Structuring Element Representation of an Image and Its Applications

Transcription

1 Iteratioal Joural of Cotrol Structurig Automatio Elemet ad Represetatio Systems vol. of a o. Image 4 pp. ad Its Applicatios December Structurig Elemet Represetatio of a Image ad Its Applicatios Jisug Oh Abstract: I this paper we preset the liear combiatio of a fuzzy opeig ad closig filter with locally adaptive structurig elemets that ca preserve the geometrical features of a image. Based o the adaptatio algorithm of liear combiatio of the fuzzy opeig ad closig filter the optimal structurig elemet for image represetatio is obtaied. The optimal structurig elemet is a idicator of the shape ad directio of a object s image which is useful i filterig multi resolutio segmetatio ad recogitio of a image. Keywords: Fuzzy morphological filter image represetatio structurig elemet.. INTRODUCTION Sice a morphological filter ca aalyze the geometric features of a image it is widely used i image filterig segmetatio ad edge detectio. Compared with classical morphology fuzzy morphology [] provides a more ituitive ad less restrictive fittig. Recetly a despeclig techique usig fuzzy morphology has bee proposed i []. I [] it is show that fuzzy morphology ca cope with the ambiguous ad obscure ultrasoic image ad further that liear combiatio of fuzzy opeigclosig ad fuzzy closigopeig (LFCO) is used for specle reductio. Sice the structurig elemet is large eough to iclude oise features ad its shape is adapted to the geometry of the image features to be preserved the selectio of the structurig elemet with optimal shape is importat. That is the desig of a morphological filter is eeded i selectig the shape of structurig elemet that is adapted to the local geometry of the image beig processed. The structurig elemet for a morphological filter ca be directly obtaied from iput image [3] or weighted oes ca be used [4]. Recetly implemetatio of the adaptatio of the structurig elemet i fuzzy morphology has bee proposed i [5]. Based o the result i [5] liear combiatio of a adaptive fuzzy opeig ad closig (LAF) filter is proposed. The optimal structurig elemets obtaied from LAF filter represet the shape ad directio of a object s image which is useful i the applicatios of filterig segmetatio ad recogitio of a image. I this paper the fuzzy morphology [] ad its Mauscript received May 6 003; revised Jue 3 004; accepted October 004. Recommeded by Editorial Board member I So Kweo uder the directio of Editor Keum Shi Hog. Jisug Oh is with the School of Electrical Egieerig Halla Uiversity Sa 66 Heugup Woju Gagwo 07 Korea ( jsoh@hit.halla.ac.r). adaptatio [5] are briefly reviewed. The the liear combiatio of a fuzzy opeig ad closig filter with a locally adaptive structurig elemet is proposed. Usig the optimal structurig elemets obtaied from the adaptatio algorithm a image is represeted i form of the set of optimal structurig elemets. Simulatio results of structurig elemet represetatio of images are give.. FUZZY MORPHOLOGICAL FILTER.. Fuzzy morphology I this sectio some basic defiitios of morphological operators are itroduced. The maximum ad miimum operators will be deoted as ad respectively. Fuzzy mathematical morphology [] has bee developed usig the otio of fuzzy fittig. Fuzzy fittig of a fuzzy set A ito a fuzzy set B is characterized by a iclusio idicator I A B 0. [ ] I( A B) = x X ( A( x) + B( x) ) where ( ) deotes a membership fuctio. The value of I ( A B) betwee 0 (o fit) ad (perfect fit) idicates the degree of fittig of A ito B. Usig I ( A B) operators such as erosio ( Θ ) dilatio ( ) opeig ( ) ad closig( ) are defied as follows: fθ ( ) = m K + m F ( ( m) + f ( + m) ) f ( ) = m K m F 0 ( ( m) + f ( m) ) f ( ) = ( fθ) ( ) f ( ) = ( f ) Θ ( ) where f ( ) ad ( ) with support regios F ad K are membership fuctios of sigal f ( ) ad structurig elemet ( ) respectively.

2 50 Jisug Oh.. Adaptatio of structurig elemet i fuzzy morphology The desig of fuzzy morphological filters cosists i adaptig the shape of structurig elemets to the local geometry of a image. Recetly the optimizatio method i fuzzy morphology is proposed i [5]. More details ca be foud i [5]. By miimizig the iequality idex J ( ) = t ( ) f ( ) the algorithm for updatig the structurig elemet is provided by ( i+ ) ( i) f = +η sg t ( ) f ( ) where ( ) t deotes target membership fuctio deotes morphological operator sg () deotes the sig fuctio i is iteratio umber ad η is traiig rate. The gradiet for the erosio fθ ( ) filter e is f ( ) T e = e ( M) e ( 0 ) e ( M) f f f f ( i) where e m = U f ( m) f ( + m) if ( ( m) + f ( + m) ) is miimum 0 otherwise ad { M 0 M } K. The gradiet f ( ) for the dilatio filter d f ( ) is also give by T d = d ( M) d ( 0 ) d ( M) f f f f ( i) where d m = U ( m) + f ( m) if f ( ( m) f ( m) ) () 0 + is maximum 0 otherwise. Note that U [] a = for a 0 0 for a < LINEAR COMBINATION OF ADAPTIVE FUZZY MORPHOLOGICAL OPERATORS 3.. Liear combiatio of adaptive fuzzy opeig ad closig The liear combiatio of the fuzzy opeig ad closig filter is defied as ( ) = ( ) ( ) f f + f. () The liear combiatio of morphological filters ca elimiate the bias of the idividual morphological (a) Iequality idex. (b) Updated SEs per iteratio. Fig.. Covergece of LAF for D cocave sigal. filter. For a flat structurig elemet i.e. ( m) = m () becomes a pseudomedia filter [6]. From () the algorithm for updatig the structurig elemet of LAF ca be easily derived as ( i+ ) ( i) η f f = + sg t ( ) f ( ). + The gradiets of opeig ad closig are give by ( ) () () = ( E f R +Ι) d f Θ f i i ( ) () () = ( Ι D f R) e f f i i (3) where E f = e e e ( i) D f M f f + M f = d d d R ad Ι are f M f f + M ( M + ) ( M + ) reflectio ad idetity matrices respectively. 3.. Covergece Fig. illustrates the shape of updated structurig elemets (SE) for the LAF filter usig a iitial flat structurig elemet for a Dimesioal cocave sigal. As oe ca see the shape of the updated structurig elemet for the cocave sigal is cocave which is optimal for both opeig ad closig. Fig.

3 Structurig Elemet Represetatio of a Image ad Its Applicatios 5 also illustrates fast covergece with less iteratio. 4. SIMULATION RESULTS To obtai locally optimized structurig elemets the image is partitioed ito regios ad a optimal structurig elemet correspodig to each regio is obtaied by (3). I this simulatio a 5x5 flat structurig elemet is used as the iitial SE i the adaptatio process. 4.. Experimet : figerprit image The proposed filter is tested o a figerprit image as illustrated i Fig. (a). Fig. (b) ad (c) idicate the optimal structurig elemet represetatio of a figerprit image. The set of optimal structurig elemets obtaied from the proposed method is a idicator of the directio of the figerprit ad very similar to the figerprit directioal map [78]. We thus develop a structural represetatio of figerprit images that might be useful i the recogitio of figerprits. 4.. Experimet : ultrasoic image Fig. 3 shows structurig elemet represetatio of a ultrasoic image. This example also idicates that the set of optimal structurig elemets as prior iformatio provides a structural characterizatio of the images that is useful i the removal of specle oise []. To evaluate the performace of the LAF filter the smoothig measuremet proposed i [] is used. Figure 4 shows a scatter plot of the gradiet magitude of the origial ultrasoic image (xaxis) versus the gradiet magitude of the filtered image (yaxis). Sharpeed ad smoothed pixels are grouped ito two sets C (above the lie y = x ) ad D (below the lie y = x ) respectively. Note that i geeral C << D. The lie y = ad x + bd ca be obtaied from the curve fittig method. The slope a D offers a idicatio of the smoothig iduced by the filter. Also the offset b D gives a idicatio of the bias iduced by the filter. The smoothig measuremet is provided by D Smoothig = a D C + D. As show i Fig. 4 ad Table compared to media ad pseudomedia filters the LAF filtered image is less smoothed i.e. the detail edges are preserved. I additio the filtered image by the LAF filter is less biased Experimet 3: Noflat area detectio I A B idicates the degree of fittig Sice betwee two sets it ca be exteded to a averaged degree of fittig betwee A set ad other sets B l l = L as follows: ( A) I ( A B ) I l l = L (a) Figerprit image. (b) 8x8 regio. (c) 6x6 regio. Fig.. Optimal structurig elemet represetatio..

4 5 Jisug Oh (a) Ultrasoic image. (a) Media filter ( a D = b D =0.874). (b) 8x8 regio. (b) Pseudomedia filter ( a D = b D =.576). (c) 6x6 regio. Fig. 3. Optimal structurig elemet represetatio. Table. Smoothig results. Media filter Pseudomedia filter LAF filter (c) LAF filter ( a D = b D =.466). Fig. 4. Scatter plot of gradiet magitude. That is the value of I ( A) is idicative of how well the graph ( x) fits beeath the graph ( x) l A Bl

5 Structurig Elemet Represetatio of a Image ad Its Applicatios X X X3 X4 R R R3 R4 Fig. 6. Morphological multi resolutio decompositio. (a) Average iclusio idicator. (a). (b). (c) 3. (d) 4. Fig. 7. Decompositio SEs for ultrasoic image. (b) Noflat area. Fig. 5. I ( l ) map of ultrasoic image ad area detectio. i average sese. Usig the set of optimal structurig elemets obtaied from each regio deoted as l for the l th regio each I ( l ) is calculated ad displayed i Fig. 5(a). As oe ca see the values of I correspodig to flat SEs are low while the ( l ) values of I l correspodig to detailed SEs are high. That is the average iclusio idicator idicates the degree of relative flatess or extet of detail of a object s image. The oflat area of a image is easily detected by the thresholdig the value of ( l ) Fig. 5(b)). The flat area detected by I (see I l or the set of optimal structurig elemets is also useful i flat zoe filterig [9] Experimet 4: morphological multi resolutio decompositio I mathematical morphology a multi resolutio aalysis decomposes a image ito differet sub images where each sub image cotais objects of a specific size. Fig. 6 shows the stadard morphological decompositio [0] where stads for opeig followed by closig operatio. I geeral the size of the structurig elemet is icreased i the subsequet S + S decompositio stage s + i.e.. Istead of usig the predefied structurig elemet [0] the set of optimal structurig elemets that cotai the geometric features of a image ca be used for decompositio. To acquire the structurig elemet for 4stage decompositio i.e. 4 the values of ( l ) { I ( ) l ( L) } = [ r r ] ad 3 I are raed as follows: R l r L where R deotes ascedig ra operator. Ad the l l = for l { I( l) rl/4} l l = for l { rl/4 I( l) rl/} < 3 l l = for l { rl/ I( l) r3 L/4} < 4 l l = for l { r3 I( l )} <. Fig. 7 depicts the structurig elemet for decompositio obtaied from Fig. 3(c) ( L = 6 6 ) by the above equatios. Clearly the structurig elemets betwee stages are show to have the S + S icreasig relatioship. As oe ca see

6 54 Jisug Oh optimal structurig elemet from the adaptatio algorithm of a liear combiatio fuzzy opeig ad closig filter represets the geometric features of a image. The average iclusio idicator also illustrates the degree of relative flatess or detail of a object i a image. Therefore the structurig elemet represetatio of a image promises to be very suitable for future wor i filterig multi resolutio segmetatio ad recogitio of image. (a) R. (b) R. (c) R3. (d) R4. (e) X4. Fig. 8. Morphological multiresolutio decompositio of ultrasoic image. the structurig elemets for decompositio have horizotal directio with differet sizes. The morphological decompositio result is show i Fig. 8. This decompositio result is revealed to filter out small objects of a certai size while preservig horizotal details at each stage. 5. CONCLUSIONS I this paper we preseted the liear combiatio of a fuzzy opeig ad closig filter with locally adaptive structurig elemets. It is show that the REFERENCES [] D. Siha ad E. R. Dougherty Fuzzy mathematical morphology J. Visual Comm. Image Represetatio vol. 3 o. 3 pp Sep. 99. [] E. Aviato ad M. Ito Specle reductio for ultrasoic images usig fuzzy morphology IEICE Tras. If. Syst. vol. E84D o. 4 pp Apr. 00. [3] M. Tsubai ad M. Ito Edge ehacemet of ultrasoic images by morphological operatios based o locally variable structurig elemets IEICE Tras. If. Syst. vol. E85D o. pp Ja. 00. [4] M. H. Sedaaghi ad Q. H. Wu Weighted morphological filter Electroics Letters vol. 34 o. 6 pp Aug [5] J. Oh ad L.F. Chaparro Adaptive fuzzy morphological filterig of impulse oise i images Multi. Systems Sigal Process. vol. pp [6] M. A. Schulze ad J. A. Pearce Liear combiatios of morphological operators: The midrage pseudomedia ad LO filters IEEE Iter. Cof. o ASSP vol. 5 pp [7] V. S. Spriivasa ad N. N. Murthy Detectio of sigular poits i figerprit images Patter Recogitio vol. 5 o. pp [8] M. M. S. Chog T. H. Ngee L. Ju ad R. K. L. Gay Geometric framewor for figerprit image classificatio Patter Recogitio vol. 30 o. 9 pp [9] P. Salembier ad J. Serra Flat zoe filterig coected operators ad filters by recostructio IEEE Tras. o Image Process. vol. 4 o. 8 pp Aug [0] Z. Zhou ad A. N. Veetsaopoulos Morphological methods i image codig IEEE Iter. Cof. o ASSP vol. 3 pp Mar. 99. [] J. Dij D. Ridder P. W. Verbee J. Walrave I. T. Youg ad L. J. Vliet A quatitative measure for the perceptio of sharpeig ad smoothig i images Proc. 5 th Aual Cofer. Advaced School for Computig ad Imagig Delft pp. 998 Mar. 999.

7 Structurig Elemet Represetatio of a Image ad Its Applicatios 55 Jisug Oh received the B.S. ad M. S. degrees i Electrical Egieerig from Yosei Uiversity Korea i 987 ad 989 respectively ad the Ph.D. degree i Electrical Egieerig from the Uiversity of Pittsburgh U.S.A. i 998. He is curretly a Professor i the School of Electrical Egieerig at Halla Uiversity Korea. His research iterests iclude image processig ad multimedia systems.