A Computational Method for Subdivision Depth of Ternary Schemes

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

Quantum (q, h)-Bézier surfaces based on bivariate (q, h)-blossoming

We introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.

An improved high-precision subdivision algorithm for single-track absolute encoder using machine vision techniques

In order to achieve high-precision and robust measurement for a single-track absolute encoder, an improved subdivision algorithm based on machine vision technology is proposed. First, the composite subdivision algorithm combining RANdom SAmple Consensus and least square estimation is introduced. Second, the proposed algorithm is proved to be high-precision and effective to remove fault error from signal noise and spot by simulation. Finally, real test results show that the algorithm output angle stably achieves precision within 2.5 arc-seconds and accuracy up to 1.6 arc-seconds. Research results provide a high-precision and robust subdivision algorithm that is improved by modern machine vision technology of RANdom SAmple Consensus and it can be applied in the field of absolute angle sensor in the future.