A note on least-norm solution of global WireWarping

WireWarping [1] is a fast surface flattening approach, which presents a very important property of length-preservation on feature curves. The global scheme of WireWarping formulates the warping problem into an optimization in angle space and solves it by using the Newton’s method. However, some diverged examples were found in our recent tests. This technical note presents a least-norm solution in terms of angle-error for the global WireWarping. The experimental tests show that the least-norm solution is more robust than the Newton’s algorithm. 1. Problem The Newton’s method solves a constrained optimization problem by converting the objective function and the constraints into an augmented objective function J(X) with X as the variable vector. Then, the update vector δ in each iteration step is computed by the linear system, ∇2J(X)δ = −∇J(X), which is formed by the Hessian matrix ∇2J(X) and the gradient ∇J(X). However, the Newton’s algorithm has no control over the magnitude of δ. There, vibration is easily generated when the status variable X is near optimum. In some extreme cases, such vibration may move the system to a status that can hardly converge. Fig.1 shows such a vibrated example when using Newton’s method to compute the global WireWarping. To make the Newton’s method more robust, the soft-line-search strategy [2] is always employed to determine the actual update step size αδ (0 < α ≤ 1) (e.g., [3]). However, such a line-search introduces additional sub-routine of iterations so that actually slows down the computation. Stimulated by the recent work of least-norm solution of angle-based parameterization in [4], a least-norm solution is proposed in this note to increase the robustness of optimization while remaining the same efficiency as Newton’s method in each iteration step. The constrained optimization problem to be solved is Eq.(13) in [1]. minθi ∑ i (θi − αi) s.t. npπ − ∑np b=1 θΓp(b) ≡ 2π (∀p = 1, ...,m) ∑np b=1 lb cosφb ≡ 0, ∑np b=1 lb sinφb ≡ 0 (∀p = 1, ...,m) ∑ qk∈v θk ≡ 2π (∀v ∈ Φ) (1) Here, we adopt the same nomenclature. Φ represents the collection of interior vertices on accessory feature curves, θi is the 2D angle associated with the wire-node qi to be computed, αi represents its optimal angle (i.e., the 3D angle employed in [1]), and lb denotes the length of an edge on wires. To simplify the expression, a permutation function Γp(b) is used to return the global index of a wire-node on the wire patch Pp with the local index b, and its inverse function Γ−1 p (j) that gives the local index of a wire-node qj on the wire-patch Pp.