A surprising Radon transform result and its application to motion detection

An elliptical region of the plane supports a positive-valued function whose Radon transform depends only on the slope of the integrating line. Any two parallel lines that intersect the ellipse generate equal line integrals of the function. We prove that this peculiar property is unique to the ellips e; no other convex, compact region of the plane supports a nonzero-valued f unction whose Radon transform depends only on slope. We motivate this problem by considering the detection of a constant-velocit y moving object in a sequence of images, in the presence of additive, white , Gaussian noise. The intensity distribution of the object is known, but th e velocity is only assumed to lie in some known set, for example, an ellips e or a rectangle. The object is to find a space-time linear filter, operati ng on the image sequence, whose minimum output signal-to-noise ratio (SNR) for any velocity in the set is maximized. For an ellipse (and its special c ases, the disk and the line-segment) the special Radon transform property o f the ellipse enables us to obtain a closed-form, analytical solution for t he minimax filter, which significantly outperforms the conventional three-d imensional (3-D) matched filter. This analytical solution also suggests a c onstrained minimax filter for other velocity sets, obtainable in closed for m, whose SNR can be very close to the minimax SNR.