Discrete polynomial moments for real-time geometric surface inspection

Abstract

A thorough analysis of discrete polynomial moments and their suitability for application to geometric surface inspection is presented. A new approach is taken to the analysis based on matrix algebra, revealing some formerly unknown fundamental properties. It is proven that there is one and only one unitary polynomial basis that is complete, i.e., the polynomial basis for a Chebychev system. Furthermore, it is proven that the errors in the computation of moments are almost exclusively associated with the application of the recurrence relationship, and it is shown that QR decomposition can be used to eliminate the systematic propagation of errors. It is also shown that QR decomposition produces a truly orthogonal basis set despite the presence of stochastic errors. Fourier analysis is applied to the polynomial bases to determine the spectral distribution of the numerical errors. The new unitary basis offers almost perfect numerical behavior, enabling the modeling of larger images with higher-degree polynomials for the first time. The application of a unitary polynomial basis eliminates the need to compute pseudo-inverses. This improvement in numerical efficiency enables real-time modeling of surfaces in industrial surface inspection. Two applications in industrial quality control via artificial vision are demonstrated.