Adaptive Window Size Image De-Noiizing Baed on Interection of Confidence Interval

Transcription

1 Journal of Mathematical Imaging and Viion 16: 223±235, 2002 # 2002 Kluwer Academic Publiher. Manufactured in The Netherland. Adaptive Window Size Image De-noiing Baed on Interection of Confidence Interval (ICI) Rule VLADIMIR KATKOVNIK Department of Mechatronic, Kwangju Intitute of Science and Technology Kwangju , Republic of Korea KAREN EGIAZARIAN & JAAKKO ASTOLA Signal Proceing Laboratory, Tampere Univerity of Technology, FIN-33101, Tampere, Finland Abtract. We decribe a novel approach to olve a problem of window ize (bandwidth) election for filtering an image ignal given with a noie. The approach i baed on the interection of confidence interval (ICI) rule and give the algorithm, which i imple to implement and nearly optimal in the point-wie mean quared error rik. The local polynomial approximation (LPA) i ued in order to derive the 2D tranform (filter) and demontrate the efficiency of the approach. The ICI rule give the adaptive varying window ize and enable the algorithm to be patially adaptive in the ene that it quality i cloe to that which one could achieve if the moothne of the etimated ignal wa known in advance. Optimization of the threhold (deign parameter of the ICI) i tudied. It i hown that the cro-validation adjutment of the threhold ignificantly improve the algorithm accuracy. In particular, imulation demontrate that the adaptive tranform with the adjuted threhold parameter perform better than the adaptive wavelet etimator. Keyword: local adaptive window ize tranform, local polynomial approximation, window ize election 1. Introduction A noie removal (de-noiing) i one of the important problem in image proceing. Among other approache to thi problem the local polynomial approximation (LPA) can be treated a probably one of the mot theoretically jutified and well tudied one. Originally LPA wa propoed and developed in tatitic for proceing calar and multidimenional noiy data. It i a powerful nonparametric technique which provide etimate in a point-wie manner baed on a mean quare polynomial fitting in a liding window e.g. [1, 3, 5, 6, 13, 16]. In term of the ignal and image proceing the LPA i a flexible tool to deign 2D tranform with precribed reproductive propertie with repect to polynomial (mooth) component of ignal. The invariant and variant optimal window ize election ha been tudied thoroughly by many author. Thee optimal, in particular, varying data-driven window ize method are of pecial interet for the problem where the piecewie mooth approximation are the mot natural and relevant one. Some image de-noiing problem provide good example of thee cae. A crucial difference between the nonparametric LPA etimate and the more traditional parametric one, ay the polynomial mean quared etimate, i that the latter are formed a unbiaed one while the former are biaed on the definition and the reaonable choice of the biaedne controlled by the value of the window ize i of importance. It can be emphaized that the problem of the optimal window ize election admit an accurate mathematical formulation in term of the nonparametric approach, where the optimal window ize i defined by a compromie between the bia and the variance of etimation.

2 224 Katkovnik, Egiazarian and Atola Two main idea are exploited for adaptive (datadriven) window ize election. The firt one i baed on etimation of the biaedne and the variance of the etimate of the ignal with the correponding optimal window ize calculation baed on theoretical formula. However, the bia depend on the derivative of a given ignal. Thu, we need to etimate thee derivative and for thi purpoe to elect auxiliary window ize. Actually thi ort of method, known a ``pilot etimate'', i quite complex in implementation and have quite a few deign parameter. The econd alternative idea doe not have to deal with the bia etimation. Thi group of method i baed on the quality-of-fit tatitic uch a the cro-validation, generalized cro-validation, C p, Akaike criteria, etc., which are applied for direct optimization of the accuracy. A number of publication concerning the window ize election problem i very large and growing quickly. A review on the field or even it brief analyi i far beyond of the goal of thi paper. Here we give only few reference illutrating the baic progre in different direction. A ucceful implementation of the firt approach baed on the pilot etimate ha been reported by everal author. An automatic local window ize elector with etimation of the higher order derivative of y(x), which are plugged into the local rik expreion, wa developed in [3]. The empirical-bia window ize election [19] ue the etimate at everal window ize in order to approximate the bia, and reult in quite an efficient adaptive moother for etimation of the function and it derivative. The imilar idea have been exploited in the adaptive moother decribed in [20]. Mot of publication concerning the econd approach are related to a data-baed global (contant) window ize election e.g. [6, 7, 13]. The linear LPA with the varying window ize found by minimization the o-called ``peudo-mean quared error'' i conidered in [18]. The target point i left out of the averaging in the peudo-mean quared error what differ thi method from the tandard mean quare method. It i reported that the propoed peudo-mean quared error work better then the local cro-validation [18]. Thi paper i baed on a quite recent new development. The interection of confidence interval (ICI) rule originally wa propoed and developed in [4] and [8] for de-noiing of 1D obervation and hown to be quite efficient in particular for many ditinct application e.g. [10, 12, 14]. Firt reult on application of the ICI rule to image de-noiing have been reported in [9, 15]. Thi paper preent a ytematic development of the ICI rule for image de-noiing including baic idea, algorithm, theoretical performance analyi and imulation reult. We introduce adaptive 2D tranform for moothing an image intenity function given with an additive noie. Thee tranform are able to produce a piece-wie mooth urface with a mall number of dicontinuitie in the intenity function or it derivative. Thi allow certain deirable feature of image uch a jump or intantaneou lope change to be preerved. The further paper i organized a follow. In ection 2 the LPA technique i decribed in detail with theoretical reult concerning the accuracy analyi. The idea and algorithm implementation of the ICI window ize election i decribed in ection 3 a well a the data-driven adjutment of the threhold parameter and the multiple window LPA. Simulation, given in ection 4 illutrate a good performance of the propoed method a well a ome modification of the baic algorithm. 2. Local Polynomial Approximation 2.1. Tranform Baed on LPA We tart with a dicuion of 2D linear tranform derived on the bae of the LPA method. Suppoe that we are given by noiy obervation of the image intenity y(x); y 2 R 1 ; x 2 R 2 ; on the regular or irregular grid of argument value x() ˆ (x 1 (); x 2 ()) being two dimenional vector with component x 1 () and x 2 () and the parameter indicating correponding th pixel of the image. Then, the noiy data can be given a: z(x) ˆ y(x) "(x) (1) where "(x) are independent and identically ditributed (i.i.d.) random Gauian error, E["(x)] ˆ 0; E[" 2 (x)] ˆ 2 : It i aumed that y(x) belong to a nonparametric cla of piece-wie continuou r differentiable function F r ˆfjD r r 1; r 2 y(x)j L r (x) L r ; 8 r 1 r 2 ˆ rg; (2)

3 Adaptive Window Size Image De-Noiing 225 r1 r2 where D r 1; r 2 r 1 r r 1 x r 2 x 2 i a differentiation operator and L r i a finite contant. Our goal i to etimate y(x) depending on the obervation fz(x)g with the point-wie mean quared error (MSE) rik which i a mall a poible. The LPA of y(x) i applied in the following form. Firt, a part of the truncated Taylor erie i ued in order to approximate the varying intenity y(x); and econd, thi expanion i exploited locally in a comparatively mall area. In fact, the local expanion i applied in order to calculate the etimate for a ingle ``central'' pixel only. For the next pixel the calculation hould be repeated. Thi point-wie procedure determine a nonparametric character of etimation. The following criteria function i applied in the LPA e.g. [1, 3, 6, 13, 16]: J h (x) ˆ w h (x() x)(z(x()) C T (x() x)) 2 (3) (x) ˆ ( 1 (x); 2 (x); :::; M (x)) T ; C ˆ (C 1 ; C 2 ; :::; C M ) T ; x() ˆ (x 1 (); x 2 ()); (4) where (x) i a et of linear independent 2D polynomial of the power from 0 to m with 1 ˆ 1. A total number of thee polynomial i equal to M ˆ C2 2 m (2 m)! (2 m)(1 m) ˆ 2m! ˆ 2. The term ``coordinate function'' i ometime ued for k. The window w h (x) ˆ w(x=h)=h formalize the localization of fitting with repect to the centre x, while the cale parameter h > 0 determine the ize of the window. The windowing weight w h (x) ˆ w(x=h)=h i a function atifying the conventional propertie: w(x) 0; w(0) ˆ max x w(x), RR w(x 1 ; x 2 ) dx 1 dx 2 =1. A multiplicative window w(x) ˆ w 1 (x 1 )w 2 (x 2 ), where w 1 (x 1 ) and w 2 (x 2 ) are function of calar argument, i ued in many application. If the window i rectangular all obervation enter in the criteria function with equal weight. Nonrectangular window uch a triangular, quadratic, Epanechnikov and o on [1, 3, 6, 13, 16] uually precribe higher weight to obervation which are cloer to the centre x: The point-wie LPA inure the reproduction propertie of the etimate with repect to the polynomial component of y(x). But it hould be emphaized that the LPA etimate of y(x) i not a polynomial function. Thi i a principal difference between the nonparametric LPA and the correponding parametric model. For m ˆ 2 a full et of the linear independent polynomial i of the form 1 ˆ 1; for m ˆ 0; 2 ˆ x 1 ; 3 ˆ x 2 ; for m ˆ 1; 4 ˆ x 1 x 2 ; 5 ˆ x 2 1 =2; 6 ˆ x 2 2 =2; for m ˆ 2; (5) 2 m! with M ˆ 2m! ˆ 6: Minimizing J h (x) with repect to C; ^C(x; h) ˆ arg min J h; (6) C2R M give ^y(x) ˆ ^C 1 (x; h) a an etimate of y(x); and ^C l (x; h); l ˆ 2; :::; M; a etimate of the derivative of y(x). For the polynomial (5) thee etimated derivative are D 1 1; 0 y(x); D1 0; 1 y(x); D2 1; 1 y(x); D2 0; 2 y(x); D 2 2; 0 y(x) repectively. Recall that the firt and econd order derivative (D k 1 k 2 k 1 ; k 2 y(x); 1 k 1 k 2 2) are ued a a tool for image egmentation and enhancement. However, in thi paper we concern with the etimation of the intenity function only. Thee etimate can be repreented in the form of a linear tranform (filter) ^y(x; h) ˆ g 1 (x; x(); h)z(x()); (7) where g 1 i a firt element of the vector g given by the equation g ˆ 1 w h (x() x)(x() x); ˆ w h (x() x)(x() x) T (x() x): (8) It can be verified that for any polynomial y(x) of the power m the etimate (7) i accurate. In particular, for the polynomial (5) it mean that g 1 (x; x(); h) k (x()) ˆ k (x); for k ˆ 1; :::; M; 8x: (9) It how that the tranform with the weight g 1 ha an accurate reproductive propertie for the 2D polynomial component of the intenity up to the power m ˆ 2. The linear tranform (7)-(8) can be applied to data given on any regular or irregular grid, in particular, to data with lot obervation and for data interpolation problem when x doe not belong to the grid fx()g. It

4 226 Katkovnik, Egiazarian and Atola i aumed in the formula (7)-(8) that the ummation i alway performed within boundarie of the image frame. For the regular infinite grid and x belonging to thi grid the linear etimator (7)-(8) can be written a a homogeneou (tationary) tranform: ^y(x; h) ˆ g 1 (x(); h) z(x x()); (10) where g 1 i a firt element of the vector g g ˆ 1 w h (x()) (x()); ˆ w h (x()) (x()) T (x()): (11) Then, the equation (9) for the polynomial (5) have a form of moment retriction on the weight function g 1 g 1 (x(); h) ˆ 1; g 1 (x(); h)x 1 () ˆ 0; g 1 (x(); h)x 2 () ˆ 0; g 1 (x(); h)x 1 ()x 2 () ˆ 0; g 1 (x(); h)x 2 1 () ˆ 0; g 1 (x(); h)x 2 2 () ˆ 0: (12) An important difference between the etimate (7)- (8) and (10)-(11) i that the latter aume that a number of obervation in the etimate doe not depend on the location of x in the image. A a reult the function z(x) hould be defined beyond the boundarie of the image frame. Naturally, the accurate polynomial fitting i not fulfilled in thi cae for pixel in a neighborhood of the boundarie of the frame. The etimate in the form (7)-(8) i free from thi boundary effect. Actually, the deign of the weight g 1 by the formula (7)-(8) or (10)-(11) i quite of a general nature. Say, we can tart from the zero order approximation with the only one coordinate function 1 ˆ 1. In thi cae the fitter give an approximation by a contant into the liding window. It i able to guaranty the accurate unbiaed reproduction only for the contant component of the intenity. Further, the linear function 2 ˆ x 1 and 3 ˆ x 2 can be added to 1 ˆ 1 a element of the vector ˆ ( 1 ; 2 ; 3 ) T : The correponding tranform i able to obtain the accurate reproduction of contant and linear on x 1 and x 2 component of the intenity. A a next tep, the cro term 4 ˆ x 1 x 2 can be involved into the LPA by including in ˆ ( 1 ; 2 ; 3 ; 4 ) T : Further, one or both of the quadratic function 5 ˆ x 2 1 =2; 6 ˆ x 2 2 =2 can be ued in the LPA: Thu, it i not neceary to ue a full et of the coordinate function k of ome particular power m. The conidered deign of the tranform can be ued in order to obtain deirable propertie with repect to any component of the intenity function. Naturally, non-polynomial coordinate function can be applied in the local approximation in a traightforward manner. The linear etimator (7)-(8) and (10)-(11) have a very long prehitory e.g. [1, 3, 5, 6, 13, 16]. They are a very popular tool in tatitic and ignal proceing with application to a wide variety of the field for moothing, filtering, differentiation, interpolation and extrapolation Accuracy of the LPA It i well known that window ize election i a crucial point of the efficiency of the local etimator. When h i relatively mall, the LPA give a good approximation of y(x) but then fewer data are ued and the etimate are more variable and enitive with repect to the noie. The bet choice of h involve a trade-off between the bia and variance, which depend on the order of the derivative being involved in the LPA, a ample period, the noie variance, and value of the derivative of y(x) beyond the order ued in the LPA: The etimation error can be repreented a follow e(x; h) ˆ y(x) ^y(x; h) ˆ y(x) g 1 (x; x(); h)z(x()) ˆ E(e(x; h)) e 0 (x; h); (13) where E(e(x; h)) ˆ y(x) P g 1(x; x(); h) y(x()) = P g 1(x; x(); h)[y(x) y(x()] and e 0 (x; h) ˆ P g 1 (x; x(); h) "(x()) are the bia and random component of the etimation error. The Taylor erie of the power r with the reidual term in Lagrangian' form i ued for the difference y(x) y(x()) :

5 Adaptive Window Size Image De-Noiing 227 y(x) y(x()) ˆ S 1 () S 2 (); S 1 () ˆ S 2 () ˆ r 1 k 1 k 2 ˆ 1 D k 1 k 2 k 1; k 2 k 1 k 2ˆr 1 k 1!k 2! [x 1 x 1 ()] k1 [x 2 x 2 ()] k2 y(x); 1 k 1!k 2! [x 1 x 1 ()] k1 [x 2 x 2 ()] k2 D r k 1; k 2 y( x() (1 ) x); 0 1: (14) Aume that in (7)-(8) i a full et of 2D polynomial of the power m and r ˆ m 1: Then, according to (9), g 1 (x; x(); h)s 1 () ˆ 0 and the bia i defined by the (m 1) th derivative Dk m 1 1 ; k 2 y. Subtituting (14) into the formula for the bia-error, we obtain 1 je(e(x; h))j L m 1 k k 1 k 2 ˆ m 1 1!k 2! jg 1 (x; x(); h)jjx 1 x 1 ()j k 1 jx 2 x 2 ()j k 2 ; where according to (2) we ued that jd m 1 k 1 ; k 2 yjl m 1. The variance of the random component i given by the equation var(x; h) ˆ E(e 0 (x; h) 2 ) ˆ 2 (g 1 (x; x(); h)) 2 : (15) In order to derive the formula which provide a clear dependence of the accuracy on the window width parameter h we aume that the ampling period i mall,! 0, and the 2D ampling grid i regular. The um in the all above formula can be tranformed into integral, and, after ome manipulation, we arrive to the following expreion: Z Z ^y(x; h) ˆ g 1 (u)y(x hu) du 1 du 2 ; g ˆ 1 w(u)(u); Z Z ˆ w(u)(u) T (u) du 1 du 2 ; g ˆ (g 1 ; :::; g M ) T ; u ˆ (u 1 ; u 2 ): (16) Then, the formula for the bia and variance can be given in the explicit analytical form je(e(x; h))j h m 1 L m 1 (x)a; Z Z 1 A ˆ jg 1 (u)jju 1 j k 1 ju 2 j k 2 k 1!k 2! k 1 k 2 ˆ m 1 du 1 du 2 ; var(x; h) ' 2 2 Z Z B; B ˆ jg 1 (u)j 2 du 1 du 2 : (17) h 2 Thu, the point-wie mean quared rik r(x; h) in aymptotic with a mall can be repreented a follow r(x; h) ˆ E(e(x; h)) 2! 2 (x; h) 2 2 B ˆ r(x; h); h 2!(x; h) ˆ h m 1 L m 1 (x)a; (18) where!(x; h) denote the upper bound of the bia. Minimizing on h the upper bound r(x; h) ofthe mean quared rik give for the ideal value of the window ize and the rik upper bound: h (x) ˆ 2 2 1=(2m 4) B A 2 (L m 1 (x)) 2 2 ; 2 ˆ 1 m 1 (19) and r (x) ˆ r(x; h (x)) ˆ var (x)(1 2 ); var (x) ˆ var(e 0 (x; h (x))); ˆ! (x)=td (x); p! (x) ˆ!(x; h (x)); td (x) ˆ var (x); 20 where the contant which i not depending on x; how a proportion between the upper bound of the bia and tandard deviation of the etimation error at the ideal window ize h (x): The formula (19) and (20) demontrate that the ideal window ize h (x) depend on the (m 1) th derivative of y(x) and the ideal variance-bia trade-off i achieved when the ratio between the bia and tandard deviation! (x)=td (x) i equal to. It can be een that!(x; h) < td(x; h) if h < h > td(x; h) if h > h (21) In what follow thi inequality i ued in order to tet the hypothee: h > < h :

6 228 Katkovnik, Egiazarian and Atola 3. Adaptive Window Size Selection 3.1. The idea of the ICI The etimation error of the LPA can be repreented in the form je(x; h) j ˆ jy(x) ^y(x; h) j!(x; h) e 0 (x; h) ; (22) where!(x; h) i the upper bound of the etimation bia and e 0 (x; h) i a random error with the probability denity N(0; td 2 (x; h)): Then e 0 (x; h) 1 a=2 td(x; h) hold with the probability p ˆ 1 a, where 1 a=2 i (1 a=2) th quantile of the tandard Gauian ditribution, and with the ame probability je(x; h) j!(x; h) 1 a=2 td(x; h): (23) It follow from (21) that the inequality (23) can be weakened to je(x; h) j ( 1 a=2 ) td(x; h): (24) Now let u introduce a finite et of window ize: H ˆfh 1 < h 2 < :::: < h J g; tarting with a quite mall h 1 ; and, according to (24), determine a equence of the confidence interval D j) of the biaed etimate a follow D( j) ˆ [^y(x; h j ) td(x; h j ); ^y(x; h j ) td(x; h j )]; (25) The following i the ICI tatitic, which i ued in order to tet the very exitence of thi common point and in order to obtain the adaptive window ize value: Conider the interection of the interval D( j); 1 j i; with increaing i, and let i be the larget of thoe i for which the interval D( j); 1 j i; have a point in common. Thi i define the adaptive window ize and the adaptive LPA etimate a follow 3.2. Algorithm ^y (x) ˆ ^y(x; h (x)); h (x) ˆ h i : (28) The following algorithm implement the procedure (28). Determine the equence of the upper and lower bound of the confidence interval D(i) a follow D(i) ˆ [L i ; U i ]; U i ˆ ^y(x; h i ) td(x; h i ); L i ˆ ^y(x; h i ) td(x; h i ); (29) where i given by (26). Let L i 1 ˆ max[l i ; L i 1 ]; U i 1 ˆ min[ U i ; U i 1 ]; i ˆ 1; 2; :::; J; L 1 ˆ L 1 ; U 1 ˆ U 1 30 where 1 ˆ p 1 a=2 (26) m 1 i the threhold of the confidence interval. Then for h ˆ h j (24) i of the form y(x) 2D( j); (27) and we can conclude from (23) and (24) that while h j < h hold for h ˆ h j ; 1 j i; all of the interval D( j); 1 j i; have a point in common, namely, y(x). In the oppoite cae, when the interection of the confidence interval i empty it indicate that h j > h : Thu, the interection of the confidence interval can be ued in order to verify the inequalitie (21). Figure 1. Graphical illutration of ICI rule.

7 Adaptive Window Size Image De-Noiing 229 then the adaptive window length h i i the larget i when L i U i (31) i till atified. Thi i i the larget of thoe i for which the confidence interval D(i) have a point in common a it i dicued above. We wih to emphaize that thi window ize ICI election procedure require a knowledge of the etimate and it variance only. The ICI rule i graphically illutrated in Fig. 1, where the vertical line with arrow how the ucceive interection of the confidence interval (1, 2), (1, 2, 3), and (1, 2, 3, 4). Auming that the interection with the forth confidence interval (correponding h ˆ h 4 ) i empty, we obtain the ``optimal'' adaptive window ize h ˆ h 3 : 3.3. Adjutment of the Threhold The threhold parameter in (29) play a crucial role in the performance of the algorithm. Too large or too mall reult in overmoothing and undermoothing data. Let u preent ome figure for following from the theoretical analyi. Auming a ˆ 0:05 or 0:01, then 1 a=2 ˆ 2 or 3 repectively, we obtain for the threhold: 3:0; for p ˆ 0:05; ˆ for m ˆ 0; 4:0; for p ˆ 0:01; 2:7; for p ˆ 0:05; ˆ 3:7; for p ˆ 0:01: for m ˆ 1; Remind that the formula for (19) determining the optimal threhold i obtained for the aymptotic a! 0 and provided that the intenity y(x) imooth enough in the neighborhood of x, i.e. the firt and econd order derivative exit repectively when the zero order (m ˆ 0) or the firt order (m ˆ 1) LPA i applied. In practice, the moothne of y(x) can not be guaranteed for every x and can be not mall. Another uncertain point concern the confidence level a and quantile 1 a=2 which are ued in calculation of the threhold. Thu, we may conclude, that there are ambiguitie, which influence a election of the threhold and thee ambiguitie cannot be reolved in term of the theoretical analyi only. However, the threhold i a natural invariant deign parameter of the algorithm, which can be ued in order to refine the algorithm and to adjut it to the available obervation. We produced a number of Monte-Carlo imulation experiment in order to verify a role of : In particular, the MSE of de-noiing i minimized on in every Monte-Carlo imulation run. The optimal value of found in thi way are random but have very mall variation. Actually, it mean that thee optimal value of depend on tatitical propertie of the noie but not particular ample ued in the Monte- Carlo run. Thee optimal are quite robut with repect to random noie component of obervation. The cro-validation (CV ) i one of the popular tool developed in quality-of-fit tatitic for model election and adjutment e.g. [7]. For the linear etimator in the form (7) the CV lo function can be repreented a a weighted um of quared reidual e.g. [8]: I CV ˆ z ^y(x(); h 2 (x())) 1 g 1 (x(); x(); h : (32) (x())) Thu, the procedure (29)-(31) i aumed to be repeated for every 2 G; G ˆf 1 ; 2 ; ::: NG g; and ^ ˆ arg min 2G I CV (33) give the adjuted threhold parameter value. The cro-validation in the form (32) preent quite a reaonable and efficient elector for. Our attempt to ue intead of the cro-validation another qualityof-fit tatitic, in particular the C P, Akaike criteria and it modification (ee e.g. [7]), which are different from I CV only by the ued weight of the reidual, have not hown an improvement in accuracy. The adjuted adaptive LPA etimation conit of the following baic tep: 1. Set ˆ l, l ˆ 1; 2; :::; N G and x ˆ x(); ˆ 1; 2; ::; N: 2. For h ˆ h i, i ˆ 1; :::; J; calculate the etimate ^y(x(); h), the adaptive window ize h (x()) and the etimate ^y(x(); h (x())): 4. Repeat Step 2 for all x(), ˆ 1; 2; ::; N; and l ; l ˆ 1; 2; :::; N G : 5. Find ^ from (33) and elect etimate ^y(x(); h (x()) correponding to ^ a the final one.

8 230 Katkovnik, Egiazarian and Atola The tandard deviation ued in td(x; h) i etimated by ^ ˆ fmedian( jz z 1 j : p ˆ 2; ::; N)g=( 2 0:6745): (34) The average 1 N 1 P N n ˆ 2 (z z 1 ) 2 could alo be applied a an etimate of 2. However, we prefer a median (34) a a robut etimate Multiple Window Etimation Different idea can be ued for a deign of the varying window for proceing of 2D image ignal. The implet and tandard one aume that a ymmetric quare window i applied for every pixel and the ize of the window i the only varying parameter to be found. A more complex approach aume that the varying window i compoed from a number of eparate egment, ay from four quadrant hown in Figure 2. The centre of the window i the initial point of the Carteian coordinate ytem (0; 0). Each egment i a quare covering a part of the correponding quadrant. It i aumed that thi initial point (0; 0) i the centre of the LPA etimate for each quare egment. The ize of thee quare are the parameter of the combined window. The ICI rule i ued for independent election of the ize of thee eparate four window. There are a number of way to fue etimate obtained for the eparate window egment into the one final etimate. Some of our imulation reult preented in thi paper are obtained for the following final etimate: ^y(x) ˆ k j^y j (x; h j (x)); j ˆ [1; 2; 3; 4] k j ˆ td 2 j td 2 ; td2 ˆ j ˆ [1; 2; 3; 4] td 2 j ; (35) where ^y j (x; h j (x)) are the etimate with the ICI rule adaptive window ize, j ˆ [1; 2; 3; 4]; obtained repectively for the window 1; 2; 3; 4 in Figure 2. Further k j and td j are the weight and the tandard deviation of thee etimate ^y j (x; h j (x)). In the etimate (35) we ue a linear fuing of the etimate with the invere tandard deviation of the etimate Figure 2. Four quadrant window 1, 2, 3 and 4 ued for directional window ize election by the ICI rule. a weight. Similar multiple combined window etimate have been applied in [4] and [8] for 1D function etimation. 4. Algorithm and Simulation Reult The ICI rule for window ize election and the multiple window etimate introduced in Section 3 define a baic algorithm developed for noie reduction. A number of modification of thi algorithm ha been developed and tudied. Thee modification ue different method in order to form the combined window, pecial correction of the adaptive window ize given by the ICI rule and the different etimation method applied to the data in thee varying window. In particular, an adaptive ize and hape window growing pixel-by-pixel uing the ICI rule i propoed in [2]. In thi paper the ICI rule i ued for the pointwie varying window ize egmentation of the image only, while the orthogonal tranform etimator (e.g. wavelet and dicrete coine tranform (DCT)) different from the LPA are applied for de-noiing in thi varying ize egment. The median filter equipped with the ICI for the varying window ize election are reported in [11, 14]. It wa noticed that the ICI adaptive window ize, in particular for mall ; can be corrupted by pike which erroneouly iolate mall value to the window ize [8]. The preliminary filtering of h j (x) conidered a a function of x, ay by a imple median filter, i able to improve the quality of de-noiing.

9 Adaptive Window Size Image De-Noiing 231 In thi ection we preent imulation reult which illutrate the efficiency of the ICI rule and give an inight into behavior of the etimate. We conider alo ome modified verion of the baic algorithm. 1. Let u tart from a imple binary image. Figure 3 how a true image, noiy image and de-noied image. The baic LPA tranform with m ˆ 0 and ˆ 4:0 i ued. The adaptive window ize for window 1, 2, 3 and 4 are given in Figure 4. Small and large window ize are hown there by black and white, repectively. Iolated black point in Figure 4 are pike correponding to random mall window ize given by the ICI rule. It deerve to be mentioned that thee iolated pike have different location into four different window and do not influence the final de-noied image hown in Figure 3. Actually thee pike can be eliminated by increaing the value of. However, it reult in increaing of the value of the root mean quared error (RMSE). Preenting Figure 4 we emphaize that the obtained window ize actually correpond to the intuitively clear behavior of the varying window ize relevant to the moothing of the data if the true image i known. Thu the window ize delineate the true image of the quare and the variation of the window ize provide a hadowing of the image from different ide of the image in full agreement with the directional window ued for moothing (ee Figure 2). 2. Now let u demontrate a different algorithm of uing the ame ICI rule. The developed algorithm comprie of the following two part. The firt part i applied for a point-wie image egmentation. Thi egmentation aume that the LPA with ICI rule i ued for every pixel in order to find the adaptive ize of four directional rectangular window a hown in Figure 2. A a reult, every pixel can be an entry of many different etimate obtained for adaptive varying ize window with different center. The econd part of the algorithm aume that the DCT tranform filtering [2] i applied for every of thee adaptive ize window. All obtained etimate are accumulated in a buffer and averaged in order to produce the final etimate for every pixel. Experiment were performed on the tet image ``Cameraman'' (8 bit gray-cale image) corrupted by different type of noie. The reult are compared with the wavelet tranform baed (Haar, Symmlet, Coiflet, Tranlation Invariant [17]) and Wiener filter. The new algorithm howed a valuable ignal-to-noie ration (SNR) improvement (more than 4-5 db) for mot of the cae. Some illutrative image are given in Figure 5. Figure 5a,b how the original and noiy image, while the DCT etimate decribed above i given in Figure 5d. The RMSE value how a valuable original noie reduction. The viual quality i quite acceptable for thi level of the noie. In Figure 5c we how a an intermediate reult the filtering obtained from the zero order LPA (ample averaging). The etimate obtained for four adaptive varying window are averaged with the weight reciprocal to the variance of thee True Image Noiy Image, σ= Etimate, RMSE= Figure 3. True image, noiy image and denoied image. The baic LPA filter with m ˆ 0 and = 4.0.

10 232 Katkovnik, Egiazarian and Atola Adapt Win I Adapt Win II True image Noiy image, σ= a) b) Adapt Win III Adapt Win IV LPA den, RMSE= DCT den, RMSE= c) c) d) Figure 4. Adaptive window ize for window I, II, III and IV. The LPA filter with m = 0 and = 4.0. Black and white correpond to mall and large window ize repectively. Iolated black point are pike in the window ize correponding to random mall window ize by the ICI rule. etimate. Figure 6 how the varying adaptive window ize obtained repectively for the window I, II, III and IV (Figure 2, h k ˆ 2 k ; k ˆ 0; 1; :::; 6). Here black and white area correpond, repectively, to mall and large window ize. Again, the adaptive window ize delineate contour of the image and demontrate a very reaonable performance of the ICI rule a a window ize elector. Let u dicu on a role of the threhold. Simulation with ˆ 1:5; 2:0; 2:5; 3:0; 3:5; 4:0 (36) wa produced with uing DCT tranform filtering and ICI window ize election a it i decribed in Section III. Reulting RMSE are obtained RMSE ˆ 0:0806; 0:0605; 0:0706; 0:0811; 0:0954; 0:1102; (37) repectively. Thu, the bet performance with RMSE ˆ 0:0605 wa achieved for ˆ 2:0: It i the ideal reult a it aume that the true image, ued for Figure 5. a) True image, b) Noiy image, c) LPA denoiing, d) DCT denoiing with ICI adpative window ize. RMSE calculation, wa known. Comparion of RMSE value how that the improvement up to 5 db can be obtained by optimization. The image preented in Figure 5d i given for the value of the threhold ˆ 2:5 obtained by the CV adjutment of a it i decribed Section 3. The grid (36) wa ued in the optimization problem (33). Thi reult i quite cloe to the optimal ˆ 2:0. It i well known that an improvement of qualitative criteria, uch a RMSE, doe not guaranty a viual improvement of image. However, the imulation confirm that in term of thi ort of criteria the CV can be applied for the adjutment of the de-noiing algorithm with varying data-driven window ize. 3. Here we conider the DCT tranform filtering equipped with varying adaptive window ize. Thu, we apply the DCT tranform for image de-noiing intead of the LPA tranform. Further, we apply it in the ICI rule for the varying window ize election. The latter i done by uing in (25) the tandard deviation td(x; h j ) of the DCT tranform. More detail on thi algorithm a well a it tatitical jutification can be found in [2].

11 Adaptive Window Size Image De-Noiing 233 Opt win I Opt win II Opt win III Opt win IV Figure 6. Adaptive window ize obtained by ICI with = 2.5. Figure 7. True ``Montage'' image; noiy image; the etimate with the varying adaptive window ize ( = 2.0); the etimate with the fixed ideal ize window. Reult of the local adaptive DCT filtering a well a the filtering with the ideal fixed window ize DCT tranform (5 5) are preented in Figure 7. The ``Montage'' image, compoed from different type of ubimage, i ued in thee experiment. RMSE of the filtered ignal i equal to 0:028 and 0:037 for the adaptive varying and ideal fixed window ize filter, repectively. The filter with the varying window ize yield better reolution of detail a well a le value of RMSE. Table 2 below preent accuracy reult obtained for thi montage image by different filter. We compare the local DCT filter conidered above veru the Wiener filter and different wavelet baed filter. All accuracy reult are in favor of the local DCT with adaptive varying window ize. 5. Concluion A novel approach to olve a problem of varying adaptive window ize election for filtering a noiy image i preented. The LPA i ued in order to demontrate the efficiency of the approach, while a poible development to another linear and nonlinear filter (tranform) can be given. The algorithm i Table 2. Comparative reult Ued Filter RMSE MAE Local DCT with adaptive tranform ize Local DCT with ideal tranform ize Wiener Filter(5x5 ) Wavelet Package Haar Wavelet Package Wavelet PO Haar (4 level) Wavelet PO 8 (4 level) Wavelet TI Haar (5 level) Wavelet TI 8 (5 level) RMSE=Root Mean Square Error, MAE=Mean Abolute Error, PO= Periodic, TI= Tranlation invariant imple to implement and require calculation of the etimate and their tandard deviation for a et of the window ize value. The adaptive tranform i built a J parallel filter, which are different only by the window ize h j ; j ˆ 1; 2; :::; J; and the elector, which determine the bet window ize h (x()) and the correponding etimate ^y(x(); h (x())) for every pixel x(): Thi elector ue the ICI tatitic. In can be proved in a imilar way a it i done in [4] for 1D regreion de-noiing, that the adaptive algorithm i

12 234 Katkovnik, Egiazarian and Atola nearly optimal in the point-wie rik for etimating the ignal and it derivative. In imulation the ICI adaptive window ize algorithm demontrate an improved performance a compared with their nonadaptive window ize counterpart. Reference 1. W.S. Cleveland and C. Loader, ``Smoothing by local regreion: principle and method,'' in Statitical theory and computational apect of moothing (Ed.. W. Hardel and M. Schimek), pp. 10±49, Phyica-Verlag, K. Egiazarian, V. Katkovnik, H. Oktem, and J. Atola, ``Tranform-baed denoiing with parameter adaptive to unknown moothne of the ignal,'' in Ed. Creutzburg and Egiazarian: Spectral Technique and Logic Deign for Future Digital Sytem, Proceeding of International Workhop SPECLOG'2000, Tampere, TTKK, Monitamo, Finland. 3. J. Fan and I. Gijbel, Local polynomial modelling and it application, London: Chapman and Hall, A. Goldenhluger and A. Nemirovki, ``Adaptive de-noiing of ignal atifying differential inequalitie,'' IEEE Tran. Inf. Theory, Vol. 43, No. 3, pp. 873±889, W. Hardle, Applied nonparametric regreion. Cambridge, Univerity Pre, Cambridge, T. Hatie and C. Loader, ``Local regreion: automatic kernel carpentry'' (with dicuion), Statitical Science, Vol. 8, No. 2, pp. 120±143, C.M. Hurvich and J.S. Simonoff, ``Smoothing parameter election in nonparametric regreion uing an improved AIC criterion,'' Journal of the Royal Statitical Society, Ser. B, Vol. 60, pp. 271±293, V. Katkovnik, ``A new method for varying adaptive bandwidth election,'' IEEE Tran. on Signal Proceing, Vol. 47, No. 9, pp. 2567±2571, V. Katkovnik, H. Oktem, and K. Egiazarian, ``Filtering heavy noied image uing ICI rule for adaptive varying bandwidth election'', ISCAS'99, Orlando, Florida, May 30 ± June 2, V. Katkovnik, A. Gerhman, and L.J. Stankovic, ``Senor array ignal tracking uing a data-driven window approach'', Signal Proceing, Vol. 80, No. 12, pp. 1507±2515, V. Katkovnik, K. Egiazarian, and J. Atola, ``Median filter with varying bandwidth adaptive to unknown moothne of the ignal,'' International Conference on Circuit and Sytem (ISCAS'2000), May 28±31, 2000, Geneva, Switzerland, Proceeding of ISCAS'2000, Vol., pp. 519± V. Katkovnik, and LJ. Stankovic, ``Periodogram with varying and data?driven window length,'' Signal Proceing, Vol. 67, No. 3, pp. 345±358, V. Katkovnik, Nonparametric identification and moothing of data (Local approximation method). Nauka, Mocow, 1985 (in Ruian). 14. V. Katkovnik, K. Egiazarian, and I. Shmulevich, ``Adaptive varying window ize filtering baed on the interection of confidence interval rule,'' 2001 IEEE - EURASIP Workhop on Nonlinear Signal and Image Proceing, June 3±6, 2001, Baltimore, Maryland, USA. 15 V. Katkovnik, K. Egiazarian, and J. Atola, ``Local tranformbaed image de-noiing with adaptive window ize election,'' EOS/SPIE Sympoium, Image and Signal Proceing for Remote Sening, September 25±29, 2000, Barcelona, Spain. 16 C. Loader, Local regreion and likelihood, Serie Statitic and Computing, Springer, N.Y., 1999, pp S. Mallat, A Wavelet Tour of Signal Proceing, Academic Pre, J.A. McDonald and Owen A.B., ``Smoothing with plit linear fit,'' Technometric, Vol. 28, No. 3, pp. 195±208, D. Ruppert, ``Empirical-bia bandwidth for local polynomial nonparametric regreion and denity etimation,'' JASA, Vol. 92, No. 439, pp. 1049±1062, W.R. Schucany, ``Adaptive bandwidth choice for kernel regreion,'' JASA, Vol. 90, No. 430, pp. 535±540, Vladimir Katkovnik received the M.Sc.(1960), Ph.D.(1964) and Doctor of Sc.(1974) in Technical Cybernetic from Leningrad Polytechnic Intitute. From 1961 to 1991 he held poition of Aitant, Aociate and Profeor at the Department of Mechanic and Control Procee of thi Intitute. From 1991 until 1999 he wa a Profeor of Statitic Department of the Univerity of South Africa, Pretoria. During 2000 he wa a Viiting Profeor at Signal Proceing Laboratory, Tampere Univerity of Technology, Tampere, Finland. Currently he i with Mechatronic Department of Kwangju Intitute of Science and Technology, South Korea. Hi reearch interet include linear and nonlinear filtering, nonparametric and robut etimation, image proceing, nontationary ytem, time-frequency analyi, tochatic optimization and adaptive tochatic control. He publihed five book and more than 150 paper. Karen Egiazarian wa born in Yerevan, Armenia, in He received the M.Sc. degree in mathematic from Yerevan State Univerity in 1981, and the Ph.D. Degree in phyic and mathematic from Mocow M.V. Lomonoov State Univerity in 1986.

13 Adaptive Window Size Image De-Noiing 235 In 1994 he wa awarded the degree of Doctor of Technology by Tampere Univerity of Technology, Finland. He ha been a Senior Reearcher at the Department of Digital Signal Proceing of the Intitute of Information Problem and Automation, National Academy of Science of Armenia. He i currently a Full Profeor in the Intitute of Signal Proceing at Tampere Univerity of Technology. Hi reearch interet are in the area of applied mathematic, digital logic, ignal and image proceing. He ha publihed more than 200 article in thee area, and i coauthor (with S. Agaian and J. Atola) of the book ``Binary Polynomial Tranform and Nonlinear Digital Filter'', publihed by Marcel Dekker, Inc. in 1995, and three book chapter. From 1976 to 1977, he wa a Reearch Aitant at the Reearch Intitute for Mathematical Science of Kyoto Univerity, Kyoto, Japan. Between 1979 and 1987, he wa with the Department of Information Technology, Lappeenraanta Univerity of Technology, Lappeenraanta, Finland, holding variou teaching poition in mathematic, applied mathematic, and computer cience. From 1988 to 1993, he wa an Aociate Profeor in applied mathematic at Tampere Univerity, Tampere, Finland. Currently, he i a Profeor of digital ignal proceing at Tampere Univerity of Technology, the head of Signal Proceing Laboratory and the Director of the Tampere International Center in Signal Proceing (TICSP). Hi reearch interet include ignal proceing, coding theory, and tatitic. Jaakko Atola wa born in Helinki, Finland, in He received the B.Sc., M.Sc., Licenciate, and Ph.D. degree in Mathematic from Turku Univerity, Turku, Finland, in 1972, 1973, 1975, and 1978, repectively.