Optimal detection of known moving objects in a noisy image sequence with velocity uncertainty

The optimal algorithm for detecting a moving object in an image sequence that is corrupted by additive, white, Gaussian noise depends on the available prior knowledge of the object´s velocity and its intensity distribution. This paper addresses the optimal detection problem where the object intensity distribution in known, but the object velocity is assumed only to be contained in some set, V. We formulate an optimal 3-D detection filter as the solution of a convex mini-max optimization problem, such that for all velocities contained in V, the minimum signal-to-noise ratio (SNR) improvement of the filter is maximized. The optimal filter is expressed in terms of a 2-D Lagrange multiplier function, that in turn satisfies a 2-D integral equation. For the case where V is a circular disk one can obtain a closed-form analytical solution for the optimal filter that gives a greater than 6.7 dB improvement in mini-max SNR compared with an optimized 3-D matched filter that is designed for the center velocity of the disk. Equivalently, for the same guaranteed SNR the optimal filter can cover a region of the velocity plane having an area 22 times greater than that covered by the 3-D matched filter. Put still another way, one optimal filter gives the same performance as a bank of 22 3-D matched filters