Improvement of the resolution of an instrument by numerical solution of an integral equation

AbstractA method of data analysis has been developed which has the same effect as a substantial increase in the resolving power of the instrument which produced the data. The output of a physical instrument is represented by the known function in an integral equation which, if solved, would yield the input which the instrument was intended to measure. But this presents a classic example of an “improperly posed” problem: one for which even small errors in the given data make solution impossible or grossly inaccurate. Earlier investigators concluded noise (including computational error) could be overcome, or a solution could be obtained with detail beyond the Rayleigh limit of resolution of the instrument, but not both at once. Recent approaches require not only statistical information on the background- and instrument-generated noise, but the expected value of the input itself. In many technologically important problems it is the latter which is unavailable. Therefore, a procedure was developed which, from known data only, generates a serviceable starting estimate of the input; iteratively refines the estimate so that, if numerical errors did not accumulate, the sequence would converge to that (computationally inaccessible) estimate which minimizes the rms error; detects the onset of excessive computational error accumulation; and at that point stops with the “best accessible” estimate. Application to computer-simulated experimental data has consistently yielded resolution far beyond the Rayleigh limit, in the presence of substantial noise, with quite modest demands on computer time