Mathematical Sciences: Statistical Methods for Incomplete and Biased Data with New Applications

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The problem of recovering an input signal from a blurred output, in an input-output system with linear distortion, is ubiquitous in science and technology. When the blurred output is not corrupted by statistical noise, the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters, subject to positivity con- straints on the solution. By showing that all such linear inverse problems with positivity constraints can be interpreted as statistical estimation problems, extremely simple algorithms can be developed for finding the problems' maximum likelihood solutions. Extensions to specific problems like noise degrada- tion, estimation of blurring parameters (parameters of the point spread function) and regularization techniques will follow. The methodology will be applied to new problems including restoration of images blurred by relative motion between the subject and the camera lens. The blurring of a signal (sound, image, etc.) is common in science and technology, for example the systematic distort observed signal becomes what mathematicians call linear inverse problems with positivity constraints. This research will develop methods of solution which are broadly applicable to such problems. The basis for this approach is to draw on statistical principles and exploit the connection between such linear inverse problems and statistical estimation problems for incomplete data where methods already exist.