Low Space-bandwidth Product Signal Extrapolation Using a New Robust Iterative Random Grain Allocation Scheme
M. Stojancic, G. Eichmann · Journal of Modern Optics · 1989
The extrapolation in the space domain of a partially observed low space-bandwidth product (SBP) sequence or, equivalently, the resolution of the Fourier spectra in the frequency domain, in the presence of appreciable noise, is considered. The unknown sequence estimate is based on a number of acquired samples on a given measurement interval and the prior knowledge of the signal frequency bandlimit. Using an approach similar to a Monte-Carlo method, the extrapolated sequence samples are constructed from variably sized elementary grains. The new iterative algorithm, at each iteration step, based on a random-number generator, decides both the sample position to be considered and the sign of a grain that might be added to the current sample value. A sample update in each iteration step is either accepted or rejected in accordance with an appropriate decision rule. While exploiting the extrapolated sequence frequency bandlimit as a constraint, this decision rule is based on a non-increasing l1-norm of a cumulative error vector. As the extrapolated sequence approaches its final form, the elementary grain size is decreased to allow for subtle sample updates. A heuristic schedule for gradual change of the elementary grain size, similar to the temperature schedule of the simulated annealing method, is used. A pre-processing, which compensates for the model inconsistency that is due to either the presence of noise and/or the lack of precision of the linear degradation operator, is also introduced. Furthermore, it is shown that an additional constraint, such as a given signal upper bound, greatly improves the quality of reconstruction. Several simulation examples, for the extrapolation of low-SBP sinusoidal and other arbitrary sequences and in the presence of a high level of noise, are presented.