A Note on Least-norm Solution of Global WireWarping

Transcription

1 A Note o Least-orm Solutio of Global WireWarpig Charlie C. L. Wag Departmet of Mechaical ad Automatio Egieerig The Chiese Uiversity of Hog Kog Shati, N.T., Hog Kog cwag@mae.cuhk.edu.hk Abstract WireWarpig [1] is a fast surface flatteig approach, which presets a very importat property of legth-preservatio o feature curves. The global scheme of WireWarpig formulates the warpig problem ito a optimizatio i agle space ad solves it by usig the Newto s method. However, some diverged examples were foud i our recet tests. This techical ote presets a least-orm solutio i terms of agle-error for the global WireWarpig. The experimetal tests show that the least-orm solutio is more robust tha the Newto s algorithm. 1. Problem The Newto s method solves a costraied optimizatio problem by covertig the objective fuctio ad the costraits ito a augmeted objective fuctio J(X) with X as the variable vector. The, the update vector δ i each iteratio step is computed by the liear system, 2 J(X)δ = J(X), which is formed by the Hessia matrix 2 J(X) ad the gradiet J(X). However, the Newto s algorithm has o cotrol over the magitude of δ. There, vibratio is easily geerated whe the status variable X is ear optimum. I some extreme cases, such vibratio may move the system to a status that ca hardly coverge. Fig.1 shows such a vibrated example whe usig Newto s method to compute the global WireWarpig. To make the Newto s method more robust, the soft-lie-search strategy [2] is always employed to determie the actual update step size αδ (0 < α 1) (e.g., [3]). However, such a lie-search itroduces additioal sub-routie of iteratios so that actually slows dow the computatio. Stimulated by the recet work of least-orm solutio of agle-based parameterizatio i [4], a least-orm solutio is proposed i this ote to icrease the robustess of optimizatio while remaiig the same efficiecy as Newto s method i each iteratio step. The costraied optimizatio problem to be solved is Eq.(13) i [1]. s.t. mi θi i (θ i α i ) 2 p π p b=1 θ Γ p (b) 2π ( p = 1,..., m) p b=1 l b cos φ b 0, p b=1 l b si φ b 0 ( p = 1,..., m) q k v θ k 2π ( v Φ) Here, we adopt the same omeclature. Φ represets the collectio of iterior vertices o accessory feature curves, θ i is the 2D agle associated with the wire-ode q i to be computed, α i represets its optimal agle (i.e., the 3D agle employed i [1]), ad l b deotes the legth of a edge o wires. To simplify the expressio, a permutatio fuctio Γ p (b) is used to retur the global idex of a wire-ode o the wire patch P p with the local idex b, ad its iverse fuctio Γ 1 p (j) that gives the local idex of a wire-ode q j o the wire-patch P p. (1) 1

2 Figure 1: Numerical vibratio occurs whe usig the global WireWarpig to flatte the frot piece of a shirt as show i the zoom-widow, uwated curve distortio is geerated. This is because that the umerical computatio vibrates see the chart of δ 2. Here, gree ad red lies represet the key feature curves ad the accessory feature curves followig [1]. 2. Least-orm Solutio As stated i [4], carefully selectig alterative variables could make the liearizatio more accurate so that the computatio coverges faster tha the Newto steps. We reformulate Eq.(1) by chagig the variables from θ i to the agle estimatio error e i = θ i β i, where β i is the curret agle at wire-ode i ad θ i is the optimal agle to be computed. The optimizatio problem i Eq.(1) is reformulated as s.t. mi ei i e2 i p π p b=1 (β Γ p (b) + e Γp (b)) 2π ( p = 1,..., m) p b=1 l b cos φ b 0, p b=1 l b si φ b 0 ( p = 1,..., m) q k v (β k + e k ) 2π ( v Φ) (2) For the costraits with φ b, as φ i = π (e i + β i ) + φ i 1 ad φ 1 = π (e i + β 1 ) accordig to Eq.(3) i [1], φ i = ca be expressed as φ i = iπ i h=1 (e h + β h ) = i + ξ i with i = iπ i h=1 β h ad ξ i = i h=1 e h. Usig Taylor expasio cos( i + ξ i ) = cos i + ( si i )ξ i + ( 1 2 cos i)ξ 2 i + si( i + ξ i ) = si i + (cos i )ξ i + ( 1 2 si i)ξ 2 i + we the trucate the series by retaiig the liear terms oly (i.e., with the approximatio error O(ξi 2)). Therefore, the o-liear costraits p b=1 l b cos φ b 0 ad p b=1 l b si φ b 0 are liearized ito l i si i ) = l i cos i (3) ad (e Γp (b) b=1 i=b (e Γp (b) b=1 i=b i=1 p l i cos i ) = l i si i (4) i=1 2

3 respectively. I summary, Eq.(2) is coverted ito mi r 2 s.t. Cr = b (5) where C is a co var matrix. As discussed i [1], if there are l wire-odes with their 2D agles locked by the key feature curves, the umber of variables for the above problem is var = ( m p=1 p) l. If there are r iterior vertices o the accessory feature curves, the total umber of costraits is co = 3m + r. I geeral, co < var, there are multiple solutios for Cr = b. Amog the, we eed to seek oe that leads to a miimal orm r 2 o the variable vector r. This is a least-orm problem. For a full rak coefficiet matrix C, the least-orm problem has a uique solutio (c.f. [5]) r = C T (CC T ) 1 b (6) The value of r ca be solved by fidig a solutio to the ormal equatio (CC T )x = b followig by a substitutio that r = C T x. The matrix C has full rak as the costraits i Eq.(1) are idepedet. Startig from lettig β i = βi 0, we iteratively update the value of β i by solvig e i i Eq.(5) ad update its value with β i = β i + e i i each step. The iteratio is stopped whe 1 var r 2 < 10 8 is satisfied. The resultat optimal agle for each wire-ode is the determied. Why such a least-orm solutio i each iteratio step is more robust? The major reaso is that amog all possible solutios, the oe with miimal estimatio error is adopted. While the Newto s update just move the system variables alog the optimal directio but ot determie a optimal step size whe ot coductig the soft-lie-search strategy, the least-orm solutio actually mimics the soft-lie-search. Aother mior beefit of the least-orm solutio is that we do ot eed to compute the secod derivative of J(X). Although the case of co var is ever foud i our tests, the liearizatio of global warpig problem as Eq.(2-4) gives a possible solutio to compute the update vector r by which is i fact a least-square solutio. C T Cr = C T b (7) 3. Iitial Value Same other optimizatio techiques, the least-orm solutio still relies o good iitial agle values o wire-odes. I the origial publicatio [1], 3D agles are employed to be the iitial agle values o wire-odes. However, this does ot give satisfactory results whe flatteig some highly curved surfaces (e.g., the pats of wet-suit i Fig.2 which are also show i [1]). For those surfaces without key feature curves determied (e.g., Fig.2), the warpig of feature curves are flexible. Thus, the Least-Square Coformal Map preseted i [6] is used to pre-flatte the surface ito plae, ad the 2D agle at each wire-ode is tha adopted as the iitial value of iteratio. For those surfaces with the shape of key feature curves specified (e.g., the perpedicular key feature curves are specified i Fig.1), the iitial value βi 0 at a wire-ode q i is determied as { βi 0 αi (2π q = j v αl j )/ q k v α k q i v, v Φ otherwise α i (8) where q k ad q j are the wire-ode associated with the same vertex as q i. q j is the ode o key feature curves with its 2D agle specified as α L j, q k is o a accessory feature curve, ad Φ is the set of iterior vertices o accessory feature curves. The iitial value of agles determied by the above methods esure that the agle compatible costrait q k v θ k 2π ( v Φ) has bee satisfied at the begiig, which makes the computatio easier to coverge. 3

4 Figure 2: Differet iitial values will lead to differet results: (a) with 3D agles as iitial values uwated overlappig is geerated at the darts, ad (b) usig the result of least-square coformal map [6] as iitial values the result has bee improved. Both results are geerated by the least-orm solutio of global warpig. Figure 3: The flatteig result by the least-orm solutio of global WireWarpig the iitial value is computed by Eq.(8). 4. Experimetal Results The least-orm solutio of the global WireWarpig itroduced i this techical ote has bee tested o several examples. The first test is give to the shirt model show i Fig.1. The result by usig the least-orm solutio is give i Fig.3, ad it is easy to fid that the computatio coverges 1 very fast (i.e., var r 2 0 after two steps of iteratio). A more fair compariso is give o the orm of residual vector σ of the costrai equatios i Eq.(1). More specifically, the residual vector is σ = ( p 2)π p b=1 θ Γ p (b) p b=1 l b cos φ b p b=1 l b si φ b 2π q k v θ k. (9) Figure 4 ad 5 show the compariso, which is cosistet with the orm of update vector i.e., the least-orm solutio coverges faster ad is also with less vibratio. The least-orm solutio solves a liear equatio system with dimesio co co (i.e., CC T ) 4

5 Figure 4: Chart for comparig the covergecy o the orm of residual vector by the Newto s method vs. the least-orm solutio. i each iteratio plus oe step of substitutio. For the umerical computatio proposed i [1], i Eq.(23)-(24), the update vector i each step is also determied by computig a (3m + r) (3m + r) liear system followed by substitutio. As co = 3m + r, the computatios i each step by the method of [1] ad the least-orm solutio are the same. Therefore, the computatio speed of leastorm solutio is faster as it usually eeds less steps tha Newto s method does to coverge. Ackowledgmets The author would like to thak TPC (HK) Ltd for providig the 3D surface patches of garmet suits. Refereces [1] C.C.L. Wag, WireWarpig: A fast surface flatteig approach with legth-preserved feature curves, Computer-Aided Desig, vol.40, o.3, pp , [2] K. Madse, H.B. Nielse, ad O. Tigleff, Optimizatio with Costraits, Lecture Notes, [3] Y. Liu, H. Pottma, J. Waller, Y.-L. Yag, ad W. Wag, Geometric modelig with coical meshes ad developable surfaces, ACM Trasactios o Graphics, vol.25, o.3, pp , [4] R. Zayer, B. Lévy, ad H.-P. Seidel, Liear agle based parameterizatio, Proceedigs of Eurographics Symposium o Geometry Processig, [5] L. Vadeberghe, Applied Numerical Computig, Lecture Notes, [6] B. Lévy, S. Petitjea, N. Ray, ad J. Maillot, Least squares coformal maps for automatic texture atlas geeratio, ACM Trasactios o Graphics, vol.21, o.3, pp ,

6 Figure 5: Aother example for comparig the covergecy o the orm of residual vector by the Newto s method vs. the least-orm solutio. 6