Surface Modelling Using Discrete Basis Functions for Real-Time Automatic Inspection

Abstract

This chapter presents an introduction to discrete basis functions and their application to real-time automatic surface inspection. In particular, discrete polynomial basis functions are analyzed in detail. Emphasis is placed on a formal and stringent mathematical background, which enables an analytical a-priori estimation of the performance of the methods for specific applications. A generalized synthesis algorithm for discrete polynomial basis functions is presented. Additionally a completely new approach to synthesizing constrained basis functions is presented. The resulting constrained basis functions form a unitary matrix, i.e. are optimal with respect to numerical error propagation and have many applications, e.g. as admissible functions in Galerkin methods for to solution of boundary value and initial value problems. Furthermore, a number of case studies are presented, which show the applicability of the methods in real applications.