REPRESENTING 2D and 3D data sets with implicit polynomials

Transcription

1 An Adaptve and Stable Method for Fttng Implct Polynomal Curves and Surfaces Bo Zheng, Jun Takamatsu, and Katsush Ikeuch, Fellow, IEEE Abstract Representng 2D and 3D data sets wth mplct polynomals (IPs) has been attractve because of ts applcablty to varous computer vson ssues. Therefore, many IP fttng methods have already been proposed. However, the exstng fttng methods can be and need to be mproved wth respect to computatonal cost for decdng on the approprate degree of the IP representaton and to fttng accuracy whle stll mantanng the stablty of the ft. We propose a stable method for accurate fttng that automatcally determnes the moderate degree requred. Our method ncreases the degree of IP untl a satsfactory fttng result s obtaned. The ncrementablty of QR decomposton wth Gram-Schmdt orthogonalzaton gves our method computatonal effcency. Furthermore, snce the decomposton detects the nstablty element precsely, our method can selectvely apply rdge regresson-based constrants to that element only. As a result, our method acheves computatonal stablty whle mantanng fttng accuracy. Expermental results demonstrate the effectveness of our method compared wth pror methods. Index Terms Fttng algebrac curves and surfaces, Implct polynomal (IP), Implct shape representaton (a) (c) (b) (d) I. INTRODUCTION REPRESENTING 2D and 3D data sets wth mplct polynomals (IPs) s currently attractve for vson applcatons such as fast shape regstraton [3], [2], [29], [25], [26], [], [28], [2], recognton [], [25], [24], [9], [8], [33], [36], smoothng and denosng [28], [22], 3D reconstructon [9], mage compresson [6], and mage boundary estmaton [27]. In contrast to other functon-based representatons such as B-splne, Non-Unform Ratonal B-Splnes (NURBS) [2], Ratonal Gaussan [5], and radal bass functon (RBF) [32], IPs are superor n such areas as fast fttng, few parameters, algebrac/geometrc nvarants, and robustness aganst nose and occluson. To acheve the representaton of 2D/3D shapes usng an IP, the optmzaton of usng classcal least squares methods that mnmze the dstance between the data set and the zero level set of polynomal s advocated n recent lterature [23], [29], [3], [2], [2], [], [5], [6], [7], [28], [7], [4], [37], []. Another fttng method n [35] has the potental of offerng an accurate model by convertng Fourer seres to IP form. However, the pror methods greatly suffer from two major ssues as follows. ) Degree-fxed fttng procedure Before handlng a fttng procedure, the degree of IP frst must be determned accordng to the complexty of the object, and the degree wll be fxed durng the procedure. Ths s no problem when fttng a smple object; for example, t s easy to determne a 2D IP of degree 2 to ft a 2D shape that looks lke an ellpsod. B. Zheng and K. Ikeuch are wth Insttute of Industral Scence, the Unversty of Tokyo, 4-6- Komaba, Meguro-ku, Tokyo, , JAPAN. E-mal: {zheng, k}@cvl.s.u-tokyo.ac.jp J. Takamatsu s wth NAIST, Takayama, NARA JAPAN. Manuscrpt receved March, 28; revsed Dec. 8, 28. (e) Fg.. IP fttng results of Orgnal data (a): (b) 4-degree IP, (c) 6- degree IP, (d) -degree IP, (e) -degree IP stablzed by conventonal RR regularzaton, (f) The result of our method. However, the determnaton confuses a user who needs to ft a complex object such as the bunny n Fg. (a); determnng an over-low degree leads to undesred naccuracy (Fg. (b)), whereas determnng an over-hgh degree leads to the over-fttng problem (Fg. (d)). Therefore, the user s forced to waste much tme n fndng a moderate degree by tryng dfferent degrees several tmes and selectng the best one from the results (Fg. (b) (d)). 2) Global fttng nstablty Despte the over-fttng problem, there s no doubt that the hgher degree IP s very convenent for a complex object. Fortunately, several research groups have already proposed the method, rdge regresson (RR) regularzaton, to overcome over-fttng problems, that s, to remove the extra zero sets by mprovng the numercal stablty of the fttng procedure. Fg. (e) shows an example where the RR regularzaton was used to mprove the global stablty of the -degree IP shown n Fg. (d). The method s obvously useful for over-fttng, but, on the other hand, the method produces deteroraton n local accuracy. In ths paper, we wll solve these two ssues wth a novel (f)

2 2 method based on QR-decomposton wth Gram-Schmdt orthogonalzaton and a novel RR regularzaton. The method s capable of: ) adaptvely determnng the moderate degree for fttng that depends on the complexty of objects, n an ncremental manner wthout greatly ncreasng the computatonal cost; 2) retanng global stablty whle also avodng excessve loss of local accuracy. As an example, shown n Fg. (f), our method adaptvely determnes an IP of moderate degree for fttng the bunny object, and the result shows better global and local accuracy than the pror method (Fg. (e)). The frst advantage of ths method s ts computatonal effcency because the ncrementablty of Gram-Schmdt QR decomposton dramatcally reduces the burden of the ncremental fttng by reusng the calculaton results of the prevous step. The second advantage s that the method can successfully avod global nstablty, and furthermore mantan local accuracy, because constrants derved from the RR regularzaton are selectvely utlzed n our ncremental scheme. Fnally, a set of stoppng crtera can successfully stop the ncremental scheme at the step where the moderate degree s acheved. Ths paper s organzed as follows: after revewng some pror work on solvng IP fttng problem n Secton II, we present key components of our algorthm n Sectons III, IV, and V. Secton VI reports expermental results followed by dscusson and concluson n Sectons VII and VIII. A. Mathematcal Formulaton II. BACKGROUND IP s the mplct functon defned n a multvarate polynomal form. For example, the 3D IP of degree n s denoted by: f n(x) a jk x y j z k,j,k;+j+k n ( } x {{... z n } )(a a... a n ) }{{} T, () m(x) T a where x (x y z) s one data pont n the data set. An IP can be represented as an nner product between the monomal vector and the coeffcent vector as m(x) T a. The order for monomal ndces {, j, k} s a degree-ncreasng order named nverse lexcographcal order [3]. The homogeneous bnary polynomal of degree r n x, y, and z, +j+kr a jkx y j z k, s called the r-th degree form of the IP. B. Fttng Methods The objectve of IP fttng s to fnd a polynomal f(x), of whch the zero set {x f n (x) } can best represent the gven data set. Ths fttng problem can be formulated n least squares optmzaton manner as: M T Ma M T b, (2) where M s the matrx of monomals whose -th row s m(x ) (see Eq. ()); a s the unknown coeffcent vector; and b s a zero vector. Note, Eq. (2) s just transformed from the least squares result, a M b, where M (M T M) M T s the so-called pseudo-nverse matrx. Note, many efforts such as [], [28], [7] have been made for solvng the lnear system of equatons (2) by frst overcomng the sngularty of M T M and b. In ths paper, we suppose that Eq. (2) has been modfed by mprovng sngularty of M T M and makng b by those technques. C. Rdge Regresson (RR) Regularzaton for Stable Fttng Although the lnear methods mprove the numercal stablty to some extent, they stll suffer from the dffculty of achevng global stablty. Especally whle modelng a complex shape wth a hgh-degree IP, many extra zero sets are generated. One mportant reason for global nstablty s the collnearty of column vectors of matrx M, causng the matrx M T M to be nearly sngular (see [28]). Addressng ths ssue, Tasdzen et al. [28] and Sahn and Unel [22] proposed usng rdge regresson (RR) regularzaton n the fttng, whch mproves (that s, decreases) the condton number of M T M by addng a term κd to the dagonal of M T M, where κ s a small postve value called the RR parameter, and D s a dagonal matrx. Accordngly Eq. (2) can be modfed as See the detal n [28], [22]. (M T M + κd)a M T b. (3) D. Shortcomngs of Pror Methods As mentoned n Secton I, the frst shortcomng of the pror methods s that none of them allow changng the degree durng the fttng procedure. Because of ths, solvng the lnear system (2) derved from the least squares method requres that the fxed IP degree must be assgned to construct the nverse of the coeffcent matrx M T M. The second shortcomng s related to the local naccuracy resultng from the conventonal RR regularzaton proposed n [28], [22]. Snce the methods cannot rule out the ll condton for the covarant matrx M T M n tme, they have to employ the RR term κd to operate the whole matrx M T M, whch leads to losng much local accuracy whle global stablty s mproved. In ths paper, we wll present a method to solve both these problems. III. INCREMENTAL FITTING In ths secton, we present an ncremental scheme usng QR decomposton for the fttng methods, whch allows the IP degree to ncrease durng the fttng procedure untl a moderate fttng result s obtaned. The computatonal cost s also saved because each step can completely reuse the calculaton results of the prevous step. In ths secton, frst we descrbe the method for fttng an IP wth the QR decomposton method. Next, we show the ncrementablty of Gram-Schmdt QR decomposton. After that, we clarfy the amount of calculaton needed to ncrease the IP degree. A. Fttng usng QR decomposton In place of solvng the lnear system (2) drectly, we frst carry out QR decomposton on matrx M as: M Q N m R m m, where Q satsfes: Q T Q I (I s an m m dentty matrx), and R s an nvertble upper trangular matrx. Then, substtutng M QR nto Eq. (2), we obtan: R T Q T QRa R T Q T b Ra Q T b Ra b. (4) After upper trangular matrx R and vector b are calculated, the upper trangular lnear system can be solved quckly n O(m 2 ).

3 3 L ( ) -th step -th step ( +) -th step L.25.2 Our method ICCG LU Fg. 2. The ncremental scheme that teratvely keeps solvng an upper trangular lnear system. R a b ~ R + ~ a+ b + CPU tme (s).5..5 r, j b ~ r, j b ~ Num. of monomals -th step ( +) -th step Fg. 4. A comparson between our method and the pror methods wth respect to calculaton tme. Fg. 3. In order that the trangular lnear system grows from the -th to ( + )-th step, only the calculaton shown n lght gray s requred; other calculaton s omtted by reuse at the -th step. B. Gram-Schmdt QR Decomposton Our ncremental algorthm depends on the QR-decomposton process. We found that the Gram-Schmdt QR Decomposton s very sutable and powerful for savng the computatonal cost for our algorthm (dscussed n the next subsecton). Now, frst let us brefly descrbe the QR decomposton based on the Gram-Schmdt orthogonalzaton. Assume that matrx M consstng of columns {c, c 2,, c m } s known. The Gram- Schmdt algorthm orthogonalzes {c, c 2,, c m } nto the orthonormal vectors {q, q 2,, q m } that are the columns of matrx Q, and smultaneously calculates the correspondng upper trangular matrx R consstng of elements r,j. The algorthm s wrtten n an nductve manner. Intally let q c / c and r, c. If {q, q 2,, q } have been computed at the -th step, then the ( + )-th step for orthonormalzng vector c + s r j,+ q T j c +, for j, q + c + r j,+ q j, j r +,+ q +, q + q + / q +. (5) Wth the Gram-Schmdt algorthm, matrx M s successfully QRdecomposed as M QR, and thus the problem of solvng Eq. (2) can be transformed to solve a lnear system wth an upper trangular coeffcent matrx. C. Incremental Scheme The dea of our ncremental scheme s to contnuously solve upper trangular lnear systems (4) n dfferent dmensons where the QR Decomposton wth Gram-Schmdt orthogonalzaton (5) s utlzed. Ths process s llustrated n Fg. 2, where the dmenson of the upper trangular lnear system ncreases, and thus the coeffcent vectors of dfferent degrees can be solved. We desgned ths ncremental scheme not only because, at each step, solvng an upper trangular lnear system s much faster than solvng a square one, but also because the calculaton for dmenson ncrement between two successve steps s computatonally effcent. Fg. 3 llustrates ths effcency by clarfyng necessary calculaton from the -th step to the ( + )-th step n our ncremental process. For ths calculaton, n fact, t s only necessary to calculate the parts that are llustrated wth gray blocks n Fg. 3. For constructng the ( + )-th upper trangular lnear system from the -th one, we are concerned wth two types of calculaton: ) How to calculate the upper trangular matrx R +, 2) How to calculate the rght-hand vector b+. Takng advantage of Gram-Schmdt QR Decomposton, we fnd that, for the frst calculaton, that s, growng from R to R +, we only need to calculate the rghtmost column of R +, and the other elements of R + are not changed from R ; for the second calculaton, that s, growng from b to b+, we only need to calculate the bottom element of b+, and the other elements of b + are not changed from b. For the frst calculaton, the result can be smply obtaned from Gram-Schmdt orthogonalzaton n Eq. (5). For the second calculaton, lettng b+ be the bottom element of vector b+, the calculaton of b+ can be stated as b+ q T + b after carryng out the (+)-th step of Gram-Schmdt orthogonalzaton n Eq. (5). In order to clarfy the computatonal effcency, let us assume a comparson between our method and a brute-force method, such as the 3L method [], that teratvely calls the lnear method at each step for obtanng the coeffcent vectors of dfferent degrees. It s obvous that, for solvng coeffcent a at the -th step, our method needs nner-product operatons for constructng the upper trangular lnear system (see (5)), and O( 2 ) for solvng ths lnear system; whereas the latter method needs 2 nnerproduct operatons for constructng lnear system (2), and O( 3 ) for solvng Eq. (2). We show a smple example n Fg. 4 to compare the actual calculaton tme between our method descrbed above and two other methods that solve the lnear system (2) ndependently at each step wth two famous lnear system solvers: ) LU decomposton

4 4 and 2) ncomplete Cholesky conjugate gradent (ICCG). The result was taken from the mean of, calculatons for solvng the same 2D fttng problem that ncrementally fts an IP to a certan data set untl achevng the th degree; practcally the IPs of about the th degree are often requred to represent complex shapes. As shown n ths fgure, our ncremental scheme performs much faster than the other two methods durng the dmensongrowng process. Note that, although the ICCG algorthm s very effcent for solvng a large-scale sparse lnear system, matrx M T M n lnear system (2) s often a dense matrx, and therefore ICCG cannot result n good performance. IV. GLOBAL/LOCAL STABILIZATION As descrbed n Sectons I and II, lnear fttng methods usually suffer from the dffculty of achevng global stablty. Although rdge regresson (RR) regularzaton can be adopted to acheve global stablty [28], [22], local accuracy mght deterorate too much. Addressng ths problem, n ths secton we present a computatonally smple procedure to detect whether a specfc monomal wll contrbute to the accuracy of IP representaton durng the ncremental steps. If t wll not, RR s used to stablze the fttng at the current step. To explan the procedure, we frst clarfy how RR regularzaton acheves global stablty and why t sacrfces local accuracy. Then we propose a new RR regularzaton-based stablzaton technque: our regularzaton can concentrate on the mprovement of stablty by selectvely applyng RR regularzaton to our ncremental scheme. A. RR Constrants Frst let us analyze why the numercal nstablty occurs. In fact, an mportant reason s the collnearty of matrx M, whch causes ts covarance matrx M T M to be nearly sngular. The collnear columns of M are degenerated to contrbute very lttle to the overall shape of the ft (see [28]). But ths lttle contrbuton may result n the senstvty for a hgh-degree IP, e.g., dvergence of some coeffcent values. As a result, there are extra undesred solutons generated. Now let us nterpret RR regularzaton n [4], [28], [22] to be some ndvdual constrants that we call RR constrants hereafter. Accordng to the conventonal defnton n [4], the formula of RR regularzaton shown n Eq. (3) can be equvalently transformed as ˆM T ˆMa ˆM T ˆb, (6) where ˆM s the matrx combnng matrx M and the square roots of dagonal elements of D, and vector ˆb s from the extenson of b wth zeros as follows: ˆM κd M κd b and ˆb,. κdll where d ll s the l-th dagonal element of D. Eq. (6) s smlar to Eq. (2). The only dfference between them s that there are some addtonal row vectors at the bottom of matrx ˆM, and actually these addtonal row vectors act as the lnear constrants n fttng. Let us call the constrants RR constrants. These RR constrants overcome the sngularty of matrx ˆM T ˆM and thus keep the IP presentaton globally stable, but an excessve brute-force constrant manner causes all the zero set to be closed to ts orgn (see [28]), and thus local accuracy deterorates. Our clam s that f we can apply the RR constrants only to the necessary part, such as the part correspondng to the monomal that weakly contrbutes to the whole IP representaton, the RR regularzaton can resst deteroraton of local accuracy. B. Proposed RR Regularzaton Frst, t s necessary to detect global nstablty. Fortunately, n our ncremental process, snce matrx M has been QRdecomposed as M QR, we can observe that M T M R T R, and thus nstablty of M T M can be determned from the egenvalues of R. We can easly evaluate the sngularty of R by observng only the dagonal values at each step, snce upper trangular matrx R s egenvalues always le on ts man dagonal. If the value of the dagonal element r at the -th step s relatvely too small, we can assume the current column c of M mght be nearly collnear to the prevously generated orthogonal space of {c, c 2,, c } so as to have lttle contrbuton to the overall shape of ft. Thus, we apply the -th RR constrant only to ths part to concentrate on solvng ths collnearty. The acton s as follows. Suppose matrx ˆM denotes the matrx M after havng added the -th RR constrant as: ˆM M... κd. We assume ts QR decomposton s ˆM ˆQ ˆR and each element after the constrant s denoted wth ˆ. The dfference between M and ˆM s only the last row whose last element s only non-zero. From Eq. 5, ths effect propagates only r, q, and b. The norm of ˆq before normalzng s r 2 + κd, where r s the norm of q. Therefore, ˆr s derved as follows: ˆr r 2 + κd. (7) From the dervaton, ( ) b ˆb ˆq T r q T r 2 b r. (8) + κd ˆr b From Eq. (7) we can see obvously that once the -th egenvalue r s relatvely too small, t can be mproved to be a larger one as ˆr by addng the -th RR constrant. Followng ths stablty mprovement, coeffcent â at ths step s thus calculated as â ˆb ˆr r r 2 + κd b (< b r a ). (9) Also we can see that the dvergence of a s restraned by addng the -th RR constrant. In short, at an ncremental step, once a column vector c s detected to be lnearly dependent on the precedng columns, the computaton n Eqs. (7), (8) and (9) should be done.

5 5 C. Normalzaton for the Lnear System However, checkng the value of r mght be unfar for judgng the collnearty. In the result obtaned from Gram-Schmdt orthogonalzaton, r mght be related not only to the degree of the collnearty but also to the norm of the correspondng column of M. In order to remove only the effect of the norm, t s necessary to normalze the lnear system (2). Normalzng the lnear system (2) s performed as follows: M T M a M T b, () where M s the normalzed matrx of M and b s the normalzed vector of b n Eq. (2) descrbed as { } M c c, c 2 c 2,..., c n, c n b b b, () assumng M {c, c 2,, c n }. Therefore the orgnal coeffcents can be obtaned as a { c b a, c b 2 a 2,..., c b n a n}, assumng a (a, a 2,..., a n). Consequently the RR regularzaton s formulated as (M T M + κd )a M T b. (2) Here we choose the same selecton method for determnng RR matrx D as Tasdzen s method descrbed n [28], [22]. Namely, each dagonal element of D s chosen as d ll d ll c l, where the 2 detal for calculatng dagonal elements d ll refers to [28], [22]. An mportant property of the normalzaton process s that, f M s QR-decomposed as M Q R, then all the man dagonal elements of R vary between and, and the elements of vector Q T b vary between and. Ths benefcal pre-process helps us to make a far judgment to check out the exstence of collnearty n the ncremental process, by just checkng whether r s relatvely too small n the range: [ ]. D. Achevng Eucldean Invarant Method Unfortunately, the above method s not Eucldean nvarant, although t effectvely mproves the stablty for fttngs. That mght restran further applcaton to vson applcatons such as object recognton and pose estmaton. For keepng the fttng method Eucldean nvarant, the dagonal matrx D n Eq. (3) s desgned to satsfy V DV T D, where V s a block transformaton matrx for IP transformaton as: a V a (see [28] and [22]). Accordng to [28], [22], f we let D KD, where K s a dagonal matrx and ts dagonal elements vary the same as the blocks of V (e.g., K dag{k k k k 2 k 2 k 2... k... k n} n 2D), then D s also Eucldean nvarant. The mportant pont s that k s correspond to elements n the -th form. Therefore, to extend our method to be a Eucldean nvarant verson, the ncremental fttng procedure can just be modfed to ncrease oneform by one-form, and once the unstable stuaton s elmnated, then we can mprove the egenvalues correspondng to the whole form. V. FINDING THE MODERATE DEGREE Now the coeffcents of varous degrees can be worked out stably by the ncremental method descrbed above. The remanng problem s how to measure the moderaton of degrees for these resolved coeffcents. In other words, when should we stop the ncremental procedure? n f ( x ) f ( x ) IP x N Data set x e x + f ( x ) f ( x ) Fg. 5. Defnton of two types of our smlarty measurement: one s based on the dstance between data pont and IP zero set, and the other s based on the dfference between normal drectons assocated wth data pont and IP zero set. A. Stoppng Crteron For ths problem, we frst need to defne a smlarty metrc that s capable of measurng the dstance between the IP and the data set. Then, based on ths smlarty metrc, we must defne a stoppng crteron for termnatng the ncremental procedure. Once ths stoppng crteron s satsfed, we consder the desred accuracy s reached and the procedure wll be termnated. As shown n Fg. 5, a set of smlarty functons can be wrtten as follows: D dst N e, D N smooth N (N n ), (3) N where N s the number of ponts, e f(x ) f(x ), N s the normal vector at a pont x obtaned from the relatons of the neghbors (here we refer to the Sahn s method [22]), and n f(x ) f(x ) s the normalzed gradent vector of f at x. Note e s also vewed as the approxmaton for the Eucldean dstance from x to the IP zero set n [29]. D dst and D smooth can be consdered as two measurements on dstance and smoothness between the data set and the IP zero set. Therefore, the closer D dst and D smooth are to and respectvely, the better the fttng become. Then, we can smply defne a stoppng crteron as (D dst < T ) (D smooth > T 2 ), (4) where T and T 2 are two tolerances beng close to zero and one respectvely. For the data set wth varaton of nose level, an alternatve smple way can be consdered as: (R dst < T ) (R smooth < T 2 ). (5) where R dst and R smooth are resduals of D dst and D smooth respectvely, namely we only see the dfference between the errors of current and prevous steps. An example of usng ths stoppng crteron s shown n Secton VI E. B. Algorthm for Fndng the Moderate IPs Gven the above condtons, our algorthm can be smply descrbed as follows: ) Constructng the upper trangular lnear system wth column vectors of M n Eq. (2) wth the method descrbed n Secton III; 2) Stablzaton wth the method descrbed n Secton IV B;

6 6 (a) (b) (c) (d) D dst D smooth D dst D smooth.85 -degree 2-degree 8-degree 2-degree.2.8 Fg. 7. IP fts. IP fttng results n 2D/3D. Frst row: orgnal objects; Second row: α (e) Fg. 6. (a) Orgnal mage. IP fttng results at dfferent coeffcent numbers: (b) α 28 (sx-degree). (c) α 54 (nne-degree). (d) α 92 (thrteendegree). (e) Graph of D dst. and D smooth vs. coeffcent number α. Note o symbols represent the boundary ponts extracted from the mage, and sold lnes represent the IP zero set n (b)-(d). Orgnal Objects Degree-fxed fttng wth 2-degree 3) Solvng current lnear system to obtan coeffcent vector a; 4) Measurng the smlarty for the obtaned IP; 5) Stoppng the algorthm f the stoppng crteron (4) or (5) s satsfed; otherwse gong back to ) and ncreasng the dmenson by addng the ( + )-th column of M. Degree-fxed fttng wth 4-degree Our method (Degree-unfxed fttng) 2-degree IP 6-degree IP 2-degree IP VI. EXPERIMENTAL RESULTS Our experments are set n some pre-condtons. ) As a matter of convenence, we employ the constrants of the 3L method [] that takes two addtonal data layers at a dstance ±ε outsde and nsde the orgnal data as the optmzaton constrant. 2) All the data sets are regularzed by centerng the data-set center of mass at the orgn of the coordnate system and scalng t by dvdng each pont by the average length from pont to orgn, as done n [28]; 3) We choose T and T 2 n Eq. (4) wth about. and.95 respectvely, except for the experment n Secton VI E. 4) The RR parameter κ n Eq. (7) s emprcally set to ncrease the orgnal dagonal element r about %. Note: we also refer the nterested reader to the dscusson on settng κ n [28]. A. Examples In ths experment, we ft an IP to the boundary of a cell object shown n Fg. 6(a). The moderate IP s found out automatcally based on our stoppng crteron (see Fg. 6(d)). To gve a comparson, we also show some fts before reachng the desred accuracy (see Fg. 6(b) and (c)). We also show the convergence of D dst and D smooth n Fg. 6(e), whch proves the stoppng crteron n Eq. (4) can effectvely measure the smlarty between IPs and data sets. Some 2-D and 3-D experments are shown n Fg. 7. The result shows our method s adaptvty for dfferent complextes of shapes. In our experment, the layer dstance of the 3L method n [] s set as ε.5. Fg. 8. Comparson between degree-fxed fttng and adaptve fttng. Frst row: orgnal objects. Second row: IP fts resultng from two-degree degreefxed fttng. Thrd row: IP fts resultng from four-degree degree-fxed fttng. Fourth row: IP fts n dfferent degrees resultng from adaptve fttng settng parameters as T. and T B. Degree-fxed Fttng vs. Adaptve Fttng Fg. 8 shows some comparsons between the degree-fxed fttng methods and our adaptve fttng method. Compared wth degree-fxed methods, n the results of our method, there s nether over-fttng nor nsuffcent fttng. Ths shows that our method s more meanngful than the degree-fxed methods, snce t fulflls the requrement that the degrees should be subject to the complextes of object shapes. To clarfy agan, as descrbed n Secton III-C, our method saves much computatonal tme despte ts determnaton of the moderate degree by an ncremental process. C. Comparson of Fttng Stablty Fg. 9 shows a comparson between three methods: ) 3L method [], 2) 3L method [] + 2D RR regularzaton [28] and 3L method [] + 3D RR regularzaton [22] and 3) our method. As a result, our method shows better performance than the others, n both global stablty and local accuracy. D. An Example for Nosy Data Fttng We generated some synthetc nosy data sets shown n the top row of Fg.. For overcomng the effect on the varaton of nose,

7 Objects 3L method 3L + RR method our method Fg. 9. Comparson of fttng results: frst column: orgnal 2D/3D objects, second column: results obtaned by 3L method [], thrd column: results obtaned by 3L method + RR method [28], [22] (for 2D and 3D respectvely), last column: results obtaned by our method. Results n top row are 2th degree and n bottom row are 8th degree. B. IP vs Other Functons Fg.. Top row: Nosy data sets made by addng Gaussan nose to the orgnal model wth standard devatons: from left to rght.,.5,.2; Bottom row: the correspondng fttng results. In contrast to other functon-based representatons such as B- splne, NURBS [2], Ratonal Gaussan [5], radal bass functon [32] and topology-preservng polynomal model [], IP representaton cannot gve a relatvely accurate model. But ths representaton s more attractve for the applcatons of fast regstraton and fast recognton (see the work n [28], [25], [26], [3], [34], [38]), because of ts algebrac/geometrc nvarants [3], [28]. And Sahn and Unel also showed IP s robustness for mssng data n [22]. For a more accurate representaton of a complex object, t may be requred to segment the object shapes and represent each segmented patch wth an IP, whch also has been consdered n our prevous work [39]. It s worth notng that another polynomal representaton method, topology-preservng polynomal model [], has the potental of offerng an accurate model, but t has not yet been wdely used n recognton. we adopted the second stoppng crteron (5) where parameters wth respect to resduals are set as T. and T Then we obtaned the fttng results as shown n the bottom row of Fg.. A. QR Decomposton Methods VII. DISCUSSION Other famous algorthms of QR decomposton have been presented by Householder and Gvens [8]. They have proved more stable than the conventonal Gram-Schmdt method n severe cases. But n ths paper, snce our target s only the regularzed data set, we gnore the mnor nfluence of roundng errors. Here we would just lke to take advantage of the propertes of QR decomposton that orthogonalze vectors one by one, whch s the ndspensable aspect to acheve our proposed adaptve-degree fttng algorthm. 2 Snce the normals often vary more strongly than vertces, we set the smoothness threshold T 2 to be more tolerant than the dstance threshold T C. Applcablty for Recognton An mportant advantage of IP for recognton s the exstence of algebrac/geometrc nvarants, whch are functons of the polynomal coeffcents that do not change after a coordnate transformaton. The algebrac nvarants that are found by Taubn and Cooper [3], Teral and Cooper [25], Keren [] and Unsalan [34] are expressed as smple explct functons of the coeffcents. Another set of nvarants from the covarant conc decompostons of IP s found by Wolovch et al. [36]. A stable coeffcent measurement called PIM usng IP-nterpolated data set s proposed by Le et al. [7]. Our method s able to obtan the coeffcents of multple IP degrees lower than the fnal fttng degree durng the ncremental process. Therefore, combnng the pror methods descrbed above wth these multple-degree coeffcents enables us to extract rcher nvarant nformaton. Thus t s expected that the nvarants have better dscrmnablty; the low-degree coeffcents encode shape nformaton roughly, but wth less senstvty to nose, whle the hgh-degree coeffcents can encode shape nformaton n detal.

8 8 VIII. CONCLUSIONS Ths paper brngs two major contrbutons. Frst, t proposes a degree-ncremental IP fttng method. The ncremental procedure s very fast snce the computatonal cost s saved by the process n whch we keep solvng upper trangular lnear systems of dfferent degrees, and effectvely employ the ncrementablty of the Gram- Schmdt QR decomposton. Thus the method s able to adaptvely determne the moderate degree for varous shapes. Second, based on the benefcal property of the upper trangular matrx R n QR decomposton that egenvalues are the elements on ts man dagonal, the fttng nstablty can be checked out easly and quckly from the dagonal values. By selectvely utlzng the RR constrants, our method not only mproves global stablty, but also mantans local accuracy. ACKNOWLEDGMENTS Ths work s supported by the Mnstry of Educaton, Culture, Sports, Scence and Technology Japan under the Leadng Project: Development of Hgh Fdelty Dgtzaton Software for Large- Scale and Intangble Cultural Assets. 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