Curve and Surface Fitting Techniques in Computer Vision

Curve and surface-fitting are classic problems of approximation that find use in many fields, including computer vision. There are two broad approaches to the problem — interpolation, which seeks to fit points exactly, and regression, which seeks a rougher approximation which is more robust to noise. This survey looks at several techniques of both kinds, with a particular focus on applications in computer vision. We make use of an empirical first-level evaluation approach which scores the techniques on multiple features based on how important they are to users of the technique and developers. This provides a quick summary of the broad applicability of the technique to most situations, rather than a deep evaluation of the performance and accuracy of the technique obtained by running it on several datasets.

Progressive Iterative Approximation with Preconditioners

The progressive iterative approximation (PIA) plays an important role in curve and surface fitting. By using the diagonally compensated reduction of the collocation matrix, we propose the preconditioned progressive iterative approximation (PPIA) to improve the convergence rate of PIA. For most of the normalized totally positive bases, we show that the presented PPIA can accelerate the convergence rate significantly in comparison with the weighted progressive iteration approximation (WPIA) and the progressive iterative approximation with different weights (DWPIA). Furthermore, we propose an inexact variant of the PPIA (IPPIA) to reduce the computational complexity of the PPIA. We introduce the inexact solver of the preconditioning system by employing some state-of-the-art iterative methods. Numerical results show that both the PPIA and the IPPIA converge faster than the WPIA and DWPIA, while the elapsed CPU times of the PPIA and IPPIA are less than those of the WPIA and DWPIA.

Upon Element-Free Galerkin Method and its Using in Static and Dynamic Structure Analysis

This paper presents, in a synthetically way, the fundamentals of the element-free Galerkin (EFG) method - a meshfree method - under development but with many capabilities for solving complex problems in mechanical engineering, like impact problems etc. For interpolation, the EFG method uses moving least-squares (MLS) interpolants in curve and surface fitting. Unlike other interpolants, the MLS interpolants do not pass through the data because the Dirac function properties are not available. This aspect could be a disadvantage of the EFG method but next to it, there are many advantages. Upon these issues a discussion exists in this paper. Finally, some applications of the EFG method are presented referring to static and dynamic analysis of structures. The examples and conclusions can be useful for knowing and using of the EFG method