Increasing the spatial resolution of signals in optical systems

Abstract

Reconstruction of the signal in the intervals between discrete values is of great importance in solving the problem of spatial superresolution in optical microscopy and digital holography. The article deals with the issue of restoring high-resolution image elements from a certain number of raster images displaced by a sub-pixel shift. The numerical values of the image samples are obtained by spatial integration over some finite area of regular rasters. The spatial resolution is increased using an analytical expression for the spectrum of discrete signals obtained using the apparatus of generalized functions. Unlike ideal sampling, the spectrum of the function is supplemented by a multiplier, whose form depends on the type of aperture. To obtain high-resolution image elements, it is necessary to divide the Fourier spectrum of the sampled image by a factor depending on the selected aperture. The spectrum of the aperture is usually used, therefore, if the spectrum of the image obtained by averaging over a certain aperture is known, then the spectrum of the original image can also be obtained. Apertures of various shapes are used, for example, elliptical, diamond-shaped, hexagonal, but most often rectangular apertures. The simulation results are presented for a rectangular aperture but in the case of its substitution with, for example, a set of regular apertures in the form of a circle, the expression will be true for regular circular rasters. The analytical expression for the spectrum of the image obtained by averaging over a certain aperture can be used to reconstruct the spectrum of the original image. Having received the inverse Fourier transform from it, it is possible to obtain the original image. With an increase in the spatial resolution it becomes possible to carry out studies by methods of digital holography of volumetric diffuse objects while retaining the quality of analog holography (when recording in photographic media) and create optical superresolution systems based on optical microscopes.

Keywords

• sampling
• sampling rate
• spatial frequencies
• generalized functions
• kotelnikov's theorem
• fourier transform
• spectrum
• superresolution