algorith coputing, the fast algorith of the given wave atos transforation and the iproveent of the coputing speed of cycle spinning ethod will be the focus of research in the future. Wei-qiang Zhang was born in 977. He received the Ph.D. degree in applied atheatics fro the Xidian University, Xi an, Shanxi in 6. Currently, he is an associate professor in Shenzhen University. His research interests include intelligent inforation processing and iage processing. REFERENCES []. E.J. Candès, L. Deanet, The curvelet representation of wave propagators is optially sparse, Co. Pure Appl. Math., Vol.58, No., pp. 7 58, 5. []. E.J. Candès, D.L. Donoho, New tight fraes of curvelets and optial representations of obects with piecewise-c singularities, Co. Pure Appl. Math., Vol.57, No., pp. 9 66,. []. Chen Lixia,Ding Xuanhao,Song Guoxiang, Iage Denoising algorith based on total variation and wavelet transfor, Journal of xidian university, Vol.5, No.6, pp , 8. []. Lu Chengwu, Multiscale Decoposition of Iage Under (BV,E) Frae, Journal of xidian university, Vol.6, No., pp. 7-76, 9. [5]. L. Deanet, Curvelets, wave atos and wave equations, California: California Institute of Technology, 6. [6]. Deanet L, Ying L X, Wave Atos and Sparsity of Oscillatory Patterns, Appl.Coput. Haron. Anal., Vol., No., pp , 7. [7]. Coifan R R, Donoho D L, Translation-invariant denoising, Lecture Notes in Statistics: Wavelets and Statistics, New York: Springer-verlag, pp. 5-5, 995. [8]. J.P. Antoine, R. Murenzi, Two-diensional directional wavelets and the scale-angle representation, Signal Processing, Vol.5, No., pp. 59 8,996. [9]. Guoun Liu, Perona-Malik Model Based On Wave Atos, Proceedings of the rd International Congress on Iage and Signal Processing, pp. 6-66,. []. Ke Ding, Wavelets, Curvelets and Wave Atos for Iage Denoising, Proceedings of the rd International Congress on Iage and Signal Processing, pp ,. []. Anil A. Patil, Jyoti Singhai, Iage denoising using curvelet transfor: an approach for edge preservation, Jounal of Scientific & Industrial Research, vol. 69, pp. -8,. []. M.O. Ulfarsson, J.R. Sveinsson, and J.A. Benediktsson, Speckle Reduction of SAR Iages in The Curvelet Doain, Proceeding of the International Geoscience and Reote Sensing (IGARSS), vol., pp. 5-7, Toronto, Canada,. []. G. Plonka, J. Ma. Nonlinear regularized reaction-diffusion filters for denoising of iages with textures, IEEE Trans. Iage Process., 7, pp. 8-9, 8. []. J. Ma, G. Plonka. Cobined curvelet shrinkage and nonlinear anisotropic diffusion, IEEE Trans. Iage Process.,6, pp. 98-6, 7. Ji-qiang Feng was born in 979. He received the Ph.D. degree in signal and inforation processing fro the Shenzhen University, Shenzhen, Guangdong in. Currently, he is an assistant professor in Shenzhen University. His research interests include intelligent inforation processing and optiization theory. ACADEMY PUBLISHER 6