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2D Fourier Transforms

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This is an extremely brief introduction to 2D transforms. This is mostly to help you understand telescopes, interferometers and beamforming techniques that is covered in another tutorial series.

The 2D Fourier transform is defined exactly like its 1D equivalent. The mathematical equations are given below:

$$\begin{align*} F(u,v) &= \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} f(x,y) e^{-2\pi j (ux+vy)} \; \rm{d}x \rm{d}y \\[2em] f(x,y) &= \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} F(u,v) e^{2\pi j (ux+vy) \; \rm{d}u \rm{d}v} \end{align*}$$

The 2-dimensional exponents in the equations correspond to a combination of 2D sine and cosine waves. An illustration of these basis vectors for an 8x8 pixel image is shown below:

One extremely useful 2D-transform is that of a circular disk or aperture. The Fourier transform is called an Airy disk.

You will encounter this pair in optics– Fraunhofer diffraction pattern of a circular aperture, and in the theory of radio telescopes. The Airy Disk is the 2D equivalent of the sinc-function. The central portion of the Airy Disk can be approximated to a Gaussian profile.