Refereed full papers (journals, book chapters, international conferences)

1984

A size- and position-invariant description of an image function can be obtained via the absolute value of the Mellin transform of its Fourier amplitude spectrum. If the transform is implemented on a digital computer via a discrete Fourier-Mellin transform, these exact invariances are not preserved due to sampling- and border-effects. In this paper these effects are discussed, and an alternative correlation method is proposed. The method consists of calculating the normalized absolute magnitude of the discrete Fourier transform (DFT) of the image function (which gives invariance to translation and multiplicative amplitude changes) and a subsequent logarithmic distortion in x- and y-direction, which converts scaling to translation. Two such transforms are compared by calculating the normalized Euclidean distances between both for all possible relative shifts along the main diagonal. If, for some shift, the distance has a minimum below a similarity threshold, the underlying image functions will probably differ only by translation and scaling. The magnitude of this shift is related to the scale factor between the objects. Good separation between similar and nonsimilar objects is possible if two size criteria imposed by the DFT are met: the total object size must not exceed fracN 4 (N is the number of image points in each dimension), and object details have to be larger than about 4 image points. As a consequence, N increases with object complexity and desired scale range. With N = 64, only a limited object manifold can be handled; with N = 256, useful results are found for quite complicated forms. The method described has the additional advantage that the magnification factor is accessible. Its disadvantage, the requirement that two transforms have to be compared at many relative positions, can be avoided by an independent evaluation of the scale factor via the normalized central second moments of the image functions. These moments can be calculated from the normalized absolute magnitude of the discrete Fourier transform. If the size ratio of image and sample is thus known, the relative position can be estimated, where the logarithmically distorted spectra would coincide in case of similarity. Correct scale factors are obtained if the object size is between 4 and about fracN 2 image points. Since one comparison suffices for every stored sample, the proposed method is even faster than a discrete Fourier-Mellin transform.