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Fourier-Mellin Transform

FMT as an Image Processing Tool

The Fourier-Mellin transform is a useful mathematical tool for image recognition because its resulting spectrum is invariant in rotation, translation and scale. The Fourier Transform itself (FT) is translation invariant and its conversion to log-polar coordinates converts the scale and rotation differences to vertical and horizontal offsets that can be measured. A second FFT, called the Mellin transform (MT) gives a transform-space image that is invariant to translation, rotation and scale.

The Fourier-Mellin Transform

 

Normalized Cross Correlation

NCC as an Image Processing Tool

The correlation between two signals (cross correlation) is a standard approach to feature detection as well as a component of more sophisticated techniques. Textbook presentations of correlation describe the convolution theorem and the attendant possibility of efficiently computing correlation in the frequency domain using the fast Fourier transform. Unfortunately the normalized form of correlation (correlation coefficient) preferred in template matching does not have a correspondingly simple and efficient frequency domain expression. For this reason normalized cross-correlation has been computed in the spatial domain. Due to the computational cost of spatial domain convolution, several inexact but fast spatial domain matching methods have also been developed. An algorithm for obtaining normalized cross correlation from transform domain convolution has been developed, see Lewis [1]. The new algorithm in some cases provides an order of magnitude speedup over spatial domain computation of normalized cross correlation.