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2 Chapter 3 Adaptive Wavelet Packet Trasform Zuofeg Zhou ad Jiazhog Cao Additioal iformatio is available at the ed of the chapter Abstract Two-dimesioal over-complete wavelet packet trasform ca better represet the texture ad log oscillatory patters i atural images. I this chapter, combiig the doubly Wieer filterig algorithm ad Wieer cost fuctio, a ew best wavelet packet decompositio scheme for image deoisig applicatios is proposed. The experimet results for the test image database show the effectiveess of the proposed image deoisig algorithm compared to some existig image deoisig methods. Keywords: daptive wavelet packet trasform, Wieer cost fuctio, image deoisig 1. Itroductio Wavelet with vaishig momets are very effective for represetig piecewise smooth images [1]. However, two-dimesioal separable wavelets are ill-suited to represet log oscillatory patters i images with abudat textures, partly owig to their poor directioal selectivity i frequecy domai. These oscillatory variatios of itesity ca oly be represeted by smallscale wavelet coefficiets. I some image-processig applicatios, such as image deoisig or image compressio, those small-scale coefficiets are quatized to zero i the low bit rate image compressio ad are thresholded or shruke to zero i image deoisig, which degrades compressio ad deoisig performace sigificatly. To overcome this circumstace, oe way is to fid the more suitable image represetatio techiques such as curvelet, cotourlet [], ad ridgelet [3]. ut these methods eed the researchers to desig the ew directioal compact filter or multichael filter baks, which is also challegig i filter desig area. other way is to improve the idea of wavelet desig method to accommodate the ew requiremet where the over-complete wavelet packet decompositio is proposed. 015 The Author(s). Licesee ITech. This chapter is distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.

3 54 Wavelet Trasform ad Some of Its Real-World Applicatios Over-complete wavelet packet cotais a mass of libraries of waveforms, from which the best wavelet packet base ca be selected to efficietly represet log oscillatory patters give the correspodig criterio. For example, i image compressio applicatio, the best wavelet packet base is usually foud by pruig the full wavelet packet decompositio give a user predefied cost fuctio. Recetly, a large variety of cost fuctio have bee proposed, such as Shao s etropy cost fuctio ad vector etropy cost fuctio, which are used i the rate distortio cotrol strategy. O the other had, differet from fidig the best tree structure of the wavelet packet decompositio, some researchers devoted themselves for fidig the best wavelet packet decompositio base [4] which is equivalet to fid the best filter baks of wavelet packet decompositio. I image deoisig applicatio, due to ukow oiseless image as well as a great diversity of filterig methods i the trasform domai, selectig the best wavelet packet base is always a difficult problem. Elighteed by the idea of doubly local Wieer filterig method [5] ad spatially adaptive wavelet domai shrikage image deoisig algorithms, the best wavelet packet bases selectio method usig the local Wieer cost fuctio ad its applicatio i image deoisig is proposed. I this chapter, we will first give the detail of the best wavelet packetbased selectio algorithm ad the discuss its applicatio i image deoisig.. Best wavelet packet base selectio Let ψ 0 (x), ψ 1 (x) be the wavelet fuctio ad scalig fuctios, the basic idea of wavelet packet fuctio ca be defied as: y ( x) = h( k) y ( x - k), y + 1 å k å ( x) = g( k) y ( x - k), for ³ 1 k (1) where h (k), g(k) are the orthoormal filters, respectively. Give a sigal s() which is satisfied the Nyquist samplig criterio, subspace V 0 spa{ ψ 0 (x l):l Z} is usually assumed to be the first-level subspace of sigal s(). The subspace with depth is defied as V spa{ψ,l (x)= / ψ ( x l), l Z}. To uderstad this subspace easily, the dyamic iterval, ( + 1)) ca be associated with the subspace V. For example, the iterval 0, 1) is equivalet to V Two-dimesioal wavelet packet bases The two-dimesioal separable wavelet packet fuctios are defied as the tesor products of two oe-dimesioal wavelet packet fuctios, that are, V 0 =spa{ ψ 0 (x l)ψ 0 (y p):(l, p) Z } ψ m, (x, y)=ψ m (x)ψ (y), m, 0, 1, ()

4 Adaptive Wavelet Packet Trasform 55 Similar to the oe-dimesioal case, the subspace V m, spa{ψ m, ;l, p (x)= ψ m ( x l)ψ ( y p), (l, p) Z } (3) is referred to as the subspace with depth ad a wavelet packet fuctio ψ m, (x, y). Figure 1 gives the subspace of D wavelet packet base diagram. It ca be see that each ode is divided ito two braches. Let us associate the dyadic square m, (m + 1)), ( + 1)) with the subspace V m,, for example, V 0 associates with the square 0, 1), V 1 1,0 associates with the rectagle 1 /, 1) 0, 1 / ), ad V 1 1,1 associates with the square 1 /, 1). It has bee proved that: if the squares m, (m + 1)), ( + 1)), (m,, ) Ω are a partitio of the uit square 0, 1), the the family of fuctios {Ψ m, ;l, p (x, y):(l, p) Z, (m,, ) Ω} costitutes a orthogoal wavelet packet base of V 0. Figure 1. The subspace of D wavelet packet base.. Best wavelet packet base selectio uder wieer cost fuctio Similar to the tree structure illustrated i Fig. 1, D wavelet packet decompositio ca be easily expressed by the quad-tree structure with the root ode {0, 0, 0}. Each ode (m,, ) (except the last level ode) has four child odes (m, ; + 1), (m, + 1, + 1), (m + 1,, ) ad (m + 1, + 1, + 1). This quad-tree structure facilitates the best wavelet packet decompositio procedure. Give the image ad the oise level σ (if ukow, the oise level ca be estimated by the M D estimator i the wavelet domai), the wavelet packet coefficiet at the ode (m,, ) uder J-level wavelet packet decompositio is represeted as y m,(p, q), m, =0, 1,, 1; =0, 1,, J. For each ode, its Wieer cost fuctio is computed by m, éy ( p, q) ù, ; = s ë û,, = 0,1,, - 1; = 0,1,,. p q m, éy ( p, q) ù + s ( ) å å K K (4) J m m J ë û

5 56 Wavelet Trasform ad Some of Its Real-World Applicatios ssumig all the ode i the quad-tree decompositio forms a set Class (A)={Ω={(m,, )}, our task is to fid a subset Ω * from set Ω which have the smallest total cost fuctio (the summatio of the Wieer cost fuctio at all wavelet packet decompositio ode). Similar to most of the search algorithms of the best wavelet packet base, the best wavelet packet base uder Wieer cost fuctio is obtaied by pruig a J-level full quad-tree from bottom to top. The searchig algorithm ca be described as follows: i. Let S(m,, J )={ (m,, J ) }, m, =0, 1,, J 1 represet J leaf odes i the J-level full quad-tree. ii. For 0 < J ad each ode (m,, ) i the -th level, calculate the correspodig Wieer cost fuctio (, ; ) J m m, éy ( p, q) ù = s ë û åå (5) p q m, éy ( p, q ) ù ë û + s as well as the summatio of the Wieer cost fuctios of its four child odes by J% ( m,, ) = J( S( m,, + 1)) + J( S( m, + 1, + 1)) + + J( S(m + 1,, + 1)) + J( S(m + 1, + 1, + 1)) iii. If J (m,, )< J (m,, ), the U U U S( m,, ) = S( m,, + 1) S( m, + 1, + 1) S(m + 1,, + 1) S(m + 1, + 1, + 1) otherwise, S(m,, )={(m,, )}. iv. Calculate the Wieer cost fuctio of each set S(m,, ) by ( ( )) å ( ) J S m,, º J u, v, r, m, = 0,1, K, - 1, ( u, v, r) ÎS( m,, ) (6) v. If >0, set 1 ad retur the step (ii); otherwise, output Ω * =S(0, 0, 0). The set Ω * is composed of all leaf odes of the best quad-tree decompositio structure. I this way, give the iput oisy image ad the oise level, the optimal orthogoal wavelet packet decompositio structure ad the miimal total Wieer cost fuctio ca be obtaied. Figure shows a demo of best wavelet packet decompositio where Fig. (a) is its square represetatio ad Fig. (b) is the correspodig tree structure of this best wavelet packet decompositio.

6 Adaptive Wavelet Packet Trasform 57 Figure. demo of best wavelet packet decompositio 3. Image deoisig algorithms based o best wavelet packet decompositio This sectio gives two image deoisig algorithms based o the best wavelet packet decompositio. I the above sectio, the optimal wavelet packet decompositio structure is obtaied uder the Wieer cost fuctio. Whe computig the Wieer fuctio, the oiseless image ad the oise level are assumed as kow. I practice, this assumptio is ureasoable. It is well kow that the oise level ca be accurately estimated by the M D estimator from the iput oisy image. The key problem is how to estimate the oiseless image. Elighteed by the empirical Wieer filterig ad the doubly local Wieer filterig image deoisig algorithms, we first filter the oisy image to get the pilot image, ad the the pilot image is used to get the best wavelet packet decompositio structure. It is well kow that udecimated wavelet packet decompositio ca achieve better image deoisig performace tha the decimated wavelet packet decompositio. This is owig to the property that they are shift ivariace ad robustess. So, the udecimated wavelet packet decompositio ca ameliorate some upleasat pheomeo that appears i the maximally decimated wavelet packet decompositio, such as Gibbs-like rigig aroud edges ad specks i smooth regio. I what follows, we will give the detail of the image deoisig algorithm usig the udecimated wavelet decompositio. Let x(p, q) be a iput oisy image of size N N, the operator Shift k,l (x) deotes circularly shiftig the iput image x(p, q) by k idices i the vertical directio ad l idices i the horizotal directio, ad the operator Ushift k,l (x) is a similar operatio but i the opposite directio. I the proposed algorithm, we first use the decimated best wavelet packet decompositio algorithm to deoise all possible shift versios of the oisy image to get a set of deoised images ad the ushiftig these deoised images ad average

7 58 Wavelet Trasform ad Some of Its Real-World Applicatios them to get the fial deoised image. The ew algorithm is referred to as the local Wieer filterig usig the best udecimated wavelet packet decompositio (LWF- UWPD), which ca be summarized as follows: i. Usig the local Wieer filterig image deoisig algorithm to each shift of iput oisy image shift (x(p, q)) to get a pilot image s^(p, q) ; ii. iii. Give the pilot image ad oise level, the best wavelet packet decompositio structure ca be obtaied by usig the searchig algorithm i sectio. Give the best wavelet packet decompositio structure, the empirical eergy distributio of the pilot image ca be estimated by E m, 1 ( p, q ì ) ˆ (, ), (,, ) ( s, t) W, m, # s p s q t ü W m * = í å + + ÎW Î ý (7) î þ where W ad s^,m, (p, q) represet the directioal widow ad the pilot image s best wavelet packet decompositio coefficiets, respectively. + iv. Usig the estimated eergy distributios ad oise level, the local Wieer filterig is operated o the base wavelet packet decompositio coefficiets of the oisy image, that is, E ( p, q) s% ( p, q) = y ( p, q), ( m,, ) ÎW m, m, m, m, E ( p, q) + s * (8) where y m, (p, q) are the best wavelet packet decompositio coefficiets of the oisy image i the subspace V m,. v. Ushift all the shifted deoised images ad average them to obtai the fial deoised image s (p, q) J J s% ( p, q) = ( )(, ) J å Ushift s p q k= 0 å % l= 0 k, l k, l (9) 4. Experimetal results We choose the 8-bit grayscale images Lea ad arbara, ad a texture image, as the test images. I the proposed image deoisig algorithm, the best wavelet packet decompositio structure is varied with differet shift oisy images ad differet oise level. For better illustratio, the best wavelet packet decompositio structure for differet oise level is show i Fig. 3 for arbara image.

8 Adaptive Wavelet Packet Trasform 59 Figure 3. The best wavelet packet trees for the arbara image with differet oise levels: (a) σ =10 ; (b) σ =15 ; (c) σ =0 ; (d) σ =5. I Table 1 ad Fig. 4, we give the deoisig performace of the image deoisig algorithms usig the udecimated best wavelet packet decompositio. The experimetal results show that for images of structural textures, for example, arbara ad texture images, the proposed algorithm greatly improves deoisig performace as compared with the existig state-of-theart algorithms. Test image Boat Figerprit House Noise level DLWFDW DF -GSM LWF- UWPD Test Images Lea arbara Texture Noise level DLWFDW DF -GSM LWF- UWPD Table 1. The performace compariso of the LWF- UWPD ad several state-of-the-art image deoisig algorithms

9 60 Wavelet Trasform ad Some of Its Real-World Applicatios (a) (b) Figure 4. (a) The oiseless image (the left top corer), the oisy image (the right top corer, oise level 0), the deoised image by the DF -GSM algorithm (the left bottom corer, PSNR = d ), ad the deoised image by the LWF- UWPD algorithm (the right bottom corer, PSNR = d ); (b). Zoomed local regios of the four images i (a). Author details Zuofeg Zhou * ad Jiazhog Cao * ddress all correspodece to: zfzhou@opt.ac.c Ψi a Istitute of Optics ad Precisio Mechaics of Chiese cademy of Scieces, Ψi a, Peoples Republic of Chia Refereces [1] Shui, P.-L., Zhou, Z.-F., ad Li, J.-Ψ., Image deoisig based o adaptive wavelet packets usig wieer cost fuctio, IET Image Processig, 007, 1(3), pp: [] Mih, D., ad Marti. V., The cotourlet trasform: a efficiet directioal multiresolutio image represetatio, IEEE Trasactios o Image Processig, Dec. 005, 14(1), pp:

10 Adaptive Wavelet Packet Trasform 61 [3] Zhag,., Jalal M.F., ad Starck J.-L., Wavelets, ridgelets, ad curvelets for poisso oise removal, IEEE Trasactios o Image Processig, 008, 17(7), pp: [4] Mahalakshmi,.V., ad ad, M.J., daptive wavelet packet decompositio for efficiet image deoisig by usig eighsure shrik method, Iteratioal Joural of Computer Sciece & Iformatio Techology 014, 5(4), pp: [5] Shui, P.-L., Image deoisig algorithm via doubly local Wieer filterig with directioal widows i wavelet domai, IEEE Sigal Processig Letters, 005, 1(10), pp:

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