A description is given of the relationship between morphologically filtering and then sampling vs. sampling, and then morphologically filtering in the sampled domain. The authors also describe the relationship between morphologically filtering vs. sampling, morphologically filtering in the sampled domain, and then reconstructing. Unlike the standard communication sampling theory where for appropriately low-pass filtered images there is commutivity between sampling and filtering, this is not the case for appropriately morphologically simplified images. The relationship which does exist shows that the commutivity holds to within one sampling interval distance in the unsampled domain and to within two sampling intervals in the sampled domain.>

A real-time scheme is introduced that meets a need for fast, reliable, multiple-target motion detection without accurate object velocity or structure measurements. A multiresolution approach decomposes images into a pyramid with several spatial frequency bands to selectively detect motion of interest. Multiple moving targets are detected using a multiple window, coarse-to-fine focus of attention scheme to reconstruct motion energy and search for targets. Real-time sensor motion (translation and rotation) compensations use hierarchical correlation and minimum perturbation. Results are shown for multiple moving targets in FLIR (forward-looking infrared) image sequences from both stationary and moving sensors.>

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras. We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.) The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, proving Theorem. Every group of bounded order is a direct sum of cyclic groups, and Theorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.