Integral logarithmic transforms are defined for both one-dimensional and two-dimensional input functions. These have the desirable properties of linearity and invariance to scale change of the input. Two-dimensional integral logarithmic transform is additionally invariant to rotation. The integral logarithmic transforms are conveniently inverted by simple differentiation. Second, a new approach is given for the problem of reconstruction of phase from modulus data. A set of Wiener-filter functions is formed that multiply, in turn, displaced versions of the modulus data in frequency space such that the sum is a minimum L₂-error norm solution for the object. The required statistics are power spectra of the signal and noise, and correlation between modulus data at a given frequencies and complex object spectral values at adjacent frequencies. Finally, a new technique is proposed to reconstruct a turbulent image from a superposition model. Imagery through random atmospheric turbulence is modeled as a stochastic superposition process. By this model, each short-exposure point spread function is a superposition of randomly weighted and displaced versions of one intensity profile. If we could somehow estimate the weights and displacements for a given image, then by the superposition model we would known the spread function, and consequently, could invert the imaging equation for the object.
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