Signal and Image Synthesis: Alternating Projections Onto Convex Sets

Abstract

Alternating projections onto convex sets (POCS) [319, 918, 1324, 1333] is a powerful tool for signal and image restoration and synthesis. The desirable properties of a reconstructed signal may be defined by a convex set of constraint parameters. Iteratively projecting onto these convex constraint sets can result in a signal which contains all desired properties. Convex signal sets are frequently encountered in practice and include the sets of bandlimited signals, duration limited signals, causal signals, signals that are the same (e.g., zero) on some given interval, bounded signals, signals of a given area and complex signals with a specified phase. POCS was initially introduced by Bregman [156] and Gubin et al. [558] and was later popularized by Youla & Webb [1550] and Sezan & Stark [1253]. POCS has been applied to such topics as acoustics [300, 1381], beamforming [426], bioinformatics [484], cellular radio control [1148], communications systems [29, 769, 1433], deconvolution and extrapolation [718, 907, 1216], diffraction [421], geophysics [4], image compression [1091, 1473], image processing [311, 321, 470, 471, 672, 736, 834, 1065, 1069, 1093, 1473, 1535, 1547, 1596], holography [880, 1381], interpolation [358, 559, 1266], neural networks [1254, 1543, 909, 913, 1039], pattern recognition [1444, 1588], optimization [598, 1359, 1435], radiotherapy [298, 814, 1385], remote sensing [1223], robotics [740], sampling theory [399, 1334, 1542], signal recovery [320, 737, 1104, 1428, 1594], speech processing [1450], superresolution [399, 633, 654, 834, 1393, 1521], television [736, 786], time-frequency analysis [1037, 1043], tomography [1103, 713, 1212, 1213, 1275, 916, 1322, 1060, 1040], video processing [560, 786, 1092], and watermarking [19, 1470]. Although signal processing applications ofPOCS use sets of signals,POCSis best visualized viewing the operations on sets of points. In this section, POCS is introduced geometrically in two and three dimensions. Such visualization of POCS is invaluable in application of the theory.