Rotationally invariant correlation filtering for multiple images

Abstract

Numerous problems in optical pattern recognition require the classification of a finite number of images regardless of their orientation angles. A method is presented for designing translationally invariant optical correlation filters that produce a specified response to rotations of each of the input images. The correlation filter is formed as an infinite linear combination of the angular Fourier harmonics of the n input images. Imposing the requirements for the specified rotational response of the filter to each input image leads to a vector-matrix convolution equation that is to be solved for the unknown angular weighting coefficients. Expansion into vector-matrix angular Fourier series and term-by-term solution for the Fourier coefficients gives a number of systems of n equations for n unknowns. The elements of the matrix describing each system of equations are the correlations between the angular harmonics of the input images. Inverting this matrix decorrelates the angular harmonics of the input images and allows an arbitrary response to be obtained. Choosing the special case that the rotational responses consist of only one nonzero Fourier series coefficient gives the generalization of the circular harmonic filter to n images. We call this the multiple circular harmonic filter. The classification of n input images can be achieved by choosing the rotational response amplitude to be a different constant value for each image. As shown by examples, this allows multiple input images to be distinguished while preserving rotational invariance.