Improving the Modeling of the Plantar Surface via Bezier Networks and Laser Projection

Transcription

1 ISSN 19 14X (print), ISSN 5 884X (online) INFORMATION TECHNOLOGY AN CONTROL, 014, T. 4, Nr. 4 Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion J. Apolinar Muñoz Rodríguez Centro de Investigaiones en Optia A.C. Apartado ostal 1-948, León, GTO, 7000, Méio munoza@oton.io.m ttp://d.doi.org/ /01.it Abstrat. An automati tenique to improve te plantar surae model via ezier networks is presented. Tis tenique generates te oot sole model via ezier networks based on surae points, wi are retrieved via line proetion. Te surae model is deined b means o network weigts and te ontrol points, wi are measured via line position b a ezier network. Tus, te model represents te plantar surae wit ig aura. It is beause te model passes troug b all ontrol points o te psial surae. Furtermore, te network redues operations and memor size to alulate te surae. It is beause te model is implemented wit less matematial terms tan te traditional models. Also, te alibration o vision parameters is perormed via laser line to avoid eternal alibration, wi determines vision parameters outside o te vision sstem and inreases te surae representation inaura. Additionall, te soe-last bottom is adusted to plantar surae model. Te viabilit o te proposed surae modeling is orroborated b an evaluation based on te speed, aura and memor size o te traditional surae models. Tus, te ontribution o te proposed tenique is eluidated. Kewords: Foot sole model; laser line proetion; ezier networks. 1. Introdution Nowadas, te oot sole data plas an important role in te oot are, gait arateristis and ootwear manuature [1], [17], [6]. Tus, omputational models ave been implemented to represent te plantar surae []. In tis ield, algoritms ave been implemented to onstrut te plantar surae model [], [7]. Tis model provides data to perorm te diagnoses o te gait arateristis and te bod untionalit [8]. Te standard metod tat onstruts te surae model is te non-uniorm rational -Splines (NURS) [19]. Tis metod onsumes long time and uge memor to represent te plantar surae. It is beause te model is generated b a great amount o terms [5]. Moreover, tis model does not interpolate all surae points. Also, tis metod perorms te surae measurement via eternal alibration, wi inreases 0. % te surae representation [9]. Te same riterion is orroborated b te models generated via -Splines and least squared. Tereore, te improvement o plantar surae model still represents a allenge task. To improve te plantar surae modeling, it is neessar to develop a model tat interpolates all surae points b using small memor size in sort time. Te proposed tenique onstruts te plantar surae model via ezier networks to improve te aura, speed and memor size o te traditional models. Te surae measurement is perormed b a ezier network via line proetion. Also, te vision parameters are alibrated inside o vision sstem via laser line to avoid psial measurements o eternal alibration. Tus, te surae points are measured wit ig aura to build te plantar surae model. Tis model is deined via network weigts, wi are omputed b an equation sstem to interpolate all ontrol points. Tus, te plantar surae is represented wit ig aura. Also, tis model redues operations and memor size to ompute te surae. It is beause te network is implemented wit less matematial terms tan te traditional models. Additionall, te soe-last bottom is adusted to te plantar surae model. Te viabilit o tis model is eluidated b an evaluation based on model aura, representation aura, operations number and memor size o te traditional models. Te paper desribes te surae model in Setion, te surae measurement in Setion, te alibration in Setion 4, te plantar surae modeling in Setion 5 and te evaluation in Setion 6. 59

2 J. Apolinar Muñoz Rodríguez. asi teor Te proposed ezier network onstruts te plantar surae model via ontrol points 00, 01, 0,...,, wi are sown in Fig.1 as a 44 surae segment. Te positions o te points are depited b (u 00, v 00), (u 00, v 01), (u 00, v 0),..., (u, v ), respetivel. Tus, te surae model is onstruted b te net network m 0 n S( u, v) W ( u) ( v), i0 i i 0 u 1, 0 v 1, (1) n i i ni i ( u) u (1 u), i n n!. i i!( n i)! For tis equation, W i are te weigts and i(u) (v) are te ezier untions [16]. Tus, te network given in Eq.(1) generates ea model b te net equation S(u, v)=(1-u) (1-v) W (1-u) u(1-v) W ,..,+u (1-v) W 0 0+(1-u) (1-v) () vw ,..,+ u (1-v) vw 1 1 +,.., + u v W. Tis network provides ontinuit in te internal ontrol point via ezier ontinuit [0]. Te ontinuit in te edge points is dedued based on te suraes S(u,v) and (u,v), wi are sown in Fig.. Te ontinuit C 0 is reaed wen S(1,v)=(0,v) or te interval 0 v 1. ased on Eq. (), te ontinuit C 0 is dedued b te net equation o ommon edge 0 ', ( v), W0, ( v) 0 W. () 0, Tis ontinuit is aieved b using te last points o S(u,v) as te irst points o (u,v): 0= 00, 1= 01, = 0, = 0. Te ontinuit G 1 is dedued based on te plane tangent to te surae via derivatives: S(u,v)/u and (u,v) /u. Tus, te ontinuit G 1 is desribed b te net equation S( u, v) Figure 1. Surae generated via ezier network S(u,v) based on ontrol points S( u, v) u = 10 1 = 0 = 1 0= 0 1 S( u, v) ( u, v) Figure. Connetion o te suraes S(u,v) and (u,v) via G 1 u 1 ( u, v). (4) u u 0 To aieve tis ontinuit, a ezier untion (u)(v)f i, is added in Eq.() beore te edge points. Tus, a network o it degree is obtained and its derivatives are S(u,v)/u=5-5F and (u,v)/u =5F To satis Eq.(4), te values 0 = F = F 0 = are establised. Tis means tat F, 0= and F 0 are equall spaed in straigt line and S(u,v)/u = (u,v)/u. ased on tese ontinuit parameters, te ollowing network is obtained S(u,v)=(1+4u)(1-u) 4 [(1+4v)(1-v) (1-v) vw (1-v) v W 0 0+(5-4v)v 4 0] + 10(1-u) u [(1+4v)(1-v) 4 W (1-v) v W (1-v) v W 1 1+ (5-4v)v 4 W 1 1]+ 10(1-u) u [(1+4v)(1-v) 4 W (1-v) v W (1-v) v W + (5-4v)v 4 W ]+ (5-4u)u 4 [(1+4v)(1-v) (1-v) v W (1-v) v W +(5-4v)v 4 ]. (5) Tus, te network o Eq.(5) preserves ontinuit G 1, wi is enoug to onnet two suraes in a good manner [1]. Te interpolation is obtained via weigts W i,. Te weigts W 00=W 0= W 0= W =1 are initiall establised. Ten, te point S(u, v)= 10 and its position (u 10=1/, v 10=0) are replaed in Eq.(5) to obtain Eq.(6). Also, S(u,v)= 0 and (u,v)=(u 0=/, v 0=0) are replaed in Eq.(5) to obtain Eq.(7): 10= (1+4u 10)(1-u 10) (1-u 10) u 10 W (1-u 10) u 10 W 0 0+(5-4u 10)u 104 0, (6) 0= (1+4u 0)(1-u 0) (1-u 0) u 0 W (1-u 0) u 0 W 0 0+(5-4u 0)u (7) Solving tese equations, W 10 and W 0 are obtained. In te same manner, W 01 and W 0 are omputed b substituting te points {(u,v)=(u 01=0, v 01=1/), S(u,v) = 01} and {(u,v)=(u 0=0, v 0=/) and S(u, v)= 0} in Eq.(5). Also, tis proedure alulates W 1,, W,, W 1 and W. Ten, te points {(u 11=1/, v 11= 1/),

3 Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion S(u,v)= 11}, {(u 1=/, v 1=1/), S(u,v)= 1}, {(u 1= 1/, v 1=/), S(u,v)= 1} and {(u =/, v = /), S(u, v) = } are substituted in Eq.(5) to obtain te net equation sstem 11 R11 ( u11) ( v11) 1 R1 ( u1) ( v1) 1 R1 ( u1) ( v1) R ( u) ( v) ( u11) ( v11) ( u11) ( v11) ( u11) ( v11) W1111 ( u) ( v1) ( u1) ( v1) ( u1) ( v1) W11 ( u1) ( v1) ( u1) ( v1) ( u1) ( v1) W11 ( u) ( v) ( u1) ( v) ( u) ( v) W. (8) For tis equation sstem, R 11 is obtained b te sum o terms o te rigt and o Eq.(5) eept te terms [10(1-v 11) v 11 W (1-v 11) v 11 W 1 1] + [10(1-v 11) v 11 W (1-v 11) v 11 W,]. In te same manner, R 1, R 1 and R are obtained b substituting (u=u 1, v=v 1), (u=u 1, v=v 1), and (u=u, v=v ), in Eq.(5), respetivel. solving Eq.(8), W 11, W 1, W 1, W are determined. Tus, te model as been ompleted b solving a 44 matri and our matries. Te model aura is eluidated b omputing surae points via position (u, v). For instane, te positions (u 00, v 00), (u 10, v 10),,..., (u, v ) are substituted in te network (5) and te results are 00, 10,...,, respetivel. Tus, all ontrol points ave been interpolated. Tis means tat te network passes troug b all ontrol points. Tus, te ezier network (5) provides interpolation and preserves ontinuit. Te model aura is alulated b means o relative error, wi is omputed b rms(%)=rms 100/ m. Here, rms is te root mean squared Eq.(17) and m is te mean o te surae i. Te surae representation aura is given b te sum o model error and measurement error. Tus, te surae sown in Fig. is represented wit a relative error o %. Te operations to ompute te surae S(u,v) are dedued via network (5). For instane, te term (1+4v)(1-v) 4 W is alulated b one sum, one rest and seven multipliations. Tus, te siteen terms o Eq.(5) are determined b 00 operations: 160 multipliations, 0 sums and 0 rests. Tis model is deined b te terms W 00 00, W 01 01,..,W, and te untions i(u) (v) are te same or all segments. Tus, te network uses siteen memor loalities to save ea surae model. Tpiall, te viabilit o a surae model is dedued via aura, speed and memor size [6], [4], [0]. To eluidate te viabilit o te ezier network, it is eamined based on te standard model NURS. Tis metod is desribed b means o te ollowing epression N0 ( u) N0 ( v) W0000 N0 ( u) N1 ( v) W0101. (9)... N ( u) N ( v) W S( u, v) N ( u) ( v) W u) N ( v) W 0 N0 00 N0 (... N ( u) N ( v) W For tis equation, W i are te weigts and N i p (u) N q (v) are te -spline basis untions [4]. Te weigts W i are alulated b means o te knot vetor, wi adds points to move te surae toward te ontrol points [8]. For a surae wit degree (p, q), tere are at most (p+1)(q+1) non-zero untions to be omputed 1 01 [15]. Te operations to ompute a surae point b NURS are dedued via Eq.(9). Ea -spline produes an epression su as N i (u)=(au +bu +u+d), wi is alulated b sums and 8 multipliations. Tus, te siteen terms N i (u)n (v)w i i are determined b 415 operations: 111 sums and 04 multipliations. Tis result orresponds onl to te numerator o Eq.(9), wi is bigger tan te 00 operations o te ezier network. Te NURS aura indiates tat te model does not passes troug b all ontrol points [5]. Te aura reported b tis model is a relative rms around 0.000% [0]. Te NURS saves te model b means o siteen memor loalities or te terms W i i, siteen loalities or W i, siteen loalities or te values u tat produe N i p (u) and siteen loalities or te values v tat produe N q (v). Te above results indiate tat te ezier network improves te traditional surae models. For instane, te network improves te model aura. It is beause te network passes troug b all ontrol and preserves ontinuit. Tus, te surae representation is aieved wit a ig aura. It is an improvement respet to NURS, wi do not interpolate all surae points. Also, te ezier network generates te model in ast orm b solving a 44 matri and our matries. In te NURS metod, te model is generated via knot vetor, wi adds points to move te surae toward te ontrol points. Furtermore, te network omputes ea surae point via 00 operations. Also, te ezier network saves ea segment model via siteen memor loalities wereas NURS uses sit our memor loalities. Tus, te proposed metod improves te aura, speed and memor size o te traditional surae model. Te optial setup to perorm te surae measurement is desribed in Setion.. Optial setup or surae measurement Te surae representation inludes two stages: surae measurement and model implementation. Tereore, te aura o te plantar surae representation is obtained b te sum o measurement error and model error. Te surae points or te surae model are measured via line sanning. Tis proedure avoids eternal measurements to te vision sstem. Tus, te measurements (,, z) lead to obtain a ig aura or te surae representation. It is beause te measurement error is inluded in te aura o te plantar surae representation. Te optial setup to measure te plantar surae is sown in Fig.. Tis arrangement inludes a laser line, a CC amera, a glass platorm, an eletromeanial devie and a omputer. In tis setup, te -ais and -ais are loated in te glass platorm, wi is perpiular to z- ais. Te geometr o line proetion in -ais is sown in Fig. 4, were is te oal lengt, is te image enter, L is te distane between te laser line and te optial ais, is te distane rom te lens to te glass platorm and te surae is indiated b i. 61

4 J. Apolinar Muñoz Rodríguez Te sanning o plantar surae is perormed b moving te laser line in -ais via eletromeanial devie, wi provides te oordinate. Te oordinate is dedued via line proetion in -ais, wi is sown in Fig. 4, were te oordinate is alulated b te equation =(+ i)/( - ). In tis epression is te piel size. Te vision parameters,, L,,, are alibrated in Setion 4. Te surae z is determined via Fig.4 b i=[l/( i- )]-. Tis equation indiates tat te surae i is diretl proportional to te line siting s i, wi is alulated b Figure. Optial setup to perorm te sanning o oot sole i Glass platorm i z-ais z-ais Laser diode Laser line CC Foot sole L Optial ais CC Foot sole t Lens Lens Optial ais 0 i s i 0 -ais Glass -ais Figure 4. Geometr o te line proetion in -ais. Geometr o te line proetion in -ais s i = 0 i. (10) Te line position i is determined based on maimum intensit via ezier urves [11]. To arr it out, a ezier urve is built via piel position i and piel intensit I i b te net two equations (u) =(1- u) n 0 + n (1- u) n-1 u n (1- u) n- u +. + u n n, 0u1, (11) I(u) =(1- u) n I 0 + n (1- u) n-1 ui n (1- u) n- u I +. + u n I n, 0 u 1. (1) substituting te piel position i in Eq.(11) and te piel intensit I i in Eq.(1), a onave urve is obtained. Tus, te line position is omputed based on irst derivative I (u)=0 via bisetion metod [9]. Ten, value u were I (u)=0 is replaed in Eq.(11) to obtain te position (u). And te position i=(u) is replaed in Eq.(10) to ompute te line siting s i. Te surae i is omputed b a ezier network [10], wi is desribed b te net epression n i0 m H ( u, v) w ( u) ( v), (1) 0 0 u 1, 0v 1. i i Tis equation is applied to ompute te ontrol points, wi are used to onstrut te surae model. Tereore, te notation o Eq.(1) is dierent to te notation o Eq.(1). In tis ase, i(u) and (v) are related wit s i and, respetivel. Tus, te line siting is onverted to a value u b means o te epression u= a 0 +a 1s i. (14) For tis equation, te values a 0 and a 1 are obtained b solving te equations a 0+a 1s 00=0 and a 0+a 1s nm=1. Te oordinate is onverted to a value v b v= b 0 + b 1. (15) For tis equation, te values b 0 and b 1 are obtained b solving te equations b 0+b 1 0=0 and b 0+b 1 n=1. Ten, te values i are substituted in Eq.(1) to obtain te net network H(u,v)=w (u) 0(v)+ w (u) 1(v)+,...,+w 1m 1m 1(u) m(v)+,...,+w nm nm n(u) m(v). (16) For tis equation, te weigts w i are omputed b substituting te values (u, v) o ea i in Eq.(16) to obtain an equation sstem []. Here, te proedure to obtain Eq.(6) to Eq.(8) is applied. solving te equation sstem, te weigts w i are obtained to omplete te network H(u, v). Tis network is applied to ompute te surae rom te line sown in Fig. 5. In tis proedure, te line siting s i is deteted in ea oordinate. Ten, te values (u, v) o te siting are replaed in Eq.(16) to alulate te i 6

5 Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion surae sown in Fig. 5, were te smbol "" represents te data measured b a oordinate measure maine (CMM). Te measurement aura provided b te network (16) is determined via root mean squared. Tis rms value [] is omputed b te net epression rms 1 n n i1 ( o i ) i, (17) For tis equation, o i is te surae data z i measured b a oordinate measure maine (CMM), i is te surae data z i =H(u, v) alulated b te network (16) and n is te number o points z i. Te error o te slider position is mm, wi is added to Eq.(17) to obtain a rms= mm or te proile sown in Fig. 5. Tus, te network provides te measurement o plantar surae as z i = H(u, v). Te alibration to determine te oordinate is desribed in Setion Calibration o vision parameters Te alibration o vision parameters is perormed inside o te vision sstem to omplete te oordinates (,, z) to onstrut te surae model. Tus, errors o eternal measurements to te vision sstem are avoided. Te traditional models perorm an eternal alibration to determine te vision parameters. Te eternal alibration is perormed outside o te vision sstem via measurement o eternal reerenes [1]. Tis eternal alibration transorms te world oordinates Surae dept (,) in mm Surae widt in mm (-ais) Figure 5. Laser line proeted onto te plantar surae. Surae dept omputed b te network orm te laser line w=( w, w, z w) to te image oordinates (X u, Y u) b means o te epression = R w+t, were =(,, z ) are te amera oordinates, R is te rotation matri and t is te translation vetor. Tis alibration is suitable or a stati setup. ut, te alibration is not aieved wen te setup is modiied in te vision proess. It is due to te lak o reerenes ( w, w, z w) during te vision task. In te proposed setup, te vision parameters are modiied wen te vision sstem is moved in z-ais. Tereore, te alibration is perormed via mobile setup, were te oordinate is provided b te eletromeanial devie and te oordinate z is given b Eq.(16). Te oordinate = (+ i)/( - ) is determined via alibration o te parameters,, and. ased on Fig.6, te radial distortion is dedued via line position i= (X i+ i),were X i is te distorted position and i is te distortion. Tus, te undistorted siting is given b te equation s i=(x 0+ 0)- (X i + i) and te distorted siting is given b te equation S i=(x 0-X i). Te distortion is dedued b te equation i=(x 0-X i) + 0 -s i = S i-s i+ 0. In tis ase, te irst line siting is deined witout distortion, tereore, 0=0, 1=0 and s 1=S 1. Tereore, te undistorted siting s i is represented based on te irst distorted siting S 1 b means o te epression s i=i*s 1. Tus, te distortion in -ais is alulated b i=i*s 1-(X 0 X i) or i=1,,,,n and =1,,,,m. Te same proedure is perormed to determine te distortion based on Fig. 6 via equations t =( 0- ) and T =(Y 0-Y ). Tus, te distortion in -ais is omputed via equation = *T 1-(Y 0 Y ). Te image plane o te amera is plaed parallel to te glass platorm sown in Fig.. To orroborate tat te image plane is parallel to te glass window in -ais, te proetion k i o Fig. 6 is alulated. Tis proetion is omputed b te net epression ( i ) i [( 0 ) si] i ki. (18) For tis equation,,, are onstants and i is omputed via i. Tus, te epression k i given b Eq.(18) is a untion o irst order and te derivative dk/ds is a onstant. To orroborate tat te image plane is parallel to te glass platorm in -ais, te proetion q o Fig. 6 is alulated. Tis proetion is omputed based on t =( 0- )-( - ) b means o te net epression ( ) i [( 0 ) t ] i q. (19) For tis equation, 0 are onstants and Eq.(19) is a untion o irst order. Tereore, te derivative dq/dt is a onstant. Tus, te image plane is parallel to glass platorm wen te derivatives dk/ds and dq/dt are a onstant. Te image entre is dedued via Fig. 6 b te equation L=( i- )(+ i), wi generates te sequene ( 1 )( + 1)=( )(+ ) = ( )(+ ) and te epression [ ( - )- 1( 1-6

6 J. Apolinar Muñoz Rodríguez )]( 1- ) =[ ( - )- 1( 1- )]( 1 - ). Tus, te image entre is deined b ( 1 1 ( ( 1 ( )( 1 ) )( ). (0) )( 1 )( 1 1 ) For tis equation, i is alulated b te network via s i. Ten, te distane is determined b ) ( ) 1 ( 1 ). (1) ( ) 1 Te oal lengt is dedued via Fig. 7 b moving te amera lens rom to 1. Tus, te line is moved rom i to α i and te distane 1 is alulated b n 1 0 k 1 k k n Laser line Reerene plane - ais Optial ais 1= i(α i- )/(α i-α 0). Ten, L=( 0- ) and 1L = 1(α 0- ) are dedued rom Fig. 6 and Fig. 7, respetivel. From tese terms and te equation + = 1+ 1, te oal lengt is omputed b 1 ( 1 0 ( 1 ) ) / ( 0, () ) 1= () Tus, te oal lengt and its modiiation 1 are obtained. From Fig. 6, te piel size is dedued via distanes a =( 0- ) +, b =[( 0- )/] + and = [( 0- )/ 0 + ( 0- )] + (+). From tese distanes and te relationsip =a+b, te net equation is obtained ( 0 ( ) 0 ) ( 0 ) 0 (4) Tis equation is solved b te bisetion metod via interval 0η 1 to obtain η. Ten, te oordinate is dedued based on Fig. 6 b solving te equations: t 1=( 0 )-( 1 ) and t =( 0 )-( ). From tis alibration, te oordinate =(+ i)/( - ) is omputed to omplete te oordinates (,, z) o te plantar surae. Te surae modeling is desribed in Setion 5. n n 1 0 q 1 L q q n n 1 0 Reerene plane - ais Optial ais Laser line L Reerene plane n - ais Optial ais Figure 7. Geometr o te optial setup at 1 in -ais 1 0 n 0 1 Laser line Figure 6. Geometr o te optial setup in -ais. Geometr o te optial setup in -ais 5. lantar surae modeling Te plantar surae model is onstruted via oordinates (,, z). Te slider maine provides te oordinate were te line is aptured. From te line siting values (u,v), te network in Eq.(16) alulates te surae z=h(u, v). Te oordinate =(+ i)/ ( - ) is determined based on line position. From all images, te network H(u,v) omputes te plantar surae sown in Fig. 8. Te aura o tis measurement is a relative rms(%)= 0.48 %. Tis 64

7 Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion values is alulated b te epression rms(%)=rms 100/ m, were te rms= mm and n=400. Ten, te surae is divided in 44 segments to onstrut te model. Tis proedure is perormed via points sown in Fig. 8, wi represent te eel area. Tus, te surae positions (, ) o ea segment are onverted to values (u, v) b te ollowing equations u=(-p)/t, p = int(/t)*t, 0 u 1, (5) v=(-q)/t, q = int(/t)*t, 0 v 1. (6) For tese equations, T= is te period o ea segment and te values (p, q) are te period number. Ten, te ontrol points i = i are replaed in Eq.(5) to build te model S(u,v)=(1+4u)(1-u) 4 (1+4v)(1-v) 4 W ,...,+10(1-u) u (1+4v)(1-v) 4 W ,...,+(5-4u)u 4 (5-4v)v 4 W. (7) For tis equation, te weigts W 01, W 0, W 10, W 0, W 1, W, W 1, W are omputed based on te proedure desribed in Eqs. (6) and (7). Tese weigts are put in Eq.(7) to obtain Eq.(8), wi is solved to obtain W 11, W 1, W 1, W. Tus, te model o ea segment is generated via siteen terms W i i. Tereore, plantar surae model is a matri, wi ontains te terms W i i o all surae segments. Te steps to onstrut te surae model are indiated in te low art sown in Fig. 9. In tis low art, te values L 1 and L are te lengt and te widt o te measured surae data, respetivel. Te untion i() provides te integer part o a division. Te plantar surae model is deined b te matri M(i, ), wi stores all terms W i* i. Te terms G i in te low art orrespond to G 01=[ 01-(1+4v 01) (1- v 01) 4 00-(5-4v 01)v ]/10(1-v 01) v 01, G 10=[ 10- (1+4u 10)(1-u 10) (5-4u 10)u 104 0] / 10(1-u 10) u 10,...,G =[ - (1+4v )(1-v ) (5-4v )v 4 ]/ 10(1-v ) u. Te surae model determines te surae z= S(u, v) via position (, ). Also, te network provides equations or te surae position (, ) via net epressions (u, v) = W (u) 0(v) + W (u) 1(v)+,...,+W 0 0 0(u) (v)+,...,+w (u) (v), (8) (u, v) = W (u) 0(v) +W (u) 1(v) +,...,+W 0 0 0(u) (v)+,,+w (u) (v). (9) For tese equations, te proedure to determine te weigts W i is applied to ompute te weigts W i and W i. Tus, Eq.(8) and Eq.(9) alulate te position =(u,v) and =(u,v). Start =0; Measured surae data Read ( i,); i=0; ( i,); ( i, ); Compute p= i( ( i, ) / T) * T; q=i( ( i, )/T)*T; u( i, )=[ ( i,)- p]/ T; v( i,)=[ ( i,)- q]/t; i= i+1; =+1; Yes i > No > No Yes Compute ui= u( i, ); vi= v( i, ); W00= W0= W0= W=1; W01= [ G0v 01- G01v 0 ]/[(1- v 0) v 01- (1- v01 v ) 0 ] 01 ; W0= [ G 01-(1- v 01) W 01 01]/ v 010 ; W10= [ G0u10 - G10u 0 ]/[(1- u 0 u ) 10 -(1- u 10 u ) 0 ] 10 ; : : W = [ G -(1- v ) W ]/ v ; Solve te equation sstem 11 R11 ( u11 ) ( v11 ) ( u11 ) ( v11 ) ( u11 ) ( v11 ) R ( u1) ( v1) ( u) ( v1) ( u1) ( v1) R1 ( u1 ) ( v1 ) ( u1 ) ( v1 ) ( u1 ) ( v1 ) R ( u ) ( v) ( u) ( v ) ( u1) ( v ) =0; ( u11 ) ( v11 ) W11 11 ( u1) ( v1) W 1 1 ( u1 ) ( v1 ) W 11 ( u) ( v) W Figure 8. lantar surae retrieved via network H (u, v). Surae points to ollet 44 segments No Yes i > i=0; M( i+p*t, +q*t)= W i * i i= i+1; = +1 End No > Yes No i+ p*t>l1 && + q*t>l Figure 9. Flow art to onstrut te surae model Yes 65

8 J. Apolinar Muñoz Rodríguez To represent te surae (,, z), te networks Eq.(7), Eq.(8) and Eq.(9) are evaluated or ea segment in te intervals 0u1and 0v 1. In te surae edge, te period T varies in -ais and -ais aording to te segment points. Tus, plantar surae sown in Fig. 10 is omputed b z=s(u, v), =(u, v) and = (u, v). Te aura o tis surae is determined via rms Eq.(17), were o i is provided b H(u, v), i is te surae omputed b S(u, v) and n= 400. Tus, an rms= mm is obtained and te relative error is rms(%)=0.00% based on m=.08 mm. Te proedure to ompute obet surae via network (5) is desribed b te algoritm sown in Fig. 11. Te algoritm to onstrut te surae model and te algoritm to ompute te obet surae are implemented in C programming language. Figure 10. lantar surae generated b te network S(u, v) Te surae model provides te data to adust te soe-last bottom sown to te plantar surae. To arr it out, te soe-last bottom sown in Fig. 1 is sanned via setup sown in Fig. 4. In tis proedure, te line siting s i is deteted via Eq.(10), Eq.(11) and Eq.(1). Ten, te siting is onverted to values (u, v), wi are replaed in te network Eq.(16) to ompute te surae z=h(u, v). Te oordinate is provided b te slider devie, wereas te oordinate is omputed via equation =(+ i)/( - ). Ten, te model o te soe-last bottom is onstruted via Eq.(7), Eq.(8), Eq.(9). Tese equations are evaluated in te intervals 0u1 and 0v1to obtain ea surae segment o soe-last bottom sown in Fig. 1. Ten, te morpologial parameters are dedued rom te diagram sown in Fig. 1 [11]. Te oot lengt is te distane between te eel point A and te maimum point in -ais. Te enter line is te line passing te enter eel A and te seond metatarsal C. Te oot widt is te distane between te points E and in -ais. Te mid oot widt is te breadt at ½ o te oot lengt in -ais. Te eel widt is te maimum widt in te eel area. Te positions o irst and it metatarsal eads are pointed b and J, respetivel. Tus, te surae is moved toward te plantar surae in -ais and -ais. To arr it out, te maimum plantar surae d outside o soe-last bottom is deteted via equation d = s-, were is te ontour position o te plantar surae in -ais and s is te ontour position o te soe-last bottom. Ten, te ontour o te soe-last bottom is magniied based on te sale ator ε=( s+d )/ s. Tus, te surae position (, ) is re-omputed via equations =ε and =ε. Ten, te position o te morpologial parameters o te soelast bottom is moved toward te morpologial parameters o te oot sole. Tese two steps are sown in Fig. 1 b te green das line. Ten, te ontour is smooted to obtain ontinuit G 1 via derivatives o te neigbor points. Te ontour is smooted in -ais b te epression i=( i-1,+ i+1, )/, were te indees (i, ) orrespond to te row and olumn number, respettivel. Te smooted ontour is sown in Fig. 1 b te blak ontinuous line. ased on tese steps, te eine T=, L, L, r, s, t, E[1]=E[]=A[1]=A[]=10; 1 or p=0 to L1 or q=0 to L u=0; or k=0 to r v =0; or t=0 to s Suv=(k+r*p,t+s*q)=0; E[0]=1+4* v; E[]=5-4* v; A[0]=1+4* u; A[]=5-4* u; or i=0 to SUM=0; or =0 to SUM=SUM+E[ ]*pow(1- v,4-i( /)* pow( v,* +i(* /))* W( i+q*t, + p*t)* ( i+q*t, +q*t); SUM=A[ i]*pow(1- u,4-i-i( i/))*pow( u,* i+i(* /))*SUM; i Suv(k+r*p,t+s*q)=Suv(k+r*p,t+s*q)+SUM; v= v+1/s; u= u+1/r; Figure 11. Algoritm to ompute te obet surae S(u,v) Figure 1. Soe-last bottom to be adusted to te plantar surae. Surae o te soe-last bottom represented b te network S(u,v) 66

9 -ai -ai Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion lantar surae lengt ( ) E J d Foot widt C Mid oot widt G F Foot lengt lantar surae lengt ( ) E J G Foot widt C F Foot lengt z ai mm Soe-last eigt ( - ) 6 0 d z Ar eig Soe-last lengt ( - ais) mm I Heel widt H 6 I H 0 A A lantar surae widt ( -ai ) lantar surae widt ( -ai ) () Figure 1. Morpologial parameters o te oot sole. Mating o te soe-last bottom wit te morpologial parameters. () Adusted soe-last bottom in -ais and -ais soe-last bottom is adusted in -ais and -ais. In tis proedure, te position (, ) is magniied b te equations =ε and =ε. Ten, te ontour is mated and smooted wit te morpologial parameters. Te adustment o te soe-last bottom in -ais and -ais is sown in Fig. 1(). Te adustment in z-ais is perormed based on te diagram sown in Fig. 14, were te surae d z outside o te ontour is determined b d z=z -z s. For tis equation z is te plantar surae position and z s is te position o te soe-last bottom. Ten, te ontour o te soe-last bottom is magniied in z-ais via sale ator ζ=(z s+d z)/z s. Tus, te surae is re-omputed b i= iζ. Te result o tis proess is sown in Fig. 14 b te green das line. Ten, te ar eigt is replaed in te surae o te soe-last bottom, wi is smooted b means o te equation i=( i+ i+1,+ i,+1+ i+1,+1)/4. Tis adustment is sown b te ontinuous blak line in Fig. 14. Tus, te adustment in z-ais is perormed b deteting d z to magni te surae via epression i= iζ. Ten, te surae o te soe-last bottom is mat- () Figure 14. iagram to adust te plantar surae in z-ais. Adusted surae in z-ais. () Adusted soe-last bottom in -ais, -ais and z-ais ed wit ar eigt. Tus, te soe-last bottom is smooted based on neigbor points and te result is sown in Fig. 14. Tus, te adustment o te soelast bottom as been ompleted and it is sown in Fig. 14(). Te evaluation o te plantar surae model is desribed in Setion Evaluation o te plantar surae model Te plantar surae model as been onstruted via ezier networks. Tis model interpolates all ontrol points and preserves ontinuit. Moreover, te alibration does not add errors to te surae representation. It is beause te vision parameters are determined based on te laser line. Tus, surae representation error is minor to 1%. Te surae representation unertaint u is dedued b te measurement unertaint u and te model unertaint u M. ased on te propagation law, te measurement unertaint is dedued via line position (u, v) and te oal lengt 1 b te ollowing epression 67

10 J. Apolinar Muñoz Rodríguez u = u u u + u v v + u 1, (0) were / u, / v, / are te sensitivit oeiients. Te unertaint u v=( v/ ) (u ) is dedued via Eq.(15) and u =0.570 piels is omputed b te standard deviation o siteen measurements o te laser line in -ais. Tus, te sensitivit oeiient is v/ =b 1=0.8 1 /piel and te unertaint is u v= Te unertaint u u=( u/ s) (u s) is dedued via Eq.(14), were u s = piels is te standard deviation o siteen measurements o te line siting in -ais. Tus, te sensitivit oeiient is u/ s= /piel and u u= Te unertaint u 1=( 1/ ) (u ) is determined via Eq.(), were (u ) =( / i) (u i) and u i=0.8 piels is te standard deviation o siteen measurements i. Te sensitivit oeiients are / i= /piel, 1/ = /piel. Tus, te unertainties are u =0.04 and u 1= Ten, te sensitivit oeiients / u= 0.55mm, / v=0.594 mm and / 1 = 0.54mm are omputed based on te network Eq.(16). Tese values are substituted in Eq.(0) to alulate te measurement unertaint u = mm. Ten, te model unertaint u M is determined via surae position (u, v) b te ollowing epression u = S M u + S u u. (1) v u v Te unertaint u v= ( v/ ) (u ) is dedued via Eq.(6), were u =0.570 piels is te standard deviation o siteen measurements o te laser line in -ais. Tus, te sensitivit oeiient is v/ = b 1= /piel and te unertaint is u v= Te unertaint u u is dedued b moving te laser line via slider devie rom a reerene 0. ased on te displaement s=( 0), te unertaint is deined b u u=( s/ ) (u ), were u =0.868 piels is te standard deviation o siteen displaements. Tus, sensitivit oeiient is s/ =0.106 mm/piel and u u= mm. Te sensitivit oeiients S/ u= 0.41mm and S/ v= mm are alulated via network Eq.(7). Tese values are substituted in Eq.(1) to ompute te model unertaint u M= mm. Ten, te unertaint u is determined b u=(u +u M )1/ [18]. Tus, te unertaint o te surae representation is u=0.064 mm. Te proposed model preserves bot interpolation and ontinuit. Tus, a ontribution is aieved to improve aura, speed and memor size o te plantar surae representation. Tpiall, -splines and NURS do not interpolate to preserve ontinuit. Moreover, te interpolation provided b tese metods introdues disontinuities near o te interpolated points [5], [14]. Te interpolation and ontinuit as a great inluene in te surae representation aura. For instane, te network interpolates all ontrol points b solving a small equation sstem. Tus, a ig aura or surae representation is aieved. Also, te model improves te speed to ompute ea surae point via 00 operations. Tis number o operations is minor tan te number o operations perormed b te traditional metods. Moreover, te metod saves te terms W i i o ea surae segment b means o via siteen memor loalities. Te ontribution o te proposed surae modeling is eluidated b an evaluation based on model aura, representation aura, number o operations, memor size and onsumed time. Te evaluated metods are Least squared, -Splines and NURS, wi are indiated in te irst olumn o Table 1. Tese metods onstrut te plantar surae model. Te aura o ea model is determined via relative rms and it is sown in te seond olumn in millimeters and in perentage. Te model aura o te traditional metods is a relative rms over 0.01%. Tis aura is reported b te traditional metods in te reerenes [1], [4], [7]. Te surae representation aura is alulated b te sum o te model error and te measurement error. Te representation aura o te traditional metods is a relative rms over 0.86%, wi is sown in te tird olumn. To represent te plantar surae sown in Fig.10, 7840 points are omputed. To alulate tese points, te ezier networks perorm (784000)= operations and -splines perorm ( )= operations. Te ourt olumn sows te number o operations to represent te plantar surae. Te ezier network uses 160 btes to save ea segment model via siteen loalities. Te plantar surae model is omposed b 870 segments, wi lead to use 1900 memor btes. Te -splines use 480 btes to save ea segment model b means o ort-eigt memor loalities or te terms N i p (u), N q (v) and W i i. Tus, te -splines onsume memor btes to save te plantar surae model. Te memor size used to save te plantar surae model is sown in te it olumn. Te time taken to represent te plantar surae inludes surae measurement, model implementation and surae omputation. Te time is sown in te sit olumn. Te results presented in Table 1 indiate tat te ezier network improves te model aura and surae representation aura. It is beause te model interpolates all surae points and preserves ontinuit. Also, te speed and memor size ave been improved via ezier network. Tese results provide a ontribution o te proposed metod in te plantar surae modeling. For instane, te ezier networks represent te plantar surae wit a relative rms=0.484% and redue te number o operations and memor size. Tus, tat te model o te plantar surae o te traditional metods as been improved. Te improvement is orroborated b te results presented in Table 1. Te omputer used to perorm te surae modeling is a C o 1.8 GHz. In tis omputer, te surae modeling is implemented in C programming language. Te slider moves te vision sstem at 4 steps per 68

11 Improving te Modeling o te lantar Surae via ezier Networks and Laser roetion Table 1. Evaluation o te plantar surae model. Metod Model aura. Relative rms % Aura o te plantar surae representation. Relative rms % Operations to represent te plantar surae. Memor size to save te plantar surae model. tes Consumed time to represent te plantar surae. Seonds Least squared 0.5mm (1.46 %) 1.15 mm (5.09%) 0.19 mm (0.86%) 0.18mm (0.97%) mm ( 0.464%) s -Splines 0.007mm (0.01%) s NURS 0.009mm (0.04%) mm 0.00% s ezier Networks s seond. Te rame rate o te amera is 4 ps. Ea image o laser line is aptured via language C based on XCLI sotware librar o EIX. Ea laser line image is proessed b te network in se. Te step resolution o te slider maine is 0.04 mm. Tis maine as a position error o mm. Tis error is added to te network H(u,v), wi provides te surae measurement. Tus, a ig aura or surae representation is aieved via ezier networks. Tus, te plantar surae modeling is perormed in good manner. 7. Conlusions A tenique to onstrut te plantar surae model via ezier networks as been presented. Te proposed tenique provides a valuable tool or te eamination o te plantar surae via omputational models. Tis tenique avoids measurements on te setup, as is ommon in te optial metods. Te vision parameters are obtained automatiall b omputational proess to improve representation aura. It is beause te errors o eternal measurement to te vision sstem are not added to te surae representation. Also, te proposed model redues te number o operations and memor size to represent te plantar surae. Tus, omputational-optial setup onstruts te model o te plantar surae, wi provides morpologial parameters to perorm te adustment o te soelast bottom. Tereore, te modeling o te plantar surae is arried out in good manner. Aknowledgments Te autor would like to tank te inanial support b CONACYT Meio. Reerenes [1] A. esatoire, A. Tévenon,. Moretto. aropodometri inormation return devie or oot unloading. Medial Engineering & sis, 009, Vol. 1, [] A. Agić. Foot struture desriptors, Ergonomia IJE&HF, 006, Vol. 8, [] A. S. Rodrigo, R. S. Goonetilleke, C.. Witana. Model based oot sape lassiiation using oot outlines. Computer-Aided esign, 01, Vol. 44, [4] A. Krisnamurt, R. Kardekar, S. MMains. Optimized GU evaluation o arbitrar degree NURS urves and suraes. Computer-Aided esign, 009, Vol. 41, [5] C. G. rovatidis, S. K. Isidorou. Solution o one-dimensional perboli problems using ubi -Splines olloation. International Journal o Computer Siene and Appliation, 01, Vol. 1, [6] C. M. Cuang, H. T. Yua. A new approa to z-level ontour maining o triangulated surae models using illet mills. Computer-Aided esign, 005, Vol. 7, [7]. Cotoros, M. aritz, A. Staniu. Coneptual analsis o orrespondene between plantar pressure and orretive Insoles. World Aadem o Siene, Engineering and Tenolog, 011, Vol. 59, [8] G. V. Ravi Kumar,. Srinivasan, K. G. Sastr,. G. rakas. Geometr based triangulation o multiple trimmed NURS suraes. Computer- Aided esign, 001, Vol., [9] H. Frederik, G. J. Lieberman. Introdution to operations resear. MGraw-Hill, USA, 198, p [10] J. A. Muñoz Rodriguez, F. Caves. Foot sole sanning b laser metrolog and omputer algoritms. Laser Sanner Tenolog, Inte, 01, p [11] J. A. Muñoz-Rodriguez. Computer vision o te oot sole based on laser metrolog and algoritms o artiiial intelligene. Optial Engineering, 009, Vol. 48, [1] J. L. Vilaa, J. C. Fonsea, A. M. ino. Calibration proedure or measurement sstem using two ameras and a laser line. Optis and laser Tenolog, 009, Vol. 41, [1] J. Wang, H. Zang, G. Lu, Z. Liu. Rapid parametri design metods or soe-last ustomization. International Journal o Advaned Manuaturing Tenolog, 011, Vol. 54, [14] L. J. Ferrer-Arnau, R. Reig-olaño,. Marti-uig, A. Manabaas, V. arisi-aradad. Eiient ubi spline interpolation implemented wit FIR ilters. International Journal o Computer Inormation Sstems and Industrial Management Appliation, 01, Vol. 5,

12 J. Apolinar Muñoz Rodríguez [15] L. iegl, W. Tiller. Te NURS ook, Springer. nd Ed., U.S.A. 1997, p [16] M. E. Mortenson. Geometri Modeling. Wille, nd Ed.; U.S.A. 1997, p [17] M. Inatouski, A. Sviridenok, V. Laskovski and. Krupiz. iomeanial analsis o antropometri and untional zones on uman plantar at walking. Ata meania et automatia, 008, Vol., 19-. [18] M. Mamud,. Joanni, M. Ro, A. Iseil, J. F. Fontaine. part inspetion pat planning o a laser sanner wit ontrol on te unertaint. Computer-Aided esign, 011, Vol. 4, [19] M. Moimaru, M. Koui, M. oi. Analsis o - uman oot orms using te Free Form eormation metod and its appliation in grading soe lasts. Ergonomis, 000, Vol. 4, [0] N. E. Leal, O. Ortega Lobo, J. W. ran. Improving NURS Surae Sarp Feature Representation. International Journal o Computational Intelligene Resear, 007, Vol., No., [1] O. Czarn, G. Husmans. ezier suraes and inite elements or MH simulations. Journal o Computational sis, 008, Vol. 7, p []. H. Winston. Artiiial Intelligene, Addison- Wesle, Tird Ed, U.S.A., p , 199. [] R. C. Gonzalez,. Wintz. igital image proessing, Addison-Wesle, nd Ed; U.S.A., 1987, p [4] S. H. Kim, K. H. Sin. A metod or deorming a surae model using guide suraes in soe design. International Journal o Advaned Manuaturing Tenolog, 009, Vol. 4, [5] S. Yuwen, G. ongming, J. Zenuan, L.Weiun, - spline surae reonstrution and diret sliing rom point louds. International Journal o Advaned Manuaturing Tenolog, 006, Vol. 7, [6] T. W. Weerasinge, R. S. Goonetillenke. Getting to te bottom o ootwear ustomization. Journal o Sstems Siene and Sstems Engineering, 011, Vol. 0, 10-. [7] W. Cung, S. H. Kim, K. H. Sin. A Metod or planar development o suraes in soe pattern design. Journal o Meanial Siene and Tenolog, 008, Vol., [8] Y. Cong, W. Lee, M. Zang. Regional plantar oot pressure distributions on ig-eeled soes-sank urve eets. Ata Meania Sinia, 011, Vol. 7, [9] Y. F. Li, S. Y. Cen. Automati realibration o an ative strutured ligt vision sstem. IEEE Transations on Robotis and Automation, 00, Vol. 19, Reeived June