A Neighbor-finding Algorithm Involving the Application of SNAM in Binary-Image Representation

Transcription

1 TELKOMNIKA, Vol. 3, No. 4, December 205, pp. 39~329 ISSN: , accredted A b DIKTI, Decree No: 58/DIKTI/Kep/203 DOI: /TELKOMNIKA.v A Neghbor-fndng Algorthm Involvng the Applcaton of SNAM n Bnar-Image Representaton Je He, Hu Guo, Defa Hu* 2 School of Informaton and Electronc Engneerng, Wuzhou Unverst, Wuzhou , Guang, Chna 2 School of Computer and Informaton Engneerng, Hunan Unverst of Commerce, Changsha 40205, Hunan, Chna *Correspondng author, e-mal: hdf666@63.com Abstract In vew of the low eecuton effcenc and poor practcablt of the estng neghbor-fndng method, a fast neghbor-fndng algorthm s put forward on the bass of Square Non-smmetr and Antpackng Model (SNAM) for bnar-mage. Frst of all, the mproved mnor-dagonal scannng wa s appled to strengthen SNAM s adaptablt to varous tetures, thus reducng the total number of nodes after codng; then the storage structures for ts sub-patterns are standardzed and a grd arra s used to recover the spatal-poston relatonshps among sub-patterns, so as to further reduce the complet of the neghbor-fndng algorthm. Epermental result shows that ths method s eecuton effcenc s sgnfcantl hgher than that of the classc Lnear Quad Tree (LQT)-based neghbor-fndng method. Kewords: Neghbor-Fndng; SNAM for Bnar-Image; Mnor-Dagonal Scannng Mode; Grd Arra Coprght 205 Unverstas Ahmad Dahlan. All rghts reserved.. Introducton As the bass of a varet of mage operatons such as the calculaton of mages geometrc propertes, the labelng of connected domans, mages edge detecton, etc. [], neghbor-fndng algorthm can drectl affect the eecuton effcenc of these operatons. The epressve methods adopted n dgtal mage and ther adacent defntons decde the performance and space-tme complet of neghbor searchng algorthm. The orgnal neghborfndng algorthm s eecuted n the two-dmensonal arra of mage, however, due to the large storage space the two-dmensonal arra occupes and the wde search scope nvolved n the pel-pel traversal, the neghbor-fndng algorthm based on pel arra has an unsatsfactor effcenc. The wdel used Methods such as quadtree [2], lnear quadtree [3] can reduce basc mage representaton unts, save the storage space and enable the rapd block-block eecuton of mage operaton. Lterature [4-6] separatel puts forward neghbor-fndng algorthms based on the quadtree and lnear quadtree representatons, whch have sgnfcantl mproved the effcenc of some mage processng operatons [7-9]. However, because the neghbor relatonshp s not a one-to-one relatonshp, ths knd of method cannot be easl performed n the actual mage operaton. Therefore, as new and more smplfed mage epressve methods emerge contnuousl, people are stll seekng for hgher-effcent neghbor searchng methods. The method of non-smmetr and ant-packng mage representaton acheves optmal asmmetrc mage segmentaton so as to obtan a hgher effcenc of representaton than that of the quadtree method [0], and derves a seres of hgh-effcent mage processng algorthms [-4]. Lterature [5] proposes a bnar mage representaton method whch uses square, solated pont, lne segment as the basc representaton unts n the NAM method. Wth regular shape and smple structure, these basc representaton unts can serve as ecellent bass for further developng other mage processng algorthms. However, there reman some defcences wth ths method, restrctng ts supportve ablt for other mage processng procedures: Its raster scannng means s prone to producng more sub-schemes when performng reverse arrangement n mages wth rch massve tetures; the separate standalone storage structure of the three sub-schemes also ncreases the algorthm s space-tme complet. Besdes, although the coordnates of the sub-schemes have been recorded, there s a lack of descrpton about the Receved August 26, 205; Revsed October 25, 205; Accepted November 3, 205

2 320 ISSN: relaton of spatal poston among them, whch tpcall entals traversal over all storage queues n the dervatve algorthm. Snce the effcenc of neghbor searchng algorthm depends drectl on the relatve postonal relaton between and quantt of basc representatve unts, solutons to the above problems are the ke to hgh-effcent neghbor searchng on SNAM. The author begns b utlzng the structural characterstc of square to nclude sub-scannng along the back-dagonal drecton n raster scannng and enhance the adaptablt of reverse arrangement to massve tetures; and then uses a same data structure to record three dfferent sub-schemes so that the storage and operatng mode can be unformed across dfferent sub-schemes; net the author constructs grd matr, an ntermedate structure, to restore the postonal relaton among the sub-schemes and reduce the searchng range for neghbor. Through the above measures, the neghbor searchng s mplemented on the result of SNAM codng, and compared epermentall wth the LQT-based neghbor searchng method to demonstrate the effcac of ths work. 2. SNAM for Bnar-Image and Its Optmzaton 2.. Square NAM-Based Representaton Method of Bnar Image The mage epressve method of transform doman usuall has hgher compresson rato than methods of spatal doman [6], but the latter can reserve the mage s orgnal local tetural characterstcs and postonal relaton more effectvel. SNAM for bnar-mage s a nondestructve representaton method, whch s based on the space doman and uses three tpes of sub-patterns - square, lne segment and solated pont - as the basc mage representaton unt. Its process s as follows: square, lne segment and solated ponts at dfferent scales are etracted n a raster scan mode from a packed bnar mage through the ant-packng algorthm, and correspondng parameters of these sub-patterns are recorded and stored, thus formng the representaton of the gven bnar mage. Fgure shows the result of ant-packng, codng and storng a bnar mage wth SNAM. Fgure (a) s the gven orgnal bnar mage T wth a sze of 2 n 2 n (n =3). Fgure (b) shows the result of SNAM ant-packng n the raster scan mode. S square sub-patterns (s, s 2, s 3, s 4, s 5 and s 6 ), two lne segments (l and l 2 ) and two solated ponts (p and p 2 ) are etracted from the orgnal mage. Fgure (c) s the representaton of T b usng the above-mentoned sub-patterns. (a) An 8 8 bnar mage T (b) The SNAM ant-packng T={s, s 2, s 3, s 4, s 5, s 6, l,l 2, p, p 2 } (c) The result of of T Fgure. The process of ant-packng a bnar mage wth SNAM Accordng to SNAM, the tems need to be recorded are: for square, the coordnates sp of the upper left verte (.e., the startng pont) and the sde length sde; for lne segment, the coordnates p of the startng pont and the coordnates p 2 of the end pont; for solated pont, onl the coordnates of the verte. The storage structures for all sub-patterns are as shown n Fgure 2. TELKOMNIKA Vol. 3, No. 4, December 205 :

3 TELKOMNIKA ISSN: sp sde p p 2 _ p (a) square (b) lne (c) solated pont Fgure 2. The process of ant-packng a bnar mage wth SNAM SNAM can also brng some problems for mage processng. Lterature [7] notes that the raster scan mode has a good performance toward mages wth unobvous block features, but shows low effcenc toward mages wth obvous block features. In addton, the dverst of SNAM s storage structures for three tpes of sub-patterns and ts orgnal ant-packng algorthm lead to the emergence of both horzontal and vertcal lne segments n the codng result, further ncreasng the complet of subsequent mage processng Optmzaton of the Scannng Mode Raster scan s characterzed b the lne-b-lne horzontal scan, and a combnaton of raster scan and block scan wll produce a stronger adaptablt to mages wth block teture. The horzontal drecton s stll the man scannng drecton, but when unmarked black pont s detected, the structural characterstcs of square are used and determnng the pel values of pels on the mnor-dagonal s regarded as the core algorthm - thus the mnor-dagonal scannng mode s formed. Its prncple s shown n Fgure 3. Theren, Fgure 3(a) s the dagram of the mnor-dagonal scannng drecton, where the dotted lne ndcates that after these ponts form a square, the needn t go through horzontal scan an more. Fgure 3(b) shows the seral numbers of all pels n the mnor-dagonal scannng process. Ths mode not onl has a stronger adaptablt, but onl lowers the complet of SNAM b onl producng horzontal lne segments. (a) Drecton for mnor-dagonal scan (b) Pel sequence n mnor-dagonal scan Fgure 3. Prncple of mnor-dagonal scan The specfc procedure of the SNAM algorthm based on mnor-dagonal scan (Mnor Dagonal-SNAM, or MD-SNAM) s as follows: Input: a bnar mage T wth a sze of n n. Output: V_s the set of square sub-patterns, V_l the set of lne segment subpatterns, V_p the set of solated pont sub-patterns. Step Search for the unmarked pont wth zero pel value b startng from the pel frst seen when accessng T n ths crcle, and use the found pel as the upper verte of the square sub-pattern, then record ts coordnates (sp_, sp_), and ntalze the sde length of the square sub-pattern wth sde. Step 2 Search for unmarked pels on the back dagonal of the square wth sde-long sdes, and udge whether ther pel values are 0: f the are not 0, swtch to Step. A Neghbor-fndng Algorthm Involvng the Applcaton of SNAM n Bnar-mage (Defa Hu)

4 322 ISSN: Step 3 Make sde=sde+, and contnue to perform Step 2 untl no more square subpattern forms; store the found square n V_s, and mark all the pels covered b the square wth sde-long sdes. Step 4 Rotate Steps ~3 untl no new square sub-patterns can be etracted out. Step 5 Search for the unmarked pont wth zero pel value b startng from the pel frst seen when accessng T n ths crcle, and record ts coordnates (l_, l_ ). Step 6 Judge whether the value of the neghborng pel at the horzontal rght sde of the pont (l_, l_ 2 ) s zero: f t s, contnue the search toward the horzontal rght sde untl the lne segment stops enlargng, then record the lne segment s rght endpont s coordnates (l_ 2, l_ 2 ), and store the lne n V_l, and then mark all the pels that t covers; f t s not zero, go to Step 5. Step 7 Rotate Step 5 and Step 6 untl no new lne segment sub-patterns can be etracted out. Step 8 Search for the unmarked pont wth zero pel value b startng from the pel frst seen when accessng T n ths crcle, and record ts coordnates (p_, p_); then store t n V_p and mark ths pont. Step 9 Rotate Step 9 untl no new solated-pont sub-pattern can be etracted out. Step 0 Output V_s, V_l and V_p Optmzaton of the Storage Structure SNAM has desgned storage structures separatel for three dfferent tpes of basc representaton unts, whch, whle can ensure compact storage, means the same SNAM-based mage operaton algorthm needs to separatel process dfferent sub-patterns. Therefore, usng the same knd of smple data structure to record dfferent sub-patterns s the ke to lowerng the complet of the subsequent mage processng algorthm. For the three tpes of sub-patterns - square, lne segment and solated pont - nvolved n SNAM, a unform storage structure can be defned as follows: Fgure 4. The unform storage structure for SNAM sub-patterns Theren, represents the abscssa of the startng pont of the sub-pattern, represents ts ordnate, and represents the sub-pattern s sde length. In the codng, the of square and solated pont s stored as nt data, whle the of lne segment s stored as strng data. Thus the sub-patterns can be dstngushed b the value and data tpe: f the sub-pattern s square, ts must be greater than and belong to nt data; f the subpattern s solated pont, whch can be regarded as a square wth a sde of, then ts must be equal to and belong to nt data; and f the sub-pattern s lne segment, ts sde length (.e., ts length) s also greater than but belongs to stng data. In ths wa, a unform sub-pattern set can be establshed to realze rapd traversal, thus provdng more effectve supports for varous mage operatons. 3. Neghbor-Fndng Based on MD-SNAM 3.. Defnton of Neghbor The word neghbor s used for depctng the adacent relaton among mage areas that mage representaton unts correspond to, and ts defnton vares dependng on specfc mage representaton methods. Lterature [8] has presented the defnton of neghbor n the quadtree method, and consderng that both SNAM and the quadtree representaton method epress mage source as the record set of local mage blocks, a smlar defnton can be made heren: f there ests at least one pel n both sub-patterns that makes the two adon each other wthn the 4-neghbor doman, these two sub-patterns are neghbors. The basc representaton unts n SNAM are square, lne segment and solated pont. Each solated pont s essentall a square TELKOMNIKA Vol. 3, No. 4, December 205 :

5 TELKOMNIKA ISSN: wth the sde length of, thus SNAM actuall nvolves onl two tpes of sub-patterns, namel, square and lne segment. The specfc defnton s as follows: Defnton In the SNAM-based bnar mage representaton, f s ={,, } neghbors s ={,, } n the drecton D=(EAST,SOUTH,WEST,NORTH): () When s s square and s s square, ther neghbor relatonshp s shown n Fgure 5 and follows the followng formula: () (2) (a) The scope of s s east neghbors (b) The scope of s s west neghbors (c) The scope of s s south neghbors (d) The scope of s s north neghbors Fgure 5. The process of ant-packng a bnar mage wth SNAM (3) (4) (2) When s s square or pont and s s lne segment, ther neghbor relatonshp s shown n Fgure 6 and follows the followng formula: (a) The scope of s s east neghbors (b) The scope of s s west neghbors (c) The scope of s s south neghbors (d) The scope of s s north neghbors Fgure 6. The neghbor relatonshp between square s and lne segment s A Neghbor-fndng Algorthm Involvng the Applcaton of SNAM n Bnar-mage (Defa Hu)

6 ISSN: TELKOMNIKA Vol. 3, No. 4, December 205 : (5) (6) (7) (8) (3) When s s lne segment and s s square or pont, ther neghbor relatonshp follows the followng formula: (9) (0) () (2) (4) When s s lne segment and s s lne segment, because the mnor-dagonal scannng mode onl generates horzontal lnes whch onl have south and north neghbors, ther neghbor relatonshp follows the followng formula: (3) (4) 3.2. Grd Arra SNAM does not nvolve the recordng of the spatal-poston relatonshp wth codes, thus neghbor-fndng becomes an neffcent process of repeated defnton-based searches

7 TELKOMNIKA ISSN: throughout the code arra. Therefore, the neghbor-fndng process can be smplfed b constructng an ntermedate structure and restorng the poston relatonshps among subpatterns. A n n matr, namel grd matr, can be constructed for a n n bnar mage. Each unt s frst assgned wth a value of ; then the SNAM codng result of the mage s read, and the poston that each sub-pattern corresponds to n the grd matr, the structure of whch s shown n Fgure 7, s re-flled wth a value, whch s equal to the storage seral number for the correspondng sub-pattern + 2 because the seral number of the SNAM code arra begns from 0 and also has been used for pre-fllng; and then the SNAM code and the flled value for each sub-pattern make up a grd arra, wth the flled value actng as the correspondng nde. Thus the onl step for fndng the neghbor of some block s to scan the pel values surroundng ths block - f a dfferent value that s not and detected, the sub-pattern that ths nde value corresponds to n the grd matr s the neghbor that s beng searchng for. (a) SNAM-based segmentaton of an 8 8 mage (b) The correspondng grd matr Fgure 7. The neghbor relatonshp between lne segment s and lne segment s Wthout the applcaton of the grd matr method, neghbor-fndng would be a ver tme-consumng process of traversng the entre SNAM code lst to make one-b-one udgment as to whether the currentl accessed SNAM sub-pattern meets the condton for neghbor relatonshp. Wth grd matr, the operaton scope of neghbor-fndng n certan drecton can be lmted to the pels on the boundar block of sub-patterns, thus the operaton load can be sgnfcantl cut Grd Arra After the storage queue V of the SNAM algorthm for bnar mage as well as the V- generated grd matr M and grd arra T s gven, the search for the neghbor R of a gven subpattern r= (,, ) wll become ver eas. Take the search for east neghbor for eample: frst of all, udge the sub-pattern tpe of r accordng to ts ; then determne whether r s close to the rght boundar of the mage accordng to the abscssa + of r s endpont, and f t s, r s east neghbor R s an empt set, otherwse the values for the rght boundar of r subpattern n the grd arra are ndees of all ts neghbors - then, compare them wth the values on the rght boundar of the grd arra one b one, and add these poston-storage ndees to the neghbor set R as long as there s no -value nde. Because each nde n the SNAM code lst corresponds to one and onl sub-pattern, the ndees can lead to the neghborng models. The algorthm prncples for neghbor-fndng n varous drectons are bascall the same, thus the eastward neghbor-fndng s used heren agan as an eample to eplan the specfc algorthm eecuton steps as follows: Input: grd arra T, the gven coordnates (, ) of the startng pont of sub-pattern r, mage resoluton n, and code arra V={V_s,V_l,V_p}, where V_s, V_l and V_p respectvel represent the code arra set for square, lne segment and solated pont. Output: the set E_R of sub-patterns of r s east neghbors Step Customze a set E_R and ntalze t to be empt. A Neghbor-fndng Algorthm Involvng the Applcaton of SNAM n Bnar-mage (Defa Hu)

8 326 ISSN: Step 2 Input the gven coordnates (, ) of the startng pont of sub-pattern r that s used for neghbor-fndng, and match the startng pont s coordnates wth the code arra to produce the basc nformaton and seral number (nde) of sub-pattern r. Step 3 Use the basc nformaton of the sub-pattern to udge the basc tpe of t: f t s square or solated-pont, ump to Step 4; f t s lne segment, whch means t has no east or west neghbors, ump to Step 5. Step 4 Va the coordnates (, ) of r s startng pont, fnd r s rght-most endpont (+length,) n the grd matr M. If r s close to the rght boundar of the mage, that s, +length s equal to n, then t does not have east or west neghbors, thus go to Step 5; otherwse, thoroughl scan M for pel unts located wthn the area (+length+, +length-) on the rght sde of the mage rght boundar, and udge whether ts numercal value (nde) m s (representng whte pont): f t s not, use m to fnd the correspondng sub-pattern n the grd arra T, and record t n E_R. Step 5 Output the E_R. 4. Eperments and Analss The epermental envronment s made up b Intel Core 3-30M 2.4GHz processor, 2G memor, Mcrosoft Wndows 7 Professonal + SP3 operatng sstem and MATLAB7.0-based IDE. Fgure 8 shows eght bnar mages wth dfferent completes to be tested n the eperment. The defnton of mage complet s as follows: Defnton 2 Complet C of an bnar mage can be defned n the followng wa: calculate, based on the gven mage representaton method, the total number of ts nodes N Q as well as the total pel number N f, and put them n the formula of: C= N Q / N f (5) In ths paper, the LQT representaton method s regarded as a benchmark for comparson, and mage complet s measured b the number of LQT segmentaton blocks. Table s the comparson of the codng performances of LQT, SNAM and MD-SNAM. Theren, N represents the number of sub-patterns or nodes, SNAM s the orgnal square NAM-based representaton method, MD-SNAM s SNAM based on the mnor-dagonal scan, LQT _ T s the rato of the total data amount of LQT to that of TNAM, LQT _ R s the rato of the total data amount of LQT to that of RNAM, LQT _ S s the rato of the total data amount of LQT to that of SNAM, and LQT _ MDSNAM s the rato of the total data amount of LQT to that of MD-SNAM. The specfc data n the epermental result s shown n Fgure 8. Table shows that, for the above eght mages, SNAM and MD-SNAM have outperformed LQT, though these two also have sgnfcant dfferences n performance. The orgnal SNAM needs three felds - coordnates of the startng pont and sde length - for recordng square, four felds - coordnates of the startng pont and of the endpont - for recordng lne segment, and onl 2 felds, namel the pont coordnates, for recordng solated pont. B comparson, MD-SNAM provdes a unform storage structure for all these tpes: for recordng square, t s the same as SNAM; because t onl generates horzontal lnes, t onl needs three felds - coordnates of the startng pont and the length - for recordng lne segment; and t also needs three felds for recordng solated pont, whch s regarded as square wth a sde length of. Obvousl, MD-SNAM nvolves one less feld than the orgnal SNAM for lne segment, but one more feld for solated pont. After codng, the numbers of lne segments and solated ponts var form one mage to another, leadng to the dfference n the data amount n the codng result. Therefore, the dfference n storage structure actuall does not have a decsve mpact on the codng performances of MD-SNAM and SNAM. Furthermore, compared wth the orgnal SNAM, MD-SNAM not onl has an optmzed storage structure, and also adopts an mproved scannng mode - namel, the mnor-dagonal scannng mode, provdng t wth a stronger adaptaton to mages local block tetures, so as to ncrease squares and decrease nodes. Accordng to the epermental result, regardless of mage complet, MD-SNAM has TELKOMNIKA Vol. 3, No. 4, December 205 :

9 TELKOMNIKA ISSN: fewer nodes, hgher compresson rato and better spatal effcenc than the orgnal SNAM, thus t can further reduce the tme complet of the subsequent mage processng. (a) Lena (b) Flght (c) Buldng (d) Boat (d) Map (e) Pctures (f) Baboon (g) Brdge Fgure 8. Eght bnar mages Table. Comparson of the performances of LQT, SNAM, MD-SNAM algorthms Image C N LQT SNAM MD-SNAM LQT _ T LQT _ S LQT _ MDSNAM Map Pctures Baboon Brdge Lena Flght Buldng Boat For square and lne segment, MD-SNAM gves dfferent defntons of neghbor, but the prncples of searchng for ther neghbors n the grd matr are dentcal. In fact, the operaton amount of fndng the neghbor of a lne segment, whch s pel n wdth, s less than that of fndng a square s neghbor. Snce LQT s basc representaton unt s square mage block, a square sub-pattern s chosen as a benchmark for MD-SNAM n the eperment. The eperment s amed at comparng the performances n an eastward neghbor-fnd process, and the result s shown n Table 2, where, T LQT and T MD-SNAM represent the eecuton tme of neghbor-fndng based respectvel on LQT and on MD-SNAM, and t L-MS represents the speed-up rato of MD- SNAM-based neghbor-fndng to LQT-based neghbor-fndng. It can be seen from the Table 2 that hgher mage complet can result n longer eecuton of the eastward neghbor-fndng wth the two algorthms, ndcatng that mage complet s proportonal to the eecuton tme of the correspondng algorthm. The average eecuton tme of LQT-based neghbor-fndng s 4.275~3.76 tmes that of neghbor-fndng wth SNAM based on grd arra, whch full demonstrates that the neghbor-fndng algorthm based on MD-SNAM s smpler, faster, and more supportve for other subsequent mage processng. A Neghbor-fndng Algorthm Involvng the Applcaton of SNAM n Bnar-mage (Defa Hu)

10 328 ISSN: Table 2. Comparson of the eecuton tme of an eastward neghbor-fndng process respectvel wth the LQT-based algorthm and wth MD-SNAM-based algorthm Image C Eecuton tme (ms) Speedup rato T LQT T MD-SNAM t L-MS Map Pctures Baboon Brdge Lena Flght Buldng Boat Concluson Ths paper, frstl, mproves the scannng mode of the orgnal SNAM so that t can handle mages wth varous teture characterstcs; secondl, optmzes the storage structure for sub-patterns so as to elmnate the storage dfference n varous sub-patterns; thrdl, realzes the descrpton of the poston relatonshps among sub-patterns wth the grd arra; and fnall, uses the grd arra to lmt the range of one-drectonal neghbor-fndng to the boundar of the grd arra that the benchmark blocks correspond to, thus creatng a new neghbor-fndng algorthm, whch has hgher eecuton effcenc than the classc LQT-based method and can be appled n mage operatons such as calculaton of geometrc propertes and moment generaton. Acknowledgements Ths work was supported b Guang Natural Scence Foundaton Program (Grant No. 203GXNSFBA09275, Grant No. 203GXNSFBA09276), Guang Unverst of Scence and Technolog Research Program (Grant No. 203YB227, Grant No. 203YB228) and Natonal Natural Scence Foundaton of Chna (Grant No: ). References [] Xng Z, Shu L. Image Edge Feature Etracton and Refnng Based on Genetc-Ant Colon Algorthm. TELKOMNIKA (Telecommuncaton Computng Electroncs and Control). 205; 3(): [2] JJ Laguarda, E Cueto, M Doblare. A Natural Neghbor Galerkn Method wth Quadtree Structure. Internatonal Journal for Numercal Methods n Engneerng. 2005; 63(6): [3] I Gargantn. An Effectve Wa to Represent Quadtrees. Communcatons of the ACM. 982; 25(2): [4] V Khanna, P Gupta, JC Hwang. Mantanng Connected Components n Quadtree-Based Representaton of Images. Iranan Journal of Electrcal and Computer Engneerng. 2003; 2(): [5] BP Kumar, P Gupta, CJ Hwang. An Effcent Quadtree Data Structure for Neghbor Fndng Algorthm. Internatonal Journal of Image and Graphcs. 200; (4): [6] P Bhattachara. Effcent Neghbor-Fndng Algorthm n Quadtree and Octree. Kanpur: Indan Insttute of Technolog, [7] S De Brun, W De Wt, J Van Oort, et.al. Usng Quadtree Segmentaton to Support Error Modelng n Categorcal Raster Data. Internatonal Journal of Geographcal Informaton Scence. 2004; 8(2): [8] CL Wang, SC Wu, YK Chang. Quadtree and Statstcal Model-Based Lossless Bnar Image Compresson Method. Imagng Scence Journal. 2005; 53(2): [9] G Mnglun, Y Yee-Hong. Quadtree-Based Genetc Algorthm and Its Applcatons to Computer Vson. Pattern Recognton. 2004; 37(8): [0] Chuanbo C. Stud on A Non-Smmetr and Ant-Packng Pattern Representaton Method. Wuhan: Huazhong Unverst of Scence and Technolog, [] Peng W, Chuanbo C, Yunpng Z. Image Set-Operaton Algorthm based on NAM and Its Epermental Research. Journal of Yangtze Unverst (Natural Scence Edton). 2009; 6(2): [2] Weu H. Stud on Multple Sub-Pattern Image Representaton and Retreve Methods based on NAM. Wuhan: Huazhong Unverst of Scence and Technolog, TELKOMNIKA Vol. 3, No. 4, December 205 :

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