Multiresolution morphing for planar curves

Abstract

We present a multiresolution morphing algorithm using "as-rigidas-possible" shape interpolation combined with an angle-length based multiresolution decomposition of simple 2D piecewise curves. This novel multiresolution representation is defined intrinsically and has the advantage that the details' orientation follows any deformation naturally. The multiresolution morphing algorithm consists of transforming separately the coarse and detail coefficients of the multiresolution decomposition. Thus all LoD (level of detail) applications like LoD display, compression, LoD editing etc. can be applied directly to all morphs without any extra computation. Furthermore, the algorithm can robustly morph between very large size polygons with many local details as illustrated in numerous figures. The intermediate morphs behave natural and least-distorting due to the particular intrinsic multiresolution representation.

An Implicit Method for Interpolating Two Digital Closed Curves on Parallel Planes

“…In what follows we have taken the digital contours Γ and ∆ to lie on the planes z = r Γ and z = r ∆ , with r Γ < r ∆ and the coefficient α in equation (15) to be equal to 0.1. The grids are relatively coarse, ranging from 50 ÷ 70 nodes in the x and y directions and 20 nodes in the z direction.…”

“…Considering the characteristics of the equation (9) we obtain that the absolute values of Ψ i,j,k decrease, as we move from the boundary to the zero-level set of Ψ , provided that the coefficient α is greater than zero, thus in each step we update the values of Ψ i,j,k on a narrow band of nodes, using the values of Ψ i,j,k that have already been calculated (solved), starting from the boundary of the domain, via equation (15). Then, we consider as solved the point, whose value is closest to solved points, i.e., the one with the maximum absolute value in the narrow band.…”

Multiresolution Analysis

“…In [46] a curvature-based MR representation for 2D polygonal curves was introduced. This MR representation is based on an intrinsic parameterization of both the coarse shape and the detail coefficients.…”

G 1 -smooth splines on quad meshes with 4-split macro-patch elements

“…In the last decades, several works have been focusing on the construction of G 1 surfaces, including [CC78], [Pet95], [Loo94], [Rei95], [Pra97], [YZ04], [GHQ06], [HWW + 06], [FP08], [HBC08], [PF10], [BGN14], [BH14]. Some of these constructions use tensor product b-spline elements.…”

“…In [PF10], it is shown that bicubic G 1 splines with linear transition maps requires at least a 9-split of the parameter domains. In [HBC08], bicubic 4-split macro-patch elements are used. They are represented by 36 control coefficients.…”