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15. The Imaging Principles
Stéphane
Guilloteau
guillote@iram.fr
IRAM, 300 rue de la Piscine, F-38406
Saint Martin d'Hères, France
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Assuming identical antennas, we have shown in previous lectures that an
interferometer measures the visibility function
![$\displaystyle V(u,v) = \ensuremath{\int\!\!\int}B(x,y) I(x,y) \ensuremath{e^{-2i\pi(ux+vy)}} dx dy$](img1535.png) |
(15.1) |
over an ensemble of points
, where
is the power
pattern of the antennas and
the sky brightness distribution.
The imaging process consists in determining as best as possible the sky brightness
. Since Eq.15.1 is a convolution, the imaging process will involve
deconvolution techniques.
Let
be the sampling (or spectral sensitivity) function
![$\displaystyle S(u,v) \neq 0 \Longleftrightarrow \exists i \in 1,n (u_i,v_i) = (u,v)$](img1540.png) |
|
|
|
![$\displaystyle S(u,v) = 0 \Longleftrightarrow \forall i \in 1,n (u_i,v_i) \neq (u,v)$](img1541.png) |
|
|
(15.2) |
The spectral sensitivity function
contains information on the
relative weights of each visibility, usually derived from noise
predicted from the system temperature, antenna efficiency,
integration time and bandwidth.
Let us define
![$\displaystyle I_w(x,y) = \ensuremath{\int\!\!\int}S(u,v) W(u,v) V(u,v) \ensuremath{e^{2i\pi(ux+vy)}} du dv$](img1542.png) |
(15.3) |
where
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,
can be identified with
![$\displaystyle I_w(x,y) = (B(x,y) I(x,y)) \ensuremath{\ast\!\ast}(D_w(x,y))$](img1545.png) |
(15.4) |
where
![$\displaystyle D_w(x,y) = \ensuremath{\int\!\!\int}S(u,v) W(u,v) \ensuremath{e^{2i\pi(ux+vy)}} du dv = \widehat{S W}$](img1546.png) |
(15.5) |
is called the dirty beam, and is directly dependent on the
choice of the weighting function
, as well as on the spectral sensitivity
function
.
is usually called the dirty image.
Fourier Transform, which allows to directly derive
from the measured
visibilities
and spectral sensitivity function
, and Deconvolution,
which allows to derive the sky brightness
from
, are thus two key
issues in imaging (see Eq.15.4).
Subsections
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Anne Dutrey