Approximation by NURBS with free knots

Transcription

1 pproximtion by NURBS with free knots M Rndrinrivony G Brunnett echnicl University of Chemnitz Fculty of Computer Science Computer Grphics nd Visuliztion Strße der Ntionen 6 97 Chemnitz Germny Emil: mhrvo@informtiktu-chemnitzde bstrct his pper dels with pproximting noisy smples by NURBS with specil emphsis on free knots We consider the knots s unknown prmeters so s to find their optiml positions he three unknowns (the weights the control points nd the knots) re therefore mixed together in nonliner wy fter reclling briefly the fixed knot problem we show wy to discrd the first two unknowns he problem becomes then simpler nd we show how to solve the resulting problem with only one prmeter which is the free knots Modifiction of the lgorithm is lso treted so s to void undesired knot position such s non-incresing knot sequence Implementtion spect nd numericl results re given t the end to support the ide Introduction NURBS settings re pprecited in mny theoreticl nlyses becuse they llow flexible description of both free form surfces nd usul geometries such s conic sections Indeed the set of rtionl functions is much lrger thn tht of polynomil functions so NURBS give minly better pproximtion thn their B-spline counterprts do nother reson for the pprecition of NURBS is tht it is supported by mny softwres For instnce OpenGL nd CIS ([] []) hve built-in commnds for drwing NURBS by only giving the required prmeters pproximtions with NURBS hve lredy been treted in mny documents (see mong others [] [5]) In [] the uthor uses n itertive segment determintion in order to estblish the positions of the knots In [5] In the context of B-splines the use of free knots hs lredy been investigted by severl uthors ([6] [7] [8]) he purpose of this pper is the description of NURBS pproximtion with free knots ht mens we im prticulrly t finding the optiml knot position while determining the other prmeters which re the control points nd the weights One of the min difficulties of NURBS over B- spline is the estblishment of the weights which should be positive On the other hnd since we del with rtionl functions mny of the formultions led to nonliner problems In section we formulte in detil the mening of free knots We recll lso briefly the method used for fixed knots ([]) he min ide for solving the free knot problem is given in section Numericl results re given in the lst section In this document we tret only curves the cse of surfces will be done in future pper Problem setting nd nottions NURBS NURBS (nonuniform rtionl B-splines) with weights nd control points is given by: $! * ' where &(' ) ' () is the usul ([]) B-spline bsis defined on knot sequence: + + '-/ + ' ' Remrk For our cse we del only with the cse + + ' ' 76 VMV Erlngen Germny November

2 U : * M M d d d5 d Figure : NURBS with In the sequel we will denote: d () () + ' + () Free knot problem Suppose we re given noisy smples is supposed to be very lrge Like d d d7 in other fitting we wnt to find the NURBS curve 5 which best fits these dt Becuse we wnt to find the optiml positions of the knots we put them s vribles ht mens we hve the following problem: * $ (5)! his problem is very difficult becuse the unknowns ( ) re ll mixed in ' nonlinerly Furthermore we need to dd some constrints bout the positivity of the weights In the next section we will show how to simplify this problem Brief recll of the fixed knot problem If we re given knot sequence then the problem ( *! )!!& will be refered to s fixed knot problem It is investigted in [] where it is shown to be equivlent to solving liner system: +*-/ - (6) where nd B re given in block structure: * 65 * ) * * 95 : 7 )DC ;< EFEFE ) GC ;< *;< >=?@B C ;< EFEFE C ;< >HI C'L NM OP SR C L C L NM SR C L 6 E&EFE ) ) =?@B E&EFE HI VU U ) XWYR Z Z ' O identity mtrix of order[ ]\ _\ _\ ) ) W b&c ' b ' c 6ed 6f K (7) (8) (9) () () K () ' WR In * - ll these expressions is understood to be nd is positive constnt which should be chosen lrge enough (see []) in order to ensure positivity of the weights Free knots Simplifiction of the problem he position of the knots is very importnt in NURBS fitting bd plcement of knots my led to remrkble distortion of the geometry One needs therefore wy to find the best position of

3 the knots In this section we intend to simplify the problem (5) which hs three prmeters First we cn reduce it into problem with one prmeter only Indeed ccording to section for given knot we cn solve the subproblem (6) in order to determine the corresponding weights nd the control points In other words nd re functions of ie _ 7_ + + ) Problem (5) is therefore simplified into: $! L M L M $!& () Penliztion of undesired knot position L M L M From now on we will write only insted of in order to simplify the nottion We hve then $! $! () his problem still llows the presence of the situ- is not incresing In this section we will modify this problem so tht only knots with + tion where + ' + ' + my hppen By denoting: we hve ) ) )) 7 Y $ $ nd by defining Y Y $ 5 $ $ $ Y nd $ (5) Note tht (5) is nothing else but sclr version of problem () which involves vector vlued expressions We introduce now the function nd we define if if + + ' il + ' 7 + ' + + / (6) + ' We modify then the problem (5) into * il + W (7) where is very lrge positive number Let us see the reltion between (5) nd (7) by remrking the following two properties of eqution (7) If we hve + ' + ' + then + ' ; + ' ; / for ll + nd therefore il hus * + il W $!6 If there is some + ' such tht ; + ' ; / then + ' ; + ' ; nd so il + is nonzero Becuse of our ssumption tht is very lrge number we cn expect tht $ + il W is lso very lrge Since we re serching for the minimum of * + il W the preceeding two points show tht with + ' ; + ' ; cn never relize this minimum ht mens tht the integrtion of the triling term in (7) penlizes those + ' with ; + ' ; / + + Remrk Sometimes it is lso desirble to hve! In this cse we need only to replce (6) into for ' il + '! 7 + '! + + /! + ' (8)

4 _ + il nd [ the problem is then usul nonliner lest squre problem which is _ W Implementtion By denoting * Such problem cn be solved by nonliner lest squre solvers like Levenberg-Mrqurdt nd Guss-Newton (see [9] []) Note tht for ech evlution of the function _ we need to solve the subproblem (6) in order to know the corresponding + + We remrk tht the order of the liner system (6) is smll It does not depend on the number of dt points It depends exclusively on the degree of the NURBS Furthermore ' + tking into ccount tht the support of is + ' we conclude tht the mtrices in (7) re bnded More precisely we hve: $ bfc for s consequence we need only to compute few entries nd the mtrices re sprse On the other hnd we must note tht the computtion of one entry of this system involves ll dt points he remedy to tht problem is to ssemble the mtrix nd vector =?@ 7 R R HI K =?@ R (9) only once nd store them in rries so tht they cn be red nd need not be recomputed in subsequent computtions Note tht nd re independent of hey depend only on the initil dt points he only things which need to be updted in ech itertion of the nonliner lest squre re the vlues of which re mostly zero except for some few vlues hey cn be computed with fst lgorithm (see []) Numericl results he numericl results in this pper were ll done with the Levenberg-Mrqurdt lgorithm Note tht HI K this lgorithm is itertive nd so it needs some initil guess he initil guess tht we hve tken here is equidistnt knots he first numericl test tht we perform is the reconstruction of W-form curve We hve dt which were dded with rndom noise of mplitude 6 he curve is reconstructed with the help of the formerly described lgorithm he overll time for the reconstruction is seconds In Figure we see grphicl illustrtion of the dt the initil curve nd the reconstructed curve he second test is free-form curve he time needed for the reconstruction is like in the first test grphicl illustrtion cn be found in Figure Figure : Initil curve reconstructed curve nd 75 smples with noise mplitude= Figure : Initil curve reconstructed curve nd 75 smples with noise mplitude= 6

5 5 Future work ll the computtions tht we hve performed shows tht the most expensive prt of this lgorithm is the ssembly of the mtrices in (6) he ssembly tkes in generl more thn of the whole computtionl work On the other hnd we cn see tht those mtrices cn be ssembled in prllel becuse they consist only of sums of some terms which cn be distributed on ech processor he first future work will del with the prlleliztion of this lgorithm in which ech processor will hve lmost the sme number of dt points so s to ensure lod blncing he second future work will consist in extending this lgorithm for surfces References [] G Frin Curves nd surfces for computer ided geometric design cdemic Press Boston ed 99 [] C de Boor prcticl guide to splines Springer New York 978 [] B Elsässer pproximtion mit rtionlen B- Spline Kurven und Flächen PhD thesis Drmstdt 998 [] Corney D modeling with CIS kernel nd toolkit ohn Wiley & sons Chichester 997 [5] P Lurent-Gengoux nd M Mekhilef Optimiztion of NURBS representtion Computer ided Design pp vol 5 No 99 [6] H Schwetlick nd Schütze Lest squres pproximtion by splines with free knots BI pp 6-8 vol 5 No 995 [7] D upp pproximtion to dt by splines with free knots SIM Numer nl pp 8- vol 5 No 978 [8] Schütze Diskrete udrtmittelpproximtion durch Splines mit freien Knoten PhD thesis Dresden 998 [9] Nocedl nd S Wright Numericl Optimiztion Springer Series in Opertion Reserch New York 999 [] R Fletcher Prcticl methods of optimiztion ohn Wiley & Sons Chichester ed 987 [] Neider Dvis nd M Woo OpenGL progrmming guide ddison-wesley publishing compny Reding 99