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In the problems of Fourier synthesis encountered in astronomy,
the object function of interest,
, is a
real-valued function of an angular position variable
. The geometrical elements
under consideration are presented in Fig. 14.1.
Figure:
Traditional coordinate systems used to express the
relation between the complex visibilities and the brightness distribution
of a source under observation. Here, the two antennas Aj and Akpoint toward a distant radio source in a direction indicated by the
unit vector
s, and
b is the interferometer baseline
vector. The position pointed by the unit vector
so is commonly
referred to as the phase tracking center or phase reference
position:
.
![\begin{picture}(100,140)(0,0)
%\put(0,0){\line(1,0){100}}
%\put(0,140){\line(1,0...
...} \put(30,110){\vector(-1,
-4){5}} \put(22,98){$\mathbf{\sigma}$ }
\end{picture}](img1024.gif) |
The object model variable
lies in some
object space Ho whose vectors, the functions
,
are defined at a high level of resolution. This space
is characterized by two key parameters: the extension
of its field, and its resolution scale
.
To define this object space more explicitly, we first
introduce the finite grid (see Fig. 14.2):
![\begin{displaymath}\text{\boldmath$G$\unboldmath } = \text{\boldmath$L$\unboldma...
...boldmath } : -\frac{N}{2} \leq p \leq \frac{N}{2}-1
\right\},
\end{displaymath}](img1028.gif) |
(14.1) |
where N is some power of 2.
On each pixel
, we
then center a scaling function of the form
![\begin{displaymath}e_{\mathbf{p}}(\mathbf{x}) = e_{\mathbf{0}}(\mathbf{x}-\mathb...
...}{\delta x}\bigr)
\text{sinc} \bigl( \frac{y}{\delta x}\bigr).
\end{displaymath}](img1030.gif) |
(14.2) |
Figure:
Object grid
(left hand)
and Fourier grid
(right hand) for N=8.
The object domain is characterized by its resolution scale
and the extension of its field
,
where N is
some power of 2 (the larger is N, the more oversampled is the object
field). The basic Fourier sampling interval is
,
the extension of the Fourier domain is
.
![\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}
(110,60)(0,0)
%%
\p...
...ector(2,1){9.8}}
\put(96,41){$\mathbf{q}\delta u$ }
%%
\end{picture}\end{figure}](img1033.gif) |
It is easy to verify that these functions form an orthogonal set.
In this presentation of WIPE, the object space Hois the Euclidian space generated by the basis vectors
ep,
p spanning
(see Fig. 14.2). The dimension of
this space is equal to N2: the number of pixels in the
grid
. The functions
lying in Ho can therefore
be expanded in the form
![\begin{displaymath}\phi(\mathbf{x}) = \sum_{\mathbf{p}\in\text{\boldmath$G$\unboldmath }}
a_{\mathbf{p}} e_{\mathbf{p}}(\mathbf{x}),
\end{displaymath}](img1035.gif) |
(14.3) |
where the
ap's are the components of
in the
interpolation basis of Ho.
The Fourier transform of
is defined by the relationship
where
u is a two-dimensional angular spatial frequency:
u = (u,v). According to the expansion of
we therefore
have:
![\begin{displaymath}\widehat{\phi} (\mathbf{u}) = \sum_{\mathbf{p}\in\text{\boldm...
...ldmath }}
a_{\mathbf{p}} \widehat{e}_{\mathbf{p}}(\mathbf{u}),
\end{displaymath}](img1037.gif) |
(14.4) |
where
![\begin{displaymath}\widehat{e}_{\mathbf{p}}(\mathbf{u}) = \widehat{e}_{\mathbf{0...
...c{u}{\Delta u}\bigr)
\text{rect}\bigl(\frac{v}{\Delta u}\bigr)
\end{displaymath}](img1038.gif) |
(14.5) |
and
.
The dual space of the object space,
, is the
image of Ho by the Fourier transform operator:
is
the space of the Fourier transforms of the functions
lying
in Ho.
This space is characterized by two key parameters: its
extension
, and the basic Fourier sampling
interval
(see Fig. 14.2).
Next: 14.3 Experimental data space
Up: 14. Advanced Imaging Methods:
Previous: 14.1 Introduction
S.Guilloteau
2000-01-19