Implicit Randomized Progressive-Iterative Approximation for Curve and Surface Reconstruction

Abstract

In this paper, we propose the implicit randomized progressive-iterative approximation (IR-PIA) method for curve and surface reconstruction. By introducing the effective probability criterion, the IR-PIA method selects the working elements of the collocation matrix to adjust the control coefficients. It is proved that the sequence of curves and surfaces generated by the control coefficients converge to the least-norm results in expectation. The IR-PIA method reduces the computation complexity and speeds up the curve and surface reconstruction compared with the implicit progressive-iterative approximation (I-PIA) method (Hamza et al., 2020). Numerical examples show that the IR-PIA method can be more efficient than the I-PIA method (Hamza et al., 2020). • The IR-PIA method converge to the least-norm solution in expectation. • The IR-PIA method eliminates the extra zero-level sheets effectively. • The working elements are selected to update the control coefficients. • The larger entries of the difference vectors can be annihilated as far as possible.