3D Imaging

The most straight forward method suggested by Eq. 14.2.5 for recovering the sky brightness distribution, is to perform a 3D Fourier transform of $V(u,v,w)$. This requires that the $w$ axis be also sampled at least at Nyquist rate. For most observations it turns out that this is rarely satisfied and doing a FFT on the third axis would result into severe aliasing. Therefore in practice, the transform on third axis is usually done using the direct Fourier transform (DFT), on the un-gridded data.

For performing the 3D FT (FFT on the $u$ and $v$ axis and DT on the $w$ axis) one would still need to know the number of planes needed along the $n$ axis. This can be found using the geometry as shown in Fig. 14.2. The size of the synthesized beam in the $n$ direction is comparable to that in the other two directions and is given by $\approx \lambda/B_{max}$ where $B_{max}$ is the longest projected baseline length. Therefore the separation between the planes along $n$ should be $\le \lambda/2B_{max}$. The distance between the tangent plane and points separated by $\theta$ from the phase center is given by $1-cos(\theta)\approx \theta^2/2$. For critical sampling then would be

$$N_n = B_{max}\theta^2/\lambda.$$ (14.2.7)

At 327 MHz for GMRT, $B_{max} \approx 25$ km. Therefore, for mapping $1^\circ$ field of view without distortions, one would required 8 planes along the $n$ axis. With central square alone however, one plane should be sufficient. At these frequencies it becomes important to map most of the primary beam since the number and the intensity of the background sources increase and the side lobes of these background sources limit the dynamic range in the maps. Hence, even if the source of interest is small, to get the achievable dynamic range (or close to it!), one will need to do a 3D inversion (and deconvolution).

Another reason why more than one plane would be required for very high dynamic range imaging is as follows. Strictly speaking, the only point which completely lies in the tangent plane is the point at which the tangent plane touches the celestial sphere. All other points in the image, even close to the phase center, lie slightly below the tangent plane. Deconvolution of the tangent plane then results into distortions for the same reason as the distortions arriving from the deconvolution of a point source which lies between two pixels in the 2D case. As in the 2D case, this problem can be minimized by over sampling the image and that, in this case, implies having at least 2 planes in the $n$ axis, even if the Eq. 14.2.7 tells that 1 plane is sufficient.

Figure 14.3: Approximation of the celestial sphere by multiple tangent planes (polyhedron imaging).