Digital Image and Spectrum Restoration by Quadratic Programming and by Modified Fourier Transformation
Issue No. 04 - April (1979 vol. 1)
Johan Philip , Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.
We consider the convolution equation f * h + e = d, where f is sought, h is a known ``point spread function,'' e represents random errors, and d is the measured data. All these functions are defined on the integers mod(N). A mathematical-statistical fonnulation of the problem leads to minff * hdA, where the A-norm is derived from the statistical distribution of e. If f is known to be nonnegative, this is a quadratic progamming problem. Using the discrete Fourier transforms (DFT's) F, H, and D of f, h, and d, we arrive at a minimization in another norm: minF F · H-D ¿. A solution would be F = D/H, but H has zeros. We consider the theoretical and practical difficulties that arise from these zeros and describe two methods for calculating F numerically also when H has zeros. Numerical tests of the methods are presented, in particular tests with one of the methods, called ``the derivative method,'' where d is a blurred image.
Johan Philip, "Digital Image and Spectrum Restoration by Quadratic Programming and by Modified Fourier Transformation", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 1, no. , pp. 385-399, April 1979, doi:10.1109/TPAMI.1979.4766947