Coarse-fine bimodality analysis of circular histograms

Abstract The bimodality of a population P can be measured by dividing its range into two intervals so as to maximize the Fischer distance between the resulting two subpopulations P 1 and P 2 . If P is a mixture of two (approximately) Gaussian subpopulations, then P 1 and P 2 are good approximations to the original Gaussians, if their Fisher distance if great enough. For a histogram having n bins this method of bimodality analysis requires n − 1 Fisher distance computations, since the range can be divided into two intervals in n − 1 ways. The method can also be applied to ‘circular’ histograms, e.g. of populations of slope or hue values; but for such histograms it is much more computationaly costly, since a circular histogram having n bins can be divided into two intervals (arcs) in n ( n − 1)/2 ways. The cost can be reduced by performing bimodality analysis on a ‘reduced-resolution’ histogram having n / k bins; finding the subdivision of this histogram that maximizes the Fisher distance; and then finding a maximum Fisher distance subdivision of the full-resolution histogram in the neighborhood of this subdivision. This reduces the required number of Fisher distance computations to n ( n − 1)/2 K 2 + O( k ). For histograms representing mixtures of two Gaussians, this method was found to work well for n / k as small as 8.