Improved Computing Efficiency of Cross-correlation in the Fourier Domain

Calculation of the cross-correlation can be carried out in both the spatial and Fourier domain. The relative efficiency of the two techniques is dependent on the search area and target area sizes. The Fourier Transform method becomes more efficient as the target area size approaches the search area and with larger search and target areas. This paper outlines techniques which have been investigated into improving the run-time performance of cross-correlation in the Fourier domain. Implementations of cross-correlation in the Fourier Domain generally use Fast Fourier Transform (FFT) algorithms, which require the search and target data sets to be extended with zeros so that their sizes are a common power of 2. These algorithms are referred to as Radix-2 transforms, and work by combining transform levels at successively increasing powers of 2. A potentially more efficient FFT algorithm, called a Mixed Radix FFT, avoids the artificial increase of the data sets and consequently the number of calculations. The method works by calculating transforms at levels equal to the prime factors of the data set size and combining them at levels of the prime multiples. The algorithm’s efficiency is heavily dependent on the highest prime factor, being more efficient the lower this factor is. The Mixed Radix algorithm has been compared to the Radix-2 algorithm for a number of search/target data set sizes in which the highest prime factors are 3 or 5 and significant improvements in run-time performance have been achieved. The results show that performance can be optimised by careful selection of the data set sizes. The paper also presents some results based on real imagery data, comparing the displacement vectors derived with a traditional spatial cross-correlation technique to those derived with the fast Mixed Radix FFT.