# Interpolation formula for two dimensional signal reconstruction in the frequency domain from polar samples

In the book, Advanced Topics in Shannon Sampling and Interpolation Theory by Robert J. Marks II, one may find an interpolation formula for reconstructing a two dimensional signal from regular polar samples in the frequency domain:

Let $f(x,y)$ be space-limited to $2A$. Let its Fourier transform in polar coordinates $F(\rho,\phi)$ be angularly band-limited to $K$. Then $F(\rho,\phi)$ can be reconstructed from its polar samples via

$$F(\rho,\phi)=\sum_{n=-\infty}^\infty\sum_{k=0}^{N-1}\widetilde{F}\left(\frac{n}{2A},\frac{2\pi k}{N}\right)\operatorname{sinc}\left[\frac{2A(\rho-n)}{2A}\right]\frac{\sin\left[\frac{1}{2}(N-1)\left(\phi-\frac{2\pi k}{N}\right)\right]}{N\sin\left[\frac{1}{2}\left(\phi-\frac{2\pi k}{N}\right)\right]},$$ where $N$ is assumed even and $$\widetilde{F}\left(\frac{n}{2A},\frac{2\pi k}{N}\right)=\begin{cases}F\left(\frac{n}{2A},\frac{2\pi k}{N}\right),&n\ge 0, \\ F\left(-\frac{n}{2A},\frac{2\pi k}{N}+\pi\right),&n<0.\end{cases}$$

I figured that the argument of the cardinal sine function contained a typo, since the $2A$ in the numerator and denominator would appear to cancel. Thus I would guess that the author actually had

$$\operatorname{sinc}\left[\frac{\rho-2An}{2A}\right]$$

in mind. Correct me if I'm wrong?

Then the last quotient with the sine functions looks very similar to the Dirichlet kernel:

$$D_N\left(\phi-\frac{2\pi k}{N}\right)=\frac{\sin\left[\left(N+\frac{1}{2}\right)\left(\phi-\frac{2\pi k}{N}\right)\right]}{\sin\left[\frac{\phi-\frac{2\pi k}{N}}{2}\right]}$$

but is not quite the same.

Incidentally, my attempts to implement the interpolation formula in the theorem via Matlab for specific examples have failed completely with regards to reconstructing a signal. Perhaps someone can point me in the direction of a correct formula for signal reconstruction in the frequency domain from polar samples? suggests that $$\operatorname{sinc}\left[2A(\rho-\frac{n}{2A})\right]$$ might be the correct form of the $\operatorname{sinc}$ term.
• Then this is the same as $$\operatorname{sinc}\left[\frac{\rho-n\Delta\rho}{\Delta\rho}\right],$$ with $\Delta\rho=\frac{1}{2A}$. Interestingly I also tried to implement this formula on Matlab too, but with no success. – Jason Born Jun 9 '17 at 18:08