Image Enhancement by Wavelet Multi-scale Edge Statistics

Transcription

1 Image Enhancement by Wavelet Mult-scale Edge Statstcs Author Lew, Alan Wee-Chung, Jo, Jun Hyung, Chun, Yong-Sk, Ahn, Tae-Hong, Chae, Tae Byong Publshed 0 Conference Ttle Proceedngs of the st Internatonal Conference on Pattern Recognton Copyrght Statement 0 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works. Downloaded from Lnk to publshed verson Grffth Research Onlne

2 Image Enhancement by Wavelet Mult-scale Edge Statstcs Alan Wee-Chung Lew, Jun Jo, Yong-Sk Chun, Tae-Hong Ahn 3, Tae Byong Chae School of Informaton and Communcaton Technology, Gold Coast campus, Grffth Unversty, QLD4, Australa Dept. of KOMPSAT-5 System Engneerng & Integraton Korea Aerospace Research Insttute POB 3 Yusung Daeeon , Korea sk@kar.re.kr 3 Dept. of Computer Game Development Chonnam Techno Unversty Korea ath95@hanmal.net Abstract The dstrbuton of wavelet modulus maxma across wavelet scales can be used to characterze edges n an mage. In ths paper, we present a novel algorthm that performs mage enhancement by mappng the dstrbuton of the wavelet modulus maxma of the blurred mage to that of a generc sharp mage. Expermental results confrm that the proposed algorthm s able to perform mage enhancement wthout ntroducng unpleasant vsual artfacts.. Introducton Edges are mportant features that affect the perceptual qualty of an mage. Blurred mages usually have edges that appeared overly smooth. One way to mprove the perceptual qualty of an mage s to mprove the sharpness of edges n the mage. Unsharp maskng s one of the man technques for mage enhancement [-5]. The basc dea of unsharp maskng s to add to the blurred mage a hgh-pass fltered verson of the mage to emphasze hgh frequency regons such as edges n an mage. Wavelet allows a mult-resoluton decomposton of a sgnal. Recently, an unsharp maskng algorthm based on wavelet transform has been proposed for mage sharpenng [6]. The algorthm makes use of the correlaton between dfferent wavelet scales to retan wavelet coeffcents related to edges but remove noserelated coeffcents. The retaned edge-related coeffcents are then used to obtan the correcton mage whch s then added back to the orgnal mage to enhance the perceptual qualty. In ths paper, we ntroduce a novel mage sharpenng algorthm based on wavelet mult-scale edge decomposton. Our algorthm s based on the fact that the mult-scale edges extracted by a partcular nondecmated wavelet scheme follow a specfc dstrbuton accordng to the degree of regularty of the edges n the mage. The novelty of our approach les n that we explot the statstcal relatonshp between edges at multple wavelet scales of the blurred and sharp mage to enhance the blurred mage. Ths s done by applyng a non-lnear mappng to the wavelet coeffcents related to mult-scale edges n the blurred mage.. Wavelet Mult-Scale Edge Analyss Ponts of sharp varatons are usually one of the most mportant features for analyzng the propertes of sgnals or mages. In mages, local sharp structures such as edges contan rch nformaton and are mportant for many applcatons such as obect segmentaton and scene analyss. However, edges n an mage could exst n dfferent scales and we need a range of scales to descrbe these structures effectvely. In [7], Mallat et al. proved that the wavelet modulus maxma of a certan dyadc wavelet transform can detect the locaton of the sngular structures n a sgnal. Consder a wavelet constructed n such a way that t s the frst order dervatve of a smoothng functon, the To appear n ICPR 0

3 local modulus maxma of the dyadc WT by such a wavelet then characterze sharp varatons or edges n a sgnal at multples of dyadc scale. Let θ(t) be the smoothng functon that s used to smooth a sgnal at varous scales. Let ψ(t) be the frst dervatve of θ(t),.e., ψ(t)=dθ(t)/dt, and let the wavelet functon and the correspondng smoothng functon be defned by / ψ ( t) = ψ ( / t) θ ( t) = θ ( t) () The WT of a sgnal f (t) s then gven by the convoluton of f (t) wth ψ (t) dθ W f ( t) = f ο ψ ( t) f ( )( t) = ο () dt For D mage f(x,, the dyadc WT s gven by the D convoluton of f(x, wth the D scalng functon φ and the wavelets ψ : W f ( x, = f οψ ( x, W S J f ( x, = f οψ ( x, (3) f ( x, = f οφ J ( x, where the wavelets are partal dervatve of a -D cubc splne smoothng functon. The -D WT of an mage gves the gradent of the mage at multples of dyadc scales,.e., W f ( x, and W f ( x,, whch gve the vertcal and horzontal gradent at scale, respectvely. To detect the wavelet modulus maxma at each wavelet scale k, we frst compute the gradent magntude mage and the gradent normal mage at scale k. The wavelet modulus maxma at scale k are then obtaned by searchng along the gradent normal drecton such that the gradent magntude reaches a local maxmum. The locatons of these local maxma correspond to the mult-scale decomposton of edges n the mage. In Fg., we show the dstrbuton of the wavelet modulus maxma across three wavelet scales for a blurred mage and a sharp mage. The blurred mage s obtaned by convolvng the orgnal 5x5 mage wth a Gaussan flter of σ = 3. Row of Fg. shows the dstrbuton of the wavelet modulus maxma at scale (y-axs) vs scale (x-axs). Row 3 shows the dstrbuton of the wavelet modulus maxma at scale 3 (y-axs) vs scale (x-axs). We can see from column that when the mage s sharp, the magntudes of the wavelet modulus maxma reman farly constant across all 3 scales. In contrast, when the mage s blurred, the magntudes of the wavelet modulus maxma decrease as one goes from coarser scale,.e. scale 3, to fner scale,.e. scale. Fg. Wavelet modulus maxma across scales for orgnal (left column) and blurred Lena (rght column). Row s the mages, row s the wavelet modulus maxma dstrbuton at scale (y-axs) vs scale (x-axs). Row 3 s the wavelet modulus maxma dstrbuton at scale 3 (y-axs) vs scale (xaxs). 3. Mult-scale Edge Enhancement Based on the observaton of the relatonshp between mage sharpness and wavelet modulus maxma evoluton across scales, we proposed to perform mage sharpenng by remappng the dstrbuton of the wavelet modulus maxma of a blurred mage M to that of a sharper mage M. We consder a WT of 3 scales. Let X, Y, Z denote the random varables that correspond to the wavelet modulus maxma at scales, and 3, respectvely, and let ρ (x,y,z) and ρ (x,y,z) be the ont probablty densty functon (pdf) for the wavelet modulus maxma of the blurred mage and the sharper mage, respectvely. We need to fnd a transformaton functon T such that the pdf of M ) = T(M ) resembles that of M,.e., we requre { u, v, w} = T( x, y, z) such that ρ u, v, w) ρ ( x, y, ). ( z

4 We realze the transformaton T by three separate transformatons, T x, T y, T z, operatng on X, Y, Z, respectvely. Let Φ be the mappng defned by Φ ( ) = C C( ) (4) where the subscrpts and n (4) denote the blurred mage and the sharp mage, respectvely. The cumulatve dstrbuton functon C (t) s gven by = t and the quantle functon C ( t) C ( q) = nft R C ( t) ρ ( τ ) dτ (5) { C ( t) q} s gven by, q = C (t) (6) The transformatons T x, T y, T z are then gven by T x = Φ(x) T y = Φ( (7) T z = Φ(z) Equaton (7) effectvely performs a remappng of the wavelet modulus maxma of the source mage at scales,, and 3 to that of the target mage by hstogram matchng on ndvdual scale. dstrbutons of the wavelet modulus maxma to that derved from an ensemble of sharp mages. Fg. shows the dstrbuton of the wavelet modulus maxma across scales before and after the remappng usng (7) for the blurred Lena Fg. 3 shows the transformatons T x, T y, T z that are appled to get the results n Fg.. Once the remappng s done, the enhanced mage can be obtaned by dong an nverse wavelet transform to the modfed wavelet coeffcents. However, to ensure that the remapped dstrbutons are preserved n the fnal reconstructon, the proecton onto convex set (POCS) algorthm s used to terate to the fnal soluton. Fg. Remappng of the wavelet modulus maxma dstrbutons of blurred Lena. Left column: dstrbutons at scale (y-axs) vs scale (x-axs) (top) and at scale 3 (y-axs) vs scale (x-axs) (bottom). Rght column: remapped dstrbutons. As the evoluton of wavelet modulus maxma across scale s related to the regularty of the edges, for mages wth smlar degree of sharpness, the dstrbutons of ther wavelet modulus maxma across scale would be smlar. Ths has been observed to be the case. Hence, to sharpen an mage, we remap the Fg. 3 The transformaton functons T x, T y, T z that were used to perform the wavelet modulus maxma remappng. Sold dot s T x, astersk s T y, crcle s T z. In POCS-based mage reconstructon [8-], every known property of the orgnal mage f s formulated as a correspondng convex set n a Hlbert space H. The orgnal mage s then assumed to le n the ntersecton of these convex sets, Ι k C = f C = (8) 0 Let Q = Pk Pk Κ P, where P are the proecton operator n onto C, then x Q x converges to an element n C 0 as n = n = 0, for any startng pont x C 0. In our reconstructon algorthm, Q = P P. The proecton operator P s realzed by the settng the value of W f ( x, ) and W f ( x, ) at the modulus maxma y y postons to that of the remapped values. The proecton operators P s defned by P = W R W (9) where R s the truncaton operator gven by 0 x < 0 Rx = 55 x > 55 (0) x otherwse Only to 3 teratons are needed to obtan the fnal soluton.

5 4. Results In Fg. 4, we show the result of our algorthm on the blurred Lena. The reference dstrbutons are obtaned from an ensemble of sharp mages. In Fg. 5, we show the zoom-n vew of our result and the result of cubc unsharp maskng. The cubc unsharp maskng algorthm [] performs adaptve non-lnear sharpenng of an mage by modulatng the lnear unsharp flter usng a quadratc term that depends on the local gradent n the mage. It has been shown to perform sgnfcantly better than the lnear unsharp maskng algorthm [, 3]. We see that sharp edges are recovered n our result but not n cubc unsharp maskng. Fg.4 Left: Blurred Lena mage. Rght: Enhanced mage. Fg. 5 Zoom-n vew of: (left) blurred mage, (mddle) enhanced usng our algorthm, (rght) enhanced usng cubc unsharp maskng. 5. Concluson In ths paper we present a novel mage enhancement method based on mult-scale analyss of the dstrbuton of edge magntude n the wavelet space. By remappng the dstrbuton of the wavelet modulus maxma of the blurred mage to that of a generc sharp mage and reconstructng the mage usng POCS, an enhanced mage that has crsp edges can be obtaned. Our algorthm acheves sgnfcantly better result compared to a state-of-the-art adaptve unsharp maskng technque. Ths research was supported by Korean Aerospace Research Insttute (KARI) and the Australan Research Councl (ARC) under Dscovery Grant DP References [] G. Rampon, A Cubc Unsharp Maskng Technque for Contrast Enhancement, Sgnal Process, vol. 67, pp. -, 998. [] A. Polesel, G. Rampon, and V.J. Mathews, Image enhancement va adaptve unsharp maskng, IEEE Trans. Image Processng, vol.9, pp , 000. [3] Y. Yao, B. Abd, and M. Abd, Dgtal Image wth Extreme Zoom: System Desgn and Image Restoraton, n Proc. of 4 th IEEE Inter. Conf. on Computer Vson Systems (ICVS), 006, pp [4] R.C. Blcu and M. Vehvlanen, Constraned Unsharp Maskng for Image Enhancement, n Proc. of EURASIP Internatonal Conference on Image and Sgnal Processng, Cherbourg-Octevlle, France, July 008, vol.. [5] R. Lukac and K.N. Platanots, Cost-Effectve Sharpenng of Sngle-Sensor Camera Images, n Proc. of IEEE Int. Conf. on Multmeda and Expo, Toronto, Canada, 006, pp [6] Y. Lu, T.M. Ng, B.K. Lew, A Wavelet Based Image Sharpenng Algorthm, n Proc. of Int. Conf. on Computer Scence and Software Engneerng, 008. [7] S. Mallat and S. Zhong, Characterzaton of Sgnals from Multscale Edges, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 4, pp.70-73, 99. [8] X. Gan, A.W.C. Lew, and H. Yan, A POCS-based Constraned Total Least Squares Algorthm for Image Restoraton, Journal of Vsual Communcaton and Image Representaton, vol.7, pp , 006. [9] A.W.C. Lew, H. Yan, and N.F. Law, POCS-based Blockng Artfacts Suppresson usng a Smoothness Constrant Set wth Explct Regon Modelng, IEEE Trans. on Crcuts and Systems for Vdeo Technology, vol.5, pp , June 005. [0] C. Weerasnghe, A.W.C. Lew, and H. Yan, Artfact Reducton n Compressed Images based on Regon Homogenety Constrants usng the Proectons onto Convex Sets Algorthm, IEEE Trans. on Crcuts and Systems for Vdeo Technology, vol., pp , Oct 00. [] A.W.C. Lew and N.F. Law, Reconstructon From -D Wavelet Transform Modulus Maxma usng Proecton, IEE Proceedngs: Vson, Image and Sgnal Processng, vol.47, pp , Aprl 000. Acknowledgement