Transactions Briefs. Color Image Processing Using Adaptive Vector Directional Filters

Transcription

1 1414 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1998 Trasactios Briefs Color Image Processig Usig Adaptive Vector Directioal Filters K. N. Plataiotis, D. Adroutsos, ad A. N. Veetsaopoulos Abstract A ew class of filters for multichael image processig is itroduced ad aalyzed i this brief. This class costitutes a geeralizatio of vector directioal filters. The proposed filters use fuzzy trasformatios of the agles amog the differet vectors to adapt to local data i the image. The priciple behid the ew filters is explaied ad comparisos with other popular oliear filters are provided. The specific case of color image processig is studied as a importat example of multichael image processig. Simulatio results idicate that the ew filters offer some flexibility ad have excellet performace. Idex Terms Adaptive filters, color image processig, fuzzy membership fuctios, vector directioal filters. I. INTRODUCTION Filterig of multichael images has received icreased attetio recetly due to its importace i processig color images. Numerous filterig techiques have bee proposed to date for multichael image processig [1] [3]. Noliear filters applied to images are required to preserve edges ad details, ad remove impulsive ad Gaussia oise. O the other had, vector processig is oe of the most effective methods available to filter oise ad detect edges o multichael images. Rak, ad especially media filters, have bee used extesively as multichael image filters [4]. A ew class of filters for processig vector-valued sigals was itroduced i [5]. The so-called vector directioal filter (VDF) uses the agle betwee the image vectors as a orderig criterio. The VDF operates o the directio of the image vectors, separatig i this way the processig of vector data ito directioal processig ad magitude processig. Thus, it preserves chromaticity while removig impulsive oise [6]. I this brief, a ew class of VDF s is itroduced. Fuzzy membership fuctios based o a agle criterio are adopted to determie the weights of a adaptive weighted mea filter [1], [7], [8]. Our obective is to develop a computatioally efficiet multichael filter, which will have good performace without requirig ay a priori kowledge about the sigal ad oise characteristics. II. ADAPTIVE VDFS A. The Filter Structure Let y(x): Z l! Z m, represet a multichael image ad let W 2 Z l be a widow of fiite size (filter legth). Usually we cosider a square widow ( = N 1 2 N 1) cetered aroud the Mauscript received August 23, 1995; revised September 11, This paper was recommeded by Associate Editor W.-C. Siu. K. N. Plataiotis was with the Departmet of Electrical ad Computer Egieerig, Uiversity of Toroto, Toroto, ON, M5S 3G4, Caada. He is ow with the Departmet of Mathematics, Physics ad Computer Sciece, Ryerso Polytechic Uiversity, Toroto, ON, M5B 2K3 Caada ( kostas@dsp.toroto.edu). D. Adroutsos ad A. N. Veetsaopoulos are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Toroto, Toroto, ON, M5S 3G4 Caada ( zeus@dsp.toroto.edu; av@dsp.toroto.edu). Publisher Item Idetifier S (98) curret pixel. The oisy image pixels iside the widow W are deoted as x, = 1; 2; 111;. Sice the most commoly used methodology to decrease the level of radom oise preset i the sigal is smoothig, we select a averagig operatio i order to replace the oisy vector at the widow ceter with a suitable represetative vector. The geeral form of the adaptive filter proposed here is give as a fuzzy weighted average of the iput vectors iside the widow W. Thus, the ucorrupted multichael sigal is estimated by determiig the cetroid as follows: ^y = w x = x (1) where is a parameter such that 2 [0; 1). Accordig to (1), the output of the proposed filter is a weighted average of all the vectors iside the operatio widow. The weightig coefficiets are trasformatios of the sum of distaces betwee the ceter of the widow (pixel uder cosideratio) ad all iput vectors iside the filter widow. I multichael filterig, it is desirable to perform smoothig o all vectors which are from the same regio as the vector at the widow ceter. Thus, the adaptive weights should represet the cofidece that the vectors uder cosideratio come from the same regio. It is, therefore, reasoable to make the weights i (1) proportioal to the differece i terms of a distace measure betwee a give vector ad its eighbors iside the filterig widow. The desig obective is: Assig the maximum weight to the vector which is most cetrally located iside the processig widow. Thus, the vector with the maximum weight will be the oe which has the miimum distace from all the other vectors. I this way, atypical vectors due to oise or missig compoets (outliers), which are placed far away from the cetermost vector, will be assiged smaller weights ad will cotribute less to the fial output. The weightig trasformatio is essetially a membership fuctio with respect to the specific widow compoet. I accordace to our desig obective it is reasoable to select a appropriate fuzzy trasform so that the vector with the miimum distace will be assiged the maximum weight. The membership fuctio value ca be regarded as the compariso of the vector uder cosideratio with the ideal vector which results i a distace. Thus, the degree of membership is a fuctio of the distace, which ca be defied as follows [10]: 1 = (2) 1+d(x ) where d(:) is the distace fuctio yet to be determied. If the vector uder cosideratio x has all the features of the ideal vector, the distace should be zero resultig i! 1, otherwise, if o similarity betwee the ideal ad the vector x exists, the distace shall be 1 with! 0. B. Directioal Distaces It is evidet that the value of the membership fuctio depeds o the choice of the distace criterio selected as a measure of dissimilarity. Ay distace criterio used to calculate distaces amog multichael sigals ca be utilized. Our primary obective, however, is to apply the ew filter to color images, thus a criterio suitable to /98$ IEEE

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER TABLE I ANGULAR DISTANCES membership fuctio has ot yet bee show. Sice its choice is very much problem depedet, the oly applicable a priori rule is that the desiger must cofie himself to the fuctios which are cotiuous ad mootoic [12]. measure distaces amog color vectors is used. The distace measure selected is the vector agle criterio. This criterio cosiders the agle betwee two vectors as their distace. A scalar quatity a = A(x i; x ) (3) i=1 with A(x i ; x )=cos 01 x i x t (4) x i kx is the distace associated with the oisy vector x iside the processig widow of legth. We ca cosider the distace i (3) as the sum of absolute agular distace for spherical data. This criterio was first itroduced i [5] to measure distaces amog color vectors. Sice i the RGB color space, color is defied as relative values i the trichromatic chael ad ot as a triplet of absolute itesity values, it was argued i [5] that the distace measure must respod to relative itesity differeces (chromaticity) ad ot absolute itesity differeces (lumiace). Therefore, the orietatio differece betwee two color vectors was selected i [5] as their distace measure because it correlates well with their spectral ratio differece. Based o the agle betwee two vectors, a umber of other useful distace measures ca be defied (see Table I). Ay oe of the four agular distace measures listed i Table I ca be used to geerate the weights i the adaptive desig of (1). Utilizig our geeral membership fuctio ad takig ito accout the fact that the relatioship betwee a distace measured i uits ad perceptio is geerally expoetial, a sigmoidal (piecewise) liear membership fuctio is appropriate [11]. I such a case the fuzzy weight associated with the vector x ca take the followig form: = (5) [1 + exp( )] r where ; r =1= are parameters to be determied ad i is ay agular distace measure listed i Table I. The parameter is a soft parameter used oly to adust the limit of the S-shaped membership fuctio (weight scale threshold). I our experimets we assiged the value =2, sice we decided to use membership fuctios that deliver a output i the iterval [0, 1]. The parameter r is the smoothig parameter. Sice, by defiitio, the agle distace measure i (4) delivers a positive umber i the iterval [0, ] [5], the output of the fuzzy trasformatio itroduced above produces a membership value i [=f1 + exp[()]g r ;=2]. It ca easily be see that for a moderate size widow, such a widow we ca cosider the above membership fuctio as havig values i the iterval (0, 1], e.g., [ , 1] for the agular distace of (3) with parameters r =1ad =2. As oe should expect, the fuctio used here does ot chage too much whe its values are aroud the miimum distace (regio of cofidece) ad does ot icrease quickly whe the membership fuctio s iput values are aroud 1, the regio of reectio. The fuzzy trasformatio of (5) is ot uique. Other commoly used shapes, such as triagular or Gaussia-like curves ca be used istead. However, despite past efforts, a uified form of fuzzy C. The Adaptive Filters The weighted average form of (1) it ca be see i the cotext of fuzzy systems as a geeralized defuzzificatio process, where a defuzzified value is selected as output based o a probability-like distributio obtaied i the fuzzificatio step. I fuzzy set theory defuzzificatio is realized by a decisio-makig algorithm which selects the best crisp value based o a fuzzy set. The defuzzificatio process ca be cosidered as a two step process. The first step is the coversio of the membership values associated with the fuzzy iputs ito a probability-like distributio which satisfies the idetity ad mootoicity coditios. I the secod step of the defuzzificatio process the defuzzified value is selected based o values from the probability-like distributio [13]. It is ot hard to see that through the defiitios i (1), (3) (5) the adaptive weights w = = satisfy the above coditios. Specifically: 1) 8 i;, if i = the w i = w, (idetity); 2) 8 i;, if i > the w i >w, (mootoicity). Furthermore, through the ormalizatio procedure, our weights w i (1) have the basic properties which characterize ay probability distributio, amely: 1) w > 0 ad w 2 [0; 1]; 2) w = 1. Give the satisfactio of the above coditios the defuzzificatio strategy ca be see as a form of pseudo-expectatio formula [12], [13]. Ideed, from the geeralized defuzzificatio rule of (1) if =1 the widely used Ceter of Area (COA) defuzzificatio strategy (Cetroid Defuzzifier) ca be obtaied. Accordig to the COA defuzzificatio the defuzzified value of the adaptive fuzzy filter, hereafter Adaptive Vector Directioal Filter (AVDF), is give as: ^y = w x = x : (6) The AVDF obtaied through the COA strategy geerates a vector valued sigal, which is ot icluded amog the origial set of iput vectors. Cotrollig the parameter r i (5), we ca adust the smoothess of the output. As a geeral rule, smaller values of r ca smooth out oisy vectors, while larger values ca make the overall output as oliear as required to prevet details ad to discard impulsive-type oise. Thus, the parameter should be properly chose to provide a balace betwee smoothig ad detail preservatio. Sice r is oe-dimesioal parameter, it is ot difficult to determie a appropriate value for practical applicatios. I most cases, a few trial-ad-error procedures may be eough to determie a good value. Moreover, our experiece idicates that acceptable results ca be obtaied by assigig the value r = 1. However, if the output of the adaptive fuzzy filter is required to be part of the origial iput set, a differet defuzzificatio strategy should be used. Defiig (max) as the largest membership value obtaied through (5), the adaptive weights i (1) ca be rewritte

3 1416 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1998 as follows: w = = (max) (max) Give that < (max),as!1the = w =1; = (max) (max) (max) : (7) =0; 6= (max) : (8) Equatio (8) represets the maximum defuzzifier strategy. Sice give the form of the fuctio i (3) (5) the maximum value occurs at a sigle poit oly, the maximum defuzzifier strategy coicides with the mea of maxima (MOM) defuzzificatio process. Through the maximum defuzzifier the output of the adaptive fuzzy filter is defied as: ^y = x ; = (max) : (9) I other words, we sigle out the iput vector associated with the maximum fuzzy weight. We ame this filter maximum adaptive vector directioal filter (MAVDF). It must be emphasized that through the fuzzy membership fuctio, the maximum fuzzy weight correspods to the miimum distace. If the absolute agle distace is used as the dissimilarity fuctio the ew fuzzy filter coicides with the spherical media ad delivers the same output as the basic VDF (BVDF) [5], [6]. The MADVF filter defied above is a pure chromaticity filter i the case of color image filterig. I other words, it operates o the chromaticity compoet of the color vector by filterig out color vectors with large chromaticity errors. The importace of such a image processig strategy was documeted i [5]. O the other had, the AVDF defied i (6) is ot oly a chromaticity filter sice it uses both the directioal filterig iformatio through the agle distaces for its weights as well as the magitude compoet of each oe of the color vectors. This is a feature that differetiates our desig from the chromaticity filters with gray-level processig compoets itroduced i [5]. The geeralized chromaticity filters itroduced there select a subset of the color vectors ad the apply gray-scale techiques oly to the selected group of vectors. However, if importat color iformatio was elimiated due to errors i the chromaticity-based decisio part, the filters i [5] are uable to compesate usig their gray-scale processig step. That is ot the case i the ew desig. The adaptive filter itroduced i (6) does ot discard ay magitude iformatio based o chromaticity aalysis. All the vectors iside the operatioal widow cotribute to the fial output. Simply stated, the filter assigs weights to the magitude compoet of each color vector modifyig i this way their cotributio to the output. This atural bledig of chromaticitybased weights with magitude-based iput cotributios makes the filter appropriate for color image processig. III. APPLICATION TO COLOR IMAGES A. Properties It must be emphasized that the proposed filter framework ca be applied i ay multichael sigal ad ay multivariate data with a spatial domai. The most commo multichael filterig problem, however, is that of color image filterig, therefore i this sectio we also outlie some basic properties that make the proposed filters appropriate i color image processig. We cofie ourselves maily to the MAVDF due to its ameability for mathematical treatmet. Property I. Ivariace Uder Scalig ad Rotatio: The criterio used by the MAVDF filter to sigle out oe vector is the sigmoidal trasformatio of the vector agle criterio. Fig. 1. MAVDF: Edge preservatio. From (4) it ca easily be see that the agle betwee two vectors does ot deped o the scale. Furthermore, ay rotatio of the coordiate system does ot chage the agle betwee two vectors. Thus, if the iput to the sigmoidal remai the same, the fuzzy weight selected uder scalig or rotatio is the same. Therefore, the MADVF filter is ivariat uder scalig ad rotatio, but is ot ivariat uder bias sice the additio of a costat vector could chage the sum of agles amog the vectors. Property II. Preservatio of Step Edges: It is easy to establish that a step edge is a root sigal of the MAVDF. A illustrative example is depicted i Fig. 1. A widow with size 5 is cetered aroud the vector x 4. Calculatig the absolute agle distaces amog the vectors it ca be see from the diagram that a 4 = a 5 = a 6 < a 2 = a 3, sice the vectors x 4, x 5, ad x 6 are parallel. Usig the trasformatio i (8), the higher weights will be the oes associated with these vectors. It is obvious that the MAVDF will discard the vectors x 3 ad x 2, resultig i the preservatio of the vector edge. For the adaptive filter i (6) we ca ot use the same methodology to ustify the edge preservatio property. Thus, a simple example is itroduced to illustrate the effectiveess of the proposed algorithms i filterig operatio ear oisy edges. I the experimet we use a step edge of height 2, for a two chael ivariat sigal corrupted by additive mixed Gaussia oise. The sigal descriptio is: with ad y(t) =x + w(t) (10) x = 1:5 1:75 x = 3:5 3:75 t 45 t>45 (11) w(t) =u(t)v 1(t) +[I 0 u(t)]v 2(t) (12) where u(t) = u(t)i 221 ad u(t) is a radom umber uiformly distributed over the iterval [0, 1]. The v 1 (t) results from a Gaussia distributio with zero mea ad covariace 0:05I 222. The v 2 (t) is Gaussia with zero mea ad covariace 0:25I 222. A operatioal widow of size =5was used i the experimet. Results are show i Fig. 2, where the curves deote: 1) the actual sigal; 2) the oisy iput, ad i Fig. 3, where 3) the output of the filter devised from (2) with parameters =2, r =1, ad = a; 4) the output of the Media filter; ad 5) the output of the (arithmetic) Mea filter are depicted. From the above experimet the followig coclusios ca be draw. 1) The Media algorithm works better ear the sharp edges. 2) The Mea filter works better tha the Media for the homogeeous sigal. 3) The proposed AVDF ca suppress the oise i homogeeous regios much better tha the vector Media filter ad ca preserve edges better tha the Mea filter.

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER TABLE III NOISE DISTRIBUTIONS (a) Fig. 2. (b) Simulatio experimet. (a0 Actual sigal ad (b) oisy iput. Fig. 4. Lea corrupted with (4%) impulsive oise. (a) (b) Fig. 3. Simulatio experimet. filterig results. (a) First compoet. (b) Secod compoet. TABLE II ADAPTIVE VDF S B. Simulatio Results The performace of our adaptive desigs (see Table II) is compared agaist that of popular vector processig filters, such as the widely used vector media filter (VMF), the chromaticity-based geeralized VDF (GVDF) [5] ad the arithmetic mea vector filter (AMVF). The test image selected for the compariso is the RGB color image Lea. The test image has bee cotamiated usig various oise source models i order to assess the performace of the filters uder differet oise distributios (see Table III). The ormalized mea square error (NMSE) has bee used as quatitative measure for evaluatio purposes. It is computed as N1 N2 ky(i; ) 0 ^y(i; )k 2 i=0 =0 NMSE = N1 N2 (13) ky(i; )k 2 i=0 =0 where N 1, N 2 are the image dimesios, ad y(i; ) ad ^y(i; ) deote the origial image vector ad the estimatio at pixel (i; ), respectively. Table IV summarizes the results obtaied for the test image Lea for a filter widow. The results obtaied usig a52 5 filter widow are give i Table V. A 3 i a table etry idicates the best filter performace i the correspodig row. I additio to the quatitative evaluatio preseted above, a qualitative evaluatio is ecessary sice the visual assessmet of the processed images is, ultimately, the best subective measure of the efficiecy of ay method [5]. Therefore, we preset sample processig results i Figs Fig. 4 shows the color Lea image corrupted with (4%) impulsive oise. Figs. 5 7 show results of the AVDF1, AVDF2, ad GVDF, respectively. Similarly, Fig. 8 shows the color Lea image corrupted with Gaussia oise ( =15)mixed with (2%) impulsive oise. Figs preset agai the processig of the same filters ad with the same order. C. Coclusios From the results listed above, it ca be easily see that our filters provide cosistetly good results i every type of oise, outperformig the other three multichael filters uder cosideratio. The

5 1418 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER 1998 TABLE IV NMSE (X10 02 ) FOR THE LENNA IMAGE, WINDOW TABLE V NMSE (X10 02 ) FOR THE LENNA IMAGE, WINDOW Fig. 5. ADVF1 of (4) usig widow. Fig. 7. GVDF of (4) usig widow. Fig. 6. AVDF2 of (4) usig widow. differet fuzzy filters atteuate both impulsive ad Gaussia oise with or without outliers preset i the test image. The effect of the distace measure ad the smoothig parameter selected are evidet i the results summarized above. Regardig the effect of the parameters the followig ca be cocluded. Fig. 8. Lea corrupted with Gaussia oise ( =15)mixed with (2%) impulsive oise. For small widow size, amely, widow, the filters based o the average distaces have worse performace tha the filters based o the overall distaces for all oise scearios except for the case of Gaussia oise.

6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 10, OCTOBER Fig. 9. ADVF1 of (8) usig widow. Fig. 11. GVDF of (8) usig widow. the umerical complexity of the multichael algorithm ad delivers excellet results for complicated multichael sigals, such as real color images. IV. CONCLUDING REMARKS A ew class of AVDF s for multichael image processig has bee itroduced i this brief. These filters combie, i a ovel way, fuzzy memberships, average filters ad agle-based distaces. Depedig o the criterio which the desiger uses to select the membership, a umber of differet filters ca be obtaied. Experimetal simulatio results have demostrated the efficiecy of the proposed filters. The ew filters outperform other oliear filters, such as the VMF ad the GVDF. Moreover, the ew filters preserve the chromaticity compoet, which is very importat i visual perceptio of color images. REFERENCES Fig. 10. AVDF2 of (8) usig widow. Filters based o average distaces caot smooth out impulsive oise. Despite the fact that they have relatively good performace for a larger widow size (5 2 5), they are still less effective i removig impulsive oise tha the filters based o overall distaces. By icreasig the value of the smoothig parameter r better results ca be obtaied for impulsive-type oise. This is expected sice icreased r values ehaces the oliear ature of the filter. However, the ability of the filter to smooth oimpulsivetype oise is restricted, especially for larger widows. As a geeral coclusio, the versatile desig of (1) allows for a umber of differet fuzzy filters, which ca provide solutios to may types of differet filterig problems. Simple adaptive fuzzy desigs, such as the AVDF1 ca preserve edges ad smooth oise uder differet scearios, outperformig other widely used multichael filters. If kowledge about the oise characteristics is available, the desiger ca tue the parameters of the adaptive filter to obtai better results. Fially, cosiderig the umber of computatios, the computatioally itesive part of the adaptive algorithm is the distace calculatio part. However, this step is commo i all multichael algorithms cosidered here. I coclusio, our desig is simple, does ot icrease [1] I. Pitas ad A. N. Veetsaopoulos, Noliear Digital Filters: Priciples ad Applicatios. Norwell, MA: Kluwer Academic, [2], Order statistics i digital image processig, Proc. IEEE, vol. 80, pp , Dec [3] A. N. Veetsaopoulos ad K. N. Plataiotis, Multichael image processig, i Proc. IEEE Workshop o Noliear Sigal/Image Processig, 1995, pp [4] J. Astola, P. Haavisto, ad Y. Neuvo, Vector media filter, Proc. IEEE, vol. 78, pp , Apr [5] P. E. Trahaias ad A. N. Veetsaopoulos, Vector directioal filters: A ew class of multichael image processig filters, IEEE Tras. Image Processig, vol. 2, pp , Oct [6] P. E. Trahaias, I. Pitas, ad A. N. Veetsaopoulos, Color image processig, i Advaces i 2D ad 3D Digital Processig (Techiques ad Applicatios), C. T. Leodes, Ed. New York: Academic, 1994, p. 41. [7] S. G. Tzafestas ad A. N. Veetsaopoulos, Eds., Fuzzy Reasoig i Iformatio, Decisio ad Cotrol Systems. Norwell, MA: Kluwer Academic, [8] D. R. K Browrig, The weighted media filter, Commu. ACM vol. 27, pp , Aug [9] K. N. Plataiotis, D. Adroutsos, ad A. N. Veetsaopoulos, Color image processig usig fuzzy vector directioal filters, i Proc. IEEE Workshop o Noliear Sigal/Image Processig, 1995, pp [10] H. J. Zimmerma, Fuzzy Sets, Decisio Makig ad Expert System. Bosto, MA: Kluwer Academic, [11] R. Helso, Adaptatio-Level Theory. New York: Harper & Row, [12] J. M. Medel, Fuzzy logic systems for egieerig: A tutorial, Proc. IEEE, vol. 83, pp , Mar [13] T. Jiag ad Y. Li, Geeralized defuzzificatio strategies ad their parameter learig procedures, IEEE Tras. Fuzzy Syst., vol. 4, pp , 1996.