Image Deconvolution

As mentioned in the previous section 3.3.2.2, because synthesis arrays sample the $(u,v)$ plane at discrete locations, there is incomplete knowledge about the Fourier transform of the source intensity distribution. The measured $(u,v)$ data can be thought of as the true distribution, V(u,v), in the $(u,v)$ plane multiplied by sampling function, S(u,v). The convolution theorem states that the Fourier transform of the sampled distribution (the dirty map, $I^d$) is equal to the convolution of the Fourier transform of the true source $(u,v)$ distribution (the true image, I) and the Fourier transform of the sampling function (the dirty beam,$DB$) is given by,

$$\rm {I^d = I * DB \rightleftharpoons V(u,v) \times S(u,v)},$$ (3..5)

where, the operator '*' indicates convolution, and $\rightleftharpoons$ indicates the Fourier transform. Image reconstruction algorithms attempt to recover the true image through the various deconvolution schemes. Deconvolution algorithms removes the side-lobes present in the image plane and in doing so estimates the visibility function at spatial frequencies in the un-sampled part of the $(u,v)$ plane.

In radio astronomy, the most commonly used deconvolution algorithm is CLEAN, which we briefly describe below.