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## Image Volume

Let be treated as an independent variable. Then one can write a 3D Fourier transform of with the conjugate variable for () being (), as

 (14.2.2)

Substituting for from Eq. 14.1.1 we get

 (14.2.3)

Using the general result
 (14.2.4)

we get
 (14.2.5)

This equation then provides the connection between the 2D sky brightness distribution given by and the result of 3D Fourier inversion of given by referred to as the Image volume.

 (14.2.6)

Hereafter, I would use to refer to the this Image volume.

In Eq.14.1.1, we have ignored the fringe rotation term in the exponent. This is done here only for mathematical (and typing!) convenience. The effect of including this term would be a shift of the Image volume by one unit in the conjugate axis, namely . Hence, the effect of fringe stopping is to make the top most plane of tangent to the phase center position on the celestial sphere with the rest of the sphere completely contained inside the Image volume as shown in Fig. 14.1.

Remember that the third variable of the Image volume is not an independent variable and is constrained to be . Eq 14.2.6 then gives the physical interpretation of . Imagine the celestial sphere defined by enclosed by the Image volume , with the top most plane being tangent to the celestial sphere as shown in Fig. 14.1. Eq. 14.2.6 then says that only those parts of the Image volume correspond to the physical emission which lie on the surface of the celestial sphere. Note that since the visibility is written as a function of all the three variables , the transfer function will also be a volume. A little thought will then reveal that will be finite away from the surface of the celestial sphere also, but that would correspond to non-physical emission in the Image volume due to the side lobes of the telescope transfer function (referred to by Point spread function (PSF) or Dirty beam in the literature). A 3D deconvolution using the Dirty image- and the Dirty beam-volumes will produce a Clean image-volume. Therefore, after deconvolution, one must perform an extra operation of projecting all points in the image volume along the celestial sphere onto the 2D tangent plane to recover the 2D sky brightness distribution. Fig. 14.2 is the graphical equivalent of the statements in this paragraph.

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