Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 171 Symmetrcal recursve medan flter for regon smoothng wthout edge dstorton A. Raj Laboratory of Images, Sgnals and Intellgent Systems Pars Est Cretel Unversty, F941, France raj@u-pec.fr Abstract In ths paper, we present a new medan-based operator whch assocates n a multdrectonal flterng scheme the nose suppresson ablty of the recursve medan flter wth the edge conservaton property of the standard medan flter. We show n a prelmnary comparatve study the superorty of the recursve medan flter for smoothng regons corrupted by addtve whte nose, whle the standard medan flter s showed to be better n preservng the edges wth less dstorton. A new operator called the symmetrcal recursve medan flter s then ntroduced amng to mprove nose suppresson propertes as well as to conserve accurately object contours n the fltered mages. The propertes of the proposed operator are dscussed. The operator effectveness for regon smoothng wthout edge dstorton s llustrated on real and synthetc mage processng examples. Index Terms medan flters, multdrectonal flterng, nose suppresson, edge conservaton, shape dstorton I. INTRODUCTION In mage processng applcatons, t s wshed to remove nose as well as to preserve the shape and poston of edges. Lnear flters are generally not very satsfactory because they blur unavodably sharp edges. Among nonlnear technques, order statstc have long been studed and used n dfferent dscplnes [1], [2], [3]. In partcular, many sgnal and mage processng applcatons are based on order statstcs flterng [4], [5]. For example, medan [6], [7], [8] and rank-order [9] based flters can show a great effcency for mpulsve nose suppresson. The medan flter s known to be ncreasngly effectve when the dstrbuton of nose tends towards an mpulsve one. Durng the last decades, a multtude of medanbased operators have been proposed to mprove certan aspects of medan flterng. For example, the FIR-medan hybrd flters [1], [11] assocate nose smoothng propertes of FIR lnear flters wth the edge preservaton ablty of the medan flter. In the same scope, the weghted medan flters [12], [13] affect a bg weght to the medan and non null weghts to the other order statstcs n the flter wndow. The swtched medan flter [14], [15], [16] apples a medan flter only to the pxels that are desgnated as mpulses by a pror mpulse detector, usng the assumpton that an effectve removal of nose mpulses s often accomplshed at the expense of dstorted features f the medan flter s mplemented unformly across the mage. In fact, the medan flter presents an effcent alternatve to lnear flterng when the preservaton of contours s of prmary mportance and when the nose has rather an mpulsve nature. Fg. 1. (a) regon wth sharp corners (b) result of 2D 3x3 medan Dstorton of the angular zones by 2D medan flterng However, an mportant remanng drawback of medan-based flters when appled to mages s the loss or dstorton of certan geometrcal features such as thn lnes and sharp corners, because the 2D structure of the flter wndow does not allow to preserve such features as llustrated by the example of fg.1. The multdrectonal flterng technque [17], [18] propose to combne the outputs of several basc 1D subflters along dfferent drectons n the mage n order to conserve such features. In ths artcle, we propose a new medanbased operator whch assocates the nose suppresson ablty of the recursve medan flter wth the edge conservaton property of the standard medan flter and geometrcal features preservaton by multdrectonal flterng prncple. We show that the proposed operator leads to enhanced performances compared to the classcal standard and recursve medan flters. II. ORDER STATISTIC OPERATORS AND MEDIAN FILTERS By sortng a set W = {X 1,X 2,,X M } of M ndependent and dentcally dstrbuted (..d) random varables, we obtan an ordered sequence X (1) X (2) X (M) where X (r) s called the r th order statstc n W. Let us consder an nfnte length 1D sgnal {X(), =..., 1,, 1,...} to be fltered by a sldng operator wndow W () = {X( m),..., X(),..., X( + m)} of sze M =2m +1 centered at the sample X(). The output of the order statstc flter of sze M s gven by a lnear combnaton of the order statstcs n W (): M Y () = a j X (j) () (1) j=1
172 Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 where the real coeffcents a j verfy M j=1 a j = 1 for an unbased estmaton. Assumng that {X()} s a constant sgnal corrupted by addtve whte nose {n()}, the optmal order statstc flter n the mean square error crteron s gven by: a = R 1 e e t R 1 (2) e wth a t =(a 1 a M ), a =1,..., M representng the flter coeffcents, R =(r kl )= ( E{n (k) n (l) } ),k,l=1,,m representng the M-correlaton matrx of the nose order statstcs vector and e t =(1 1) the M component untary vector. From (2) t results that the optmal quadratc order statstc flter for unform dstrbuton nose s the mddle flter Y () = 1 2 [X (1)()+X (M) ()], whle t s the averagng flter Y () = 1 M M j=1 X (j)() for gaussan nose. Furthermore, t has been shown that the more mpulsve the nose s,.e. wth less concentrated and more heavy taled dstrbuton, the more the optmal quadratc order statstc flter tends towards the standard medan (SM) flter, whch s a partcular case of order statstc flters wth a m+1 =1and a j =for j m +1: Y () =med{x( m),..., X(),..., X( + m)} = X (m+1) () (3) The medan s an effcent non parametrc estmator. For example, an mportant offset n the nput sequence,.e. a nose mpulson, has lttle effect on the output of the medan flter, whle t produces an mportant bas n the output of the averagng flter. Moreover, edges and monotonc changes are left nvarant by medan flterng whle they are smoothed by lnear flters. The medan s also the best scale estmator n the absolute mean error crteron, and the maxmum lkelhood estmator for an exponental..d nput [19]. If we replace the pont beng processed by the output of the medan operator before shftng the flter wndow to the next poston, we obtan the recursve medan (RM) flter defned by: Y () = med{y(-m),...,y(-1),x(),..., X( + m)} (4) III. NOISE SUPPRESSION AND EDGE CONSERVATION PROPERTIES In order to characterze the effect of the standard and recursve medan flters on nosy sgnals, determnstc and statstcal propertes [2] are used nstead of spectral analyss because of the nonlnearty of the medan operator. Gven the nput probablty densty functon f X and dstrbuton functon F X, the output probablty densty functon of the standard medan flter of sze M =2m +1 s gven by [19]: f Y = M. (M 1)! (m!) 2.(F X ) m.(1 F X ) m.f X In the case of the recursve medan flter, there s no analytcal expresson of the output probablty densty functon n the lterature, and t s lkely dffcult to fnd out one because of the hgh correlaton between the output samples, due to recursvty, especally for mportant flter szes. We propose hereafter an expermental comparatve study of the standard and recursve medan flters propertes. We wll consder Fg. 2..15.1.5 -.5 -.1 -.15 E { Y() } 5 1 15 2 25 3 (a) Expected RM output value, M =9, for zero mean and unt varance exponental nose as nput 14 12 1 8 6 4 2 Output Varance recursve medan standard medan 1 3 5 7 9 11 13 (b) Output varance versus flter sze - exponental nput nose: μ =, σ 2 = 4 Nose suppresson propertes of RM and SM flters n ths study the two man prmtves n mages whch are regons and edges, and wll carry out our analyss n 1D case for smplcty. So, let us frst consder, as a regon model, a fnte length constant sgnal of magntude c corrupted by addtve whte nose n(): {X() = c + n(), =, 1,..., L 1}. Snce t s well known that the medan flter s effcent for mpulsve nose, we wll use throughout ths paper an mpulsve nose model characterzed by the exponental probablty densty functon: f n (x) = α 2 exp( α x ), < x < +, α R. Fg.2-a represents expermental expected output value of the recursve medan flter of sze M =9for zero mean and unt varance exponental nose as nput. In ths fgure, we can see that the output of the recursve medan flter shows a non statonary behavor at the begnnng of the flterng operaton. Ths s due to the progressve ntroducton of recursvty startng from the sgnal border: Y (m) = med{x(),,x(m),,x(2m)} Y (m +1) = med{x(1),,x(m 1), Y(m), X(m +1),,X(2m +1)} Y (m +2) = med{x(2),,x(m 2), Y(m), Y(m + 1), X(m +2),,X(2m +2)} for 2m +1 Y () = med{y( m),, Y( 1), X(),,X( + m)} Ths phenomenon decreases for next samples such that the RM output tends towards the sgnal value and becomes statonary after a certan number of teratons. By examnng the expresson of the outputs of the standard and recursve M
Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 173 medan flters, we see n (3) that the output of the standard medan flter s computed at every pont as the medan of the M nput samples around, whle n the case of the recursve medan flter m prevous output samples Y ( m),..., Y ( 1) are used n addton to the (m+1) nput samples X(),X( +1),..., X( + m) to compute the current output Y (), as expressed n (4). From a probablstc pont of vew, the samples Y ( m),..., Y ( 1) representng prevous medan estmatons at dfferent ponts are closer to the sgnal value c than the nput samples X(),X( +1),..., X( + m). Furthermore, the changes that occur when sldng the flterng wndow from the prevous poston 1 to the current poston are: a) replacement of X( 1) by Y ( 1), b) suppresson of Y ( 1 m) from the wndow and c) ntroducton of X( + m) n the wndow. Then, only one new nput sample X( + m) s taken nto account at each teraton, so that: m prevous medan estmatons: Y ( m),..., Y ( 1), m nput samples havng served n Y () =med prevous medan estmatons: X(),..., X( + m 1), 1 new nput sample: X( + m) Thanks to the mpulsve nose rejecton property of the medan, the less the magntude of the new sample s close to the sgnal value c (.e. X( + m) represents a nose sample), the more the probablty of ts rejecton s hgh. The sample X( + m) has a chance to be preserved only f ts magntude s wthn the dynamcs of the prevous medan estmatons Y ( m),..., Y ( 1). Hence, as llustrated n fg.2-a, the successve recursve medan estmatons are closer and closer to the sgnal value c wth more and more concentrated dstrbuton around c as ncreases, whle the successve standard medan estmatons are all made from the nput samples and then have the same dstrbuton at any poston. Ths means that the recursve medan flter has less output varance than the standard medan flter whch s confrmed n fg.2-b representng expermental output varance of the standard and recursve medan flters n the case of an exponental nput nose. We can then conclude that the recursve medan flter s better than the standard medan flter for nose suppresson from regons. In order to compare the edge preservaton performances of the standard and recursve medan flters, let us consder now an nput contanng two adjacent nosy regons: { n() f < X)= n()+a f where n() s an..d zero mean whte nose, and A the magntude of the step edge. From the prevous study on a regon model we can expect that when the recursve medan flter reaches ts statonary behavor,.e. far from the sgnal borders and far from the edge, t removes nose better than the standard medan flter. However, when the fronter between the two regons s encountered, the statonarty of the flter output wll be lost before beng progressvely restored several teratons later when the flter has adapted tself to the new regon value. 1.6 E { Y() } 1.2.8.4-8 -6-4 -2 2 4 6 8 (a) Expected RM output for nosy step edge nput, M =9.6.5.4.3.2.1.6.5.4.3.2.1.6.5.4.3.2.1 Pd() standard medan -15-1 -5 5 1 15 (b) SM probablty of edge detecton Pd() recursve medan left ---> rght -15-1 -5 5 1 15 (c) Left to rght RM probablty of edge detecton Pd() recursve medan rght ---> left -15-1 -5 5 1 15 (d) Rght to left RM probablty of edge detecton Fg. 3. Asymmetrcal and non statonary behavor of the RM flter output near edges and Probablty of edge detecton wth SNR =2, M =5, th =1 Ths s llustrated n fg.3-a representng expermental expected output value of the recursve medan flter. Indeed, ths result shows an asymmetrcal behavor of the recursve medan flter and a deteroraton of ts performances near the edge. The performance deteroraton affects partcularly the second edge sde n ths example where the RM flter s appled from left to rght. In order to characterze the flter edge preservaton ablty, let us defne an expermental parameter representng the probablty of edge detecton at poston by: P d () =P { Y () Y ( 1) >th} (5) where Y () s the flter output and th s a threshold level. Ideally, P d () should be equal to 1 at =and nl elsewhere. Fg.3-b,c,d show the expermental curves of P d () for the standard medan, rght to left and left to rght recursve medan flters. In ths fgure, we can see clearly the asymmetrcal behavor of the recursve medan flter around the edge pont. As explaned above, ths s due to the loss of the statonarty of the flter output when t encounters the edge, whch mples a deteroraton of the flter performances durng several teratons. However, we can see that the recursve medan flter
174 Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 runnng from left to rght performs better than the standard medan flter on the left sde of the edge, wth hgher detecton probablty at the edge pont ( =) and lower false detecton probablty for <, whle t s less effcent than the standard medan flter on the rght sde of the edge wth hgher false detecton probablty at several postons ( >). Smlarly, the recursve medan flter runnng from rght to left performs better than the standard medan flter on the rght sde of the edge wth hgher detecton probablty at the edge pont ( =) and lower false detecton probablty for >, whle t s less effcent than the standard medan flter on the left sde of the edge wth hgher false detecton probablty at several postons ( <). The man dea n the present artcle s to combne n an effcent way recursve medan flters runnng n dfferent scannng drectons n order to mprove nose suppresson and edge preservaton performances. Image background RM flterng drectons object A I B I 1 C L Input (a) Symmetrcal recursve medan flter prncple SM flter SRM flter Output IV. SYMMETRICAL RECURSIVE MEDIAN FILTER It results from the prevous study on regon and edge models that the recursve medan flter s better than the standard medan flter for nose suppresson nsde regons, whle ths superorty s progressvely lost when edges are encountered. In ths secton, we propose a new operator called the symmetrcal recursve medan (SRM) flter, whch s mplemented as a multdrectonal recursve medan flter based on a pror estmaton of the edge ponts n the mage. The dea s that f we can do a good estmaton of the edge ponts map n the mage, then we can apply near each edge pont the recursve medan flter runnng n the drecton that preserves n the best way the contour. Fg.4-a llustrates ths flterng prncple. Along the row number, the object s delmted by ts contours at coordnates I and I 1. The proposed flterng scheme s to apply n each regon the drectonal recursve medan flter that has reached ts statonary behavor. We acheve ths by applyng the recursve medan flter runnng from left to rght to the segments [A, I ], [B,I 1 ] and [C, L], whle the recursve medan flter runnng from rght to left s appled to the segments [O, A], [I,B] and [I 1,C]. A, B and C are the mddles of the segments [O, I ], [I,I 1 ] and [I 1,L] respectvely. Snce the contour locaton consttutes the nformaton that allows to select the optmal flterng drecton, an edge ponts map must be estmated pror to the multdrectonal flterng. Classcal gradent operators are known to work well on hgh contrasted and noseless mages, but they are very weak n the presence of nose. We propose to acheve ths preprocessng phase by the separable standard medan flter snce ths operator has the nterestng property to not dsplace the contours whle t removes suffcently the nose such that an acceptable estmaton of the contour ponts can be made by means of a subsequent gradent operator. Snce the drectonal recursve medan flters have the same performance nsde regons,.e. far from the contours, where they are supposed to have reached a statonary behavor, t does not matter f the pre-processng step detects some false edge ponts. At such ponts, whchever recursve medan flter s used, t wll gve a Fg. 4. Gradent operator Contours estmaton Thresholdng (b) Synoptc dagram Prncple and synoptc dagram of the proposed method good result. Conversely, t s very mportant to detect all of the real edge ponts by the pre-processng step because at each of these ponts only one drectonal recursve medan flter gves a good result. Ths means that n the gradent thresholdng step, we should use rather a low threshold value to ensure that all of the real edge ponts are well detected. The synoptc dagram of the proposed method s shown n fg.4-b. The symmetrcal recursve medan flter s appled to mages n a separable way, along rows and columns of the mage. However, the followng lmtaton should be notced for the proposed operator. We have shown that nose suppresson from a regon by the recursve medan flter becomes optmal when the flter output reaches a statonary behavor. The flter output statonarty s broken and the flter performance s deterorated when a contour s encountered. Consequently, f the mage contans contours whch are very close to each other,.e. very small sze regons, the recursve medan flter wll not be able to reach a statonary behavor at all, such that the symmetrcal recursve medan flter whch ams to select the optmal drectonal recursve medan flter wll not be really useful. So, the proposed technque s not amed for mages contanng fne detals. In order to overcome ths lmtaton, we propose to compute the flter output when two contour ponts are very close to each other (separated by a dstance 3M) by: Y () =med(x(),y 1 (),Y 2 ()), whch s at least as effcent as the two drectonal recursve medan estmatons Y 1 () and Y 2 ().
Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 175 (a) nosy edge, SNR=2 (b) SM, M =7 (a) nosy regon, SNR=2 (b) separable SM (c) left to rght RM M=7 (d) rght to left RM M=7 (e) left to rght RM M=9 (f) rght to left RM M=9 (c) separable RM (d) 2D SM (g) SRM M=7 (h) SRM M=9 Fg. 5. Effect of SM, RM and SRM flters on a nosy step edge V. RESULTS The asymmetrcal behavor of the recursve medan flter near edges s llustrated n fg.5. The mage s fltered along rows only. The recursve medan flter (fg.5-c,d) removes nose from the two regons of the mage better than the standard medan flter (fg.5-b). However, many rows n the fltered mage show a dsplacement of the contour by several pxels n the processng drecton,.e. to the rght for the left to rght recursve medan flter and to the left for the rght to left recursve medan flter. If we apply the proposed technque, assumng that the contour poston (the mddle column) s known, we obtan the result gven n fg.5-g. In ths mage, the symmetrcal recursve medan flter removes nose from the regons and at the same tme preserves well the edge wth very lower dstorton than n the other flters cases. Fg.5 shows also that the performance deteroraton of the recursve medan flter near edges s more mportant when the flter sze ncreases (fg.5-c,e and d,f), whle the symmetrcal recursve medan flter leads to satsfactory edge preservaton performances wth dfferent flter szes (fg.5-g,h). Fg.6-a shows a nosy regon wth horzontal and vertcal edges. The separable recursve medan flter (fg.6-c) shows hgher nose smoothng ablty than the separable standard medan flter (fg.6-b). However, t shows also some dsplacement of the contours n the processng drectons whch s partcularly vsble n the top left corner of the 2D area. Usng 2D flterng wndows allows hgher nose smoothng but at the expence of dstorted geometrcal features. The angular zones of the object are dstorted n fg.6-d,e and the resultng object Fg. 6. mage (e) 2D RM (f) SRM Comparson of SM, RM and SRM flters (of sze 7) on a synthetc n fg.6-e does not present an homogeneous gray level due to the ansotropy of the recursve medan flter. In fg.6-f, the symmetrcal recursve medan flter preserves the shape and poston of the object edges more accurately compared to the standard and recursve medan operators, and at the same tme, smoothes the nose effcently from the object and the mage background regons. The contour ponts map used as the contour nformaton entry for the SRM flter n ths example has been obtaned by applyng Roberts operator to the separable SM output mage (fg.6-b) and thresholdng the resultng gradent at level 29. In the real mage example of fg.7, we can see some dscontnutes along the gull contours n the flterng drectons n the separable RM flter output. 2D RM flter output presents hgher smoothng effect but t also mples some dstortons, for example the gull eye s removed and the angular zones are rounded. The symmetrcal recursve medan flter produces better nose suppresson and edge preservaton compromse than SM and RM flters.
176 Int'l Conf. IP, Comp. Vson, and Pattern Recognton IPCV'16 (a) orgnal wth addtve exponental nose (μ =,σ =2) Fg. 7. (c) separable RM (e) 2D RM (b) separable SM (d) 2D SM (f) SRM Comparson of SM, RM and SRM flters (of sze 7) on a real mage VI. CONCLUSION In ths paper, we show that the standard medan flter produces less edge dstorton n mages whle the recursve medan flter s more effcent for nose suppresson nsde regons but loses ts performance near edges. Then, we propose a new local medan based operator called the symmetrcal recursve medan flter whch assocates the advantages of the two flters n a multdrectonal flterng scheme. The standard medan flter s used n a pre-processng step such that a contour ponts map can be obtaned by a subsequent gradent operator. The mage s then fltered by a multdrectonal recursve medan flter where the contour nformaton s used to apply near each edge pont the best drectonal recursve medan flter. The proposed method has been succesfully tested on synthetc and real mages wth nosy contours. The obtaned results showed sgnfcant enhancement n nose suppresson and edge conservaton performances n comparson to the classcal standard and recursve medan flters. We notced however that the proposed method s not applcable for mages contanng fne detals wth very small sze objects. REFERENCES [1] N. Balakrshnan and A. C. Cohen, Order statstcs & nference: estmaton methods. Elsever, 214. [2] R.-D. Ress, Approxmate dstrbutons of order statstcs: wth applcatons to nonparametrc statstcs. Sprnger Scence & Busness Meda, 212. [3] H. A. Davd and H. N. Nagaraja, Order Statstcs. John Wley & Sons, Inc., 25. [4] A. Raj, Detecton of sgnal transtons by order statstcs flterng, n Proceedngs of the Internatonal Conference on Image Processng, Computer Vson, and Pattern Recognton, IPCV 11, Las Vegas, Nevada, USA, pp. 757 761, CSREA Press, July 18-21 211. [5] C.-H. Hseh and P.-C. Huang, Adaptve rank order flter for mage nose removal, n Computer Scence and Informaton Engneerng, 29 WRI World Congress on, vol. 7, pp. 9 94, March 29. [6] S. Akkoul, R. Ledee, R. Leconge, and R. Harba, A new adaptve swtchng medan flter, Sgnal Processng Letters, IEEE, vol. 17, pp. 587 59, June 21. [7] K. Toh, H. Ibrahm, and M. Mahyuddn, Salt-and-pepper nose detecton and reducton usng fuzzy swtchng medan flter, IEEE Transactons on Consumer Electroncs, vol. 54, no. 4, pp. 1956 1961, 28. [8] A. Toprak and I. Gler, Impulse nose reducton n medcal mages wth the use of swtch mode fuzzy adaptve medan flter, Dgtal Sgnal Processng, vol. 17, no. 4, pp. 711 723, 27. [9] I. Azenberg and C. Butakoff, Effectve mpulse detector based on rankorder crtera, Sgnal Processng Letters, IEEE, vol. 11, pp. 363 366, March 24. [1] V. Lyandres and S. Prmak, The fr-medan suppresson of mpulsve nterference, Sgnal Processng, vol. 8, no. 5, pp. 883 887, 2. [11] A. Flag, G. R. Arce, and K. E. Barner, Affne order-statstc flters: medanzaton of lnear fr flters, IEEE Transactons on Sgnal Processng, vol. 46, pp. 211 2112, Aug 1998. [12] T. Chen and H. R. Wu, Adaptve mpulse detecton usng centerweghted medan flters, IEEE Sgnal Processng Letters, vol.8,pp.1 3, Jan 21. [13] G. R. Arce and J. L. Paredes, Recursve weghted medan flters admttng negatve weghts and ther optmzaton, IEEE Transactons on Sgnal Processng, vol. 48, pp. 768 779, Mar 2. [14] Z. Wang and D. Zhang, Progressve swtchng medan flter for the removal of mpulse nose from hghly corrupted mages, IEEE Transactons on Crcuts and Systems II: Analog and Dgtal Sgnal Processng, vol. 46, pp. 78 8, Jan 1999. [15] H.-L. Eng and K.-K. Ma, Nose adaptve soft-swtchng medan flter, IEEE Transactons on Image Processng, vol. 1, pp. 242 251, Feb 21. [16] S. Zhang and M. A. Karm, A new mpulse detector for swtchng medan flters, IEEE Sgnal Processng Letters, vol. 9, pp. 36 363, Nov 22. [17] X. Wang, Generalzed multstage medan flters, IEEE Transactons on Image Processng, vol. 1, pp. 543 545, Oct 1992. [18] V. R. Vjaykumar, D. Ebenezer, and P. T. Vanath, Detal preservng medan based flter for mpulse nose removal n dgtal mages, n Sgnal Processng, 28. ICSP 28. 9th Internatonal Conference on, pp. 793 796, Oct 28. [19] I. Ptas and A. N. Venetsanopoulos, Order statstcs n dgtal mage processng, Proceedngs of the IEEE, vol. 8, no. 12, pp. 1893 1921, 1992. [2] U. Eckhardt, Root mages of medan flters, Journal of Mathematcal Imagng and Vson, vol. 19, no. 1, pp. 63 7, 23.