7659, England, UK Journal of Informaton and Computng Scence Vol. 2, No. 2, 2007, pp. 119-126 Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements * R. Ng 1+, G.T.Y. Pong 2 and M. Wong 2 1 Insttute of Textles and Clothng, Hong Kong Polytechnc Unversty, Hong Kong SAR, Chna 2 Department of Appled Mathematcs, Hong Kong Polytechnc Unversty, Hong Kong SAR, Chna ( Receved 4 October 2006, accepted 4 February 2007 ) Abstract. For the applcaton of apparel ndustry, 3D body measurements are the mnmal arc length dstances between the pont couples along the body surface. The calculaton of geodescs s nontrval, and can only be approxmated for free-form surfaces, because the computaton of geodescs requres solvng a set of ordnary dfferental equatons (geodescs equatons), whch do not exhbt any closed form soluton for free-form surfaces. In ths artcle, the method of radal bass functon (RBF) s used n solvng the geodescs equatons based on the bcubc tensor product Bezer patches. Ths method does not requre any mesh to support the localzed approxmatons. The accuracy and effcency of ths meshless collocaton method are examned by comparson to the Rtz s method. Keywords: Geodescs, Radal Bass Functons 1. Introducton The ntroducton s dvded nto three parts: the background of solvng the geodescs equatons, the applcaton of geodescs equaton n the apparel ndustry, and the focus of the current research work. 1) Background of solvng geodesc equatons: Geodescs are defned n the dfferental geometry lterature as the locally shortest path between two ponts on a surface. The trajectory of the geodesc depends on the ntrnsc propertes of the surface. A local geodesc satsfes a par of ODEs, (1) and (2), defned usng the Chrstoffel symbols, Γ jk. The complexty of solvng the geodesc equatons depends on the defnton of the surface. Typcally, the solutons of geodescs on free-form surfaces do not have any closed form soluton. For ths reason, numercal method must be used to approxmate the value. GE 1 (u, v) = u" + Γ 1 11(u') 2 + 2 Γ 1 12u'v' + Γ 1 22(v') 2 = 0 (1) GE 2 (u, v) = v" + Γ 2 11(u') 2 + 2 Γ 2 12u'v' + Γ 2 22(v') 2 = 0 (2) The numercal methods beng used can be typcally classfed nto two classes accordng to whether the surface s contnuous or dscrete. In solvng the geodescs on contnuous surfaces, one usually solves the dfferental numercally. Gray [1] presented a numercal scheme of solvng the geodesc equaton (1) and (2) wth uv-parametrzaton usng the Mathematca TM software. Maekawa [2] presented another numercal scheme based on fnte dfference method. A survey of numercal schemes for solvng dfferental equatons and ntegral equatons s avalable n Cheney [3]. The other class of technques s used for the polyhedral surfaces. Polther and Schmes [4] derved the lght ray tracng method of shootng the end pont of the geodescs along the polyhedral surface. Pham-Trong, Szafran and Bard [5] searched for the dscrete geodescs usng the subdvson of patches wth unfoldng, so that the beam propagaton can reach the destnaton pont on the other end of the geodescs. Rav Kumar, Srnvasan, Devaraja Holla, Shastry, and Prakash [6] ntroduced modfed verson of Polther and Schmes work, by ncorporatng the normal nformaton of the surface. Kmmel and Sethan [7] demonstrated the applcaton of the Fast Matchng Method n fndng the geodesc paths on a trangularly meshed surface. These methods are not meshless. In terms of the meshless methods, Ng and Pong [8] reported the use of dstrbuted collocaton method, * Ths project s fnancally supported by the Polytechnc Unversty Research Grant, A/C PE04. + Contact detal: Phone: +852-2766-6532, Fax: +852-2773-1432, E-mal: roger.ng@polyu.edu.hk Phone: +852-2766-6935, Fax: +852-2362-9045, E-mal: Glory.Pong@polyu.edu.hk Publshed by World Academc Press, World Academc Unon
120 R. Ng, et al: Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements whch s a parallel verson of the collocaton method. In the current study, another meshless method, radal bass functon (RBF), s used n solvng the geodescs equatons on a famly of bcubc tensor product Bezer surface patches under uv-parametrzaton, snce nterpolaton by radal bass functons s a powerful tool n approxmaton theory. 2) Applcaton of geodescs equaton n the apparel ndustry: In the 3D garment pattern desgn, a surface mappng technque, Bjectve Pattern Map [9], unfolds the 3D surface patches nto 2D enclosed areas usng the geodesc nformaton on the 3D mannequn. Manually, the body measurement can be measured by usng a tensoned measurng tape. In a vrtual world, as a free-form surface, the 3D vrtual mannequn can be represented as an atlas of tensor product Bezer or B-Splne or NURBS surface patches. In ths case, the body measurement, such as the surface patch shown n Fg. 1, can only be calculated usng the geodesc arc length. Fg. 1. Hp to Crotch Surface Patch 3) Focus of research work: In ths artcle, the calculaton of the geodesc wll focus on a class of nonself-ntersectng surface patches, defned usng the bcubc tensor product Bezer form. Self-ntersectng surface can be reduced to non-self-ntersecton surface subpatches. Wth the scalable propertes, t s possble to reduce the problem by consderng only the normalzed patches, lyng wthn the unt cube. The numercal algorthm s based on the radal bass functon method. The choce of the tral functons wll be presented. The comparson to the Rtz s method wll be gven to llustrate the effectveness of the radal bass functon method. Fnally, the numercal result wll be compared to the data measured from the vrtual model of a real mannequn. All the programs are wrtten usng Mathematca TM verson 5.0. 2. NORMALIZATION OF SURFACE PATCHES A famly of normalzed bcubc tensor product Bezer surface patches s defned over the unt cube wth the longest sde of the patch algned wth the lne segment of [0, 1] along the x-axs. As the bcubc tensor product patch representaton s fully scalable, ths representaton can cover the whole famly of bcubc tensor product Bezer surface patches, snce any fnte bcubc tensor product Bezer surface can be enclosed by a cubc of sze equals to the longest straght lne dstance between any par of control vertces. The normalzaton process can be acheved by these transformatons: Stage 1 (Set orgn, and force all ponts nto postve zone): P = p Mn(p ) (3) Stage 2 (Scale to ft the unt cube): P n = P / Max(P ) (4) where p are the orgnal data ponts, P are the translated ponts, and P n are the normalzed ponts wth Mn(.) and Max(.) to be the mnmum and maxmum functon respectvely. It should be noted that when the Mn(.) s taken over the ordnates ndvdually, the data ponts are guaranteed to be translated nto the postve zone of {x, y, z : x 0; y 0; z 0}. However, any of the data ponts may not be located at the orgn. Yet, at least one of the data ponts can be found on the surface of the unt cube. JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence, 2 (2007) 2, pp 119-126 121 3. PREPARATION OF COLLOCATION POINTS The geodesc from a pont q 0 to another pont q 1 along the bcubc tensor product Bezer surface patch S(u, v) s ntended to be wrtten n the form of a parametrc curve α(t) wth the end ponts matchng the boundary condton of α(0) = q 0 and α(1) = q 1. The collocaton ponts are thus defned as the set of N chosen ponts {α(t ) where = 1, 2,..., N, and each t les n the range of (0, 1)}. In ths study, the collocaton ponts are evenly dstrbuted along (0, 1). Based on the experence wth usng collocaton method, the collocaton ponts must be chosen n a way to ensure the non-degenerate determnant of the matrx derved by submttng the tral functons and combnng the matchng requrement at the collocaton ponts. 4. SELECTION OF TRIAL FUNCTIONS The tral functons geo 1 (t) and geo 2 (t) are both selected to be the lnear combnaton of radal bass functons wth coeffcents {a } and {b } respectvely, n whch we match the condtons avalable, ncludng the par of geodesc ODEs, the boundary condtons and the matchng at the collocaton ponts. The geo 1 (t) corresponds to the u-parameter whle the geo 2 (t) corresponds to the v-parameter. Snce the surface s represented n the form of S(u, v), the GE 1 (u, v) and GE 2 (u, v) can n fact be wrtten as GE 1 (t) = GE 1 (u(t), v(t)) and GE 2 (t) = GE 2 (u(t), v(t)) respectvely by substtutng: geo 1 (t) = u(t) = a 0 + a 1 φ(t, c 1 ) + a 2 φ(t, c 2 ) + a 3 φ(t, c 3 ) +... (5) geo 2 (t) = v(t) = b 0 + b 1 φ(t, c 1 ) + b 2 φ(t, c 2 ) + b 3 φ(t, c 3 ) +... (6) n whch we select the radal bass functons φ(t, c ) [10] of the form: 5 2 3 4 d d d d d 1 8 + 40 + 48 + 25 + 5, 0 d r φ(, t c ) = r r r r r 0,, d > r where d = t c s the dstance between the pont t and the center c ; and r s the radus of the support for the radal bass functons centered at the center c. In ths study, the radus of the support s chosen to be 2. The advantage n usng radal bass functons s generally that they provde nterpolants rrespectve of complexty of the geometry. The compactly supported postve defnte radal bass functons φ(t, c ) are feasble n the current applcaton. From the experence of approxmatng the geodescs on the bcubc tensor product Bezer surface patches, radal bass functons must be chosen n a way to ensure the tral functons are at least twce dfferentable and wth compact support over [0, 1]. 5. FORMATION OF MATRIX EQUATION In ths study, the scope s lmted to the geodescs across the dagonal of the surface patch, because such measurements are most commonly used n the Bjectve Pattern Map. By substtutng the defnton of S(u, v), the geo 1 (t) and geo 2 (t) nto the par of geodescs equatons, two equatons are found, GE 1 (t) and GE 2 (t). Note that these two equatons wth the unknown coeffcents are very long and are omtted here. By matchng the boundary condtons, four more equatons are found, (7) to (10). u(0) = 0 = a 0 + a 1 φ(0, c 1 ) + a 2 φ(0, c 2 ) + a 3 φ(0, c 3 ) + (7) v(0) = 0 = b 0 + b 1 φ(0, c 1 ) + b 2 φ(0, c 2 ) + b 3 φ(0, c 3 ) +... (8) u(1) = 1 = a 0 + a 1 φ(1, c 1 ) + a 2 φ(1, c 2 ) + a 3 φ(1, c 3 ) +... (9) v(1) = 1 = b 0 + b 1 φ(1, c 1 ) + b 2 φ(1, c 2 ) + b 3 φ(1, c 3 ) +... 10) Fnally, requrng a perfect match of the estmaton by geo 1 (t) and geo 2 (t) at the collocaton ponts t, 2N equatons are found, (11) and (12). These equatons can be wrtten n the matrx form. GE 1 (t ) = 0, = 1,..., N (11) GE 2 (t ) = 0, = 1,..., N ) After the substtuton process s completed, they form a matrx equaton. It s mportant to check and ensure the determnant of ths matrx equaton to be non-zero, so that soluton can be guaranteed. In case, the determnant s zero, the choce of collocaton ponts must be revsed. As an example, from the experence of calculatng the geodescs on the bcubc tensor product Bezer JIC emal for subscrpton: publshng@wau.org.uk
122 R. Ng, et al: Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements surface patches when developng the Bjectve Pattern Map, two radal bass functons could be suffcent for patches wth small Gaussan curvature. In that case, sx equatons (two from (1) and (2) and four from (5) and (6)) wll be derved wth two boundary ponts and one collocaton pont. When more radal bass functons are used, more collocaton ponts are needed to solve the extra coeffcents. If the coeffcent matrx of the equaton s sngular, another set of collocaton ponts fallng n between the exstng collocaton ponts can be chosen or the number of radal bass functons can be elevated. 6. DATA PREPARATION 6.1. Testng Data The testng data s prepared based on the followng crtera: All control vertces are postoned wthn a unt cube. The startng endpont s fxed wth the heght of 0; and the termnatng endpont s wth the heght of 1. The total sample sze s 12 ponts at 6 possble heghts = 72 choces. The control vertces s randomly chosen wthn a unt cube, but wth a predetermned ncrement, say 0.0225, whle a total 1000 samples are selected. The samples are dvded nto four groups of data. In ths confguraton, some of the patches can have local optmal values, however unqueness of the geodescs s not guaranteed, as conjugate ponts may exst. Yet, as long as the control ponts do not ntersect each other, the surface patch does not ntersect tself. The surface s also ensured to be compact and contnuous. 6.2. Evaluaton Data Based on Rtz s Method The full set of testng normalzed bcubc tensor product Bezer surface patches was run agan usng another numercal method. Among many choces, the Rtz s method was chosen because t s a popular numercal technque for Calculus of Varaton [11]. The bass functon for the Rtz s method was chosen to be polynomal n the form of (13) and (14). In ths case, the v-parameter becomes a functon of u-parameter. The values of geodesc dstances were compared to those of the radal bass functon method. u(t) = t (13) v(t) = c 0 + c 1 t + c 2 t 2 + c 3 t 3 +.. + c n t n (14) 6.3. Evaluaton Data From Physcal Objects One more set of data measured from the vrtual model of a real mannequn was prepared for comparson. In prevous study, the modelng of human mannequn usng bcubc tensor product Bezer patches had been demonstrated wth reasonable accuracy f the patches were properly defned [12]. The mannequn was dgtzed by a 3D body scanner and stored n the form of a bcubc tensor product Bezer surface atlas. The lengths of the boundary curves of each patch were ensured to be equal to the physcal value. Then, both of the dagonal lengths were measured and used as the reference physcal data, and compared to the computer-calculated values. 7. SOLVING THE MATRIX EQUATION The matrx equaton derved accordng to secton V was solved usng the bult-n equaton solver of Mathematca TM 5.0. However, an adaptve scheme must be used to select dfferent collocaton ponts because t s possble that some pars of collocaton ponts do share some knd of symmetry, meanng that the rank of the matrx equaton may be reduced. In these cases, other collocaton ponts must be used. It should be noted that the matrx equaton s nonlnear wth respect to the coeffcents a and b, as the surface patches are n the bcubc tensor product form. Therefore, there can be one or more real soluton sets. Each soluton set must be testfed and compared to determne the global optmal soluton set. Durng the calculaton by Mathematca TM, the response was very slow when three or more radal bass functons were used. So, n ths study, the results are based on two radal bass functons centered at t = 1 and 1.5 wth the radus of compact support of 2. 8. EVALUATION OF RESULT The computaton speeds of both methods were measured usng the TIMING command of the Mathematca TM. JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence, 2 (2007) 2, pp 119-126 123 Snce there are many common codes shared by both methods, the tmng of the radal bass functon method ncluded the executon of the followng lnes n pseudo code, see Table I, whle the counterpart for the Rtz s Method s lsted n Table II. Table 1 KERNEL OF RADIAL BASIS FUNCTION Part 1 (Generate and solve the matrx equaton): Manually select the centers of the radal bass functon; Generate_Equatons_of_Collocaton_Ponts; Whle (Equaton_Degenerate), { Use_Other_Collocaton_Ponts; Generate_Equatons_of_Collocaton_Ponts; } Answer = Solve_Matrx_Equaton; Part 2 (Calculate the arc length): Mn_Arc_Length = Infnty; For[ = 1; < Number_of_Answers, Arc_Length = Calculate_Arc_Length[Answer[[]]]; Mn_Arc_Length = Mn[Mn_Arc_Length, Arc_Length]; ++]; (Here s the counter.) The total computatonal cost ncludes the tme n generatng and solvng the matrx equaton, and calculatng the geodesc dstance. Table 2 KERNEL OF RITZ S METHOD Intalze_Varaton_Parameter; Intalze_Max_Iteraton; Error = Infnty; Geodesc = Infnty; Whle (Error > Error_Bound && j < Max_Iteraton), { Increase_Order_of_Bass_Functon; Substtute_Dfferental_Equaton; Prepare_Matrx_Equaton; Solve_Varaton_Parameter; Arc_Length = Verfy_Varaton_Parameter; Error = Abs[Geodesc Arc_Length]; Geodesc = Mn[Geodesc, Arc_Length]; j++;} (Here j s the teraton counter.) The sample report of the result s lsted n Table III. The comparatve result between radal bass functon method and Rtz s method s summarzed n Table IV. The relatve Root-Mean-Square error (RMS), (15), s used to ndcate the dfferences n the accuracy of the approxmated geodesc dstance usng radal bass functons, gc, and the geodesc dstance from Rtz s Method, gr, whle a relatve rato (16) s used for the mprovement of the computaton tme, tr for Rtz method and tc for radal bass functon method. The result s presented n sx sgnfcant fgures n the normalzed unt. Sgn(gC gr) * Sqrt( (((gr gc) / gr) 2 )) (15) [Note: Sqrt s the square root functon.] tr / tc (16) JIC emal for subscrpton: publshng@wau.org.uk
124 R. Ng, et al: Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements Table 3 SAMPLE DATA OUTPUT TABLE Radal Bass Functon Rtz s Method Normalzed Unt Second Normalzed Unt Second 1.57495 0.141 1.56924 0.125 1.37739 0.141 1.37116 0.110 1.41368 0.125 1.40248 0.109 1.59956 0.140 1.59710 0.125 Table 4 COMPARISON OF RADIAL BASIS FUNCTION METHOD AND RITZ METHOD Set Mean Accuracy Loss Mean Tme Improved 1 2.89% 101% 2 3.01% 101% 3 3.22% 98% 4 3.17% 105% Fnally, a set of sample geodesc dstances approxmated usng the radal bass functons accordng to the scanned patches was compared to the geodesc dstances measured from the vrtual model of a real mannequn. The calculated geodesc dstances are rounded up to sx sgnfcant fgures. The relatve error was measured accordng to (17), wth gp representng physcal geodesc. The result s shown n Table V. Sgn(gC gp) * Sqrt((gP gc) / gp) 2 ) (17) 9. COMMENTS ON USING RADIAL BASIS FUNCTIONS Table 5 COMPARISON OF CALCULATION BY RADIAL BASIS FUNCTION METHOD AND PHYSICAL DATA Data Calculated by RBF Method Physcal Data (cm) Relatve Error (%) 1 13.1038 13.0-0.8 2 6.18354 6.25 1.06 3 10.2788 10.25-0.28 4 8.84254 8.75 1.06 Accordng to the current plot study, the advantages of usng the radal bass functons n calculatng geodesc dstances nclude: Effcent method wth reasonably accuracy even when small set of radal bass functons to be used. Ease of software mplementaton. Yet, ths method has the followng dsadvantages: JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence, 2 (2007) 2, pp 119-126 125 Effcency drops when more collocaton ponts are used. The convergence of the matrx equaton solvng may take much longer tme when more radal bass functons are used. The typcal computaton of radal bass functon method based on based on two radal bass functons, mplemented on a dual Xeon TM 1.8 GHz personal computer under Mathematca TM Verson 5.0. Ths method can be very tme consumng as the number of the radal bass functons ncreases. 10. CONCLUSION In the plot study, the geodesc equatons on the subset of bcubc tensor product Bezer patches have been solved numercally by the nterpolaton usng radal bass functons. The results of computng the geodescs by ths radal bass functon method are compared to that by the Rtz s method. The agreement between these two methods s acceptable, whle the computaton effcency of the radal bass functon method s farly good. Moreover, the accuracy of approxmatng geodescs on the scanned patches s found excellent when compared wth the measurements on the vrtual model of a real mannequn. The future work s to extend the study to at the full scale, wth the theoretcal analyss, and weak-form solutons. 11. References [1] A. Gray. Modern Dfferental Geometry of Curves and Surfaces. CRC Press, 2006. [2] T. Maekawa. Computaton of shortest paths on free-form parametrc surfaces. Journal of Mechancal Desgn. 1996, 118: 499-508. [3] W. Cheney. Analyss for Appled Mathematcs. New York: Sprnger-Verlag, 2001. [4] K. Polther, M. Schmes. Straghtest geodescs on polyhedral surfaces, In: H. C. Hege, H. K. Polther (Eds.). Mathematcal Vsualzaton. Sprnger-Verlag, 1998. [5] V. Pham-Trong, N. Szafran, L. Bard. Pseudo-geodescs on three-dmensonal surfaces and pseudo-geodesc meshes. Numercal Algorthms. 2001, 26: 305-315. [6] G.V.V. Rav Kumar, P. Srnvasan, V. Devaraja Holla, K.G. Shastry, B.G. Prakash. Geodesc curve computatons on surfaces. Computer Aded Geometrc Desgn. 2003, 20: 119-133. [7] R. Kmmel and J.A. Sethan. Computng geodescs paths on manfolds. Appled Mathematcs. 95: 8431-8435. [8] R. NG, G.T.Y. Pong. Measurng Geodesc Body Measurements wth Dstrbuted Collocaton Method. IMAC 2005. T1-I-45-0374, 2005, [9] R. NG, C. K. CHAN, R. AU, T.Y. PONG. Analytcal soluton to the Bjectve Mappng Problem n pattern desgn. Journal of Chna Textle Unversty. (Englsh Edton), 1996, 13: 94-98. [10] Z. Wu. Compactly supported postve defnte radal functons. Advances n Computatonal Mathematcs. 1995, 4: 283-292. [11] I. M. Gelfand, S. V. Fomn. Calculus of Varaton. New Jersey: Prentce Hall, 1963. [12] R. NG, C. K. CHAN, R. AU, T.Y. PONG. Automatc generaton of human model from lnear measurements - Algebrac Mannequn. Ergon-Axa 98. 1998, 267-270. 12. TRADEMARK Mathematca TM s a trademark of Wolfram Research Inc. Xeon TM s a trademark of Intel Corporaton. JIC emal for subscrpton: publshng@wau.org.uk
126 R. Ng, et al: Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements JIC emal for contrbuton: edtor@jc.org.uk