1 July 2006 Annular Radon transform and axicon transform
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Optical Engineering, 45(7), 078201 (2006). doi:10.1117/1.2226990
We consider integral transforms that can describe ideal imaging optical systems with annular impulse response. The integral transform, referred to as the annular Radon transform (ART), is considered as an average taken along all circles of fixed radius in the plane. The inversion formula for the ART is given. It is shown that for a function of two variables to be reconstructed uniquely, the averaging functions along all circles with at least two different radii need to be known. The ART can be optically realized using a spatial filtering system (SFS) with amplitude filter, its transmittance being proportional to the Bessel function of zero order. Also, an axicon transform (AT) is considered that can be optically realized using a SFS with axicon in the spatial-frequency plane. The inverse and direct ATs are shown to be of the same form (up to a constant). A numerical comparison of the AT and ART has been made when both were used for reconstruction of transformed and distorted images. The relation between the distortion variances of the output functions and the SFS reconstructed input function is derived.
Viktor Viktorovich Kotlyar, Alexander Alexeevich Kovalev, "Annular Radon transform and axicon transform," Optical Engineering 45(7), 078201 (1 July 2006). http://dx.doi.org/10.1117/1.2226990


Radon transform


Bessel functions

Spatial filters

Optical filters

Optical engineering

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