Polyhedron Imaging

As mentioned above, emission from the phase center and from points close to it lie approximately in the tangent plane. Polyhedron imaging relies on exploiting this fact by approximating the celestial sphere by a number of tangent planes as shown in Fig. 14.3. The visibility data is phase rotated to shift the phase center to the tangent points of the various planes and a small region around the tangent point is then mapped using the 2D approximation. In this case however, one needs to perform a joint deconvolution involving all tangent planes since the sides lobes of a source in one plane would leak into other planes as well.

The number of planes required to map an object of size $\theta$ can be found simply by requiring that maximum separation between the tangent plane and the region around each tangent point be less than $\lambda/B_{max}$, the size of the synthesized beam. As shown earlier, the separation of a point $\theta$ degrees away from the tangent point is $\approx \theta^2$. Hence for critical sampling, the number of planes required is equal to the solid angle subtended by the sky being mapped ($\theta_f^2$) divided by the solid angle of the synthesized beam ($\theta^2$)

$$N_{poly}~=~2\theta_f^2 B_{max}/\lambda~=~2B_{max}\lambda/D^2~~({\rm for}~\theta_f~=~{\rm full}~{\rm primary}~{\rm beam}).$$ (14.2.8)

Notice that the number of planes required is twice as many as the number of planes required for 3D inversion. However since a small portion around the tangent point of each plane is used, the size of each of these planes can be small, offsetting the increase in computations due to the increase in the number of planes required. Another approach which is often taken for very high dynamic range imaging is to do a full 3D imaging on each of the planes. This would effectively increase the size of the field that can be imaged on each tangent plane, thereby reducing the number of planes required.

The polyhedron imaging scheme is available in the current version of AIPS data reduction package and the 3D inversion (and deconvolution) is implemented in the (not any more supported) SDE package written by Tim Cornwell et al. Both these schemes, in their full glory, will be available in the (recently released) AIPS++ package.