What is the need for this transform, and what benefits do we get compared to conventional Fourier transform?
In general, the fast Fourier transform (FFT) is faster than the classical Fourier transform when doing it numerically, see e.g. http://en.wikipedia.org/wiki/Fast_Fourier_transform for more.
As for the pseudo-polar FFT, see e.g. this paper A FRAMEWORK FOR DISCRETE INTEGRAL TRANSFORMATIONS I – THE PSEUDO-POLAR FOURIER TRANSFORM by A. Averbuch et al.
(http://epubs.siam.org/doi/abs/10.1137/060650283)
In general, the fast Fourier transform (FFT) is faster than the classical Fourier transform when doing it numerically, see e.g. http://en.wikipedia.org/wiki/Fast_Fourier_transform for more.
As for the pseudo-polar FFT, see e.g. this paper A FRAMEWORK FOR DISCRETE INTEGRAL TRANSFORMATIONS I – THE PSEUDO-POLAR FOURIER TRANSFORM by A. Averbuch et al.
(http://epubs.siam.org/doi/abs/10.1137/060650283)
The application of a Polar coordinate FFT to rectilinear coordinates is one type of Pseudo-Polar FFT application. This operation involves using a polar FFT on a grid of coordinates that can be easily transformed into polar coordinates. The rational is that the FFT is quicker and easier to implement. There can be application reasons for this where in the case of a polar sampled space some moments of the bessel functions can be easily determined or impervious to sampling artifacts. A simple application paper that demonstrates these points and is freely available is
"Applications of Pseudo-polar FFT in Synthetic Aperture Radiometer Imaging" by Zhang, Wu, and Sun - PIERS ONLINE, Vol 3., No. 1, 2007.
Figure 1 and 2 demonstrate the basic idea of the method while the discussion highlights some of its advantages. I've included the link as an attachment. I hope this answers your questions.
http://www.piers.org/piersonline/pdf/Vol3No1Page25to30.pdf
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