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...

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Bibliographic Details
Published in:IEEE transactions on image processing 1999, Vol.8 (8), p.1039-1049
Main Authors: Marzetta, T.L., Shepp, L.A.
Format: Article
Language:English
Subjects:
Additive noise
Additive white noise
Applied sciences
Ellipses
Exact sciences and technology
Exact solutions
Gaussian noise
Image processing
Image sequences
Information, signal and communications theory
Matched filters
Mathematical analysis
Minimax technique
Minimax techniques
Motion detection
Nonlinear filters
Object detection
Planes
Radon
Signal processing
Signal to noise ratio
Telecommunications and information theory
Transforms
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Summary: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 ellipse; no other convex, compact region of the plane supports a nonzero-valued function whose Radon transform depends only on slope. We motivate this problem by considering the detection of a constant-velocity moving object in a sequence of images. In the presence of additive, white, Gaussian noise. The intensity distribution of the object is known, but the velocity is only assumed to lie in some known set, for example, an ellipse or a rectangle. The object is to find a space-time linear filter, operating 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 cases, the disk and the line-segment) the special Radon transform property of the ellipse enables us to obtain a closed-form, analytical solution for the minimax filter, which significantly outperforms the conventional three-dimensional (3-D) matched filter. This analytical solution also suggests a constrained minimax filter for other velocity sets, obtainable in closed form, whose SNR can be very close to the minimax SNR.
ISSN:1057-7149
1941-0042
DOI:10.1109/83.777085