Finding Repetitive Patterns in 3D Human Motion Captured Data

Transcription

1 Fndng Repettve Patterns n 3D Human Moton Captured Data Ka-Ta Tang Department of Computer Scence, Cty Unversty of Hong Kong, Hong Kong tjeff@ctyu.edu.hk Howard Leung Department of Computer Scence, Cty Unversty of Hong Kong, Hong Kong howard@ctyu.edu.hk Taku Komura School of Informatcs, The Unversty of Ednburgh, Unted Kngdom tkomura@nformatcs. ed.ac.uk Hubert P.H. Shum School of Informatcs, The Unversty of Ednburgh, Unted Kngdom hubert.shum@ed.ac. uk ABSTRACT Fndng repettve patterns s mportant to many applcatons such as bonformatcs, fnance and speech processng, etc. Repettve patterns can be ether cyclc or acyclc such that the patterns are contnuous and dstrbuted respectvely. In ths paper, we are gong to fnd repettve patterns n a gven moton sgnal wthout pror knowledge about the type of moton. It s relatvely easer to fnd repettve patterns n dscrete sgnal that contans a lmted number of states by dynamc programmng. However, t s mpractcal to dentfy exactly matched states n a contnuous sgnal such as captured human moton data. A pont cloud smlarty of the nput moton sgnal tself s consdered and the longest smlar patterns are located by tracng and extendng matched posture pars. Through pattern algnment and autoclusterng, cyclc and acyclc patterns are dentfed. Experment results show that our approach can locate repettve movements wth small error rates. Keywords 3D human moton capture, pattern dscovery, repettve pattern, cyclc and acyclc patterns, pont cloud smlarty. 1. Introducton Repettve patterns are frequently appearng unts commonly found n daly lfe, for example, keywords n a text paragraph or repettve logos on the clothes. They can be ether cyclc or acyclc. A cyclc pattern repeats contnuously whle an acyclc pattern s non-contnuous and dstrbuted over the tme or regon. Fndng repettve patterns n ether dscrete or contnuous sgnals leads to the development of many mportant applcatons such as detecton of the motfs n DNA sequence, predcton of the trend of the stock prce, and mnng for desrable segments n speech or moton sgnals. Repettve patterns n dscrete sgnals such as text paragraphs, DNA sequence and musc data can be found by exact strng matchng technques [1][2][3] or nexact matchng wth dynamc Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. programmng [4][5]. Researchers beleve that repettve tandems n DNA sequence assocated wth dsease syndromes [6]. It has been a popular topc for bonformatcs researchers to repettve tandems n DNA sequences. Glbert et al. [7][8] dscover repeated patterns by extendng matched sub-strngs. To fnd repettve patterns n musc data, Hsu and Lu [9] consder exact strng matchng through the correlaton matrx of the musc notes n sequence. Some researchers attempt to match for smlar segments n contnuous sgnal such as fnancal, speech, and moton data. Snce contnuous sgnal does not contan exactly matched states, the problem s harder than that n dscrete sgnal for fndng smlar segments. Wu et al. [10] predct the trend of stock prce by matchng for hstory sgnals of smlar shapes n a contnuous fnancal data curve. In speech processng, Park [11] consders a pont cloud smlarty between an nput sgnal and a known template n order to dentfy some spoken keywords. More specfc to human moton studes, Kovar and Glecher [12] consder the pont cloud smlarty between a query and the moton data sequence and extract logcally related motons for moton blendng. They approxmate the optmal matchng paths by lne tracng technques. However, they focus n moton retreval so a known template should be provded and the length of result pattern s confned. Forbes and Fume [13] attempt to mprove the work by Kover et al. by ndexng the pont cloud by manually defnng key postures n order to speed up the searchng. However, the number of avalable moton database s growng rapdly and t s mpractcal to spend a lot of manpower to do such pre-processng. Some researchers attempt to detect cyclc patterns n captured human moton. L and Holsten [14] detect cyclc moton by constructng moton templates of standard movements lke walkng n frequency doman. Meng et al. [15] extend the work by L and Holsten. However, ths approach requres the user to know about the types of nput moton n advance. Laptev et al. [16] detect cyclc movement by algnng a sequence of space-tme correspondng ponts n vdeo frames. Gven a known cyclc moton, Ormonet et al. [17] detect the cycles by foldng and overlappng the nput moton untl the mnmal sgnal-tonose error s attaned. However, detecton of acyclc repettve patterns has receved much less attenton n the lterature. To break through the lmtatons of exstng template matchng approaches, t motvates us to solve an unsupervsed pattern dscovery problem n whch repettve patterns n 3D human moton captured data are automatcally detected wthout knowng the types of nput patterns. A pont cloud matrx of posture smlarty s consdered and the longest smlar moton segment

2 pars are located through tracng for the dagonal patterns. Cyclc and acyclc patterns are then dentfed by automatc algnment and clusterng. Fnally, the duraton of each cycle s estmated by the auto correlaton method. The robustness of our algorthm wll be shown n the experment results. Challengng cases that can be handled by our algorthm wll also be demonstrated. 2. Proposed algorthm In ths paper, repettve patterns ncludng both cyclc and acyclc patterns are dscovered n an unsupervsed way n whch no template query s needed. The nput s a 3D human moton captured data, whch s a hgh dmensonal contnuous sgnal. The pont cloud smlarty approach, whch has been used for matchng smlar regons on a contnuous sgnal, s adopted to deal wth the problem. Our algorthm ams to relax the assumpton n exstng methods that requre a known template query n order to search for smlar patterns. Fgure 1 shows the overvew of our algorthm. Frst, a grayscale pont cloud matrx of posture smlarty values s generated. Smlar postures are then clustered by turnng the pont cloud nto a bnary representaton. The local mnma of each cluster are traced dagonally to form feature ponts that approxmate the duraton of each smlar moton segments. The longest possble matchng paths are obtaned by mergng feature ponts wth the least dynamc tme warpng (DTW) cost. Fnally, the cyclc and acyclc patterns are dentfed by pattern algnment and autoclusterng technques. The perod n each cycle pattern wll then be estmated by the auto correlaton method. Generatng a posture smlarty matrx Tracng for the longest smlar segment pars Identfyng cyclc and acyclc patterns Obtanng the perod of each cycle Fgure 1. The overvew of our proposed algorthm. 2.1 Data acquston and normalzaton The moton data s captured by an optcal moton capture system as shown n Fgure 2. The capturng area s covered by seven cameras from dfferent vew ponts. Durng the capture, the actor as shown n Fgure 3 should wear a sut wth 35 optcal markers attached to dfferent body parts such as the head, the torso and the lmbs. The 3D moton of the actor s captured as a tme sequence of frames, whle each frame contans the tme stamp and the correspondng 3D posture data n terms of 3D marker coordnates. (a) Optcal markers adhered to the body (b) Captured 3D posture represented by a skeleton Fgure 3. The actor and correspondng captured skeleton. The actor s free to move around the capture area thus global translaton and rotaton are ncorporated nto the 3D coordnates of the captured moton. Normalzaton of horzontal translaton and frontal orentaton s thus needed to facltate the posture comparson. The postons of the markers p are frst translated n order to make the marker of the body center (Md-back) nvarant to the orgn of the horzontal plan. The vertcal dsplacement s allowed hence the y coordnate s not normalzed. The translaton functon s gven by Equaton (1): p Rght-pelvs marker ( x y, z ) = ( x x, y z z ) =,, Md back Md back (1) A rotaton functon s defned to ensure that the actor s always facng the front (the postve z drecton) as shown n Fgure 4(b). Three markers defnng the major orentaton of the body s shown n Fgure 4(a) and the unt vector N normal to the resultng plane s calculated. The rotaton angles θ z and θ x are obtaned by the dot product between N wth the z-axs and wth the x-axs respectvely. The coordnates of each marker p are then rotated byθ z andθ x accordngly. (a) Normalzaton of frontal orentaton Md-back marker Left-pelvs marker Fgure 2. The moton capture area. (b) Normalzed posture Fgure 4. Normalzaton of a 3D posture.

3 2.2 Generatng a posture smlarty matrx A pont cloud smlarty matrx locates smlar postures n clusters. The smlarty between every par of postures s computed by a smlarty cost functon Posture smlarty cost The perceptual smlarty between a par of postures s modeled by ther spatal dfference. A par of perceptually smlar postures show a smlar concept (e.g. the rght hand s up-lfted), and smlar normalzed coordnates n the 3D space. Fgure 5 shows two pars of spatally smlar and dssmlar postures. In the pont cloud smlarty matrx, fve types of dark pattern are observed: (a) dagonal, (b) ant-dagonal, (c) crossng, (d) V-shape, and (e) horzontal / vertcal. Dagonal pattern s runnng from bottom-left to top-rght n whch the correspondng sgnal segment pars are mapped along tme sequence. Ant-dagonal pattern s nvald because one segment of the matched par s tme-reversed. Cross and V-shape patterns are just vald for the dagonal porton. Horzontal or vertcal pattern s more or less a one-to-many mappng of postures such as a statonary posture. (a) dagonal (b) ant-dagonal (c) crossng (a) Spatally smlar postures (b) Spatally dssmlar postures Fgure 5. Smlar and dssmlar postures. Let p (x m, y m, z m ) and p j (x m, y m, z m ) be the m-th marker coordnates of the posture par (, j) respectvely, the smlarty cost C(,j) s defned by averagng the Eucldean dstances between the 3D coordnates of M pars of correspondng markers as shown n Equaton (2). In our settng, M=35 markers are consdered. The more smlar the posture par, the smaller the smlarty cost. M ( x, y, z ) p ( x, y z ) p m m m j m m, m= C, j) = 1 M ( (2) The pont cloud smlarty matrx Gven a moton sgnal that contans n frames, a n n pont cloud matrx of posture smlarty costs s constructed. The smlarty costs are normalzed to a range between 0 and 1, whch s easer to be compared and vsualzed by a grayscale btmap. Fgure 6 shows a grayscale pont cloud smlarty matrx of a moton. A pont wth a darker color exhbts a hgher smlarty between the correspondng par of postures. Only the lower half of the matrx s consdered because the matrx s symmetrc along the dagonal axs (x = y). Fgure 6. The smlarty matrx s vsualzed by a grayscale pont cloud btmap. m (d) V-shape (e) horzontal / vertcal Fgure 7. Dark patterns observed n the pont cloud. 2.3 Tracng for the longest smlar segment pars It s non-trval to search for smlar sgnal pars from the grayscale pont cloud because some regons are qute ambguous. Hence, as the frst step, a bnary pont cloud s obtaned by flterng dssmlar posture pars. Start ponts of the vald patterns descrbed n secton are then located. The possble contnuaton of each start pont s determned by dynamc tme warpng (DTW) wth a shrnkng wndow technque, whch wll be descrbed n later paragraphs Obtanng the bnary pont cloud To obtan the bnary pont cloud, only smlar posture pars are kept as pattern ponts. A classfer s traned to determne whether a par of nput postures s perceptually smlar. The ground truth smlar and dssmlar pars are determned by some users through subjectve evaluaton. Equal number of samples n each class s selected randomly and a total of 100 ground truth smlar/dssmlar pars are obtaned. From the dstrbuton of matchng costs exhbted by smlar and dssmlar pars, the false matched rate and false non-matched rate can be observed. The equal error rate, wth whch the false matched rate s equal to the false non-matched rate, s used to determne the threshold of the smlarty cost to classfy between smlar/dssmlar pars Locatng start ponts of vald patterns After the bnary pont cloud s obtaned, the start ponts of all vald patterns are then estmated. The bnary pont cloud s thnned by consderng the vertcal local mnma n terms of smlarty cost. For each frame, the local mnma are dentfed as shown n Fgure 8(a). The dark thn lnes llustrate the thnned patterns. However, we only chose the vald porton as descrbed n secton Therefore, there are a few constrants for the selecton of start ponts as shown n Fgure 8(b). Frst, a start pont should be the bottom-leftmost pont of the pattern and hence has no mnma ponts appearng n the precedng tme (regon B). Also, t should have neghborng ponts n next frames (regon A).

4 (a) Obtanng the local mnma (b) Connectvty for dagonal pattern (c) Pattern tracng (d) The optmal path of the traced pattern Fgure 8. Pattern tracng procedures Tracng patterns The pattern tracng starts from the bottom-left corner of the pont cloud. Fgure 8(c) shows the procedures of tracng a dagonal pattern of a selected regon of the pont cloud. Suppose that the start pont (n crcle) near to the bottom-left corner s consdered, possble contnuatons of a pattern are determned by DTW. A wndow of certan sze sldes across the horzontal axs wthout overlappng s used to locate canddate of contnuaton ponts by cuttng across the vertcal axs for mnma ponts. A vald canddate should make a slope wth the start pont no greater than 2 and no smaller than 1/2. There are some possble cases: (1) cuts exactly one pont, (2) cuts more than one ponts, and (3) cuts no ponts. For the frst case, the canddate s obvously the contnuaton pont. If more than one canddate s obtaned, the one wth the mnmal DTW cost s selected. If the wndow cuts no ponts, the wndow shrnks progressvely untl a vald canddate s found. Once a canddate s found, t becomes the new start pont for next teraton untl no further contnuaton ponts are found. The DTW path s safeguarded by a threshold of 0.5 (note that the DTW cost s normalzed by the length of the optmal path and falls nto the range [0, 1]). When the DTW cost s greater than the threshold, the canddate wll be gnored and ths ensures the contnuaton of pattern s vald. In our settng, an ntal wndow sze of 30 frames s used as we assume that a vald pattern should have at least 30 frames. A more aggressve wndow sze of 50 frames wll be tred for next teratons. Although there may have some gaps between pattern lnes, our algorthm can verfy whether the gaps are acceptable for a longer pattern. If yes, the start pont of the pattern next to the gap s smply bypassed. Fgure 8(d) shows the optmal path determned by DTW after the pattern tracng steps. 2.4 Identfyng cyclc and acyclc patterns Repettve patterns can be ether cyclc or acyclc and sometmes both of them may appear n the nput moton. Fgure 9 shows the character sequences that demonstrate the modes of repettve patterns. Exstng approaches assume that the nput moton should be cyclc only and they cannot detect cycles n the mxed mode because the sgnal-to-nose rato s too small. CCCCC ADBAEGH...ADCAFCCCC... (a) A cyclc pattern (b) An acyclc pattern (c) Mxed cyclc and acyclc patterns Fgure 9. Modes of repettve patterns.

5 S mn E max (a) A famly of cyclc patterns (b) Illustraton of the algned patterns Fgure 10. Algnment of patterns. The property of cyclc pattern can be observed through the optmal paths located on the pont cloud. Fgure 10(a) shows the result of a sample dance moton, whch conssts of a contnuous sequence of waltz steps. Ths moton conssts of cyclc moton only that makes the problem smpler to be llustrated. Each dagonal lne P represents a par of smlar segments. Because our algorthm obtans the segment pars as long as possble, for a cyclc moton there exsts certan degree of pattern overlappng. Moreover, the patterns form a rght-angled trangular regon and these patterns belong to the same famly of cyclc pattern. Suppose the patterns are transformed from 2D doman nto 1D tme lne as shown n Fgure 10(b). The duratons of each par of segments can be observed. It s clear that for each par of segments, the mnmum of the start ponts (S mn ) and the maxmum of end ponts (E max ) are lkely to be algned together. It gves us a good classfcaton feature to dstngush whether a pattern belongs to the same famly as others. An auto-clusterng method s ntroduced to classfy cyclc and acyclc patterns based on the algnment feature. The procedure s shown n Fgure 11. Suppose there are three resultng patterns P 1, P 2, and P 3 and ther relatonshp s unknown. One of the patterns s pcked as the ntal set, say P 2. Next, each remanng pattern s algned wth the patterns n the exstng sets. Suppose P 1 s consdered, ts S mn and E max values wll be compared wth the set n P 2. A clusterng cost C c s defned n Equaton (3) to quantfy the measurement, n whch N represent the total number of patterns n each set to be examned. If C c s smaller than 5% of the average pattern duraton D (Equaton (4)) n each set, the pattern s put nto the set. If there s more than one set, the set wth the mnmal C c s chosen to enter. C c = N E E S S = + 1 max max mn mn D = N N E S = 1 max mn N (3) (4) After the clusterng, t s ready to dstngush between cyclc and acyclc patterns. A cyclc set contans more than one patterns (N>1) whle an acyclc set contans one pattern (N=1) only. Fgure 11. Auto-clusterng nto cyclc and acyclc patterns. 2.5 Obtanng the perod of each cycle By observaton, there should be N+1 cycles for a famly havng N patterns, assumed that there are no mssng patterns and wrong patterns ncluded. However, t s not relable to smply count the number of patterns n each famly because the assumpton may not hold all the tme. Consder a snusodal sgnal of perod P, the sgnal can perfectly overlap to tself wth a phase dfference of the multples of P. Smlarly, the movement of each marker can be resolved nto x, y, and z dsplacements. If the dsplacement sgnal s cyclc, t can overlap wth tself at partcular phase dfferences. An auto correlaton method s used to detect the boundares of each cycle. Recall the 1D algnment of patterns as shown n Fgure 10(b), the cyclc patterns are overlapped at dfferent tme. The duraton of each cyclc famly s hence obtaned from the algnment result. Next, the duraton s selected as an ndependent moton sequence {x } of length L that contans only one knd of cyclc movement. The correlaton coeffcent r m (x) of the sequence {x } wth tself at dfferent lags wthout warpng {x +lag } s computed as equaton (5). The correlaton-lag nformaton of x, y and z coordnates of all 35 markers m are combned lnearly nto r wth equal weghtng.

6 r ( x) m [( x x)( x x) ] L 1 + lag = 0 = (5) 2 σ x In cyclc moton segment, the nput sgnals {x } and {x +lag } can overlap wth each other somewhere, and hence a peak wll be obtaned n the correlaton-lag graph. Fgure 12 shows the correlaton-lag graph of a moton data wth eght waltz dance steps only. Imagne that a moton trajectory sldes to the left and overlaps wth a moton trajectory wthout lag. The correlaton coeffcent decreases and then reaches a peak agan after certan lag value. The peaks wth suffcently hgh correlaton coeffcents are the boundares between every two adjacent cycles, and the number of peaks plus one s hence the number of estmated cycles. As future work, the start / end ponts of the member patterns n a cyclc famly wll be also consdered, whch approxmates the perod of each cycle and enhances the searchng for best cuttng pont of each cycle. Fnally, the cyclc patterns are cut accordngly and the perod of each cycle s determned. They are grouped together as the same repettve patterns. The dstrbuted acyclc patterns are then grouped accordng to the overall smlarty of the moton segments and ther duratons. Fgure 12. Smlarty of statng frame and the frames n the whole cyclc duraton. 3. Experments and results Our experment data contans 22 moton clps wth 4 types of dances: Waltz, Pop dance, Hp hop dance and House dance as shown n Table 1. Moton type Table 1. Experment dataset. Number of samples Average number of frames Waltz dance Pop dance Hp hop dance House dance The dance motons are performed by two dancers contanng both cyclc and acyclc movements. The moton data contans complcated movements and wth long duraton whch can test the robustness of our algorthm. To evaluate the performance of our algorthm, both type I and type II errors wll be checked. Type I error s the false postve rate that measures how many repettve patterns located by our algorthm are actually not repettve accordng to human percepton. Type II error s the false negatve rate that measures how many mssng repettve patterns that are not detected by our algorthm. Fgure 13. Tranng result of the threshold value. Fgure 13 shows the tranng result of smlarty threshold. The value 0.26 that yelds the equal error rate (EER) s chosen as the threshold T to dstngush smlar and dssmlar postures because at ths pont both the false matched rate and false non-matched rate are low. The pars wth smlarty cost smaller than T are classfed as smlar par, otherwse are dssmlar. Ths result s used n the bnary pont cloud formaton and pattern tracng. In our experment, U pars of smlar moton segments are frst dentfed by our algorthm. The anmaton of each par of moton segments s played at the same tme and evaluated by some observers subjectvely. By the human judgment, u out of U pars may be consdered as false postve. The remanng frames that are regarded as non-repettve by our algorthm are then collected nto a new sequence R. It s then segmented nto s r segments of wndow sze w. A wndow sze of w = 30 frames has been chosen because we only accept vald repettve patterns of more than 30 frames. Each wndow sldes across the entre sequence R and the cross correlaton coeffcent r R s computed by Equaton (6). The correlaton coeffcent of each segment par s gven by the average of correlaton coeffcents r x, r y, and r z of x, y, and z axs respectvely. We set a rather loose condton that the pars wth correlaton hgher than 0.5 are accepted as canddate false negatve pars. Fnally, the canddate false negatve pars are evaluated subjectvely by some observers and a total of v false negatve pars are obtaned. r R = [( R R )( R R )] 30 σ σ R R (6)

7 Frames ( ) Frames ( ) Fgure 14. A complcated movement. There are U-u+v pars of ground truth repettve pars obtaned per observer. Hence, the type I error rate s gven by u/(u-u+v) whle the type II error s gven by v/(u-u+v). Table 2 shows the experment result. The false negatve rates are 0% for all nput dance motons. It shows that our algorthm can locate all possble repettve patterns wthout pror knowledge of the nput moton. In our experment, only segment pars wth cross correlaton coeffcent greater than 0.5 are consdered as mssng cases. Ths value s suffcently low for a mssng smlar par because smlar moton segment always has a correlaton coeffcent hgher than 0.6. The 0% false negatve rate shows that our algorthm s capable to detect all possble repettve patterns wthout mssng. On the other hand, a relatvely low average false postve rate (7.7%) s obtaned. Table 2 also shows the false postve rates of each type of dance moton. Our algorthm gves the best estmaton n waltz dance (0% n both false postve and false negatve rates) whle the worse n hp-hop dance (12.8% n false postve rate and 0% n false negatve rate). Accordng to the comments from the observer, the hp-hop dance s the most complcated whle the waltz dance s qute straght forward. It shows that complcated movements are lkely to have observable dfferences. In most of the erroneous cases, the moton pars may look alke wth dfference n postons of a partcular lmb. It shows that averagng the Eucldean dstances of jont postons s not enough because t s unfar to treat the dsplacement of the jonts of end effectors equally to relatvely statc jonts such as the shoulders. Moreover, varaton n body sze and lmb lengths for dfferent dancers may lower the accuracy. To be more generc, jont angles could be consdered as a feature n the smlarty functon n the future. Also, the movements of more actve lmbs could be boasted by a larger weght n order to make the comparson more conformed to the human percepton. Our experment s ongong and more moton data of dfferent types wll be studed later. Moton type Table 2. The experment results. Type I error (False postve rate) Type II error (False negatve rate) Waltz dance 0.0% 0.0% Pop dance 12.5% 0.0% Hp hop dance 12.8% 0.0% House dance 5.5% 0.0% Fgure 14 shows a par of smlar moton wth dffcult movements, whch s a challengng case that our algorthm s able to handle. The movement s relatvely fast and t nvolves movements of the whole body. It shows that our algorthm s robust enough to catch such dffcult movement wth rapd change. 4. Concluson We proposed a method to locate repettve patterns n captured 3D human moton wthout pror knowledge about the nput pattern. Patterns of dfferent lengths can be dscovered by consderng the nput sgnal alone wthout a query. Repettve patterns are traced from a pont cloud of smlar postures. Complete pattern s obtaned by jonng ponts accordng to ther connectvty along the dagonal. Fnally, cyclc and acyclc patterns are dentfed by pattern algnment and auto-clusterng. The perod of each cycle s also estmated successfully. Experment result shows that our method has low false postve and false negatve rates and s able to handle complcated cases. As future work, the repettve patterns dscovered by our proposed method can be used for summarzng a pece of captured moton. By the way, the dscovered repettve patterns can be appled to generate dance lesson automatcally. The ubqutous dance educaton system developed by our group [18] requred teachers to desgn dance courses manually. Our algorthm can group repettve movements that are lkely the theme movements of the captured moton performed by teacher. The student can learn frequently appearng movements frst and then ther varatons. Our proposed method can also be appled n moton data retreval by consderng the repettve pattern and perodcty etc. as ndex features that may unquely defne a moton clp. 5. ACKNOWLEDGMENTS The work descrbed n ths paper was fully supported by a grant from the Research Grants Councls of the Hong Kong Specal Admnstraton Regon, Chna (Project No. CtyU 1167/05E). 6. REFERENCES [1] Qu, Y., Wang, C., and Wang, X. S Supportng fast search n tme seres for movement patterns n multple scales. In Proceedngs of the Seventh nternatonal Conference on nformaton and Knowledge Management (Bethesda, Maryland, Unted States, Nov , 1998). CIKM '98. ACM, New York, NY, [2] Agrawal, R., Psala, G., Wmmers, E. L., and Zaït, M Queryng Shapes of Hstores. In Proceedngs of the 21th nternatonal Conference on Very Large Data Bases

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