Image restoration using a discrete point spread function with consideration of finite pixel size

Objectives. The problem of restoring defocused and/or linearly blurred images using a Tikhonov-regularized inverse filter is considered. A common approach to this problem involves solving the Fredholm integral equation of the first convolution type by means of discretization based on quadrature formulas. The work sets out to obtain an expression of the point scattering function (PSF) taking into account pixel size finiteness and demonstrate its utility in application.

Methods. The research is based on signal theory and the method of digital image restoration using Tikhonov regularization.

Results. Taking into account the finiteness of the pixel size, discrete PSF formulas are obtained both for the case of a defocused image and for the case of a linearly blurred image at an arbitrary angle. It is shown that, while differences between the obtained formulas and those traditionally used are not significant under some conditions, under other conditions they can become significant.

Conclusions. In the case of restoring images at the resolution limit, i.e., when the pixel size cannot be considered negligibly small compared to the details of the image, the proposed approach can slightly improve the resolution. In addition, the derived formula for the discrete PSF corresponding to linear blur in an arbitrarily specified direction can be used to solve the problem without the need for prior image rotation and account for the blur value with subpixel accuracy. This offers an advantage in terms of improving the resolution of extremely fine details in the image, allowing the obtained formula to be used in solving the adaptive deconvolution problem, where precise adjustment of PSF parameters is required.

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