Abstract

Provided certain conditions are satisfied, the zero crossings of a signal constitute a sampling set. Indeed we have shown that images can be reconstructed from such a set. Considering this or other sampling theorems, two fundamental issues must be addressed. First, the existence of a broad class of signals satisfying the conditions in which recovery is guaranteed, and second, the existence of practical stable reconstruction algorithms. In the case of zero crossing the above conditions are not necessarily satisfied. The sine-wave-crossing (SWC) approach overcomes these difficulties. The theoretical framework, independently established by some researchers, provides several reconstruction algorithms for 1-D signals, stable in the sense that small errors in the sampling process result in relatively small errors in the reconstructed signal. Extending these results to SWC contours, we reconstruct images by applying an interpolation algorithm similar to the one used in recovering a signal sampled uniformly at Nyquist rate. Investigating the effects of crossing-location estimation on signal reconstruction, this work highlights the similarity between effects of quantization along the amplitude axis and spatial axes. Adopting a stochastic approach, we provide bounds on the reconstruction error of bandlimited and almost bandlimited signals. The bounds account for the effects of quantization levels, amplitude, and frequency of the added sine-wave and out-of-band energy, on the resultant m.s.e. Computations indicate that the bounds are tight.

© 1985 Optical Society of America