We consider the sampling of bandlimited spatiotemporal signals subject to the time-sequential (TS) constraint that only one spatial position can be sampled at any given time. Using the powerful techniques of lattice theory, we develop a new unifying theory linking TS sampling with generalized multidimensional sampling. The results have a geometric nature, involving simultaneous packing of the spectral and spatial supports in their respective domains. We provide a complete characterization of TS lattice patterns, and, extending the study to temporally nonuniform patterns, analyze their minimum and average temporal sampling rates. Unlike previous studies of TS sampling, our results apply to very general multidimensional spatial and spectral supports. We present tight bounds on the temporal parameters of those TS sampling patterns that produce zero aliasing error. The use of a TS sampler for bufferless source coding of spatiotemporal signals is considered as an attractive possibility.

For pt.I see ibid., vol.43, no.1, p.190-207 (1997). We consider the sampling of bandlimited spatiotemporal signals subject to the time-sequential (TS) constraint that only one spatial position can be sampled at any given time. Part I of this paper developed a new unifying theory linking TS sampling with generalized multidimensional sampling. It provided a complete characterization of time-sequential lattice patterns, including tight bounds on the temporal parameters of those time-sequential sampling patterns that produce zero aliasing error. In this paper we present large space-spatial-bandwidth product asymptotics for these bounds. One of the surprising results is that in many cases, there exist optimal patterns, for which, asymptotically, there is no extra penalty for lattice sampling subject to the time-sequential constraint, as compared to unconstrained multidimensional sampling. The implication to source coding is that an optimum encoder for spatiotemporal signals can be implemented with no buffering or other processing using a time-sequential sampler. The results apply to very general multidimensional spatial and spectral supports (star shaped, or at most convex).

The authors consider the tomographic reconstruction of objects with spatially localized temporal variation, such as a thorax cross section with a beating heart. The conventional scan format, in which projections are taken progressively around the object, requires high and sometimes infeasible scan rates to avoid motion artifacts in the reconstructed images. The authors formulate the problem of data acquisition as a time-sequential sampling problem of spatially and temporally bandlimited signals, where only one view can be taken at a time, but the time interval between successive views is independent of their angular separation. These conditions, naturally satisfied in magnetic resonance imaging and in X-ray CT using the Imatron system, can also be satisfied by a conventional system with a continuously and rapidly spinning gantry with source pulsing. Theoretical analysis, which includes tight performance bounds, shows that by using an optimally scrambled angular sampling order, the required scan rate can be lowered as much as four times, while preserving image quality. The analysis also greatly simplifies the design of the optimum scan pattern by reducing it to a constrained geometric packing problem.

For pt.I see ibid., vol.4, no.5, p.642-53 (1995). In pt.I the authors presented a theoretical analysis of tomographic reconstruction of objects with spatially localized temporal variation, such as a thorax cross section with a beating heart. That analysis showed that by using an optimally scrambled angular sampling order, the scan rate required to avoid motion artifacts in the reconstructed images can be lowered as much as four times while preserving image quality. Here, the authors present a simple design procedure for the optimum choice of angular sampling pattern, which depends only on pre-specified geometric and spectral parameters and the desired spatial resolution. The resulting patterns have a simple congruential structure. Reconstruction is accomplished by interpolation to standard time-invariant scan format, followed by conventional reconstruction. The interpolation only requires linear shift-invariant separable filtering, at a negligible computational cost. Simulation results demonstrate the technique and validate the analysis for both bandlimited and approximately bandlimited objects.

Sampling of signals that vary in time and have one or more spatial dimensions is studied for constrained sampling sets. The constraint imposed is that only one spatial sample can be taken at a given time, hence the name time-sequential sampling. J. Allebach (1984) has shown previously that the amount of aliasing error is not only a function of the sampling rates, but is also strongly dependent on the order in which spatial sample points are taken. A new design technique for finding optimal sampling patterns that minimize the sampling rates and produce zero aliasing error is developed by viewing time-sequential sampling as a specialized case of generalized two-dimensional sampling. New lower bounds for sampling rates required to produce zero aliasing error are given. These bounds are then used as stopping criteria for the aforementioned design technique. Theorems are given which state conditions under which the design problem reduces to a general two-dimensional sampling problem.>

Minimax-optimal interpolation algorithms minimize the error resulting from the worst signal from an allowable class. The result is presented that if this class lies in a Hilbert space, the minimax-optimal algorithm is independent of or invariant to the error norm. The result encompasses a broad class of inverse problems.>