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Alan Johnston; The harmonic vector average route to global motion calculation. Journal of Vision 2009;9(8):643. doi: 10.1167/9.8.643.
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© ARVO (1962-2015); The Authors (2016-present)
An object translating at a constant velocity gives rise to a variable local velocity field. The speed of the normal component to image contours varies with the cosine of the angle between the normal and the global motion vector. Dense arrays of randomly oriented Gabor patches with speeds chosen to conform to a single global translation can appear to move coherently with the correct global velocity. The global motion could be calculated from the distribution of pairwise intersections of the set of possible velocities consistent with the local estimates - the intersection of constraints (IOC) strategy. However, for noisy local measures there are multiple solutions leading to combinatorial complexity and problems in distribution representation and peak finding that make this approach unattractive as a biological model. The alternative of fitting a circle through the origin of velocity space is also non-trivial. The vector sum clearly does not give the global velocity and the vector average provides a vector close to the correct direction but with half the speed of the global motion. From inversive geometry we learn that a circle through the origin inverted in the unit circle maps to a line. Thus inverse speed measurements fall on a line. The correct global speed, represented by the normal to this line and its inverse magnitude, can be recovered in closed form using a simple least squares formula. Also unlike the vector average the harmonic vector average (the reciprocal of the mean of the reciprocal speeds) provides an accurate estimate of the global speed given an unbiased sample of orientations and even for a biased sample (as in a type II array) will give a velocity estimate that lies on the circle in velocity space - a valid global motion given a limited sample of orientations.
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