Paul O’Leary

Discrete polynomial moments for real-time geometric surface inspection

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.

Discrete polynomial moments and the extraction of 3-D embossed digits for recognition

We extend the theory of polynomial moments by proving their spectral behavior with respect to Gaussian noise. This opens the door to doing computations on the signal-to-noise ratios of polynomial filters and with this, the comparability to classical filters is made possible. The compactness of the information in the polynomial and Fourier spectra can be compared to determine which solution will give the best performance and numerical efficiency. A general formalism for filtering with orthogonal basis functions is proposed. The frequency response of the polynomials is determined by analyzing the projection onto the basis functions. This reveals the tendency of polynomials to oscillate at the boundaries of the support; the resonant frequency of this oscillation can be determined. The new theory is applied to the extraction of 3-D embossed digits from cluttered surfaces. A three-component surface model is used consisting of a global component, corresponding to the surface; a Gaussian noise component, and local anomalies corresponding to the digits. The extraction of the geometric information associated with the digits is a preprocessing step for digit recognition. It is shown that the discrete polynomial basis functions are better suited than Fourier basis functions to fulfill this task.

Constrained polynomial approximation for inverse problems in engineering

This paper presents the derivation, implementation and testing of a series of algorithms for the least squares approximation of perturbed data by polynomials subject to arbitrary constraints. These approximations are applied to the solution of inverse problems in engineering applications. The generalized nature of the constraints considered enables the generation of vector basis sets which correspond to admissible functions for the solution of inverse initial-, internal- and boundary-value problems. The selection of the degree of the approximation polynomial corresponds to spectral regularization using incomplete sets of basis functions. When applied to the approximation of data, all algorithms yield the vector of polynomial coefficients \(\varvec{\alpha }\), together with the associated covariance matrix \(\mathsf {\Lambda }_{\varvec{\alpha }}\). A matrix algebraic approach is taken to all the derivations. A numerical application example is presented for each of the constraint types presented. Furthermore, a new approach to performing constrained polynomial approximation with constraints on the coefficients is presented.

Hierarchical decomposition and approximation of sensor data

This paper addresses the issue of hierarchical approximation and decomposition of long time series emerging from the observation of physical systems. The first level of the decomposition uses spatial weighted polynomial approximation to obtain local estimates for the state vectors of a system, i.e., values and derivatives. Covariance weighted Hermite approximation is used to approximate the next hierarchy of state vectors by using value and derivative information from the previous hierarchy to improve the approximation. This is repeated until a certain rate of compression and/or smoothing is reached. For further usage, methods for interpolation between the state vectors are presented to reconstruct the signal at arbitrary points. All derivations needed for the presented approach are provided in this paper along with derivations needed for covariance propagation. Additionally, numerical tests reveal the benefits of the single steps. The proposed hierarchical method is successfully tested on synthetic data, proving the validity of the concept.

Analytic approach to finding lines in images and estimating their uncertainty

This paper presents a new approach to maximum likelihood fitting of lines in images. Standard approaches such as the Hough Transform are heuristic and dogged by the subtleties of quantization error. As an alternative, we take an analytic approach to the Hough Transform coupled with principles of robust maximum likelihood line fitting. We derive two line-parameter likelihood functions based respectively on the heteroscedastic Gaussian distribution and the symmetric bivariate Cauchy distribution. For these functions we establish a Nyquist rate and threshold values such that all local maxima (lines containing a minimum number of points) are found; the analytic form of the likelihood function is used to obtain precise estimates of the line parameters. Finally we obtain analytically the first order covariance of the estimated line parameters, and hence a confidence envelope about the fitted line. Contrary to popular belief this is not obtained by analyzing the shape of peaks in the Hough space. The motivation for this paper is to establish a rigorous framework for finding curves in images in a maximum likelihood sense; only armed this foundation can the more difficult problems in pattern recognition be addressed in a rigorous manner.