Shape Classification Using Contour Simplification and Tangent Function

Transcription

1 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING hape Classfcaton Usng Contour mplfcaton and Tangent Functon Ch-Man Pun and Cong Ln Abstract - In ths paper we propose a new approach for mage classfcaton by smplfyng contour of shape and makng use of the tangent functon as mage feature. We frstly extract shapes from a sample mage and connectng pxels of ts contour. The extracted contour s smplfed by our algorthm and converted nto tangent functon whch s regarded as a feature. The tangent functon represented a shape s nput nto classfed system and compared wth tangent functon from exsted classes by computng ther dstance. The nput sample mage wll fnally be classfed nto a class that has mnmum dstance wth t. The expermental results show the proposed method can acheve hgh accuracy. Keywords hape classfcaton, contour, smplfcaton, tangent functon. C I. INTRODUCTION lassfyng a group of mages, tellng out whch class they belong to, s an mportant problem n shape classfcaton and mage retreval systems. hape classfcaton systems are generally used n applcatons lke mage databases n whch the shape segmented from an mage wll be classfed nto a class that have ext n database accordng to a defned crteron, whch judge how smlar or dssmlar the sample shape to those exsted n classes. Often the crteron, or called metrc, s features based. mply comparson of pxels does not work well on most systems, snce the dfference between mages s reflected on dfferences of certan features. hape s an mportant vsual feature of an mage for content-based mage retreval or recognton [1]. Technques for shape feature extracton can be generally classfed nto two categores: boundary-based methods and regon-based methods. Boundary-based methods are wdely used. They depend so much on the boundary that a slght change can cause grave retreve errors. Compared wth boundary-based ones, regon-based technques are comparatvely more sutable for general applcatons. Zernke Moments (ZM) method s one of the most desrable regon-based methods. The modfed ZM has been adopted by MPEG-7 as a standard regon-based descrptor. The generc Fourer Descrptor s another favorable method proposed n recent years. However, most of the exstng methods cannot acheve hgh recognton rate for mages wth complex nner shapes. Therefore, researches n the feld of smlarty measurng and mage matchng manly focus on mage features nowadays[1, 2]. For example, n[3], Inner-Dstance was used as an feature of shape that descrbed the characterstcs unque from others and n[4] Radon Composte Features was employed. One wdely used approach s template matchng. However, a drect template matchng also has ts man dsadvantage: very hgh computatonal cost. Ths s owng that representaton of template from mage can be complex n processng, for nstance, the hgher resoluton or more nose, the heaver computaton s requred. A meanngful smlarty measurng should have two mportant elements: fndng a set of features that adequately contan characterstcs that can be employed to dfferentate mages and endowng the feature space wth a proper metrc.[2] Researchers have also done substantal studes on that how to choose features for a partcular problem [5, 6]. On the other hand, desgnng a proper metrc s equally mportant. Most of the tme, a metrc s desgned for a specfc features. Mathematcans have establshed the reasonable archtecture for a metrc, defnng a metrc as a cost functon d(,) [7] satsfyng condtons: 1) d( A, B) 0 for all A and B. d( A, B) 0f and only f A=B. And (, ) 0 d A A n partcular a shape resembles tself. 3) d( A, B) d( B, A) for all A and B.(ymmetr 4) d( A, B) d( B, C) d( A, C) for all A, B and C. (Trangle Inequalt Moreover, a mature hape classfcaton system s also requred to be nvarant under transton, rotaton, and change of scale. Ths paper presented an approach to obtan an approxmaton of shape contour and proposed a novel approach to classfy shapes nto multple classes based on the tangent functon. The procedure of our system s gven as followng: Ths work was supported n part by the Research Commttee of Unversty of Macau under the Grant: RG056/08-09/PCM/FT. C.-M. Pun and Cong Ln are wth the Department of Computer and Informaton cence, Unversty of Macau, Macau.A.R., Chna. (e-mal: {cmpun, ma865551}@ umac.mo). Issue 1, Volume 4,

2 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING Raw Image egmentaton mplfcaton Convert nto Tangent space mlarty Measure Fg. 1 The process of the proposed approach Result The raw mages are nput nto to a segmentaton system. The system extracts and segments the shape of objects by color, hue or even contrast etc. Extensve lterature and approaches concernng shape segmentaton are avalable.[1, 8] nce the standard dataset for mage retreval are smple shapes wth blank background and consderng ths paper manly focuses on shape rather than segmentaton, we supposed shapes have already segmented from mages. Here are workng steps n detals: 1. Extract shapes from a sample mage and connectng pxels of ts contour. 2. The smplfcaton s mposed on the shapes frstly. The output s a smplfed, de-nosed, and more lnear boundary and of less pxels whch preserves mportant and relevant shape nformaton of the orgnal mage. The remanng ponts consttute sgnfcant part of the shape. Ths step also reduces computatonal complexty for followng steps. 3. Every two consecutve ponts are vewed as a lne segment or vector. The smplfed shape s converted nto tangent space and angles of every two consecutve vectors are mapped to the range of tangent functon. 4. The array of values of Tangent functon s then nput nto classfcaton system. The system compares the tangent functon from sample wth those typcal tangent functons n each class. And fnally fnd out a class that has mnmum dstance wth the sample. We wll gve more detals n secton II n ths paper concernng contour smplfcaton. The detals of defnton and usage of tangent functon wll be provded n secton III. In secton IV, a procedure for a novel mage classfcaton system s gven and explaned n detals. An analyss on result and dscusson s presented n secton V. Fnally, a summary concluded ths approach n the last secton. II. CONTOUR IMPLIFICATION Generally, a segmented shape contans some nose or small dents whch contrbute lttle to ts features and decelerate the process of measurng. In psychology, there are strong evdence that concludes recognton from human vson manly reles on sgnfcant parts of shape.[9] Therefore, the nose and nsgnfcant nformaton on the boundary should be removed and those sgnfcant ponts on the large turnng corners should be remaned. A smplfcaton process s necessary before measurng n order to obtan a smpler shape that preserves the sgnfcant shape features. Another beneft of smplfcaton s to reduce data storage and therefore accelerate processng n later stage. The smplfcaton process s also called dscrete Curve Evoluton Procedure[10, 11]. The basc dea of smplfcaton s: we replace two consecutve lne segments on the contour wth a sngle lne segment jonng ther endponts, f jont condton of two consecutve lne segments lower than a pre-defned threshold. The jont condton s a functon whch s defned to compute the sgnfcance of the two consecutve lne segments and ther angle contrbuted to the whole shape. And our approach reduces every shape to a fxed number of ponts. To acheve ths, we frst defne the relevance functon Ks ( [10, 11]: ( s Ls ( 1) Ls ( Ks ( Ls ( ) Ls ( ) A s 1 B 1 2 s 2 ( s, s ) 1 2 Fg. 2 Turn angle Ks ( between two vectors The relevance functon Ks ( s defned on two consecutve vectors and ndcates how much sgnfcance the trangle consst of s 1 and s 2 contrbutes to the whole shape. The hgher value of Ks (, the more sgnfcance of the two consecutve vectors contrbute to the shape. If there s an, such that the Ks (, s 1) s mnmum and ponts of the shape are stll more than the pre-defned constant, the two vectors are to be replaced by one that connects the endponts. Ths operaton teratvely traverses the shape boundary untl number of ponts s reduced to our pre-defned value. From the defnton of Ks (, we can see that the larger angle and longer lne segments gves a hgher functon value so that a more sgnfcant part wll not be replaced. C (1) Issue 1, Volume 4,

3 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING Fg. 3 An mage boundary to correspondng smplfed one farthest pxel from centrod n the shape to start smplfcaton; on the other hand, the value of tangent functon correspondng to x-axs s the relatve angle change to current vector to prevous one (from the graph of step functon, that means the value of current step s assgned by the dfference that value of current step mnus value of prevous step). In ths case, the scale of y-axs represents the change of drecton from prevous lne segment. From the fgure, after smplfcaton, the result generated a smplfed shape whch s more lnear and of less nose and preserved shape nformaton as well. Note that data of orgnal shape s array of locatons of neghborng ponts, whle data of smplfed one s array of dscrete ponts n clockwse orentaton. III. TANGENT FUNCTION A tangent functon s a representaton of a shape C and nterpreted as a polygonal curve wth all vertces wthout loss of nformaton. And reconstructon of the orgnal shape s reversble f the poston of one vertex s noted down n advance. A vector on boundary s mapped to a value and the whole shape s mapped nto a step functon whch can be vewed as a sgnal and analyzed by other tools, e.g. statonary wavelet. In [7, 8], tangent functon s defned as a step functon. Let C be a polygonal curve. It s defned as a 2 functon on C :[0,1], the length of C s rescaled to 1 n order to normalze the shape so that the smlarty measure between two shapes can be nvarant under scale-of-change. The tangent functon s also called a turnng functon whch s a mult-valued functon TC ( ):[0,1] [, ] defned by TC ( )( s) C ( s) and TC ( )( s) C ( s), where TC ( )( s) C ( s) and TC ( )( s) C ( s) are left and rght dervatves of C. The value of T( p ), p s a pont on C, s angle between a reference vector and the lne segment vector whch p lay on. IV. HAPE CLAIFICATION In classfcaton system, some templates should be stored n the system n advance and samples are nput nto the system then compared wth each template to determne whch class t belongs to. In our dea, templates are preprocessed and stored n the form of tangent functons. Each class may nclude several templates that shares some part features but wth dfferent vsual shapes. In order to compare templates and samples, the system extracts the boundary of samples, smplfes the boundary and converts t nto a form of tangent functon. The smlarty metrc s defned as follows[7]: 1 (, ) T () s T () s p DAB 0 A The degree of smlarty between two shapes A and B s measured by calculatng the metrcal dstance between two tangent functons. a) B ds ( b) Fg. 4 A smplfed boundary and ts tangent functon The tangent functon TC ( )( s ) traces where the turnng takes place on the contour, ncreasng wth left turns and decreasng wth rght turns, makes t a feature frequently used as a shape sgnature. [7, 12-15] In order to acheve rotaton nvarant, we would lke to modfy and mprove the orgnal tangent functon. In our verson, on one hand, we pck up the c) Fg. 5 c) the area between two tangent functons In Fg. 5, the area n gray s the dstance between two tangent functons. mlar tangent functons generate smaller Issue 1, Volume 4,

4 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING dstance. Therefore, classfcaton can be done by comparng the template from the sample wth each template of each class and fndng out that one whch has mnmal dstance wth the sample template. And fnally, the shape wll be classfed nto the class whch the mnmal dstance template belongs to. We also present our procedure n pseudo-code as follows: INPUT ample set: { s2,..., s n } ; Class set: C { CC,,..., C }; 1 2 m A Class wth several templates ncluded: Cj { tj1, tj2,...} PROCEDURE For =1 to n egment hape and extract boundary from mplfed the orgnal boundary Generate tangent functon Ts from smplfed boundary For j=1 to m For k=1 to the sze of ( C ) Compute dstance D( T s, t jk) Note down the Class ndex of mnmal D: Mn End End ample s classfed nto C Mn End V. EXPERIMENTAL REULT An experment for evaluatng ths approach s performed on a wdely used dataset provded by. [16] There are n total 99 samples n the dataset whch belongs to 9 classes. Each classes contans 11 samples wth relatvely smlar forms but nfluenced by occluson, rotaton, artculaton and mssng parts. And ths dataset s used as templates. Other datasets wth scaled rotated varatons are generated from the orgnal templates and used as samples. j s In our experment, we generated 16 sample datasets base on the orgnal one. They are rotated 0, 30, 45 and 90 degree and scaled 2/3, 1, 4/3 and 2 tmes respectvely. The overall results are showed as follows. All the shapes are smplfed nto fxed 28 ponts. TABLE I. Overall correctly classfcaton rates for each sample dataset caled Rotated 2/3 1 4/ % 100.0% 100.0% 100.0% % 85.9% 83.8% 84.9% % 85.9% 83.8% 85.9% % 100.0% 100.0% 100.0% % 99.0% 99.0% 99.0% From table I, we can see that the performance s qute nsenstve to change of scale. Even n the worst cases when all samples are scaled to 2/3, there are only 3 samples n a sample set msclassfed. Ths s manly because, after the smplfcaton, the length of boundary s normalzed and all of the reszed samples are compared n the same scale. However, we found that rotaton of samples affects the performance of ths approach sgnfcantly when samples are not rotated wth 90*n degree and t can be attrbuted to change of nose on the boundary, snce rotaton wth 90 degree wll not change relatve poston of ponts on boundary otherwse nose may be added and the relatve poston can be changed. Although rotaton and slght change on boundary mght affect tangent functon, tangent functons from the same category mantan relatvely smlar forms. Frst column n Table 1 s the worse cases. A further nvestgaton s to fnd out classfcaton rates for each class. TABLE II Classfcaton rates n each class Rotated Class Quadrupeds Humans Arplanes Grebes Fsh Hands Rays Rabbts Wrenches Overall 86.90% 85.90% 82.80% Form Table II, the experment successfully classfed all samples n category Humans and Rays and obtan hgh correct rate n category Fsh. amples n these categores generally have smooth boundary and sharp turns. Comparson wth Composte Features by Radon Transform Fg. 6. Image dataset of 99 testng shapes provded by Kma eta al[16] In mathematcs the radon transform n two dmensons, named after the Austran mathematcan Johann Radon, s the Issue 1, Volume 4,

5 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING ntegral transform consstng of the ntegral of a functon over straght lnes xcos ysn (1). Radon transform can be represented by R (, ) f ( x, ( x cos y sn ) dxdy (3) When a generalzed mage s represented by a functon f(x,, the radon transform s used to calculate the projecton of the mage ntensty along a radal lne orented at a specfc angle. In mage processng phase, the general mage functon f(x, s replaced by bnary dscrete sgnal D ( x, wth the sze m by n, where 1, f ( x, D D ( x, 0, otherwse. The densty dstrbuton along lnes as Formula (1) can be represented by n m D (, ) D ( x, ( x cos y sn ) y1 x1 (4) where E0 s a constant determned by the sze of mage, and varable E / E0 s used to make the shape sze nvarant. Fg 7 (a) llustrates a seres of butterfles. Pcture (a) s the orgnal mage. The remanng ones n the frst column are the geometry transformatons of (a). Pcture (b) s reszed by 0.5 tme; (c) s translated by 60 by 0 pxels; (d) s rotated counterclockwse by 30 degrees. The second column s the mage densty dstrbuton D of Column One respectvely. Label x s the degree from 0 to 179, and Label y s the mage densty dstrbuton n dfferent values. The thrd column s the modfed mage densty dstrbuton D. Label x and y are the same wth Column Two. The fourth column s the curve of DF ( ). By compared (a and (b, we can see D s sze nvarant whle (a1) and (b1) reflect D s not. Where represents the lne drecton and s the dstance away from the coordnate s orgn. Usually, these two parameters and are equdstantly chosen and ther numbers are denoted as N and N. The number D s a matrx wth sze N by N representng the mage densty dstrbuton along dfferent lnes. The mage densty dstrbuton propertes of perodcty and symmetry can be easly obtaned from the defnton, D (, ) D (, 2k ) D (, ) D (, (2k 1) ) (5) Parameter defaults to 0:179, N equals to 180 and N are determned by the sze of the mage. Radon Invarant Features For an arbtrary angle, we denote N E ( ) Rs(, 1 ) (6) where E ( ) s the energy of Radon Transform along lne, and parameter N s the length of Radon Matrx R n drecton. Accordng to the conservaton of energy, we have E E ( ) E ( j ) for any gven angle. A shape n the mage can be reszed. In order to meet the need of sze nvarant, modfed mage densty dstrbuton along lnes L(, ) s defned by D (, ) E0 / E Rs( E / E0, ) (7) Fg 7: Radon Invarant Features. (a) Orgnal Image; (b) resze 0.5 tme of (a); (c) translate 60 by 0 pxels; (d) rotate 30 degrees. (*1) are mage densty dstrbutons D for correspondng (*) pctures. (* D. (*3) are curves of feature DF ( ). A shape can be shfted n the mage. Luckly the mage densty dstrbuton along lnes L (, ) s just shfted. The boundary can be easly dstngushed by judgng the frst none zero value n Radon Matrx. Thresholds, mn and max, are employed to ascertan the vald range of the shape area along drecton. mn mn D (, ) D max max D (, ) n D where Dn s a threshold controllng the range of shape area along a partcular lne wth angle ; ths parameter s used to suppress nose and outlers. n (8) Issue 1, Volume 4,

6 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING The energy of Radon Transform (6) now changes to max ( ) (, mn E D ) (9) Another dstrbuton feature can be calculated by DF E ( ) 2 (, ) ) max mn max ( ) ( D mn (10) The shape densty dscrmnaton can be normalzed through DF ( ) DF ( ) / maxdf ( ) (11) The last column n Fg. 7 llustrates the shape dscrmnaton DF ( ). By comparng the frst three ones, the translaton and sze nvarant features can be easly found out. Especally, when mages are rotated by 30 degrees, curve DF ( ) s shfted or rotated by 30 degrees. Ths characterstc nspres us to fnd a rotaton nvarant dscrmnaton, whch wll be covered n ecton 3. THE TATIONARY WAVELET TRANFORM Wavelets are mathematcal functons that cut up data nto dfferent frequency components, and then study each component wth a resoluton matched to ts scale. Classcal dscrete wavelet transform s not tme nvarant. In order to restore wavelet transform shft nvarance, the statonary wavelet transform (WT) was employed. tatonary wavelet transform (WT) s shft nvarant. Fg 8 descrbes the basc decomposton steps for one-dmensonal WT. The approxmaton and detal coeffcents at level 1 are both of sze N, whch s the sgnal length. Low-pass Lo_D Approxmaton coefs ca 1 Fg 8: Rotate Invarant Features. (a) The exact curve DF ( ) as n Fg7 (a3); (b) Level 6 sub-wavelet of (a); (c) the exact curve DF ( ) as n (b3); (d) Level 6 sub-wavelet of (c). A modfed Radon transform plus a statonary wavelet transform, called Radon Wavelet Composte Descrptor (RWCD), can be used to descrbe nvarant shapes n sze, translaton and rotaton. RWCD s defned as RWCD swc( DFs ( ), Level, wname) (1 where 0,180, Parameter Level s used to decde the total number of sub-wavelets and RWCD s a matrx wth sze 180 by Level. The dstance between hape and j can be calculated by Dstance Level 180 RWCD 1 1 mn RWCD m n jmn (13) Ths dstance s used to determne the smlarty between two shapes. In the experments phase, t wll be adopted to determne the retreve or recognton results. Fg 9 selects four dfferent levels 4, 6, 8, and 10. Ths fgure shows that when level K ncreases, the recognton rates of the nne categores rse. The reason s that when K ncreases, the number of sub-wavelets rses, and the dstance between pctures s calculated n more detal. H_D Hgh-pass cd 1 Detal coefs Fg 4: Basc decomposton step for two dmensonal sgnals DF ( ) After appled the statonary wavelet transform, can be rotaton nvarant. Fg 5 (a) and (c) are the exact curves as n Fg. 3 (a3) and (d3), and ther correspondng statonary wavelet transforms are shown n Fg 5 (b) and (d). From ths fgure, the rotaton nvarant features can be obtaned. Fg 9: Recognton Rates at Dfferent Levels hapes wth Complex Inner tructure As a regon-based analyss method, RWCD can deal wth shapes wth complex nner structures. A test shape database s Issue 1, Volume 4,

7 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING used here. Ths database conssts of 504 mages whch are selected from the MPEG-7 (CE- regon-based database. They are organzed nto 24 groups, as shown n Fg 10. In each group there are 10 smlar shapes ncludng four scaled ones, and fve rotated ones. The precson recall dagrams are presented n Fg 11 by comparng wth two classcal retreval methods, Zernke Moments [9] and Generc Fourer Descrptors[10]. VI. CONCLUION In ths paper, we proved that tangent functon can be appled on classfcaton system and our experment result turned out to be reasonably good. However, we also found that changes of nose on the contour affect the tangent functons generated from mages. That may lead to dfferent tangent functons from same shape wth dfferent degrees rotated. For nstance, B s a rotated shape from A. They may have dfferent tangent functons. And dstance between A and B may be larger than certan neghbor shape of B. Fortunately; even f mnmum dstance s from another shape from the same category, we found most shapes mantan very small dstance when rotated, whle have large dstance wth shapes n other categores. REFERENCE Fg 10: hape database selected from the MPEG-7 (CE- Fg 11 reflects that the more detal the mages s, the hgher recognton rate s. In order to rase the recognton rate, the sze of the orgnal mages can be ncreased. Fg 11: Recognton Rates to Images wth Dfferent ze Fg 12 reflects the ablty of each method n descrbng shapes wth complex structures and the overall robustness under varous geometrc transforms. It can be observed that the proposed RWCD method can handle shapes wth complex nner structures more effectvely. Fg 12: Recognton Rates of Dfferent Retreval Methods. [1] R. E. w. Rafael C. Gonzalez, Dgtal Image Processng, Englewood Clffs, N.J.: Prentce Hall, [2] M. mone antn, IEEE, and Ramesh Jan Fellow, IEEE, mlarty Measures, IEEE TRANACTION ON PATTERN ANALYI AND MACHINE INTELLIGENCE, vol. 21, no. 9, pp , [3] L. Habn, and D. W. Jacobs, hape Classfcaton Usng the Inner-Dstance, Pattern Analyss and Machne Intellgence, IEEE Transactons on, vol. 29, no. 2, pp , [4] Y. Q. C. Yun Wen Chen, Invarant Descrpton and Retreval of Planar hapes Usng Radon Composte Features, IEEE Trans. gn. Proc., vol. 56, no. 10, pp , Octorber, [5] R. K. R.Jan, and B. hunk, Machne Vson: McGraw-Hll, [6] A. K. Jan, Fundamental of Dgtal Image Processng, Englewood Clffs, N.J.: Prentce Hall, [7] L. P. C. Esther M. Arkn, Danel P. Huttenlocher, Klara Kedem, and Joseph. B. Mtchell, An Effcently Computable Metrc for Comparng Polygonal hapes, IEEE TRANACTION ON PATTERN ANALYI AND MACHINE INTELLIGENCE, vol. 13, no. 3, pp , March [8]. Xu, Robust traffc sgn shape recognton usng geometrc matchng, IET Intellgent Transport ystems, vol. 3, no. 1, pp , [9] K. t. K. sddq, and B.B.Kma, Parts of vsual form: Ecologcal and psychophscal aspects, Proc. of the IAPRs Internatonal Workshop on Vsual Form, Capr, [10] L. J. L. a. R. Lakämper, Convexty Rule for hape Decomposton Based on Dcrete Contour Evoluton, Computer Vson and Image Understandng, vol. 73, no. No.3 March, pp , [11] L. J. L. a. R. Lakämper, hape smlarty Measure Based on Correspondence of Vsual Parts, IEEE TRANACTION ON PATTERN ANALYI AND MACHINE INTELLIGENCE, vol. 22, no. 10, pp , [12] J. H. a. X. Tan, The smlarty between shapes under affne transformaton, Proc. econd Int. Conf. Computer Vson, pp , [13] J. h. a. H. J. Wolfson, And mproved model-base matchng method usng footprngts, Proc. Nnth Int. Conf. Pattern Recognton, Rome, Italy,, vol. Nov , [14] J.T. chwartz and M. harr, ome remarks on robot vson, New York Unv., Courant Inst. Math. c., Tech. Rep. 119, vol. Robotcs, no. Rep. 25, Apr, Issue 1, Volume 4,

8 INTERNATIONAL JOURNAL OF CIRCUIT, YTEM AND IGNAL PROCEING [15] H. Wolfson, On curve matchng, Proc. IEEE workshop Computer Vson, Mam Beach, FL, No.30-Dec.2, [16] J. C. Danel harvt, H. Teck, and Bejamn B. Kma, ymmetry-based Indexng of Image Databases, JOURNAL OF VIUAL COMMUNICATION AND IMAGE REPREENTATION, vol. 9, no. 4, pp , Ch-Man Pun receved the B.c. and M.c. degrees from the Unversty of Macau n 1995 and 1998 respectvely, and Ph.D. degree n Computer cence and Engneerng from the Chnese Unversty of Hong Kong n He currently s an assocate professor at the Department of Computer and Informaton cence of the Unversty of Macau. Hs research nterests nclude Content-Based Image Indexng and Retreval, Dgtal Watermarkng, Pattern Recognton, and Computer Vson. Issue 1, Volume 4,