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Digital Image and Spectrum Restoration by Quadratic Programming and by Modified Fourier Transformation

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1 Author(s)
Philip, Johan ; 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.

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Pattern Analysis and Machine Intelligence, IEEE Transactions on  (Volume:PAMI-1 ,  Issue: 4 )