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13. The Imaging Principles
Stéphane Guilloteau
IRAM, 300 rue
de la Piscine, F-38406 Saint Martin d'Hères
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Assuming identical antennas, we have shown in previous lectures that an
interferometer measures the visibility function
![\begin{displaymath}
V(u,v) = \ensuremath{\int\!\!\int} B(x,y)\, I(x,y)\, \ensuremath{e^{-2i\pi(ux+vy)}}\,dx\,dy
\end{displaymath}](img958.gif) |
(13.1) |
over an ensemble of points
(ui,vi), i = 1,n, where B(x,y) is the power
pattern of the antennas and I(x,y) the sky brightness distribution.
The imaging process consists in determining as best as possible the sky brightness
I(x,y). Since Eq.13.1 is a convolution, the imaging process will involve
deconvolution techniques.
Let S(u,v) be the sampling (or spectral sensitivity) function
![$\displaystyle S(u,v) \neq 0 \Longleftrightarrow \exists i \in 1,n \, (u_i,v_i) = (u,v)$](img959.gif) |
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![$\displaystyle S(u,v) = 0 \Longleftrightarrow \forall i \in 1,n \, (u_i,v_i) \neq (u,v)$](img960.gif) |
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(13.2) |
The spectral sensitivity function S contains information on the relative
weights of each visibility, usually derived from theoretical noise.
Let us define
![\begin{displaymath}
I_w(x,y) = \ensuremath{\int\!\!\int} S(u,v)\, W(u,v)\, V(u,v)\, \ensuremath{e^{2i\pi(ux+vy)}}\,du\,dv
\end{displaymath}](img961.gif) |
(13.3) |
where W(u,v) is an arbitrary weighting function.
Since the Fourier Transform of a product of two functions is the convolution
of the Fourier Transforms of the functions, Iw(x,y) can be identified with
![\begin{displaymath}
I_w(x,y) = (B(x,y) I(x,y)) \ensuremath{\ast\!\ast} (D_w(x,y))
\end{displaymath}](img962.gif) |
(13.4) |
where
![\begin{displaymath}
D_w(x,y) = \ensuremath{\int\!\!\int} S(u,v)\, W(u,v)\, \ensuremath{e^{2i\pi(ux+vy)}}\,du\,dv = \widehat{S W}
\end{displaymath}](img963.gif) |
(13.5) |
Dw(x,y) is called the dirty beam, and is directly dependent on the
choice of the weighting function W, as well as on the spectral sensitivity
function S. Iw(x,y) is usually called the dirty image.
Fourier Transform, which allows to directly derive Iw from the measured
visibilities V and spectral sensitivity function S, and Deconvolution,
which allows to derive the sky brightness I from Iw, are thus two key
issues in imaging (see Eq.13.4).
Next: 13.1 Fourier Transform
Up: No Title
Previous: 12.8 Self calibration by
S.Guilloteau
2000-01-19