Fair NURBS Curve Generation using a Hand-drawn Sketch for Computer Aided Aesthetic Design

Transcription

1 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, Fa NURBS Cuve Geneaton usng a Hand-dawn Sketch fo Compute Aded Aesthetc Desgn AKIRA YAMAGUCHI AND TETSUZO KURAGANO Gaduate School of Infomaton Scence, Mese Unvest 2-9 Nagabuch, Ome-Ct, Toko, 98-86, JAPAN kuagano@e.mese-u.ac.jp Abstact: - Image pocessng technques ae used to detect the edges of a hand-dawn sketch. Geneall, vaous tpes of gadaton ae epessed n the backgound of a hand-dawn sketch. Theefoe, the ognal mage s tansfomed nto a logathmc mage. The ente mage has to be bnazed fo edge detecton. To smooth the edge of the bna mage, featue based eoson and dlaton s appled. A Laplacan opeaton s appled to the bna mage to detect the edge n the bna mage. Ths edge poston coesponds to the poston of the lne n the hand-dawn sketch mage. Lne segments whch epesent the detected edges ae geneated. Usng the postons and gadents on the lne segments, a NURBS cuve s geneated. The shape of the NURBS cuve etacted fom the sketch mage s eamned b the desgne vsuall. The shape of ths NURBS cuve geneall concdes wth the ntenton of the desgne, but does not concde pecsel. Theefoe, a cuve shape modfcaton method based on the specfed s appled. The sum of the squaed dffeences between the of a cuve and the specfed of an ente cuve s lneazed b Talo s theoem, then mnmzed. The specfcaton of s pefomed b selectng a pedefned fa NURBS cuve usng the smlat of dstbuton. Eamples of NURBS cuve geneaton based on a hand-dawn sketch ae gven accodng to the geneaton pocess. Ke-Wods: - hand-dawn sketch, mage pocessng, egesson lne, NURBS cuve,,, cuve shape modfcaton, Intoducton Conventonal desgn pocedues can easl poduce smple shapes. Howeve, cuentl, ndustal desgnes ae dawn to aesthetcall pleasng feefom shapes because the have geat consume appeal, especall n a hghl compettve and techncall well-developed maket, such as that fo automobles and electcal applances. It takes a long tme to desgn aesthetcall pleasng poducts usng conventonal pocedues. A soluton to ths poblem s to establsh a method to shoten the poduct desgn peod, especall the peod fom the fst dea geneaton as t s naowed down to the fnal desgn. Theefoe, the objectve of developng ths method s to shoten the Compute Aded Aesthetc Desgn (CAAD) [] peod. When desgnes begn a poduct desgn, desgnes ceate the deas and epand them. Nomall ths pocess s pefomed on pape, and desgnes hand-dawn lnes ae called sketches. If the desgnes ough dea on the sketch can be ealzed as a eal fa cuve, t wll be effectve n shotenng the desgn peod. Thee ae two canons fo dawng technques. One s pespectve pojecton. The othe s othogonal pojecton. Geneall, t s not eas to estmate the pespectve tansfomaton mat fom a sketch dawn b pespectve pojecton. Theefoe, the sketch n ths stud s lmted to othogonal pojectons such as font vew, sde vew and top vew. Fst, hand-dawn sketches ae put nto a compute b usng a scanne. Then, mage pocessng technques such as lght ntenst tansfomaton, featue based eoson and dlaton to smooth the edge of the bna sketch mage, and edge detecton of the bna sketch mage ae eamned. In addton to these mage pocessng technques, pncpal component analss s ntoduced to geneate egesson lne segments usng the detected edges. Usng the lne segments detemned, a quntc NURBS cuve s geneated. A NURBS cuve, whch s commonl used n the aea of CAD CAM and Compute Gaphcs, s used as an epesson of a feefom cuve. A quadatc NURBS cuve s used as an epesson of a quadatc cuve usng ts weghts.

2 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, 27 3 In ths stud, a quadatc cuve s not used to epess the shape of a cuve. Theefoe, the weghts of the NURBS cuve ae not used. A cubc NURBS cuve s wdel used also, but n ths stud, ove the mult segments of the NURBS cuve s modfed based on the specfed. So, a smooth contnut s needed. Accodngl, a quntc NURBS cuve s used n ths stud. The poston of the mdponts and the gadent of the lne segments ae gven to the NURBS cuve equaton and fst devatve equaton of the NURBS cuve espectvel. Then, a NURBS cuve s geneated. Aftewads, f necessa, the shape of ths NURBS cuve s modfed accodng to the specfed adus of dstbuton. The s specfed b lnea, quadatc, cubc, quatc, quntc, and s degees functon, whle the shape of the cuve geneated fom the sketch s modfed accodng to the specfed dstbuton usng the shape modfcaton algothm. Then, b ntoducng matchng, the smlates of the geneated cuve fom the sketch to these s cuves ae eamned. Among these pedefned cuves, the hghest smlat cuve to the cuve geneated fom the sketch s selected and detemned as a fa NURBS cuve geneated fom the sketch. Thee ae man elated woks fo geneatng cuves based on a sketch. Cuve geneaton b tacng a hand-dawn sketch s avalable as a functon of a commecal compute aded aesthetc desgn sstem [2]. Cuve geneaton usng a sketch, desgn language and chaactestc lnes [3], cuve geneaton based on a hand-dawn sketch [4], cuves fo a chaacte such as a stuffed anmal desgn based on a hand-dawn sketch [], 3D shape econstucton usng hand-dawn lnes [6], and smple polgonal shape econstucton based on a hand-dawn sketch [7] have been publshed. Cuve geneaton usng a hand-dawn sketch and t s vew ponts [8], emeshng based mesh smoothng b a sketch [9], shape geneaton usng a volumetc modelng technque [], and shape geneaton usng 3D scenes [] have also been publshed. In addton to these, mechancal pats such as a pston usng a hand-dawn sketch [2], smple pats geneated b constucted sold geomet [3], and smple mechancal pats desgn usng dgtal cla [4] have been publshed. Thee ae man elated woks fo geneatng fa cuves. Fa cuve geneaton algothms elated to eneg functons have been publshed. These fnd the unfa poton of a cuve usng eneg functon [], and appl a low-pass flte to eneg functon [6]. Fa cuve geneaton algothms elated to contol have been publshed. These make monotone [7], use a clothodal cuve fo specfng the [8], and modf the cuve based on the specfed [9]. In addton to these, fa cuve geneaton b usng the second devatve values [2], usng an agument of Béze contol edges [2], mnmzng postonal, the fst, the second, and the thd devatve values [22] have been publshed. Secton 2 of ths pape descbes some technques fo mage pocessng such as lght ntenst tansfomaton, bnazaton of the sketch mage, featue based eoson and dlaton to smooth the edge of the bna mage, and edge detecton of the bna mage. Secton 3 descbes the lne segment geneaton based on the detected edges b ntoducng the pncpal component analss. Secton 4 of ths pape descbes a quntc NURBS cuve, the fst devatve of a quntc NURBS cuve, vecto,, and. Secton descbes NURBS cuve geneaton usng the postons and the gadents on the lne segments. Secton 6 descbes NURBS cuve shape modfcaton based on the specfed. Secton 7 descbes a method to specf the dstbuton b selectng a pedefned fa NURBS cuve usng the smlat of the dstbuton. In secton 8, eamples of NURBS cuve geneaton based on a hand-dawn sketch accodng to the descbed geneaton pocess ae gven. 2 Image Pocessng In ths secton, the mage pocessng technques used to detect the edges of a hand-dawn sketch ae descbed. Fst, lght ntenst tansfomaton to ovecome the gadaton gven b the desgne, bnazaton, featue based eoson and dlaton to smooth the edge of the bna mage, and edge detecton ae descbed. 2. Lght ntenst tansfomaton Geneall, vaous tpes of gadaton ae epessed n the backgound of a hand-dawn sketch [23] as shown n Fg.. In such cases, the human ee s able to ecognze lnes whch ae dawn n a dak aea as well as lnes whch ae dawn n a bght aea. Howeve, a compute can not ecognze lnes n a dak aea, because of the small dffeence n lght ntenst

3 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, 27 3 between the lnes and the backgound gadaton, whch s dak. Theefoe, the ognal mage s tansfomed nto a logathmc mage to detect edges. In the logathmc mage, the lght ntenst dffeence of adjacent mage elements n the dak aea s ve close to the dffeences n the bght aea. In othe wods, the lght ntenst of the dak aea of the ognal mage s amplfed, whle that of the bght aea of the ognal mage s educed. The logathmc mage s shown n Fg.2. Fg. Hand-dawn sketch mage Fg.2 Logathmc mage 2.2 Bnazaton of sketch mage Bnazaton s pefomed accodng to the theshold tempoal decded. The canddate ponts fo edges ae dsplaed. Ths detemnaton pocess should be done nteactvel b the desgnes. A sngle theshold s not enough n some cases to bnaze an ente mage. Theefoe, the ente mage s dvded nto mult egons, and thesholds fo each egon ae establshed ndependentl. In ths case, the ente mage s dvded nto 64 egons. The 64 thesholds fo each egon ae detemned and the ente mage s then bnazed. 2.3 Featue based eoson and dlaton to smooth the edge of the bna mage Eoson and dlaton s a well-known method to coect defects n a bna mage. The ae appled to the bna mage as a combnaton of eoson-dlaton o dlaton-eoson. Eoson emoves ganules, and solated lnes and ponts, dlaton flls holes and gaps n the bna mage. "Featue based eoson and dlaton" s used fo the pupose of smoothng the edge of the bna mage. The bna mage s tadtonall set to one fo black, and to zeo fo whte. The sum of the values of the 8 pels adjacent to the taget pel s called the neghbo [24]. It s used to detemne the taget pel value, whch s zeo o one. A neghbo vaes fom zeo to eght. It s known b ou epements that a neghbo of ove fve does not wok. In addton to ths, n case the neghbo s zeo, ths pocess s eactl the same as "pel based eoson and dlaton". Theefoe, a neghbo s detemned nteactvel usng the value fom one to fou, accodng to the vsual nspecton of the bna mage. Fo the eoson pocess, f the sum of the 8 neghbos s bgge than the neghbo, the taget pel value s set to zeo, and f the sum of the 8 neghbos s smalle than the neghbo, the taget pel value s set to one. Fo the dlaton pocess, f the sum of the 8 neghbos s bgge than the neghbo, the taget pel value s set to one, and f the sum of the 8 neghbos s smalle than the neghbo, the taget pel value s set to zeo. As mentoned above, fo the pocess of eoson and dlaton, a method to decde the taget pel value, accodng to the sum of the 8 neghbos based on the neghbo s called "featue based eoson and dlaton" [24] n ths pape. 2.4 Edge detecton Fo Laplacan opeatos, thee est 3 3,, 7 7, 3 3 and so on. Ou pupose of usng a Laplacan opeato s to detect the edge of the bna mage. Fo ths eason, a Laplacan opeato s decded as 3 3 whch s 8 neghbos. If a Laplacan opeato s appled to the bna mage, the Laplacan mage has a postve and negatve value, between these two values, the Laplacan value becomes zeo. Ths s called zeo cossng. The poston fo zeo cossng s detected as the edge of the bna mage. Assumng f (, j ) s the pel value of poston, j of bna mage, the pel value g(, j ) of the Laplacan mage coespondng to f (, j ) of the bna mage s epessed b Eq.(). g (, j) = f ( -, j - ) + f ( -, j) + f ( -, j + ) + f (, j - ) + f (, j + ) + f ( +, j - ) () + f ( +, j) + f ( +, j + ) - 8 f (, j) 3 Lne Segment Geneaton usng the Detected Edges If lnes ae dawn that oughl follow the 6 detected edge data ponts, an mage emeges as shown n Fg.3. Fg.3 Detected edge data (6 ponts) and the epesentatve lnes Fo the pncpal component analss, we have to place the data ponts nto the desable numbe of goups contanng the desable numbe of data ponts. But

4 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, we have no ule fo goupng. Theefoe, although the pncpal component analss s not the statstcal analss, we follow the pncple of Stuges ule [2]. The data ponts ae then placed nto 8 goups of 2 ponts each n ths case. The pncpal component analss s then appled to each goup. A egesson lne segment epesentng the edge data fo each goup s then obtaned. In ths manne, the numbe of goups and the numbe of detected edge data n all goups ae detemned b followng the pncple of Stuges ule. Then, egesson lne segments fo all goups ae geneated usng the detected edges fom the hand-dawn sketch. 4 NURBS Cuve Epesson A quntc NURBS cuve s used n ths stud. The objectve of feefom cuve desgn s to desgn the famewok of suface patches. Suface patches ae defned as tenso poducts, whch ae b-vaate and nomall defned b u and v. In othe wods, one knot sequence n u decton, and anothe knot sequence n v decton ae defned despte the complet of the suface patches. Theefoe, knot spacng s fed n ths stud. A quntc NURBS cuve conssts of n - segments ( n ³ 6), s composed of n contol ponts such as q,q,, qn- and n weghts such as w, w,, wn - as n Eq.(2). R = n- å N = n- å =,6 N,6 w q w whee N,6 ( =,,, n -) ae NURBS bass functons. These functons ae ecusvel defned b knot sequence t, t,, t n+ as n Eq.(3). ì ( t t < t ) ü + N, = í î othewse ï ý (3) t - t t+ M - t N = N + N ï, M, M - +, M - t+ M - - t t+ M - t ï + þ whee =,,, n -and M = 2,3,,6. The bass functons ae defned b the de Boo-Co [26] ecuson fomulas. If the knot vecto contans a suffcent numbe of epeated knot values, then a dvson of the fom N, M - /( t+ M - - t ) = / (fo some ) ma be encounteed dung the eecuton of the ecuson. Wheneve ths occus, t s assumed that / = [27]. A quntc NURBS cuve wth knot vecto (2) {-, -4, -3, -2,-,,, 2, 3, 4,, 6} s epessed as n Eq.(4). R { q 2 + t - t + t + t - t + = ( - t) ( ) q + - t + t - t ( ) q2 + t - t - t + t + t ( ) q3 + - t + t + t + t + t ( ) q4 + t q } (4) The fst devatve of a quntc NURBS cuve s epessed as n Eq.(). dr( t ) dt { 4 = - ( - t) q (2t 8t 6t 4t ) q ( t 2t 2 t) q (t 8t 6t 4t ) q ( 2t 2t 3t 2t ) q4 4 + t q } () Cuvatue vecto s epessed b Eq.(6). ( R R ) R κ = ( R ) 4 (6) whee R s the fst devatve of a NURBS cuve, and R s the second devatve of a NURBS cuve. Cuvatue s the magntude of the vecto, theefoe s epessed as n Eq.(7). k = κ (7) B defnton, the of a plane cuve s nonnegatve. Howeve, n man cases t s useful to ascbe a sgn to the cuve [28]. The choosng of the sgn s commonl connected wth the tangent otaton (n movng along the cuve n the decton of the nceasng paamete): The of the cuve s postve when ts tangent otates counte-clockwse, the of the cuve s negatve when ts tangent otates clockwse. Radus of s the ecpocal numbe of, theefoe, s epessed as n Eq.(8). = k Geneaton of a NURBS Cuve usng the Lne Segments The concept of geneaton of a NURBS cuve based on the egesson lne segments detemned usng the detected edges s llustated n Fg.4. (8)

5 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, mp d mp d d 2 mp 2 egesson lne segment mp 3 mp 4 Fg.4 Concept of geneaton of a NURBS cuve whch appomates the egesson lne segments Whee mp,mp,,mp ae the mdponts of the 4 coespondng egesson lne segments. d,d,,d ae dectonal unt vectos of the 4 coespondng lne segments. A NURBS cuve whch passes though the mdponts and has fst devatves whch ae popotonal to the dectonal unt vectos s geneated. Eq.(9) s appled to the mdponts of the lne segments b settng the paamete of Eq.(4) to zeo. R = ( ) q q+ q+2 q+ 3 q +4 (9) 2 ( =,,, m - ) whee m s the numbe of lne segments. Eq.() s appled to the dectonal unt vectos of the lne segments b settng the paamete of Eq.() to zeo whle consdeng the magntude of the fst devatves. dr = ( -q - q+ + q+ 3 + q + 4 ) () dt 24 ( =,,, m - ) whee m s the numbe of lne segments. A NURBS cuve s geneated b solvng Eq.(9) and Eq.() smultaneousl. If the numbe of lne segments s 4, the numbe of NURBS cuve equatons (Eq.(9)) s 4 and the numbe of fst devatve equatons (Eq.()) s 4. As a lnea sstem, the total numbe of equatons s 8, wheeas the total numbe of contol ponts of a NURBS cuve s 8. Theefoe, ths lnea sstem s detemned. That s, the ank of a mat of a lnea sstem s equal to the numbe of unknowns. The soluton to ths lnea sstem s eact. Howeve, n case the numbe of lne segments s 3, the numbe of equatons (Eq.(9)) whch pass though the mdponts s 3, and the numbe of equatons of the fst devatve (Eq.()) s 3. In ths case, as a lnea sstem, the numbe of equatons s 6, wheeas the numbe of contol ponts of the NURBS cuve s 7. That s, the numbe of equatons s less than the d 3 d 4 numbe of unknowns. Theefoe, ths lnea sstem s undedetemned [29]. Fo an undedetemned sstem, whle settng aula functon, the lnea sstem s solved unde the constant condton b selectng one soluton fom nfnte numbe of eact solutons usng the Lagange's method of ndetemnate multples. In case the numbe of lne segments s, the numbe of equatons (Eq.(9)) s, and the numbe of equatons of the fst devatve (Eq.()) s. In ths case, as a lnea sstem, the numbe of equatons s, wheeas the numbe of contol ponts of the NURBS cuve s 9. That s, the numbe of equatons eceeds the numbe of unknowns. Theefoe, ths lnea sstem s ovedetemned [3]. Fo an ovedetemned sstem, the dffeences of ght and left sde of all the equatons of the sstem ae mnmzed. The contol ponts calculated ae appomatons. Fo a sstem whee the numbe of lne segments s moe than, the lnea sstem s ovedetemned. Fo these sstems, n accodance wth the ncement of the dffeence between the numbe of equatons and the numbe of unknowns, the status of appomaton soluton becomes wose. 6 Cuve Shape Modfcaton based on the Specfed Radus of Cuvatue In ths secton, a method to modf a NURBS cuve shape accodng to the specfed dstbuton to ealze an aesthetcall pleasng feefom cuve s descbed. The concept of specfcaton and NURBS cuve shape modfcaton based on the specfed dstbuton s shown n Fg.. A NURBS cuve and ts plots ae shown n Fg.(a). A method to modf the shape of the NURBS cuve shown n Fg.(a) to the cuve shown n Fg.(b) s eamned. Consdeng the paamete of the NURBS cuve s dffeent fom the of the cuve, the of a NURBS cuve as a staght lne s set to the hozontal as, and the s set to the vetcal as as shown n Fg.(c). Then, the adus of dstbuton to the s dawn. Afte ths, specfed s supemposed on the cuent dstbuton. Lnea, quadatc, cubc, quatc, quntc, and s degee algebac functon s appled as specfed adus of to the cuent dstbuton to modf the shape of the NURBS cuve.

6 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, To be moe n detal, s of the algebac functon ae calculated b ntoducng the least-squaes method usng the cuent adus of dstbuton. Then, the s specfed b the detemned algebac functon. knot poston ˆ cuent NURBS cuve (a) cuent NURBS cuve and ts plots O d specfed adus of dstbuton of cuent NURBS cuve (c) dffeence between cuent and specfed As an eample, the lnea algebac functon as a specfed specfcaton s shown n Fg.(c). The th of dstbuton of a pemetcall epesented NURBS cuve s denoted as, the specfed at the same spot s denoted as ˆ, the dffeence d s shown b Eq.() and s llustated n Fg.(c). d ( ˆ,, 2,,, = q qn- q qn-2) - () Whee =,, 2,, m -, m s the numbe of specfed, and n s the numbe of NURBS cuve segments plus, whch s the degee of the cuve. S( q,, qn-2, q,, qn-2 ) whch s the sum of the squaed dffeences fo all specfed adus of s n Eq.(2) s mnmzed b ntoducng the least-squaes method. The epesson s non-lnea. Theefoe, b Talo's theoem, Eq.(2) s lnealzed as n Eq.(3). S( q,, q, q,, q ) n-2 n-2 m- 2 å é ë ( ˆ q,, qn-2, q,, qn-2) ù û = = - shape modfed NURBS cuve knot poston (b) shape modfed NURBS cuve and ts plots O specfed adus of dstbuton of modfed NURBS cuve (d) of shape modfed NURBS cuve and specfed (same as n (c)) Fg. Concept of specfcaton and NURBS cuve shape modfcaton based on the specfed dstbuton (2) S( q + D q,, q + D q, q + D q,, q + Dq ) n-2 n-2 n-2 n-2 m- é = å ê ( q,, qn-2, q,, qn-2) + D q + = ë q ù + D q ˆ n-2 + D q + + Dq n-2 - ú qn-2 q qn-2 û Eq.(3) s mnmzed b equatng to zeo all the patal devatves of S( q + D q,, q + n - 2 Dqn-2, q, +D q, q + - D wth espect to D and D ( =, 2,, n 2 q ) n - 2 n - 2) as n Eq.(4). S Dq = ( =, 2,, n - 2) S = ( =, 2,, n - 2) Dq Usng these smultaneous lnea equatons, q 2 q (3) (4) D q and D q ( =,2,, n - 2) ae calculated. Then, q and q ae detemned. 7 A Method to Specf the Radus of Cuvatue Dstbuton b Selectng a fa NURBS Cuve usng the Smlat of Radus of Cuvatue Dstbuton In ths secton, a method to specf the adus of dstbuton b selectng a pedefned fa NURBS cuve usng the smlat of adus of dstbuton s descbed. The shape of a NURBS cuve s defned b the numbe, the locaton of ts contol ponts, and the knot sequence of the knot vecto. A cuve wth a monotone dstbuton s consdeed as a fa cuve n the aea of Compute Aded Aesthetc Desgn. As a measue of cuve faness evaluaton, adus of dstbuton s used as an altenatve chaactestc of a cuve. The s specfed b lnea, quadatc, cubc, quatc, quntc, and s degee functon, whle cuve shape s modfed accodng to the specfed dstbuton usng the shape modfcaton algothm mentoned n the pevous secton. Then, s cuves, whose ae lnea, quadatc, cubc, quatc, quntc, and s degees ae geneated based on the cuve geneated fom the sketch. These s functons ae used as the specfed. Moeove, these s cuves can be consdeed as fa, snce the s ae monotone because the appled algebac functons ndependent vaables ae monotone to the dependent vaables.

7 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, 27 3 The smlat of the cuve geneated fom the sketch to these s cuves s eamned. The smlat s evaluated b. The cuve whose smlat s the hghest among these cuves s detemned as the cuve geneated fom the sketch. Fom now on, to make descpton smple, the cuve geneated fom the sketch s called cuve-s, the shape modfed cuve accodng to the dstbuton specfed b a lnea functon s called cuve-, the shape modfed cuve accodng to the dstbuton specfed b a quadatc functon s called cuve-2, the shape modfed cuve accodng to the dstbuton specfed b a cubc functon s called cuve-3, the shape modfed cuve accodng to the dstbuton specfed b a quatc functon s called cuve-4, the shape modfed cuve accodng to the dstbuton specfed b a quntc functon s called cuve-, and the shape modfed cuve accodng to the adus of dstbuton specfed b a s degees functon s called cuve-6. Cuve-S s shown n Fg.6(a), cuve- s shown n Fg.6(b), cuve-2 s shown n Fg.6(c), cuve-3 s shown n Fg.6(d), cuve-4 s shown n Fg.6(e), cuve- s shown n Fg.6(f), and cuve-6 s shown n Fg.6(g) wth the and plots espectvel. Pont maks of these fgues ndcate knot poston of a NURBS cuve. Radus of dstbuton coespondng to Fg.6(a), (b), (c), (d), (e), (f), and (g) s shown n Fg.7(a), (b), (c), (d), (e), (f), and (g) espectvel, whle s of cuves ae escaled as. The of cuve-s and cuve-, that of cuve-s and cuve-2, that of cuve-s and cuve-3, that of cuve-s to cuve-4, that of cuve-s to cuve-, and that of cuve-s to cuve-6 ae calculated. These ae summazed n Table. Fom Table, the between cuve-s and cuve- s ecognzed as the hghest. That s, the adus of cuve-s s ecognzed as beng ve smla to that of cuve- accodng to the. Theefoe, cuve- s detemned as the cuve geneated fom the sketch. In ths wa, a fa NURBS cuve s geneated fom a hand-dawn sketch. (a) cuve-s (b) cuve- (c) cuve-2 (d) cuve-3 (e) cuve-4 (f) cuve- (g) cuve-6 (a) adus of cuve-s (b) adus of cuve- Fg.6 Cuves wth and plots (c) adus of cuve-2 (d) adus of cuve-3 (e) adus of cuve-4 (f) adus of cuve- (g) adus of cuve-6 Fg.7 Radus of dstbuton and Table Coelaton cuve- cuve-2 cuve-3 cuve-4 cuve- cuve-6 cuve-s

8 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, Eamples of NURBS Cuve Geneaton based on a Hand-dawn Sketch Usng the technques mentoned n the pevous sectons, a fa NURBS cuve geneaton pocess llustated n Fg.8 s descbed accodng to the followng steps. The hand-dawn sketch mage shown n Fg.8(a) s put nto a compute b usng a scanne set at 2 dots pe nch. A gadated backgound s shown n ths hand-dawn sketch. Theefoe, tansfomaton to a logathmc mage s necessa. The logathmc mage s shown n Fg.8(b). Then, the hand-dawn sketch s bnazed. The bna hand-dawn sketch mage s shown n Fg.8(c). The edge of the bna sketch s smoothed b applng featue based eoson and dlaton. The objectve of ths pocess s to smooth the oute edge of the bna mage. Theefoe, ntenal mage contous ae elmnated. The edge smoothed bna sketch mage s shown n Fg.8(d). Then, applng a Laplacan opeaton, the edges of the smoothed bna sketch mage ae detected. The detected edges ndcated b dots ae shown wth the sketch mage as shown n Fg.8(e). Usng the detected edges, egesson lne segments ae detemned. These lne segments ae shown n Fg.8(f). Usng the detemned lne segments, a quntc NURBS cuve s geneated. The geneated NURBS cuve wth plots s shown n Fg.8(g). Radus of dstbuton and specfed adus of b quatc functon accodng to the smlat of the dstbuton ae shown n Fg.8(h). The cuve shape modfcaton algothm s then appled to ths NURBS cuve. A shape modfed NURBS cuve s shown wth ts plots n Fg.8(). Radus of a shape modfed NURBS cuve s shown wth the specfed adus of dstbuton n Fg.8(j). It s vsuall ecognzed that the the shape modfed cuve s n accodance wth the specfed. Thus, followng these steps, a fa NURBS cuve can be geneated. (a) hand-dawn sketch (b) logthmc sketch mage (c) bna sketch mage (d) edge smoothed bna sketch mage detected edges (e) edge detecton lne segments (f) lne segment geneaton (g) geneated quntc NURBS cuve wth plots (h) dstbuton and specfed () smoothed quntc NURBS cuve (j) dstbuton and specfed adus of specfed adus of b quatc functon specfed adus of Fg.8 Illustaton fo the steps fom data nput to geneaton of shape modfed NURBS cuve

9 Poceedngs of the 7th WSEAS Int. Conf. on Sgnal Pocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 24-26, Concludng Remaks The objectve of ths stud s to shoten the poduct desgn peod b etactng a feefom cuve fom a hand-dawn sketch. Edge smoothng of the bna sketch mage b featue based eoson and dlaton, and lne segment geneaton usng the detected edges ae descbed. A feefom cuve s epessed as a quntc NURBS cuve, and NURBS cuve geneaton based on the lne segments ae descbed. Then, a method to modf a NURBS cuve shape based on the specfed dstbuton s descbed. In addton to ths, a method to specf the adus of dstbuton b selectng a pedefned fa NURBS cuve usng the smlat of adus of dstbuton s descbed. Eamples of NURBS cuve geneaton based on a hand-dawn sketch ae gven accodng to the geneaton steps. We have poposed an edge smoothng method of the bna mage b featue based eoson and dlaton. 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