A Newton-Type Method for Constrained Least-Squares Data-Fitting with Easy-to-Control Rational Curves

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1 A Newton-Type Method for Constraned Least-Squares Data-Fttng wth Easy-to-Control Ratonal Curves G. Cascola a, L. Roman b, a Department of Mathematcs, Unversty of Bologna, P.zza d Porta San Donato 5, 4017 Bologna, Italy b Department of Mathematcs and Applcatons, Unversty of Mlano-Bcocca, Va R. Cozz 53, 015 Mlano, Italy Abstract Whle the mathematcs of constraned least-squares data-fttng s neat and clear, mplementng a rapd and fully automatc ftter that s able to generate a far curve approxmatng the shape descrbed by an ordered sequence of dstnct data subject to certan nterpolaton requrements, s far more dffcult. The novel dea presented n ths paper allows us to solve ths problem n effcent performance by explotng a class of very flexble and easy-to-control pecewse ratonal Hermte nterpolants that makes possble to dentfy the desred soluton wth only few computatons. The key step of the fttng procedure s represented by a fast Newton-type algorthm whch enables us to automatcally compute the weghts requred by each ratonal pece to model the shape that best fts to the gven data. Numercal examples llustratng the effectveness and effcency of the new method are presented. Key words: Least-squares data-fttng; Constraned approxmaton; Ratonal Hermte nterpolaton; Automatc selecton of weghts; Interor-pont Newton-type method. MSC: 65D10; 65D17; 41A0; 41A9; 90C51; 90C30. Correspondng author. Emal addresses: cascola@dm.unbo.t (G. Cascola), luca.roman@unmb.t (L. Roman).

2 1 Introducton In several applcatons of computer-aded geometrc desgn we have to deal frequently wth 3D pont reconstructon problems where a large number of data s gven n the form of an ordered sequence of dstnct ponts descrbng a target shape n space. Most of them usually contans measurement errors, whle only a few are rgorously generated and then turn out to be crucal to the fnal reconstructon. As regards the computaton of a surface/surface ntersecton curve, for example, the set of dscrete data s made of hghly-accurate ponts (such as border ponts, turnng ponts and cusp ponts) precsely detected on the ntersecton curve, and a sequence of marchng ponts generated from each of the above ones by gong a step n the drecton defned by the local dfferental geometry of the curve. The frst, commonly called startng ponts, wll be therefore exactly nterpolated, whle all the others wll be approxmated n order to reduce to the mnmum the sum of the squares of ther dstances from the desred curve. An analogous stuaton occurs n the pont-based constructon of a Gordon-type surface. Ths tme the nput data s gven by a set of 3D ponts, descrbng the complete cross-sectons of the target surface, whch have to be accurately approxmated to produce a far curve network to be assumed as surface skeleton. Consstency demands that cross-secton curves agree n value where an x-secton crosses a y-secton. Ths means that sequences of ponts along two transversal drectons must possess common ntersectons and these have to be assumed as postonal constrants for the fttng problem. In both the outlned crcumstances - as well as n many other applcatons - t s therefore desrable to use a fttng method that, on one hand, s able to capture the shape of the overall nput data and, on the other, to satsfy the assgned pont constrants precsely. Snce when approxmatng shapes wth a complcated behavour t always turns out to be convenent to construct the fttng curve segment-wse by means of a pecewse model defned n the space of conventonal polynomal splnes or NURBS, our dea conssts n usng the pont constrants (dentfed by the specfc applcaton we are consderng or detected from the curvature and torson nformaton of the nput dataset) to partton the gven data nto adjacent subsets that can be approxmated separately by a curve segment taken from some specfed class of approprate curves. Takng nto account that when the ponts le on the ntersecton of two analytc surfaces, dervatve constrants of any order can be easly computed by ther parametrc representaton and, when generatng Gordon-type surfaces, addtonal nformaton lke frst dervatves and/or hgher order dervatves to be assumed by the curve network n the sgnfcant locatons are also generally avalable, the most natural soluton to ths knd of constraned fttng problem can be obtaned by usng a ftter that mplements a pecewse Hermte nterpolant. Ths soluton also allows us to tremendously smplfy the computatonal process that a standard least-squares mnmzaton problem wth assocated postonal constrants would have requred n order to ensure a suffcently hgh order of contnuty at segment boundares.

3 Whle for pure nterpolaton there s probably lttle reason to use a ratonal form, for approxmaton purposes allowng weghts to be arbtrary makes t possble to produce fttng curves wth hgher accuracy and fewer control ponts. The novel soluton we are gong to propose wll rely therefore on a class of pecewse ratonal Hermte nterpolants. In partcular we wll adopt here the one ntroduced n [1,] because of ts flexblty and easness of control. Ths can also be represented n the conventonal NURBS form by assumng multple knots n correspondence of the locaton of the nterpolaton constrants and lettng the control ponts be dependent both on them and on the weghts of the ratonal representaton. In ths way, once a procedure for computng optmal weght values has been desgned, control ponts turn out to be automatcally defned and hence the best-fttng curve results completely determned. Therefore, dfferently from standard NURBS fttng procedures, whch requre a complcated and expensve teratve algorthm to mnmze wth respect to knots, control ponts and weghts, a sum of squared Eucldean norms measurng the dstance between the pont set and the curve to be generated [3,4,9,11,1,13,14,15], the least-squares fttng method we are gong to propose wll be performed exclusvely to dentfy the choce of weghts that guarantees the best reconstructon of the orgnal data. Moreover, whle the output of exstng algorthms cannot always guarantee a fttng curve wth a far shape (namely wth a curvature plot consstng of only a small number of monotone peces), due to the defnton of the novel ftter ths follows straghtforward and, whenever the degree of the curve prmtve s bgger than three, a curvature-contnuous approxmaton of the orgnal data s also ensured. The organzaton of the paper s made as follows. In Secton we ntroduce the ratonal Hermte bass to be used as novel curve prmtve for determnng the soluton of the constraned least-squares problem. Next n Secton 3 we develop a strategy for carryng out the automatc computaton of the optmal weghts to be emboded n the desred ratonal form and we descrbe the overall fttng process n all ts steps. Fnally, n Secton 4 we close the paper by showng some numercal examples that confrm the mproved performance of the nnovatve procedure relatve to conventonal and relable approaches lke the well-known lsqcurveft algorthm that s currently mplemented n MATLAB s Optmzaton Toolbox. Least-squares fttng wth a novel curve prmtve Gven a set of 3D dstnct ponts representng a target shape n space, we seek a NURBS curve that les close to the assgned data and passes through only a few of them. Let {Q k },...,M 1 denote the gven set of ponts n R 3 and I {0,..., M 1} the subset of N + 1 (N << M) ndexes specfyng all the ponts Q k that the fttng curve must nterpolate. In the followng we wll use the notaton {F } =0,...,N to dentfy the pont constrants Q I and we wll also assume that the frst and last 3

4 data ponts are always nterpolated, that s Q 0 F 0 and Q M 1 F N. Hence the pont set {Q k },...,M 1 turns out to be parttoned by Q I nto N adjacent subsets that can be approxmated separately by a curve segment c (t) taken from some specfed class of admssble curves. Snce t s requred that segments c have common endponts F, F +1 and a contact of order l at F and F +1, the C l -contnuous pecewse curve c(t) = N 1 =0 c (t) can be naturally generated through a pecewse ratonal Hermte nterpolant of degree n = l + 1. As, n practce, a reconstructon wth pecewse curves of low degree s usually preferred, due to ther smplcty and robustness, we wll confne ourselves here to consder ratonal Hermte models defned by n = 3 and n = 5 only. Ths choce s also supported by the observaton that such degrees allow us to establsh a good compromse between the number of curve peces c (t) and the accuracy of the fnal fttng c(t). By takng nto account that, f a large number of peces s used a curve wth very small fttng errors s obtaned, whle, f the number of peces s too small the fttng errors mght be very large, t s easy to understand that the number of curve segments should be controlled n such a way that the fttng errors reach a level that the user can accept. Snce the dea behnd the algorthm we are gong to propose s to ft a degree-n ratonal Hermte segment to the data confned between the nterpolatng ponts F, F +1, and, f ths cannot guarantee the desred level of accuracy, to adaptvely subdvde the pont set lmted by the assgned constrants and reconsder the subsets (repeatng the procedure untl the requred error tolerance holds), t s mportant to make a choce of n whch can guarantee a successful reconstructon that does not need too many and computatonally expensve segments. To ths am t has been proved expermentally that, f n s chosen equal to 3 or 5, a tght and economcal ft s always ensured through only a restrcted number of curve peces. From now on we wll therefore address our attenton towards the cubc and the quntc pecewse ratonal Hermte models only. Note that the same choce was also made n [7], where pecewse Hermte polynomals were orgnally adopted as fttng curve bass n alternatve to classcal B-splnes. But n ths work, to facltate local approxmaton wth endpont constrants, we wll let each sngle Hermte pece to be represented n the well-known ratonal Bézer form. Next, n order to get a standard NURBS representaton, degree-n Bézer segments wll be peced together wth n-fold knots n correspondence of data ponts {F } =0,...,N where the jons take place [1,]. As frequently done wth data-fttng procedures, the overall knot-partton {τ k },..., M 1 s obtaned by applyng the cumulatve chord length parameterzaton τ k = τ k 1 + Q k Q k 1 M 1 j=1 Q = j Q j 1 kj=1 Q j Q j 1 M 1 j=1 Q, k = 1,..., M 1 j Q j 1 (1) (wth τ 0 = 0) to the gven pont set Q k, and the subset of multple break knots {t } =0,...,N, used to dentfy the junctons between consecutve ratonal peces, s computed by selectng from the prevous one the locaton parameters of ndex I. By constructon t results therefore that t 0 < t 1 <... < t N and t 0 τ 0, t N τ M 1. 4

5 Snce any Hermte model always requres that l th -order dervatves D (l) (l 1) are assgned n correspondence of the nterpolatng ponts {F } =0,...,N, whenever they are not gven as constrants or cannot be easly obtaned (lke t happens when the gven data le on the ntersecton of two analytc surfaces), then a data-senstve dervatve estmaton must be computed as part of the fttng algorthm. Whle estmatng frst dervatves has been a subject of extensve study, an approprate estmaton of hgher order dervatves s stll consdered a dffcult task. But as t was assumed that n {3, 5} and neghborng Bézer segments are joned wth a level of contnuty related to the degree of the fttng curve through the formula l = n 1, t turns out to be suffcent to estmate frst and second order dervatves only. To ths am we are allowed to explot a 3-pont strategy, based on local quadratc nterpolaton, that enables us to derve them compatbly wth the behavour of the data. In partcular, gven the three-dmensonal pont set Q k and the sequence of locaton parameters τ k, an approprate estmaton of frst and second order dervatves {D (l) } =0,...N (l = 1, ) at ponts F, can be worked out respectvely by frstly computng the drecton vectors and (1) Q k = θ k Q k 1 + θ k 1 Q k θ k 1 + θ k () () Q k = ( Q k Q k 1 ) θ k 1 + θ k, (3) (where θ k = τ k+1 τ k and Q k = Q k+1 Q k θ k for all k = 0,..., M 1) and successvely selectng from them the N + 1 values defned n correspondence of the knot subsequence {t } =0,...,N. Remark 1 In the above strategy l th -order dervatves at τ k are estmated to be those of the quadratc whch passes through (τ j, Q j ), j = k 1, k, k + 1. Thus, n case of open curves, the auxlary pont requred by equatons () and (3) at both the endponts can be consstently derved by a quadratc extrapolatory rule. In case of closed curves, nstead, we smply assume Q 1 = Q M and Q M = Q 1. At ths pont, havng dentfed the subset of break knots {t } =0,...,N defned n correspondence of the sgnfcant ponts {F } =0,...,N and computed the assocated dervatves {D (l) } =0,...,N (for all l = 1,..., n 1 ), we am to construct N ratonal Hermte curve segments {c (t)} =0,...,N 1 of a chosen degree n (n {3, 5}) that nterpolate the assgned end constrants and best approxmate the ordered sequence of ponts lyng n between. To ths purpose t turns out to be very helpful to express each sngle Hermte pece c (t) : [t, t +1 ] R 3 nto the followng ratonal Bézer form. 5

6 Defnton Defned the overall knot-partton {τ k },...,M 1 and the related subsequence of break knots {t } =0,...,N, the degree-n (n {3, 5}) ratonal Hermte nterpolant c (t), wth t [t, t +1 ], can be convenently wrtten n the ratonal Bézer form n c (t) = P jrj,n(t), (4) j=0 where {P j } j=0,...,n denote the n+1 control ponts assocated wth the bass functons R j,n(t) = w j B j,n (t) nh=0 w h B h,n (t) (5) defned va the postve weghts w 0 = 1, { } wj, j=1,...,n 1 w n = 1, (6) and the degree-n Bernsten polynomals Bj,n(t) = n j (t +1 t) n j (t t ) j (η ) n, wth η = t +1 t. (7) Snce, once the degree n s fxed, an explct formulaton of all Bézer control ponts {P j } j=0,...,n n terms of the weghts {wj } j=1,...,n 1 and of the assgned boundary ( constrants t, F, {D (l) ) ( } l=1,..., n 1, t +1, F +1, {D (l) ) +1 } l=1,..., n 1 can be provded, we wll now formalze ther defnton both n the cubc and n the quntc cases. Defnton 3 A degree-3 ratonal Hermte nterpolant c (t) of the knd (4) s defned by control ponts {P j } j=0,...,3 havng the followng expressons: P 0 = F, P 1 = F + η D (1) 3w1, P = F+1 ηd (1) +1, P 3 = F +1. (8) 3w Analogously, 6

7 Defnton 4 A degree-5 ratonal Hermte nterpolant c (t) of the knd (4) s defned by control ponts {P j } j=0,...,5 havng the followng expressons: P 0 = F, P 1 = F + η D (1) 5w1 P 3 = F +1 (5w 4 1)η D (1) +1 10w 3 + η D() +1 0w3, P = F + (5w 1 1)ηD (1) 10w + η D(), (9) 0w, P 4 = F +1 η D (1) +1, P 5 = F +1. 5w 4 As regards the ratonal cubc prmtve, t follows by defnton that t gves C 1 smoothness, whch s suffcent for many applcatons, whle keepng computatonal costs to a mnmum. However, as for certan applcatons hgher order curves could be necessary or advantageous (n general because hgher smoothness and better approxmaton are requred), whenever C contnuty s needed the ratonal quntc model wll be able to provde the desred soluton. Despte each sngle pece s a bt more expensve than ts degree-3 correspondent, t shows excellent approxmaton propertes and makes t possble to reduce the number of peces needed to ft a target shape wthn the same tolerance. In the next sectons we wll show how the proposed ratonal models can be advantageously used for constructng an optmal fttng of 3D data ponts, such that hgh accuracy s guaranteed and a compact representaton s pursued. 3 A Newton-type optmzaton algorthm for computng best-fttng weghts We now ntroduce the notaton {Q k },...,M 1 to dentfy the M ponts Q k confned between two assgned nterpolatng ponts F, F +1 and we denote by {τ k },..., M 1 the parameters τ k assocated wth these ponts (such that Q 0 = F, Q M 1 = F +1 and τ 0 = t, τ M 1 = t +1). In ths way, the approxmatng curve c(t) = N 1 =0 c (t) wll be made of sngle peces c (t) n the form (4), defned to mnmze over each nterval [t, t +1 ] the least-squares error Φ(w 1,..., w n 1) = M 1 c (τ k) Q k (10) wth respect to the unknowns {wj } j=1,...,n 1. Snce the dependence of the th segment c (t) on the weght vector w = (w1,..., w n 1 )T of the ratonal representaton (4) s nonlnear and s not avalable n a smple analytcal form, we have to use a nonlnear optmzaton procedure that can cope wth ths problem. But as the range of postve values attanable by the parameters wj should be bounded away from zero and, for clear practcal reasons, lmted by a plausble upper bound, ndeed we have to deal wth a mnmzaton problem whose solutons can only be expected n 7

8 a partcular area. We are thus allowed to consder a numercal method for nonlnear optmzaton wth a feasble set n the form of box. However, dfferently from standard NURBS curves, we are not forced to avod usng dramatcally varyng weghts, snce, beng the control ponts defnton nfluenced by the weghts themselves, whatever ther choce s we can guarantee a good parameterzaton. In ths way we can arbtrarly set lower and upper bounds, gvng the weghts the possblty to assume also values very dstant from 1. To our purposes t has been proved expermentally that a good choce of feasble set s gven by the box Ω := {w R n 1 : l j w j u j wth l j = 10 3, u j = 10 3 j = 1,..., n 1}, where l j, u j denote the lower and upper bounds for wj, respectvely. Whle l j = 10 3 has been mposed by the condton that weghts wj should be strctly postve and suffcently far from zero, the value u j = 10 3 s dctated by a reasonable choce that allows us to smplfy our analyss and make the soluton useful n practcal problems. Indeed we have to solve a nonlnear and multvarate optmzaton problem that ams at mnmzng the objectve functon Φ : R n 1 R, defned by the followng nfntely contnuously dfferentable expresson Φ(w ) = M 1 c (τ k) Q k = M 1 E k, (11) subject to w Ω. The exstence of local mnma s generally not a real dffculty n our case, but a fast convergence towards a mnmum s certanly a crucal problem. Usually, most effcent algorthms for calculatng a local mnmum of a nonlnear functon may be consdered to be descent methods. At each step they consst n mnmzng the objectve functon along a straght lne defned by a drecton named the descent-drecton. Each method s characterzed by the way n whch ths drecton s bult. Be warned that, f t s not opportunely defned, the algorthm may become really expensve and tme-consumng snce several teratons may be needed to reach the mnmum. When the objectve functon s very regular, among the fastest descent methods we can fnd the so-called Newton-type methods. In fact, when the exact computaton of the objectve functon dervatves s possble, the strateges whch do not use these nformaton turn out to be more expensve than the ones whch use them. In prncple, for NURBS curve fttng problems, computng dervatves wth respect to weghts poses no severe dffcultes, prmarly because NURBS are a ratonal combnaton of these varables. In addton, usng the specal NURBS representaton proposed n the prevous secton, gradent and Hessan of the objectve functon turn out to possess a very smple and compact symbolc form. Thus, to perform the optmzaton process n (11) t appears to be convenent to use a Newton-type al- 8

9 gorthm that requres the computaton of both frst and second order dervatves of Φ(w ). Indeed, due to the presence of box constrants, the most adequate class of Newton-type algorthms turns out to be the one of projected affne-scalng nterorpont Newton methods. In the last ten years several papers proposng more and more effcent varants of ths type of smple-constraned optmzaton procedure have been publshed (see [8,10] and references theren). Among all possble solvers of ths class, we consder here the teratve method ntroduced n [8] and we develop some mprovements to guarantee that the desred fttng curve s obtaned wth less computaton and a fast convergence s ensured n any stuaton. In partcular, we frstly observe that there s a scalng matrx that can be cancelled on both sdes of the lnear system to be solved at each step of Henkenschloss et al. algorthm, allowng a more effcent mplementaton; secondly, snce when choosng a value of the steplength σ very close to 1 (as suggested by the authors) ther teratve method does not always converge to the optmal weght vector w,, we wll propose an nnovatve optmzaton procedure that can combne the above-mentoned smplfed verson of the algorthm n [8] wth a modfed choce of the σ parameter that s able to ensure an order-two convergence for any type of nput data. Our soluton to ths knd of problem s based on a non-statonary defnton of σ whch, although startng from a small guess n (0,1), allows us to guarantee n only a few steps ncreasng values closer and closer to 1, thus provdng an affne-scalng nteror-pont Newton-type method that s always quadratcally convergent. Before developng the novel order-two nteror-pont Newton method that enables us to compute the best-fttng values of w, we wll ntroduce the followng notaton. For the objectve functon Φ : R n 1 R, we denote by Φ(w ) R n 1 and Φ(w ) R (n 1) (n 1) ts gradent vector and ts Hessan matrx, respectvely, whle we use [ Φ(w ) ] j and [ Φ(w ) ] j 1 j for ther components. By defnton, the gradent of Φ(w ) s the (n 1) dmensonal column vector Φ(w ) = Φ(w ) w 1. Φ(w ) w n 1, (1) whle ts Hessan s the (n 1) (n 1) symmetrc matrx Φ(w ) = Φ(w ) (w 1 ).... Φ(w ) w n 1 w 1 Φ(w ) w 1 w n 1. Φ(w ) (w n 1 ). (13) 9

10 Even though provdng an explct formulaton of Φ(w ) and Φ(w ) for arbtrary objectve functons Φ(w ) s generally a dffcult and computatonally ntensve task, one of the beautes of the ratonal model proposed here for representng c (t), s that t allows us to remarkably smplfy these operatons. In fact, usng equaton (11) and the sum and product dervatve rules, we can state that Φ(w ) = M 1 E k E k w 1. M 1 E k E k w n 1 (14) where denotes the nner product of two vectors. Then, n turn, usng the dfference and quotent dervatve rules, we can assert that for a ratonal cubc Hermte element t holds E k w 1 = [F c (τ k )]B 1,3 (τ k ) 3h=0 w h B h,3 (τ k ), (15) E k w = [F +1 c (τ k )]B,3 (τ k ) 3h=0 w h B h,3 (τ k ), (16) whle for ts quntc correspondent we have E k w 1 = [F c (τk )]B 1,5 (τ k ) + η B,5 (τ k ) 5h=0 wh B h,5 (τ k ), (17) D (1) E k w = [F c (τ k )]B,5 (τ k ) 5h=0 w h B h,5 (τ k ), (18) E k w 3 = [F +1 c (τ k )]B 3,5 (τ k ) 5h=0 w h B h,5 (τ k ), (19) 10

11 E k w 4 = [F +1 c (τk )]B 4,5 (τ k ) η +1 B 3,5 (τ k ) 5h=0 wh B h,5 (τ k ). (0) D (1) Thus, nsertng equatons (15)-(16) and (17)-(0) n (14), we get respectvely Φ(w ) = M 1 M 1 E k [F c (τ k )]B 1,3 (τ k ) 3 h=0 w h B h,3 (τ k ) E k [F +1 c (τ k )]B,3 (τ k ) 3 h=0 w h B h,3 (τ k ) (1) n the cubc case, and Φ(w ) = M 1 M 1 E k [F c (τk )]B 1,5 (τ k )+ η E k D(1) B,5 (τ k ) 5 h=0 w h B h,5 (τ k ) E k [F c (τk )]B,5 (τ k ) 5 h=0 w h B h,5 (τ k ) E k [F +1 c (τk )]B 3,5 (τ k ) 5 h=0 w h B h,5 (τ k ) E k [F +1 c (τk )]B 4,5 (τ k ) η E k D(1) +1 B 3,5 (τ k ) 5 h=0 w h B h,5 (τ k ) M 1 M 1 () n the quntc one. Hence, the elements n the Hessan matrx Φ(w ) can be reduced to the smplfed expressons wrtten below. Be warned that, for ease of notaton, we wll omt the arguments of the Bernsten bass functons B j,n (τ k ) (j = 1,..., n 1) and we wll assume the followng compact forms: G 1,k := [F c (τ k)] [F c (τ k) + E k], G,k := [F +1 c (τ k)] [F +1 c (τ k) + E k], H k := [F +1 c (τ k)] [F c (τ k) + E k] + [F c (τ k)] E k, I 1,k := D (1) [F c (τ k) + E k], I,k := D (1) +1 [F c (τ k) + E k], J 1,k := D (1) [F +1 c (τ k) + E k], J,k := D (1) +1 [F +1 c (τ k) + E k], K 1 := D (1) D (1), K := D (1) D (1) +1, K 3 := D (1) +1 D(1) +1. In ths way, n the cubc case t holds 11

12 [ M 1 Φ(w ) ]11 = [ M 1 Φ(w ) ]1 = [ M 1 Φ(w ) ] = ( B 1,3) G 1,k ( 3h=0 w h B h,3), B 1,3 B,3 H k ( 3h=0 w h B h,3), ( B,3) G,k ( 3h=0 w h B h,3), whle n the quntc one we get 1

13 [ M 1 Φ(w ) ]11 = ( ) ( B1,5 G 1,k + η B1,5 B,5 I 1,k + η 4 B,5) K 1 ( 5h=0, wh h,5) B [ M 1 Φ(w ) ]1 = [ M 1 Φ(w ) ]13 = [ M 1 Φ(w ) ]14 = B 1,5 B,5 G 1,k + η (B,5) I 1,k ( 5h=0 w h B h,5), B 1,5 B 3,5 H k + η B,5 B 3,5 J 1,k ( 5h=0 w h B h,5), ( [ M 1 Φ(w ) ] = B,5) G 1,k ( 5h=0, wh h,5) B [ M 1 Φ(w ) ]3 = B,5 B 3,5 H k ( 5h=0, wh h,5) B [ M 1 Φ(w ) ]4 = B 1,5 B 4,5 H k + η (B,5 B 4,5 J 1,k B 1,5 B 3,5 I,k ) η 4 B,5 B 3,5 K ( 5h=0 w h B h,5), B,5 B 4,5 H k η B,5 B 3,5 I,k ( 5h=0 w h B h,5), ( [ M 1 Φ(w ) ]33 = B3,5) G,k ( 5h=0, wh h,5) B [ M 1 Φ(w ) ]34 = [ M 1 Φ(w ) ]44 = B 3,5 B 4,5 G,k η (B 3,5) J,k ( 5h=0 w h B h,5), ( ) ( B4,5 G,k η B3,5 B 4,5 J,k + η 4 B3,5) K 3 ( 5h=0. wh h,5) B Snce the Newton-type algorthm we are gong to descrbe s an teratve method that updates the weght vector w at each step s, untl the optmal one w, s determned, we wll denote the soluton correspondent to the s th step by w,s. Afterwards we use the explct expressons of Φ(w ) and Φ(w ) developed above to defne the followng dagonal matrces for the s th round of the algorthm: ( D(w,s ) := dag d 1 (w,s 1 ),..., d n 1(w,s n 1 ), ) where for any j = 1,..., n 1 13

14 d j (w,s j ) := d j (w,s j ), f [ Φ(w,s )] j < mn{w,s j l j, u j w,s j } or mn{w,s j 1, otherwse l j, u j w,s j } < [ Φ(w,s )] j, wth d j (w,s j ) := mn{w,s j w,s j l j, f [ Φ(w,s )] j > 0, u j w,s j, f [ Φ(w,s )] j < 0, l j, u j w,s j }, f [ Φ(w,s )] j = 0; ( G(w,s ) := dag g 1 (w,s 1 ),..., g n 1(w,s n 1 ), ) where for any j = 1,..., n 1 g j (w,s j ) := [ Φ(w,s )] j, f [ Φ(w,s )] j < mn{w,s j or mn{w,s j 0, otherwse. l j, u j w,s j } l j, u j w,s j } < [ Φ(w,s )] j, As wth any teratve scheme, to start the algorthm good ntal guesses are requred for the unknowns. Namely, we want to start wth a plausble confguraton of the varables w,0 j wth respect to the observaton that the weght vector must contan values n the range [l j, u j ]. In practce, we have found n our experments that usng the ntal vector w,0 = (1,..., 1) T s adequate for any cases. Therefore, the teratve process we are gong to llustrate always starts from a pecewse polynomal Hermte model - snce all the weghts wj are ntalzed to 1 - and proceeds updatng ther values through teratve mnmzaton of the fttng error. The other man ssue n mnmzng an objectve functon s gven, nstead, by the stop crteron. To check the convergence towards the target shape, the stop crteron Φ(w,s ) Φ(w,s 1 ) Φ(w,s ) < δ = 0,..., N 1 (3) on the squared -norms of the resduals at the current and last step s examned over each segment contemporarly wth the termnaton test on the -norm of the followng scaled gradent D(w,s ) Φ(w,s ) < ε = 0,..., N 1. (4) 14

15 The stop occurs when these norms are lower than gven accuraces. In order to see some nterestng effects when comparng ths method wth a compettve one, we have chosen for tolerances δ and ε the values and 10 5, respectvely. But these parameters mght be changed by the user f a hgher or lower accuracy s requred. Note that, for practcal reasons, the second crteron s more meanngful, but the frst one gves better nformaton about the qualty of the mnmum when the functon s flat around t. Havng fxed startng values and stoppng crterons, we are then n the poston to state the novel nteror-pont Newton-type method for the soluton of the bound constraned optmzaton problem (11). Algorthm 1: Best-Fttng Weghts Computaton (S.0) Choose tolerances δ and ε (we have set δ = and ε = 10 5 n our mplementaton). Set s := 0, w,0 := (1,..., 1) T R n 1 and Φ(w, 1 ) = Φ(w,0 ). Defne a startng parameter σ 0 (0, 1) (we have found σ 0 = 0.3 to be a good choce for any knd of data). (S.1) Compute Ψ(w,s ) := Φ(w,s ) Φ(w,s 1 ) Φ(w,s ) and b(w,s ) := D(w,s ) Φ(w,s ). If Ψ(w,s ) < δ and b(w,s ) < ε, STOP. (S.) Compute A(w,s ) := D(w,s ) Φ(w,s ) + G(w,s ). (S.3) Let x s R n 1 be the Newton-type search drecton obtaned by solvng the lnear system A(w,s ) x s = b(w,s ). (S.4) Compute P Ω (w,s x s ) usng the followng projecton mappng: P Ω : R n 1 Ω { } P Ω (w,s j x s j ) = max l j, mn{u j, w,s j x s j } j = 1,..., n 1. (5) (S.5) Compute ρ s := max { σ s, 1 P Ω (w,s x s ) w,s }, so that strct feasblty of the terates worked out n the followng step can be ensured. (S.6) Set w,s+1 := w,s + ρ s ( P Ω (w,s x s ) w,s). (S.7) Update the value of σ s through the formula 1 + σ σ s+1 s =. (6) (S.8) Set s s + 1 and go to (S.1). Remark 5 Note that the lnear system to be solved n step (S.3) contans just n 1 equatons (.e. and 4 equatons n the cubc and quntc case respectvely), that s t can be easly solved by a drect approach lke Gaussan elmnaton. 15

16 Lemma 6 Gven an ntal parameter σ 0 (0, 1), the recurrence relaton n (6) satsfes the propertes: and σ s (0, 1) s 1 (7) lm s + σs = 1. (8) Proof. Demonstraton of (7) trvally follows. To prove (8) we recall that a monotonc and bounded sequence s always convergent and, n partcular, f t s non decreasng and upper bounded, then t converges to the upper bound of the values t assumes. For the recurrence formula σ s = σ 0 (0, 1) 1+σ s 1 s 1 (9) t holds that the sequence {σ s } s 1 s non decreasng and hence convergent to 1. In fact, called γ ts lmt, we have 1 + σ γ = lm s + σs = lm s γ =. s + Thus, solvng the last equaton wth respect to γ we get γ = 1. From the above result t follows that σ s (0, 1) for any step s and the sequence generated n (S.7) converges to 1 so that Henkenschloss et al. condton on the steplength s stll satsfed. Thus the projected drecton s truncated by a coeffcent ρ s whch approaches fast enough to 1 that the followng convergence result holds. Proposton 7 The weght vector w,s converges wth order two to the best-fttng soluton w,. Proof. Snce the parameter σ s defned n (6) always stays n (0, 1) and the lnear system to be solved n step (S.3) s equvalent to the one derved by Henkenschloss et al. n [8], the same convergence result establshed for that procedure stll holds for the projected affne-scalng nteror-pont Newton method proposed n Algorthm 1. By combnng the σ parameter correcton n Lemma 6 wth the smplfed verson of 16

17 Henkenschloss et al. algorthm, t s possble to get an optmzaton method that s very effcent and extremely robust. Due to Proposton 7, the convergence theory from Henkenschloss et al. paper stll holds for ths approach. We now assemble the quadratcally-convergent teratve method n Algorthm 1 wth the steps descrbed n Secton n order to develop a constraned least-squares fttng technque that ams at generatng the pecewse ratonal Hermte curve c(t) whch passes as close as possble to the sequence of ponts Q k assgned as nput and nterpolates the set of ponts F and dervatves D (l) (l = 1, ) defned as constrants. Algorthm : Constraned Least-Squares Data-Fttng Input: - the degree n (n {3, 5}) of the desred fttng curve (remember that f C l contnuty s requred, then the chosen degree must be n = l + 1); - a 3D pont set {Q k },...,M 1 and a subset of nterpolatng ponts {F } =0,...,N wth F 0 Q 0 and F N Q M 1 ; - optonally, l th -order dervatves {D (l) } =0,...,N (l = 1, ) to be assumed by the fttng curve n correspondence of the nterpolatng ponts {F } =0,...,N ; - a parameter λ = 0, 1, specfyng the order of assgned dervatves (note that λ = 0 means no dervatves are specfed). 1. Chord Length Parameterzaton. Determne the parameters τ k to be assocated wth the ponts {Q k },...,M 1 through the chord-length method n (1) and select from them the subsequence of break knots {t } =0,...,N, fxng t 0 = τ 0 and t N = τ M 1, such that the endponts of the data set Q always match wth the endponts of the fttng curve c(t).. Check Specfed Constrants..1. If λ = 0 Compute an approprate estmaton of dervatves {D (l) } =0,...,N, for all l = 1,..., n 1, by selectng among the drecton vectors { (l) Q k },...,M 1 derved through formulas ()-(3), the ones correspondng to the subsequence of knots {t } =0,...,N... Elsef λ = 1 and n > 3 Compute an approprate estmaton of nd -order dervatves {D () } =0,...,N by selectng among the drecton vectors { () Q k },...,M 1 derved through formula (3), the ones correspondng to the subsequence of knots {t } =0,...,N. 3. Best-Fttng Weghts Computaton. For all = 0,..., N 1 approxmate the data ponts {Q k },...,M 1 lyng between two consecutve nterpolatng ponts F, F +1, through the ratonal Hermte nterpolant c (t) wth optmal parameters {w j } j=1,...,n 1 provded by Algorthm 1. Output: 17

18 the degree-n approxmatng curve c(t) = N 1 =0 c (t) n the desred form (4). To summarze, the dea at the base of our nnovatve strategy s the followng: () we defne a number of parameter values compatble wth the gven data set, () we estmate frst and, f requred, second order dervatves n correspondence of all ponts to be precsely nterpolated and () fnally, for each curve pece, we compute the weght vector whch mnmzes the fttng error n the least-squares sense, so that the ratonal Bézer form (4) can be provded. The man contrbuton of ths paper les therefore n brngng all these components together wth some nnovatons to complete a new and effcent algorthm for constraned least-squares data-fttng. Its performances n terms of accuracy, number of teratons and computng tme have been extensvely analyzed over a wde range of experments whch confrm ts superorty f compared wth exstng procedures. In the next secton we wll llustrate the results we got by comparng the proposed ftter wth MATLAB s lsqcurveft functon. 4 Comparsons and expermental results To llustrate the effcency of the fttng method derved n Secton 3, we have compared the optmal weghts computaton n Algorthm 1 wth MATLAB s lsqcurveft procedure. By default ths mplements a subspace trust regon approach based on the nteror-reflectve Newton method descrbed n [5,6], whose sngle step nvolves the approxmate soluton of a large lnear system by usng the method of Precondtoned Conjugate Gradents (PCG). The two algorthms have been mplemented n MATLAB and tested for fttng many sets of ponts wth constrants, always showng that, startng wth the same ntalzaton values and adoptng the same stoppng crterons, the novel approach allows us to reduce the number of lsqcurveft teratons and the overall computng tme sgnfcantly, whle keepng the fttng curve wthn at least the same accuracy and n good qualty. Such knds of results, provng the superor performance of our method, have been obtaned testng the two procedures on many sequences of ponts representng any type of target shape n space. Snce mportant applcatons n Computer Aded Geometrc Desgn often requre fttng ponts that le on the ntersecton of two surfaces by usng a far and accurate NURBS model, n the examples selected for ths paper we have exploted degree 3 and 5 ratonal Hermte elements to approxmate sequences of ponts {Q k },...,M 1 generated by a marchng algorthm arsng from a set of startng ponts (consstng of border ponts, turnng ponts and cusp ponts) that turns out to be crucal to the ntersecton curve reconstructon. For each segment c (t) the closeness of ft has been verfed by computng the average dstance of the ratonal curve representaton, resultng by the dentfed weghts, from the data set Q. Such a dstance has been worked out wth respect to the se- 18

19 quence of parameter values τ k prevously determned through (1), by means of the root mean square error ( 1 ERMS = M ) 1 Φ(w, ). (30) The effcency of the new procedure n terms of approxmaton accuracy, number of teratons and computng tme s confrmed by numercal results lsted n the followng tables (they all refer to tests made on a Pentum IV 3 GHz PC computer). As t appears and t was confrmed by many other experments, each round of Algorthm 1 turns out to be really effectve, so that convergence to the optmum s always acheved n a few steps whch globally requre a very short tme. More precsely, whle n the cubc case the same approxmaton accuraces reached by the lsqcurveft procedure are obtaned after not much less rounds of the algorthm, the computng tme needed by the overall process to work out the best-fttng weghts, s notceably smaller. In the quntc case the superorty of the new approach s even more evdent: more precse fttng curves are generated through a remarkably reduced number of teratons whch leads to a sgnfcant mprovement of the computatonal tme. Data ponts consdered n Tables 1-, 3-4, 5-6 and 7 are determned respectvely by a free-form surface/free-form surface, a sphere/free-form surface, a sphere/cylnder and a torus/cylnder ntersecton. In these four cases the overall set Q s made of 30, 70, 514 and 590 ponts wth Q 0 Q M 1 ; the subset of startng ponts assumed n each of these contexts s gven by 7, 5, 5 and 7 entres, respectvely. For all the data sets the sequence of break ponts F to be nterpolated by the fnal fttng concdes wth the gven startng ponts (see Fgures and Tables 1-3-5). Sometmes, however, snce the resultng number of curve peces turns out to be too small to get a suffcently accurate reconstructon, wherever requred we adaptvely subdvde the pont set {Q k },...,M 1 confned between the assgned constrants F and F +1, n such a way that the fttng error of each curve segment may reach the requred error tolerance. In the examples below ths strategy has been appled on all the 4 subsets n data set and 3 to make the ERMS errors related to the least-squares cubc procedure reach the same level of accuracy obtaned by ts quntc counterpart (see Fgures 4-6 and Tables 4-6). Parallel to ths, the frst, second, ffth and sxth of the 6 subsets dentfed by the 7 startng ponts adopted for data set 1, have been subdvded to generate 10 peces of cubc wth ERMS errors smaller than 10 (see Fgure and Table ). Analogously, the second and ffth of the 6 subsets dentfed by the 7 startng ponts adopted for data set 4, have been subdvded to generate 8 peces of quntc wth ERMS errors smaller than 10 (see Fgure 7 and Table 7). Note that the best fttng curve got by runnng the lsqcurveft algorthm on data set 4 subject to the so computed pont constrants, s not able to satsfy the requred accuracy. 19

20 Fgure 1. Least-squares fttng of data set 1 through 6 ratonal quntc Hermte elements. In both pctures the pecewse quntc that best approxmates the sequence of ponts {Q k },...,301 lyng on the free-form surfaces ntersecton s denoted by a sold lne. Control polygons of the 6 curve peces passng through the assgned break ponts {F } =0,...,6 (wth F 0 F 6 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , , , ] 86 = e-03 [ , , , ] 550 = e-03 [ , , , ] = e-03 [1.1763, , , ] 79 (sec) = e-03 [ , , , ] 37 = e-03 [ , 1.778, , ] 94 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , , , ] 13 = e-03 [ , , , ] 11 = e-03 [ , , , ] = e-03 [ , , , ] 8 (sec) = e-03 [ , , , ] 7 = e-03 [.53868, , , ] 9 Table 1. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set 1 for computng the best-fttng weghts of the ratonal quntc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 0

21 Fgure. Least-squares fttng of data set 1 through 10 ratonal cubc Hermte elements. In both pctures the pecewse cubc that best approxmates the sequence of ponts {Q k },...,301 lyng on the free-form surfaces ntersecton s denoted by a sold lne. Control polygons of the 10 curve peces passng through the assgned break ponts {F } =0,...,10 (wth F 0 F 10 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 9 = e-03 [ ,.64180] = e-03 [ , ] 9 (sec) = e-03 [ , ] 9 = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , ] 4 = e-03 [ , ] 5 = e-03 [ , ] 4 = e-03 [ , ] 5 = e-03 [ ,.64193] = e-03 [ , ] 5 (sec) = e-03 [ , ] 7 = e-03 [ , ] 4 = e-03 [ , ] 5 = e-03 [ , ] 5 Table. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set 1 for computng the best-fttng weghts of the ratonal cubc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 1

22 Fgure 3. Least-squares fttng of data set through 4 ratonal quntc Hermte elements. In both pctures the pecewse quntc that best approxmates the sequence of ponts {Q k },...,701 lyng on the sphere/free-form surface ntersecton s denoted by a sold lne. Control polygons of the 4 curve peces passng through the assgned break ponts {F } =0,...,4 (wth F 0 F 4 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , , , ] 388 = e-03 [ , , , ] = e-0 [ , , , ] 373 (sec) = e-0 [ , , , ] 6 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , , , ] 5 = e-03 [ , , , ] = e-0 [0.7843, , , ] 5 (sec) = e-0 [ , , , ] 5 Table 3. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set for computng the best-fttng weghts of the ratonal quntc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons.

23 Fgure 4. Least-squares fttng of data set through 8 ratonal cubc Hermte elements. In both pctures the pecewse cubc that best approxmates the sequence of ponts {Q k },...,701 lyng on the sphere/free-form surface ntersecton s denoted by a sold lne. Control polygons of the 8 curve peces passng through the assgned break ponts {F } =0,...,8 (wth F 0 F 8 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] = e-03 [ , ] 7 (sec) = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , ] 4 = e-03 [ , ] 5 = e-03 [ , ] 4 = e-03 [ , ] = e-03 [ , ] 4 (sec) = e-03 [ , ] 4 = e-03 [ , ] 5 = e-03 [ , ] 4 Table 4. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set for computng the best-fttng weghts of the ratonal cubc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 3

24 Fgure 5. Least-squares fttng of data set 3 through 4 ratonal quntc Hermte elements. In both pctures the pecewse quntc that best approxmates the sequence of ponts {Q k },...,513 lyng on the sphere/cylnder ntersecton s denoted by a sold lne. Control polygons of the 4 curve peces passng through the assgned break ponts {F } =0,...,4 (wth F 0 F F 4 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , , , ] 376 = e-03 [ , , , ] = e-03 [ , , , ] 74 (sec) = e-03 [ , , , ] 478 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , , , ] 6 = e-03 [ , , , 1.071] = e-03 [ , , , ] 6 (sec) = e-03 [ , , , ] 6 Table 5. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set 3 for computng the best-fttng weghts of the ratonal quntc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 4

25 Fgure 6. Least-squares fttng of data set 3 through 8 ratonal cubc Hermte elements. In both pctures the pecewse cubc that best approxmates the sequence of ponts {Q k },...,513 lyng on the sphere/cylnder ntersecton s denoted by a sold lne. Control polygons of the 8 curve peces passng through the assgned break ponts {F } =0,...,8 (wth F 0 F 4 F 8 ) are shown as dashed lnes. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] = e-03 [ , ] 7 (sec) = e-03 [ , ] 7 = e-03 [ , ] 7 = e-03 [ , ] 7 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , ] 4 = e-03 [ , ] 4 = e-03 [ , ] 4 = e-03 [ , ] = e-03 [ , ] 4 (sec) = e-03 [ , ] 4 = e-03 [ , ] 4 = e-03 [ , ] 4 Table 6. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set 3 for computng the best-fttng weghts of the ratonal cubc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 5

26 Fgure 7. Least-squares fttng of data set 4 through 8 ratonal quntc Hermte elements. In both pctures the pecewse quntc that best approxmates the sequence of ponts {Q k },...,589 lyng on the torus/cylnder ntersecton s denoted by a sold lne. The assgned break ponts {F } =0,...,8 (wth F 0 F 8 ) nterpolated by the resultng pecewse quntc are denoted by small crcles. lsqcurveft curve segment M ERMS w, nt tme = e-03 [ , , , ] = e-0 [ , , , ] 37 = e-03 [ , , , ] 169 = e-03 [ , , , ] = e-03 [ , , , ] 79 (sec) = e-03 [ , , , ] 49 = e-03 [1.8576, , , ] = e-03 [ , , , ] 133 Algorthm 1 curve segment M ERMS w, nt tme = e-03 [ , , , ] 5 = e-03 [ , , ,.98755] 13 = e-03 [ , , , ] 6 = e-03 [ , , , ] = e-03 [ , , , ] 6 (sec) = e-03 [ , , , ] 6 = e-03 [ , , , ] 7 = e-03 [0.5647, , , ] 6 Table 7. Comparson of error measures (ERMS ), optmal solutons (w, ), numbers of teratons (nt) and computng tme (tme) for lsqcurveft and Algorthm 1 when they are tested on data set 4 for computng the best-fttng weghts of the ratonal quntc model. Be warned that as regards the lsqcurveft procedure, the parameter nt ncludes also the number of PCG teratons. 6

27 In all the selected examples frst and second order dervatves at the prescrbed locatons F have been worked out through formulas ()-(3) as prevously explaned n Secton. Snce the novel ftter reles on pecewse Hermte curves, nterpolaton of computed dervatves allows us to naturally guarantee a far fttng, as proved by the curvature and torson plots n Fgures 8, 9, 10, 11. The nnovatve least-squares fttng method proposed n Secton 3 s thus optmzed for handlng arbtrary spatal data sets wth constrants n respect of great qualty and hgh speed (a) (b) (c) (d) Fgure 8. Curvature plot of the pecewse quntc fttng curve n: (a) Fg.1, (b) Fg.3, (c) Fg.5 and (d) Fg (a) (b) (c) Fgure 9. Curvature plot of the pecewse cubc fttng curve n: (a) Fg., (b) Fg.4 and (c) Fg.6. 7

28 (a) (b) (c) (d) Fgure 10. Torson plot of the pecewse quntc fttng curve n: (a) Fg.1, (b) Fg.3, (c) Fg.5 and (d) Fg (a) (b) (c) Fgure 11. Torson plot of the pecewse cubc fttng curve n: (a) Fg., (b) Fg.4 and (c) Fg.6. 5 Concludng remarks Exstng technques for computng a smooth parametrc curve that approxmates a well-ordered sequence of dstnct data satsfyng specfc requrements on ponts and dervatves to be nterpolated, rely on mathematcal models defned n the space of conventonal polynomal splnes and NURBS. Whle alternatve approaches have proposed to use pecewse Hermte polynomals of degree three and fve as possble fttng curve bass [7], ther ratonal counterparts have never been taken nto account. Ths consderaton prompted us to propose an nnovatve soluton to the problem of constraned least-squares data-fttng that s based on cubc and quntc pecewse ratonal Hermte nterpolants. The use of ths novel curve prmtve provdes an 8

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