Hierarchical String Cuts: A Translation, Rotation, Scale and Mirror Invariant Descriptor for Fast Shape Retrieval

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1 Hierarchical Sring Cus: A Translaion, Roaion, Scale and Mirror Invarian Descripor for Fas Shape Rerieval Auhor Wang, Bin, Gao, Yongsheng Published 014 Journal Tile IEEE Transacions on Image Processing DOI hps://doi.org/ /tip Copyrigh Saemen 014 IEEE. Personal use of his maerial is permied. Permission from IEEE mus be obained for all oher uses, in any curren or fuure media, including reprining/republishing his maerial for adverising or promoional purposes, creaing new collecive works, for resale or redisribuion o servers or liss, or reuse of any copyrighed componen of his work in oher works. Downloaded from hp://hdl.handle.ne/1007/6750 Griffih Research Online hps://research-reposiory.griffih.edu.au

2 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Hierarchical Sring Cus: A Translaion, Roaion, Scale and Mirror Invarian Descripor for Fas Shape Rerieval Bin Wang and Yongsheng Gao, Senior Member, IEEE Absrac This paper presens a novel approach for boh fas and accuraely rerieving similar shapes. A hierarchical sring cus (HSC) mehod is proposed o pariion a shape ino muliple level curve segmens of differen lenghs from a poin moving around he conour o describe he shape gradually and compleely from he global informaion o he fines deails. A each hierarchical level, he curve segmens are cu by srings o exrac feaures ha characerize he geomeric and disribuion properies in ha paricular level of deails. The ranslaion, roaion, scale and mirror invarian HSC descripor enables a fas meric based maching o achieve he desired high accuracy. Encouraging experimenal resuls on four daabases demonsraed ha he proposed mehod can consisenly achieve higher (or similar) rerieval accuracies han he sae-of-he-ar benchmarks wih a more han 10 imes faser speed. This may sugges a new way of developing shape rerieval echniques in which a high accuracy can be achieved by a fas meric maching algorihm wihou using he ime-consuming correspondence opimisaion sraegy. Index Terms Shape descripion; shape rerieval; hierarchical sring cus I ITRODUCTIO ow o boh quickly and effecively rerieve similar shapes Hfrom a large image daabase is an imporan and challenging problem ha coninues aracing aenions of many researchers in compuer vision and paern recogniion. Is imporance lies in ha more and more pracical applicaions, such as plan leaf image rerieval [43], fish image rerieval [0], rademark image rerieval [1], medical umor shape rerieval [], have encounered he speed and accuracy rade-off barrier. The challenges are wofold. (1) owadays, image daabases have been growing larger and larger, which make he compuaional cos of shape rerieval become prohibiively expensive (e.g. here are abou 400,000 species of plans all over he world [3] and millions of plan leaf images will be sored in he daabase). On he oher hand, many real-ime rerieval asks (e.g. online shape rerieval) require he rerieval sysems respond o he queries very quickly. Therefore, Manuscrip received December 17, 01. This work was suppored in par by he Ausralian Research Council (ARC) under Discovery Gran DP087799, and by he aional aural Science Foundaion of China under Gran B. Wang is wih Key Laboraory of Elecronic Business, anjing Universiy of Finance and Economics, anjing 10096, China ( wangbin@njue.edu.cn) and School of Engineering, Griffih Universiy, Brisbane, QLD 4111, Ausralia ( bin.wang@griffih.edu.au). Y. Gao is wih School of Engineering, Griffih Universiy, Brisbane, QLD 4111, Ausralia ( yongsheng.gao@griffih.edu.au). rerieval efficiency is becoming a more imporan issue o be addressed in real applicaions. () The shapes in daabases usually have large inra-class variaions (see Fig. 1) and small iner-class differences (see Fig. ), which make i very difficul for he rerieval sysem o achieve a desirable rerieval accuracy. Fig.1. Example leaves from he same plan species ha have large inra-class shape variaions. Fig.. Five example leaves from differen plan species ha have small iner-class differences. Shape descripion and maching are wo key pars of shape rerieval. The former exracs effecive and percepually imporan shape feaures and organizes hem in a daa srucure such as vecor, sring, ree and graph. While he laer uses he obained shape descripors o deermine he similariy (or dissimilariy) value of he wo shapes o be compared based on a shape disance measure. The performance of any shape rerieval mehod ulimaely depends on he ype of shape descripor used and he maching algorihm applied [31]. According o MPEG-7, a favourable shape descripor should have a high discriminabiliy so ha i can group similar shapes ogeher and separae dissimilar shapes ino differen groups, and a reliable shape descripor should be roaion, scaling, and ranslaion invarian [1]. In he pas decade, various mehods have been proposed for shape rerieval. To evaluae he performance of differen mehods, MPEG-7 se up a group of challenge daases (MPEG-7 Core Experimen CE-shape-1) for objecive experimenal comparison [15],[3]. Many works of recen en years [18],[13],[1],[5],[4],[3],[33],[34],[35],[37],[38] have repored good rerieval raes (>75%) on he MPEG-7 CE-1 Par B daase, wih some of he mos recen ones [4], [5], [13], [38], [37] achieved very high rerieval raes (>85%). However, heir high rerieval accuracies are obained wih high compuaional ime cos as hese algorihms mainly rely on he primiive correspondence echniques such as dynamic programming (DP). For example, he inner-disance shape conex (IDSC) [5]

3 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < achieved a very high rerieval rae of 85.40% using DP maching, whereas i only obained 68.83% when using he shape conex disance measure o replace he DP maching par. In heory, he DP based shape maching mehods have he 3 ime complexiy of O ( ) for maching wo shape conours wih sample poins [3]. Take = 100 as an example. DP based inner-disance shape conex (IDSC+DP) [5] and shape ree [13] spend 8.6 hours (on he repored.8ghz PC) and 13.9 hours (on he repored 3GHz compuer) respecively o rerieve a shape in a large daabase wih 100,000 images. Acually, almos all he exising mehods repored a rerieval rae greaer han 80% on he MPEG-7 CE-1 Par B daase adoped ime consuming DP maching. This makes he curren mehods wih high accuracies no suiable for shape rerieval in large image daabases and online rerieval, resuling in an accuracy-speed rade-off dilemma o users. The above limiaion of curren algorihms moivaes us o develop a novel shape descripion and maching mehod ha considers boh speed and accuracy in is design. A Hierarchical Sring Cus (HSC) approach is proposed in his paper, which uses a hierarchical coarse o fine cuing and coding framework o describe and mach shapes. Each subcurve of he shape is characerized by a se of geomeric and disribuion feaures ha can capure more discriminaive informaion of he shape han only using a single geomery feaure such as curvaure [39], angle paern [1][48][49], pairwise disances [46][47], inegral invarians [19][4] and riangle area [4][43][53]. The proposed HSC achieved 87.31% rerieval rae on he MPEG-7 CE-1 Par B daase, he highes accuracy on leaf rerieval, leaf classificaion and he second highes rerieval accuracy on marine animal rerieval experimen, which were achieved wih a speed of over 10 imes faser han he sae-of-he-ar benchmarks. The remainder of he paper is organized as follows: A brief review of relaed work is presened in Secion. In Secion 3, we describe he deails of he proposed Hierarchical Sring Cus (HSC) mehod. The compuaional complexiy of HSC is analysed and compared o he sae-of-he-ar benchmark mehods in Secion 4. Four groups of experimens are presened and analysed in Secion 5. Finally, we draw conclusions in Secion 6. II RELATED WORK Various shape descripion and maching mehods have been proposed in he pas decades [10], [31]. They can be broadly classified ino wo caegories, feaure meric measuremen mehod and primiive correspondence mehod. A feaure meric measuremen mehod exracs discriminaive feaures of he wo shapes o be mached o form wo numeric vecors of a specified lengh, whose similariy or dissimilariy is measured by a meric disance such as L 1 or L. In he conrary, a primiive correspondence mehod regards he shape as an aggregaion of primiives, where primiives can be poins, curve segmens, line segmens, ec. A he shape descripion sage, a se of feaures/aribues is exraced for each primiive, and a he shape maching sage, all pairs of primiives of he wo shapes are firs compared o generae a cos marix (or disance marix). The leas maching cos obained by he opimal correspondence of he primiives beween he wo shapes is considered as heir dissimilariy. A. Feaure meric measuremen mehod Fourier descripors (FDs) [40] is a classical feaure meric measuremen mehod, in which a D shape conour is firs represened as an 1D signaure, such as complex coordinaes [5], cenroid disance [41], farhes poin disance [9]. A discree Fourier ransform is hen applied o he signaure o generae a feaure vecor using Fourier coefficiens. Zhang e al. [8] conduced a large amoun of shape rerieval experimens o evaluae he performance of FDs derived from various shape signaures and repored ha he FDs derived from he cenroid disance signaure had a higher rerieval performance and is more suiable for shape rerieval han using oher shape signaures. Mos recenly, EI-ghaza e al. [6] reas he curvaure-scale-space represenaion of a shape conour as a binary image and apply a D Fourier ransform o i. The obained descripor capures he deailed dynamics of he shape curvaure and he Euclidean disance meric is used for shape maching. Oher specral ransform based shape descripors have also been proposed. Chuang e al. [17] used one-dimensional coninuous wavele ransform o creae a muli-scale shape represenaion. Yang e al. [44] proposed a wavele descripor independen of he saring-poin locaion of a conour. As he wavele coefficiens are no invarian o he roaion of a shape, Kunu e al. [7] applied Fourier ransform o he wavele coefficiens and used he obained Fourier coefficiens o describe he shape. Wang e al. [30] used sequency-ordered complex Hadamard ransform [7] o build a new shape descripor. The specral ransform based descripors are very compac. This is because he energy of he signal concenraes on he low frequency componens and only dozens of low frequency coefficiens can describe he shape effecively. In addiion, hey are also robus o noise because he noise predominanly disribues in he high frequency coefficiens which have been discarded in building he shape descripor. However, hese mehods only use a single geomeric feaure such as curvaure, cenroid disance, and farhes poin disance o consruc 1D shape signaure and lose deailed informaion ha is discarded in he ransform domain. This ofen resuls in weak discriminaion abiliy of his ype of shape recogniion mehods. According o he repored resuls on he MPEG-7 CE-1 Par B daase [17], [30] which are also consisen wih he resuls from our implemenaions of hese algorihms (see Table ), heir rerieval raes are no beer han 70%. Effors have also been made o yield shape descripor direcly from he spaial domain. These mehods can be classified ino curvaure based and boundary poin relaionship based echniques. Curvaure is a fundamenal propery of shape. The curvaure scale space mehod (CSS) [5] is a well-known curvaure based mehod, which has been recommended as one of he sandard shape descripor by MPEG-7 [11]. In CSS, Gaussian kernel wih increasing sandard deviaion is used o

4 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 3 smooh he conour a differen scales. The inflecion poins are locaed a each scale which resuls in a binary image, ermed CSS image. The maxima of he CSS images conour are used for shape maching. The ypical mehod of compuing curvaures is differenial echniques. However i amplifies noise and is no sable. To address his issue, Manay e al. [19] used inegral invarians insead of differenial invarians o compue he curvaure feaures of a shape conour. Mos recenly, Kumar e al. [4] used he area inegral invarians combined wih he arc-lengh inegral invarians for plan leaf idenificaion. The feaures of he riangles [4][53] formed by shape boundary poins are also used as an alernaive of curvaure. Rubé e al. [53] used he signed riangle area (TAR) o reflec he concave/convex properies of he shape boundary. The wavele ransform is used for smoohing and decomposing he shape boundaries ino muliscale levels. The TAR image a each scale ha is similar o he CSS image is used for shape maching. Mouine e al. [43] recenly exended he riangle area represenaion ino wo new descripors, ermed riangle oriened angles (TOA) and riangle side lenghs and angle represenaion (TSLA) which can provide more discriminaive informaion han hose of only using he area of riangles. The localiy sensiive hashing [9] echnique is used for maching. Foeini e al. [49] used he muliscale angle code sequence incorporaed wih he Muual eares Poin Disance (MPD) [8] for shape maching. The limiaion of his mehod is is expensive compuaion cos. Mos recenly, Hu e al. [48] used he angles formed by he boundary poin and is wo neighbor poins of equal disance from i o form an angle paern (AP). The relaionships of he neighbor APs are encoded ino a binary sring, ermed binary angular paern (BAP). The muliscale AP and BAP are used o build hisogram for shape maching. Boundary poin relaionship based mehods focus on capuring he geomerical properies from he space relaionship of he pairs of boundary poins. Hu e al. [47] recenly proposed a novel shape descripor, ermed muliscale disance marix (MDM). In MDM, he disances beween all possible pairs of shape boundary poins are calculaed o form a muliscale disance marix, where scale is he arc lengh. Each row of he obained marix capures cerain range of geomeric properies. Through shifing and soring operaions, an invarian marix is creaed for shape descripion. The L 1 meric is used for shape similariy measuremen. Biswas e al. [46] used, for each pair of landmark poins, he inner disance beween wo poins, relaive angle, conour disance and ariculaion-invarian cener of mass associaed wih he pair of poins o form a feaure vecor. Each of hem is quanized and is mapped on he appropriae bins in he hash able. The numbers of feaure vecors hashed o various bins are formed ino a sequence. The χ disance meric is used for shape similariy measure. B. Primiive correspondence mehod In recen years, many researchers dedicae o develop effecive shape descripion and maching mehods using correspondence based echniques o improve he recogniion accuracy. The well-known shape conex (SC) [18] builds a hisogram primiive for each conour poin o describe he relaive disribuion beween he poin and he remaining poins, which provides rich shape informaion for finding he bes poin correspondence beween wo shapes in a poin-by-poin manner. I repored 76.51% rerieval rae on he MPEG-7 CE-1 Par B daase, and achieved 86.8% when dynamic programming (DP) is used in he maching process [14]. To make he shape conex descripor robus o ariculaion, Ling e al. [5] replace Euclidean disance wih inner-disance (ID), which is defined as he shores pah beween wo conour poins wihin he shape boundary, o exend he SC mehod o a novel shape descripion mehod ermed inner-disance shape conex (IDSC). IDSC achieved a promising rerieval rae of 85.4% on he MPEG-7 CE-1 Par B daase. Inspired by he observaion ha smoohing a closed conour makes he convex and concave poins move inside and ouside he conour, respecively, Adamek e al. [3] proposed a novel muli-scale convexiy/concaviy (MCC) mehod for shape maching. MCC used he relaive displacemen of a conour poin wih respec o is posiion a he preceding scale level o measure he convexiy and concaviy properies a a paricular scale. The relaive displacemens of all he scale levels for each conour poin are capured o calculae he disance beween each pair of conour poins. The resuling disance marix is used o find he bes one-o-one dense poin correspondence. A rerieval rae of 84.93% is repored on he MPEG-7 CE-1 Par B daase. Alajlan e al. [4] proposed anoher muli-scale shape descripor, ermed riangle area represenaion (TAR). I uilizes he areas of he riangles formed by he boundary poins o measure he convexiy/concaviy of each conour poin a differen scales, where scale denoes he lengh of he arc which associaes wih he riangle formed by he conour poin and is wo neighbour poins. The muli-scale riangle areas associaed wih each conour poin are used o consruc he disance marix for finding he bes one-o-one poin correspondence. Tesed on he MPEG-7 CE-1 Par B daase, i repored a 85.03% of rerieval rae and achieved 87.3% if hree global shape feaures aspec raio, eccenriciy and solidiy are included in he shape maching. To explicily capure boh global and local geomeric properies of he shape of an objec, Felzenszwalb e al. [13] proposed a shape ree mehod, in which a shape conour is modelled o a full binary ree by recursively dividing i ino wo subcurves of he same lengh. The dividing poins are aken as he node of he ree. Each node of he ree sores he midpoin locaion of he subcurve relaive o heir sar and end poins. Those nodes a he boom of he ree capure local geomeric properies while he nodes near he roo capure more global informaion. When maching curves A and B, a shape ree is buil for A o look for a mapping from poins in A o poins in B such ha he shape ree of A is deformed as lile as possible. Shape ree is an ineresing mehod and achieved a very high rerieval rae (87.70%) on MPEG-7 CE-1 Par B daase. Primiive correspondence mehods, including [1], [33], [34], [35], [36], [37], [51], [38], and [50] are repored higher

5 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 4 discriminaive capabiliy and ofen achieved higher han 80% rerieval accuracy on he MPEG-7 CE-1 Par B daase. However he primiive correspondence mehods are compuaionally slow making hem less pracical in real world applicaions paricular for large daabase rerieval, as hey use opimisaion algorihm such as dynamic programming o look for he bes primiive correspondence beween wo shapes. Their compuaional complexiies range from O( ) o 4 O ( ). III THE PROPOSED METHOD In his secion, we firs describe he proposed sring cu process for characerizing a curve segmen, and hen inroduce he hierarchical sring cus ha can encode a shape compleely from global and coarse informaion o fine local deails. ex, he invariances of he proposed hierarchical sring cus descripor are discussed. Finally, he dissimilariy measure is presened for maching shapes. A. Sring Cu A shape can be effecively represened by a sequence of poins sampled from he conour on he objec wih uniform spacing [3], [4], [5], [13], [14]. The benefis of his represenaion is ha i is no required o seek key-poins such as poins of maximum curvaure and we can obain as good an approximaion o he underlying coninuous shapes as desired by picking he number of sample poins o be sufficienly large (see Fig. 3(a) as an example). Therefore, a shape ϕ can be described in a form of sample poin sequence φ = {p i (x i, y i ), i = 1,, }, where i is he index of he sample poin according o is order along he conour in couner-clockwise direcion, ( x i, yi ) is he coordinaes of he sample poins p i, and is he oal number of sample poins on he conour. In he design of he proposed HSC, is required o be in he power of wo (=56 in our experimens). Since he conour is a closed curve, we have p +i = p i and p i = p i, for i = 1,,, i.e. he conour is reaed as a periodic sequence wih a period of. Curve pariion by sring: Le sequence Aij = { pi, p i + 1,, p j } denoe a piece of curve segmen of he conour of he shape ϕ, which sars from he poin p i and ends a he poin p j. We define he sraigh line passing hrough he poins p i and p j as he sring ξ ii o cu he curve segmen. A can be possibly ij cu ino hree groups S r, S l and S o, where S r is he se of he poins falling o he righ side of he sring ξ ii, S l is he se of he poins falling o he lef side of he sring ξ ii, and S o is he se of he poins falling on he sring ξ ii. Obviously we have S r Sl So = Aij, S r S l = φ, S r S o = φ and S l S o = φ, i.e. { Sr, Sl, So} is a sring cu pariion of curve segmen A ij. Fig. 3 (b) illusraes an example of he sring cu process on a curve segmen of horse shape from he MPEG-7 daabase. Sring Cu Feaures: Given a piece of curve segmen A, le { S r, Sl, So} be is sring cus. The geomeric properies of he curve segmen A ij can be depiced by he disribuions of hese cus including deviaions o sring ξ ii on boh sides ( f 1 and f ), imbalance of cu ( f 3 ), and degree of bending ( f 4 ) as defined below. 1 1 f 1 = max h(, x ij ), h(, xij ), (1) T r T S l r Sl 1 1 f = min h(, ξ ij ), h(, ξij ), () T r T S l r Sl f3 = T r T l, (3) Lij f 4 =, (4) d( pi, p j ) where T r, T l and T o are he number of sample poin in S r, S l and S o respecively, L ii is he lengh of he curve segmen, h( p k, ξ ij ) is he perpendicular disance from poin p k o he sring ξ ii which can be calculaed by ( xk xi )( y j yi ) ( yk yi )( x j xi ) h(, x ij ) =, (5) ( x x ) + ( y y ) i and d( p i, p j ) denoes he Euclidean disance beween he poins p i and p j. These sring cu feaures as a whole express he configuraion and he behavior of he enire curve segmen relaive o he reference sring and also ensure heir invariance o he possible swapping of sar and end poins of he curve segmen (see invariance analysis in Secion C). j i j ij p i p i p j p j Fig. 3. Sring and sring cu. (a) a curve segmen of horse conour and is sring. (b) The sring cus he curve segmen ino hree groups; one falls on he sring, and he oher wo fall o he wo sides of he sring, respecively.

6 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 5 B. Hierarchical Sring Cus and Their Signaures The sring cus on longer curve segmens capure he more global shape informaion, while hose on shorer ones are associaed wih he finer deails of he curves. Here we propose a Hierarchical Sring Cus (HSC) mehod ha divides a closed conour ino curve segmens of differen lenghs, o provide a muliple level coarse-o-fine descripion. Fig.4. An example resul of hierarchical sring cus for a poin S a hierarchical levels = 1,, 4. A hierarchical level, for each sample poin p i, i = 1,,, we ake a piece of curve segmen A ii = {p i, p i+1,, p j } saring from p i and ending a p j. The locaion of he end poin p j and he lengh of he curve segmen are deermined by he hierarchical level using j = i +. The sring cu feaures for poin p i a level, { f1 ( i), f ( i), f3 ( i), f4 ( i)}, can be calculaed using Eqs (1-4). Thus, for each hierarchical level, we obain four -dimension Sing Cu Signaures f1 ( i), f ( i), f3 ( i), f4 ( i), i = 1,, o describe he shape in a coarse/fine level conrolled by. Each sring cu signaure is a sequence (wih a lengh of ) of sring cu feaures. As he number of sample poins on he conour ϕ is, here are log 1 levels for = 1,,log 1 ha divide ϕ ino curve segmens from he coarses ( +1) -poin lengh o he fines 3-poin lengh. Fig. 4 gives an example of he hierarchical sring cu process for a single conour poin S ha divide he shape ino curve segmens of differen lenghs a he coarses hierarchical levels = 1,,4. C. Invariance of Shape Descripor A good shape descripor should be ranslaion, scale, roaion and mirror invarian. Since he sring cu feaures of a curve segmen are calculaed solely wih respec o is sring, hey have he inrinsic invariance o ranslaion; and so do he sring cu signaures ha are derived from hese feaures. To make he sring cu signaures invarian o scaling, each signaure f g (i), g = 1,,4 is locally normalised by dividing is maximum value max ( i). I should be poined ou ha since ( ) f g i= 1,, f 3 i can ake negaive values, i is normalized by max 3 ( i). f i= 1,, When max ( i) = 0, he above normalisaion division f g i= 1,, canno be execued. In his case, he normalizaion division is omied in implemening he algorihm as he signaure wih all 0 s is already invarian o scaling. A hierarchical level = 1,,log 1, for shape signaure f g (i), g = 1,, 4, he magniudes of heir Fourier ransform coefficiens are calculaed by 1 1 jp ik Fg( k) = fg( i)exp i= 0, k = 0,, 1. (6) From heory, he above obained Fg ( k ), k = 0,, 1 are invarian o he locaion of sar poin p i of sring cu curve segmens, and hus invarian o roaion of he whole shape. To make he generaed shape descripor robus o noise and compac, he lowes M order coefficiens Fg ( k ), = 0,, M 1 where M << are used o describe he shape. Finally he hierarchical sring cus (HSC) shape descripor ψ = { ψ1,, ψlog 1} is a combinaion of descripors ψ from all hierarchical levels of defined as ψ = { Fg( k), σg g = 1,,4; k = 0,, M 1}, (7) where σ is he sandard deviaion of he sring cu feaures. I g conains supplemenary informaion o F (0) (which is he mean value) for each signaure f g. Mirror invarian maching is one of he requiremens for planar shape recogniion mehods. Mos of he exising echniques [3], [4], [5] address his invariance problem by performing he same shape maching algorihm wice, one is beween shapes A and B and he oher is beween shape A and he mirrored shape of B. To avoid he compuaional cos of applying he maching algorihm on he mirrored shape, he proposed HSC shape descripor is designed inrinsically as mirror invarian, ha is, ψ = { ψ1,, ψlog 1} remains he same if he sar poin p i and he end poin g p j are inerchanged. In Eqs (1) and (), f 1 and f are designed as he deviaions o is sring on boh sides according o heir values insead of locaions, ensuring heir invariance o he inerchange of p i and p j. Imbalance of cu f 3 changes sign when p i and p j are inerchanged o make is shape signaure 3 ( i ) funcion only have a 180 phase difference, resuling a same F 3 ( k ), k = 0,, 1. As f 4 is also independen o he direcion of shape conour, he HSC shape descripor is mirror invarian. f

7 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 6 D. Shape Dissimilariy Measure ψ ( A ) ( A ) Given wo HSC descripors ψ( A) = { F ( k), σ } and ( B ) ( B ) ( B) { Fg ( k), σg } =, exraced from shapes A and shape B, respecively, where g = 1,, 4, k = 0,, M 1 g g and = 1,,log 1, we firs compare hem in each hierarchical level = 1,,log 1 by calculae he sub-level dissimilariy a hierarchical level using Eq. (8): 4 M 1 ( A ) ( B ) ( A ) ( B ) D( AB, ) = wg Fg ( k) Fg ( k) + σg σg g= 1 k= 0, (8) where are he weighs o allow he adjusmen of wg conribuion from each sring cu feaure. The dissimilariy beween wo shapes A and B can be easily considered as an aggregaion of log 1 sub-level dissimilariies as log 1 D = 1 D ( A, B) = ( A, B). (9) However, he case of = 0 is no considered ye in he design, in which he lengh of subcurve equals o, i.e. he curve segmen is he whole shape conour. Many conour global feaures such as circulariy, eccenriciy, convexiy, raio of principle axis, ellipic variance, circular variance, have already been developed for shape descripion. To complee he design of he proposed approach, here, we ake he wo exising global conour feaures, eccenriciy (E) and recangulariy (R), as he feaures of hierarchical level = 0. Wihou losing generaliy, oher global conour feaures can be used as well. The dissimilariy of hierarchical level 0 is: ( A) ( B) ( A) ( B) D = E E + R R. (10) 0 And he overall dissimilariy beween wo shapes A and B becomes log 1 D = 0 D ( A, B) = ( A, B). (11) IV COMPUTATIOAL COMPLEXITY The compuaional cos of shape rerieval consiss of wo pars. One is he ime for compuing he shape descripor, and he oher is ha used for performing shape dissimilariy measuremen. The laer par is more imporan and plays a dominan role in deermining he rerieval speed han he former one, paricularly when he size of daabase becomes larger. This is because for a shape rerieval ask, only he query shape is required o creae is descripor online and all he descripors of gallery shapes can be calculaed offline o be sored in he daabase beforehand; whereas shape dissimilariy measuremens are conduced online beween he query descripor and all gallery descripors in he daabase. In his secion, we provide a heoreical compuaional complexiy analysis for he proposed HSC in comparison wih he sae-of-he-ar mehods (Experimenal comparisons in compuaional ime are also conduced in Secion V). In creaing he HSC descripor, for each hierarchical level = 1,,log 1, he complexiy of calculaing he sring cu signaures for he whole conour is O because he ime o compue he sring cu feaures for each curve segmen Aij = { pi, pi+ 1,, pj} is O, where is he number of he sample poins of he conour and j = i +. Then he complexiy of calculaing he signaures for all he hierarchical levels is O +,, + O 1 = O( ) 1 K = K. (1) For each obained signaure (4(log 1) signaures in oal), he complexiy of calculaing is Fourier ransform coefficiens and sandard deviaion are O ( log ) and O () respecively. Therefore he ime of compuing he Fourier coefficiens and sandard deviaions for all he signaures is O ( 4(log 1)( log + )) = O ( log + log ) = O( log ). (13) The overall complexiy of creaing he proposed HSC shape descripor is O ( + log ). A he dissimilariy measuremen par, he complexiy of calculaing Eq. (8) is O ( 4( M + 1)) = O( M ). Therefore, he complexiy of calculaing Eq. (9) is O ( M (log 1)) = O( M log ). (14) Since calculaing Eq. (10) only requires ime O (1), he ime of compuing Eq. (11), i.e. he overall compuaional complexiy of dissimilariy measuremen is O ( M log ). In Table I, we compare our mehod wih several recen represenaive mehods which are repored high rerieval accuracies (>80%) on he MPEG-7 Par B rerieval es, where is he number of sample poins of he conour, and K (for IDSC+DP [5] and SC+DP [14]) denoes he number of possible saring poins for alignmen used in heir dynamic programming par. From his able, we can see ha he proposed HSC has he lowes order of compuaional complexiy among he compeing mehods wih high recogniion accuracies. TABLE I. Comparison of he compuaional complexiy of differen mehods a he sage of shape dissimilariy measuremen. Proposed HSC MCC [3] TAR+DP [4] IDSC+DP [5] SC+DP [14] Shape Tree [13] O ( M log ) 3 O ( ) 3 O ( ) O ( K ) O ( K ) 4 O ( ) V EXPERIMETS To evaluae he effeciveness and efficiency of he proposed approach, an exensive experimenal invesigaion is conduced on MPEG-7 shape, plan leaf shape, and marine animal shape daabases. The performances of he proposed mehod are

8 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 7 compared wih he sae-of-he-ar approaches in boh accuracy and speed. In all he experimens, he same parameers ( M = 7, w 1 = 1.4, w = 0.5, w 3 = 0. 5, w 4 = 0. 4 ) ha is heurisically deermined, are used wihou uning he sysem seing o bes sui individual daase. sae-of-he-ar approaches, including he well-known inner disance (IDSC) [5], shape conexs (SC) [18] and heir varians are abulaed in Table II. Their Precision-Recall (PR) curves are ploed in Fig. 6. Fig. 7 provides 4 deailed examples ha lis he op 10 rerieved shapes and heir scores obained by he proposed HSC, IDSC [5], SC [18]. Table II. Rerieval resuls on he MPEG-7 daabase. The values wih * are from he published papers. Algorihm Rerieval rae (%) Average rerieval ime (ms) Fig. 5. MPEG-7 Par B shape daabase, which conains 70 classes wih 0 images in each class. A. MPEG-7 Shape Daabase The MPEG-7 CE-1 Par B shape daabase [15] is a widely used daase for evaluaing performances of similariy-based shape rerieval. I conains 1400 shape images, consising of 70 classes of various shapes wih 0 images in each class (see Fig. 5). The rerieval accuracy is measured using he well-known bulls-eye es [15], [3], [4], [5], [14]. In his measuremen, each shape is used in urn as a query and mached wih all he shapes in he daabase. The number of correc maches (ha is he rerieved shape and he query shape belong o he same class) in he op 0 = 40 rerieved shapes ha have he smalles dissimilariy values are couned. Since he maximum number of correc maches for a single shape is 0, he oal number of correc maches is = The percenage of mached shapes ou of 8000 is he rerieval rae of he bulls-eye es. The compuaional ime of each shape rerieval is he ime of maching he query wih all he 1400 shapes including he feaure exracion ime of he query shape. To finely compare he behavior of he proposed HSC agains benchmarks, heir Precision-Recall (PR) curves are provided. Fig. 6. Precision-Recall (PR) curves of he proposed HSC and he sae-of-he ar approaches, including TSDIZ [], TAR+WT[53], MDM [47], CBFD [6], ASD&CCD [49], FD [6][8], FPD+Global [9], CSS [5], TAR+DP [4], IDSC+DP [5], MCC [3] and SC+DP [4][18], obained on he MPEG-7 Par B shape daabase. The non-dp mehods are ploed in red, while he DP mehods are ploed in blue. * indicaes ha he resuls are from he published papers. The rerieval rae and speed of he proposed Hierarchical Sring Cus (HSC) approach ogeher wih hose of he DP on- DP MCC [3] 84.93* TAR+DP [4] 87.13* IDSC+DP [5] 85.40* SC+DP [14][18] 86.80* Shape Tree [13] 87.70* A Proposed HSC MDM [47] ASD & CCD [49] 76.0* FD[6][8] FPD [9] IDSC+SC disance measure [5] 68.83* A SC+SC disance measure [5] 64.59* A WTD [17] 67.76* A I can be seen ha he proposed approach achieved similar rerieval accuracy o inner disance wih dynamic programming (IDSC+DP), shape conexs wih dynamic programming (SC+DP), and oher dynamic programming based algorihms wih he highes repored accuracies. The 87.31% of rerieval rae of he proposed HSC is slighly higher han he renowned IDSC+DP (85.40%), SC+DP (86.80%), bu lower han ha of Shape Tree mehod [13] by 0.30%, placing i among he mos accurae shape rerieval algorihms. On he oher hand, he above accuracy is obained in a speed over 10 imes faser han hose mehods wih comparable accuracies (only 0.61%, 0.51%, 0.77% and 0.65% of he rerieval ime used by MCC [3], TAR+DP [4], IDSC+DP [5] and SC+DP [14], [18]). In our experimens, all he algorihms are wrien in Malab and are run on a PC wih Inel Core- Duo.8 GHz CPU and GB DDR RAM under Windows XP. Due o he very large compuaion cos in dynamic programming (DP) par of he benchmark algorihms, he DP par in MCC [3], TAR+DP [4], IDSC+DP [5] and SC+DP [14], [18] are implemened in C allowing he comparaive experimens o be compleed in an accepable ime. oe ha he rerieval speed of Shape Tree [13] was no repored. Bu, as is compuaional complexiy is 4 O ( ) (see Table I), i is compuaionally more expensive han he oher four benchmarks (whose complexiies are 3 O ( ) and O ( K ) respecively). The performances of some fas non-dp shape maching algorihms are also lised in Table II for comparison purpose, whose accuracies are much lower han he proposed mehod and he sae-of-he-ar benchmarks by around 0%. [14] replaced he hin plae spline maching process used in [18] wih Dynamic Programming, which increased he rerieval rae from 76.51% [18] o 86.8%.

9 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 8 Fig.7. The comparaive rerieval resuls of 4 shapes obained by he proposed HSC, SC +DP [18], and IDSC+DP [5]. Each column shows he query shape (op image) and he 10 mos similar shapes rerieved from he MPEG-7 daabase (he nd -11 h images). The number below each rerieved shape is is dissimilariy score. The incorrec his are circled in red colour. Fig. 7 gives he rerieval resuls of four ypical shapes obained using he SC+DP, IDSC+DP and he proposed approach. The resuls are lised and sored in ascending order of dissimilariy (he firs 10 ranked maches are shown). For he firs query shape wih small occlusions, he proposed mehod obains 7 correc his which is more han 5 for SC+DP and for IDSC+DP, indicaing he more robusness of he proposed approach o small occlusions han SC and IDSC. For he second shape which have larger occlusions, he performance of he proposed mehod decreases drasically (only obains correc his), while he SC and IDSC are sable which indicaes ha SC and IDSC are more robus o larger occlusions han he proposed mehod. For he hird and he fourh query shapes wih large inra-class variaions, he proposed mehod obains 9 and 7 correc his respecively which are much beer han 0 and 4 for SC, and 3 and 6 for IDSC. These comparaive resuls shows ha he proposed approach, wihou using dynamic programming, can achieve a similar/beer accuracy as SC and IDSC wih an over 10 imes faser speed for large daabase on-line rerieval. B. Plan Leaf Daabases In his secion, we apply he proposed HSC on a real and challenging applicaion, plan leaf image rerieval and classificaion, and compare is performance wih he benchmark approaches. The challenge of his applicaion lies in he high iner-class similariy beween some of he leaf shapes and he large inra-class variaions of plan leaves from he same species. Leaf rerieval The same performance evaluaion mehod as used in MPEG-7 shape daase is applied on a plan leaf image daabase colleced by ourselves. I conains 100 leaf images colleced from 100 plan species, wih 1 differen leaves from each class (see Fig. 8). Table III shows he comparaive resuls of he proposed Hierarchical Sring Cus (HSC) approach and he sae-of-he-ar benchmarks including he well-known inner disance (IDSC+DP) [5] and shape conexs (SC+DP) [14], [18]. The Precision-Recall (PR) curves of hese mehods are ploed in Fig. 9. I can be seen ha he proposed approach achieves he highes rerieval rae and obains he bes Precision-Recall (PR) curve among all he compeing mehods. The 89.40% of rerieval rae of he proposed HSC is 3.76% and.58% higher han he inner disance (IDSC) [5] and he shape conexs [14][18] respecively, and is 1.30% and 11.74% higher han he oher dynamic programming based algorihms MCC [3] and TAR+DP [4] respecively. I is worh noing ha his high accuracy is achieved wih a speed of over 15 imes faser han hose dynamic programming based mehods (only 0.61%, 0.54%, 0.80% and 0.70% of he rerieval ime used by MCC [3], TAR+DP [4], IDSC+DP [5] and SC+DP [14], [18], respecively), The performances of some fas shape maching algorihms are also lised in Table III for comparison purpose, whose accuracies are much lower han he proposed mehod by around 5%. Fig. 8. Plan leaf image daabase colleced by ourselves, which conains 100 species wih 1 images in each class. Table III. Rerieval resuls on he plan leaf image daabase. Average Algorihm Rerieval rae (%) rerieval ime (ms) MCC [3] TAR+DP [4] DP IDSC+DP[5] SC+DP [14] [18] ASD&CCD [49] on-dp FD [6][8] MDM [47] Proposed HSC

10 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 9 Fig. 9. The Precision-Recall (PR) curves of he proposed HSC and he sae-of-he ar approaches on he plan leaf daase colleced by ourselves. Leaf classificaion The publicly available Swedish leaf daabase, which comes from a leaf classificaion projec a Linkoping Universiy and he Swedish Museum of aural Hisory [45] is also used in our experimens. The daabase conains 15 differen Swedish ree species (see Fig. 10) wih 75 samples in each species. Due o he naure of his daabase (small number of classes and large number of samples in each class), he same leaf classificaion proocol and accuracy measuremen as used in [5], [13], [45] are adoped in our experimen. 5 images randomly seleced from each species are used as models and he remaining images are used as esing images. The classificaion rae is calculaed using he neares-neighbor classificaion rule. The comparaive resuls of he proposed Hierarchical Sring Cus (HSC) approach wih he sae-of-he-ar approaches are lised in Table IV. From Table IV, i can be seen ha he proposed approach mainained he highes classificaion accuracy (96.91%), consisen wih he previous resuls in rerieval seing. daase is combined wih 00 bream fish images which are frames exraced from a video clip of a swimming bream fish (examples are shown in Fig. 11 (a)) o form a daabase consising of =1300 images for esing he robusness o shape changes caused by no-rigid moion and differen viewing angle [15]. In our experimen, each of he 00 bream fish images is used as a query in urn o mach agains he 1300 images in he daabase. The number of bream fish images in he op 00 mos similar shapes is couned, and he percenage of he mached shapes ou of 00 is calculaed as rerieval rae (see also [1], [16], [17], [5], [6]). The comparaive resuls of he proposed Hierarchical Sring Cus (HSC) approach wih MCC [3], TAR+DP [4], IDSC+DP [5], and SC+DP [14],[18] are given in Table V. Considering he large shape variaions caused by non-rigid moion and differen viewing direcion (see Fig. 11 (a)), i is very encouraging o see ha he proposed HSC possess he similar level of olerance o shape disorions as he bes performing approaches, while is compuaional speed is more han 133 imes faser han he compeing algorihms (188.3, 184.7, and imes faser han MCC, TAR+DP, IDSC+DP and SC+DP respecively). The average rerieval rae (over 00 ess by using every bream fish shape as he query in each es) for HSC remains higher han MCC, TAR+DP and Shape Conex. I also worh commen ha our resul shows ha he Inner Disance approach is he mos robus o disorions (which suppor he claim in [5]) as we can see ha IDSC+DP becomes he mos accurae one in his experimen, while is accuracy is ranked he second and he hird in he leaf daabases experimens. Fig. 10. Fifeen samples (one per species) from he Swedish leaf daabase. TABLE IV. Classificaion raes on he Swedish leaf daabase. The values wih * are from he published papers. Algorihm Classificaion rae (%) MCC [3] TAR+DP [4] IDSC+DP [5] 94.13* SC+DP [5] 88.1* Shape Tree [13] 96.8* TSLA+LSH [43] 96.53* MDM+ Dimensionally reducion [47] 93.60* TAR+LSH [43] 90.40* ASD+CCD[49] Proposed HSC C. Marine Daabase The marine daabase was originally buil by Mokharian e.al. [4]. I consiss of 1100 marine animals (samples are shown in Fig. 11 (b)) which are unclassified. In MPEG-7 (Par C), his Fig. 11. Samples from 00 bream fishes (a) and 1100 marine animals (b). TABLE V. Rerieval resuls on he marine daabase. Algorihm Average rerieval Average rerieval rae (%) ime (ms) MCC [3] TAR+DP [4] IDSC+DP [5] SC+DP [14],[18] Proposed HSC D. Sensiiviy o oise To evaluae he sensiiviy o noise, a group of experimens wih differen levels of addiive noise are conduced. We adop he same noise perurbaion scheme as used in [6][9][19][50]. For each shape image in MPEG-7 Par B daase, is coordinaes (x, y) of he boundary poins are randomly varied by a Gaussian noise wih differen variance. The noise corrupion of shape is measured by he signal-o-noise raio (SR) as s SR = 10log, (16) s s n

11 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT) < 10 where s and σ n are he variances of he signal and noise sequences, respecively. In our experimens, various levels of Gaussian noise (SR=0dB-50dB) are added o he query shapes in MPEG-7 Par B daase, respecively. Fig. 1 summarizes he accuracies of rerieving shapes corruped by various levels of noise. I can be seen ha he accuracy is no affeced when he addiive noise is a he level of SR=50dB, and hen drops gracefully when noise increases o SR=30dB. When he shapes are heavily corruped o he level of SR=0dB (see he lefmos corruped shape), he proposed HSC sill mainained an accuracy over 70%. Fig. 1. Rerieval accuracy of he proposed mehod on MPEG-7 Par B shapes corruped by differen levels of Gaussian noise. VI COCLUSIO In his paper, we have presened a new hierarchical sring cus (HSC) approach o boh fas and effecively rerieve similar shapes. From a sar poin moving along he shape conour, he shape is pariioned ino muliple level curve segmens of differen lenghs, which are cu by heir corresponding srings for exracing heir geomeric and disribuion properies. A he coarse hierarchical levels, he sring cu feaures of longer curve segmens describe he global properies of he shape; while a he fine levels, he feaures exraced from he shorer ones depic he deailed informaion of he shape. The proposed HSC descripor is a ranslaion, roaion, scaling and mirror invarian shape descripor. The dissimilariy of wo shapes can be efficienly measured by comparing heir sring cu signaures of all hierarchical levels using he L1 disance wihou he need of using he ime cosly dynamic programming algorihm. The performance of he proposed HSC has been evaluaed exensively on hree well-known shape daabases and a leaf daabase of 100 plan species colleced by ourselves. 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