2D Physics-Based Deformable Shape Models: Explicit Governing Equations

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1 D Physcs-Based Deformable Shape Models: Explct Governng Equatons Stelos Krnds and Ioanns Ptas Department of Informatcs, AIIA lab Arstotle nversty of hessalonk, Box 45, 544, hessalonk, GREECE elephone: , Fax: E-mal: skrnd@zeus.csd.auth.gr, ptas@zeus.csd.auth.gr Abstract hs paper presents an accurate, very fast approach for the deformatons of D physcally based shape models representng open and closed curves. he ntroduced models overcome the man shortcomng of other deformable models,.e. computaton tme. he approach reles on the determnaton of explct deformaton governng equatons that nvolve nether egenvalue decomposton nor any other computatonally ntensve numercal operaton. he approach was evaluated and compared wth another fast and accurate physcs-based deformable shape model, both n terms of deformaton accuracy and computaton tme. he concluson s that the ntroduced model s completely accurate and s deformed very fast on current personal computers. Keywords Deformable model, Modal analyss, Real-tme deformatons, Fnte element method, Egenvalue decomposton, Deformable curves, Governng equaton.. Introducton A key problem n machne vson s how to descrbe features, contours, surfaces, and volumes, so that they can be segmented, recognzed, matched, or any other smlar underlyng process. he prmary dffcultes can be summarzed as: a) obect descrptons are senstve to nose, b) obects can be nonrgd, and c) the shape of the D obect proecton vares wth the vewng geometry. hese problems have motvated the use of deformable models []-[5], to nterpolate, smooth, and warp raw data, snce these models provde relable shape reconstructon tools that are both robust and generc. he class of deformable shape models orgnates wth the method of actve contours ( snakes ) ntroduced by Kass et al. [], that are used to locate smooth curves n D mages. Snce then, deformable models have been used for a number of applcatons n D and 3D by erzopoulos, Wtkn and Kass [6]. here have been two basc classes of deformable models: those based on parametrc sold modellng prmtves, such as those employed by Pentland et al. [], and those based on mesh-lke surface models, such as those n the work of erzopoulos et al. [6]. In the case of parametrc modellng, fttng has been performed usng the nsde-outsde functon of the modellng prmtve [7, 8], whereas n the mesh surface models, a physcalmotvated energy functon has been employed [6, 9]. Both of these approaches are qute slow, requrng dozens of teratons n the case of parametrc formulaton, and up to hundreds of teratons n the case of physcally-based mesh deformaton. Hence, all the

2 deformaton-based approaches are too computatonally ntensve so as to acheve the desred results, snce each ntermedate step turns out to be extremely tme -consumng. hus, although fnte elements are a powerful tool n computer vson applcatons, they are stll computatonally ntensve when used to deform an obect. We address ths problem by ntroducng a D physcs-based deformable approach representng open and closed curves, based on the fnte element method (FEM), usng explct functons. Hence, a very fast approach s presented n ths paper, nvolvng only the calculaton of an explct functon for the descrpton of the deformatons of the obect under examnaton, as opposed to the egenvalue decomposton that s necessary n most FEM approaches. Our approach was motvated by the technque presented n [0, ], whch conssts of analyzng nonrgd moton, wth applcaton to medcal mages. astar et al. [0] approxmated the deformng obect contour dynamcally, usng a physcally based deformable curve/surface. In order to reduce the number of parameters descrbng the deformaton, modal analyss, provdng a spatal smoothng of the curve/surface, was exploted. On the same bass, we employed modal analyss and, after smplfyng the deformaton governng equatons, we could safely nfer that the deformatons of open and closed D curves have explct governng equatons, nvolvng nether egenvalue decomposton nor any other computatonally ntensve operaton. he remander of the paper s organzed as follows. he D physcs-based deformable shape models [0] are presented n Secton. In Secton 3 the proof and the propertes of the new deformable physcs-based governng equatons are ntroduced. A comparson between the ntal deformaton models [0] and the novel ones, are presented n Secton 4, and fnal conclusons are drawn n Secton 5.. D Physcs-Based Deformable Shape Modellng Based On Modal Analyss In ths Secton, physcally based deformable shape models [0, ] explotng modal analyss are ntroduced. We consder both the surface and volume propertes of the obects at hand. We restrct ourselves to elastc deformatons,.e. we assume that the obect recovers ts orgnal confguraton as soon as all appled forces causng the deformaton are removed. Modellng an elastc D boundary can be acheved by an open or closed chan topology of masses on the contour. Each model node has a mass m and s connected to ts two neghbors wth dentcal sprngs of stffness k. he rato a k / m consttutes the so-called characterstc value of the model, whch s a constant value that descrbes the physcal characterstcs of the model. o be more detaled, t precsely determnes the model s physcal behavor. When a ncreases, the obect tends to behave as a sold one, whle when a decreases t can be treated as an elastc one. he node coordnates of the model under examnaton are stacked n vector: ( x 0 y 0 0 0,, K, ) X () 0, x y where s the number of vertces (masses) of the chan. he parameters of the elastc propertes of the model are needed (stffness constant, dumpng forces, external forces, etc.) for ts full descrpton []. he deformaton of the model s governed by a dfferental matrx equaton [3]: M & + C& + K F & () where s a vector storng nodal dsplacements of the ntal crcular chan X 0. M, C, and K [] are the mass, dampng, and stffness matrces of the model, respectvely, and F s the force vector resultng from the attracton of the model by the obect contour (usually based on the Eucldean dstance between the obect contour and the node coordnates). Equaton () s a fnte element formulaton of the deformaton process. Instead of solvng drectly the equlbrum equaton (), one can transform t by a bass change:? (3) where? s the square nonsngular transformaton matrx of order to be determned, and s referred to as the generalzed dsplacement vector. One effectve way of choosng? s settng t equal to F, a matrx whose entres are the egenvectors of the generalzed egenproblem:

3 Kφ Mφ (4) F u φ (5) Equaton (5) s referred to as the modal superposton equaton. he -th column of F, denoted by φ, s the -th vbraton mode, u (the -th scalar component of ) ts ampltude, and ts frequency. It should be noted that the vbraton modes of a generalzed egenproblem nvolvng real symmetrc matrces can be chosen to be orthonormal vectors. Furthermore, usng the standard Raylegh hypothess [0], matrces K, M and C are smultaneously dagonalzed: where F MF I F KF O O s a dagonal matrx, whose elements are the frequences and I s the dentty matrx. In practce, we wsh to approxmate nodal dsplacements by, whch s the truncated sum of the M low-frequency modes, where M. Vectors φ,, K, M M u φ (7) form the reduced modal bass of the system. hs s the maor advantage of modal analyss: t s solved n a subspace correspondng to the M truncated low-frequency vbraton modes of the deformable structure [, 0, ]. he number of vbraton modes retaned n the obect descrpton, s chosen so as to obtan a compact but adequately accurate representaton. A typcal a pror value for M, coverng many types of standard deformatons s equal to the quarter of the number of degrees of freedom of the system (.e. 5% of the modes are kept). An mportant advantage of ths formulaton s that the vbraton modes (egenvectors) and the frequences (egenvalues) of an open or closed chan topology have an explct expresson [0] and they do not have to be computed usng slow egen-decomposton technques (due to the dmensons of matrces K and M ). he frequences of the closed chan are gven by: and the vbraton modes are obtaned by: where {,, K, } and B ( ). ( ) +, K, for even, and π 4 a sn (8) π K, cos, K chan topology s very smlar, where the frequences are gven by: and the vbraton modes are obtaned by: φ (9) B s the frst Brlloun zone [0] and s equal to, K, for odd. Furthermore, the case of an open π 4a sn (0) π ( ) K, cos, K φ () (6)

4 where {,, K, } yelds: where and { 0,, K, }. Substtutng (5) nto () and premultplyng by C F CF and F F F. & + C & + O F F, () & () In many computer vson applcatons [], when the ntal and the fnal states are known, t s assumed that a constant force load F s appled to the body. hus, equaton () s called the equlbrum governng equaton and corresponds to the statc problem: In the new bass, equaton (3) s further smplfed to scalar equatons: where K F (3) u f desgnates the -th frequency (egenvalue) and the scalar vbraton mode (correspondng to egenvector (4) u s the ampltude of the correspondng φ ). Equaton (4), ndcates that, nstead of computng the dsplacements vector from equaton (3), we can compute ts decomposton n terms of the vbraton modes of the orgnal chan. he physcal representaton (fnal state) X () s fnally found by applyng the deformatons to the ntal crcular chan: X( ) X0 + F (5) 3. A Closed-Form Representaton of the D Physcs-Based Deformable Models It s obvous that the deformatons descrbed above are stll computatonally ntensve snce they requre the calculaton of a large number of summatons. In ths Secton our goal s to smplfy the deformaton process (Secton ) and to ntroduce a new way of calculatng the deformatons, achevng very fast deformaton computaton, wthout loss of accuracy. As already mentoned, physcs-based closed chan deformable model deformatons are descrbed by: [ Fnφ n( ) ] n φ ( ), B( ) ( + ) φ ( ) n n (6) where {,, K, }, wth correspondng to the total number of nodes of the model, equaton (8), ( ) φ by equaton (9), ( ) components of the force actng on node : B s the frst Brlloun zone, and fnally [ F ], F, F, F, K, F F F, x y x y x y s gven by F denotes the x and y (7) As t can be proved, the above equaton for closed models s equvalent to: + 4a F a d d µ + λ λ d he deformaton governng equaton for open chan physcs-based deformable models s proved to be: µ (8)

5 P, S a, a ( + ) ( ( + ) ( + + λ ) ) λ + µ (9) µ (0) + 4a, F µ S + P where µ and λ are two constant values equal to ( + 4a + ) / and ( 4 ) /, λ () + a respectvely, a s the model s characterstc value, s the total number of model nodes, and d s the dstance (n nodes) between the node under examnaton and the node, where force F s appled. Hence, n practce, d (only for the closed models case) s equal to mn (, ). For convenence, for the rest of the paper, our consderaton wll be focused on the closed physcs-based deformable models. hus, n practce, we wsh to approxmate nodal dsplacements by, the truncated addton of the M ' adacent nodes, where M ' : he nodes M ' to + 4a M ' + M ' F a d µ d µ + λ λ d M ' +, for each node, form the reduced nodal bass of the system. hs s the maor advantage of the new form of the deformaton governng equaton: t s solved n a nodal subspace correspondng to the M ' adacent model nodes of the deformable structure [, 0, ]. he number of nodes retaned, n the obect descrpton, for the calculatons n each step s chosen so as to obtan a compact but adequately accurate contour representaton. After performng a sequence of experments t has been observed that a typcal a pror value for M ', coverng many types of standard deformatons, s equal to ( + 4a + ) ln. hus, only a very small number of adacent nodes s taken nto account n the calculatons. In Fgure, a relatonshp between model characterstc value a, and the nodal subspace M ', wth respect to the total number of model nodes, s llustrated. hs formulaton s proved to be very fast, capable of use of real tme applcatons. () Fgure : he relatonshp between model characterstc value a, and the nodal subspace the total number of model nodes M ', wth respect to

6 4. Performance Analyss of the Proposed Method In ths Secton, a comp arson between the ntal physcs-based deformable model usng modal analyss (DMMA) descrbed n Secton and the explct formulaton deformable model (EFDM) ntroduced n Secton 3 s presented. he comparson s performed n terms of the dsplacement estmaton error and the requred computaton tme. he dsplacement error s defned as: where and E ' ' are the dsplacements calculated usng DMMA (5) and the deformable model under comparson respectvely. All experments are performed on a Pentum III (700 MHz) workstaton under Wndows 000 Professonal wthout any partcular code optmzaton. (3) Fgure : Errors regardng the modal and nodal reducton case respectvely he error comparson s llustrated n Fgure. All errors n the Fgure, are n a percentage form, due to the fact that the absolute error s not that representatve, snce t depends on the nput forces appled to the model. he error dfference of the EFDM and the DMMA s zero under any condtons. hat s, the two equatons, as expected, produce exactly the same results regardless of the deformable obect characterstcs a and the number of model nodes. Furthermo re, n Fgure, one can see the errors of the ntal deformable model n a reduced modal space (DMMA), where only 5% of the egenvalues are taken nto account. he average error s %, whch s nsgnfcant. hus, as astar et al. [0] clam, one can use the DMMA wthout any partcular loss of accuracy. Moreover, Fgure llustrates the errors for the ntroduced deformable physcs-based model n a reduced nodal space (EFMD), where only ( + 4a + ) ln adacent (neghborng) nodes are consdered. he average error s %, whch s neglgble wth respect to the global deformaton of the model, and consequently deformatons can be computed usng EFDM wthout any partcular loss of accuracy. Besdes, t s clearly depcted n Fgure that deformatons extracted from the EFDM mport an error 900 tmes less, n average, than the correspondng deformaton computed by DMMA. It must be noted that the errors do not change wth the number of nodes. Furthermore, note that error axs Y s a logarthmc one.

7 Fgure 3: Computatons of the DMMA and DMMA (5% of egenvalues) have been manually estmated. he same task s performed for the EFDM and the EFDM as well Addtonally, we have performed a computatonal complexty analyss of the methods under comparson. he DMMA (6) requres approxmately computatons (addtons and multplcatons, whch are consdered of equal complexty), where s the number of model nodes. he computatons. hus, smplfed physcs-based deformable model, EFDM, (8) requres only n theory, the explctly calculated model has approxmately tmes fewer calculatons than the ntal one, for every (Fgure 3). However, the same task was performed for the DMMA and the EFDM. he 3 correspondng computatons are gven by and ln ( + ln ) respectvely. hese curves are plotted n Fgure 3. It s worth notcng that the EFDM has computatonal complexty of order 3 O ( ), whereas the DMMA has computatonal complexty O ( ). Fgure 4: he EFDM compared wth the DMMA wth respect to computatonal tme. he same task s performed for the DMMA and the EFDM. mes are measured n seconds Moreover, we have performed computaton tme benchmarkng expermentally. Fgure 4 shows the computaton tme for all methods under comparson. he EFDM and DMMA needs approxmately the same

8 computaton tme. hs tme s almost 3 tmes less than the correspondng tme value of the DMMA. he computaton tmes of these two cases are qute satsfactory, but they are stll far from the desred e.g. for real tme use whch s acheved by EFDM. It computes the deformatons fast enough and only n the case the model nodes are more than , the computaton tme s more than second. herefore, that model can be deformed 00 tmes per second whle havng nodes and 6,5 tmes per second f t has nodes. Hence, the EFDM can be safely used n real tme applcatons. Furthermore, t must be mentoned that n Fgure 4 the computaton tmes are plotted n a logarthmc axs for better vsualzaton. In the cases of the EFDM and the DMMA we have approxmately equal computaton tme. he DMMA takes more computaton tme to acheve a result, whereas the EFDM s much faster n performng the same deformaton. We can acheve an acceleraton of 4-5 orders of magntude for almost any contour chan of length. It must noted that the calculated executon speeds shown n Fgure 4 are well n lne wth the comparng computatonal results shown n Fgure Concluson In ths artcle, we have presented a closed-form soluton for D physcs-based models deformaton, along wth ther propertes. Although all exstng physcs-based deformable models used n the lterature, are a powerful tool n a number of computer vson applcatons, they are stll computatonally ntensve when deformng an obect. he presented D physcs-based deformable model drastcally reduces the computaton tme needed to perform the deformaton. he deformaton equatons were smplfed and analyzed. As a result, a closed-form soluton can be reached for the obects deformaton. hs soluton s very useful n analyzng the deformaton behavor of the contour at hand. he extremely low computatonal tme and the low deformaton error wth respect to all the other avalable technques n the lterature, makes the ntroduced physcs-based deformable model a very promsng tool for varous mage analyss and computer vson applcatons. Acknowledgment hs work has been supported by the European non research proect IS Worlds Studo Advanced ools and Methods for Vrtual Envronment Desgn and Producton. References. Kass, M., Wtkn, A., and erzopoulos, D. Snakes: Actve contour models. Internatonal Journal of Computer Vson, p.p. 3-33, Pentland, A., and Sclaroff, S. Closed form solutons for physcally-based shape modellng and recognton. IEEE rans. on Pattern Analyss and Machne Intellgence, 3(7). p.p , erzopoulos, D., and Metaxas, D. Dynamc 3D models wth local and global deformatons: Deformable superquadrcs. IEEE rans. on Pattern Analyss and Machne Intellgence, 3(7). p.p , erzopoulos, D., Wtkn, A., and Kass, M. Constrans on deformable models: Recoverng 3D shape and non-rgd moton. Artfcal Intellgence, 36. p.p. 9-3, Sclaroff, S., and Pentland, A. Modal matchng for correspondence and recognton. IEEE rans. on Pattern Analyss and Machne Intellgence, 7(6). p.p , erzopoulos, D., Wtkn, A., and Kass, M. Symmetry-seekng models for a 3D obect reconstructon. Internatonal Journal of Computer Vson, (3). p.p. -, Pentland, A. Automatc extracton of deformable part models. Internatonal Journal of Computer Vson, 4(). p.p. 07-6, Solna, F., and Bacsy, R. Recovery of parametrc models from range mages: he case for superquadrcs wth global deformatons. IEEE rans. on Pattern Analyss and Machne Intellgence, (). p.p. 3-47, Sobottka, K., and Ptas, I. A novel method for automatc face segmentaton, facal feature extracton and trackng. Image Communcatons, Elsever, (3). p.p. 63-8, 998.

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