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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. 16.1.
Figure 16.1:
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
and
point toward a distant radio source in a direction indicated by the
unit vector
, and
is the interferometer baseline
vector. The position pointed by the unit vector
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)\{ lin...
...$} \put(30,110){\vector(-1,
-4){5}} \put(22,98){$\mathbf{\sigma}$}
\end{picture}](img1622.png) |
The object model variable
lies in some
object space
whose vectors, the functions
,
are defined at a high level of resolution. This space
is characterized by two key parameters: the extension
of its field, and its resolution scale
.
To define this object space more explicitly, we first
introduce the finite grid (see Fig. 16.2):
![$\displaystyle \mathbb{G}= \mathbb{L}\times \mathbb{L}, \quad \mathbb{L}= \left\{ p\in\mathbb{Z}: -\frac{N}{2} \leq p \leq \frac{N}{2}-1 \right\},$](img1629.png) |
(16.1) |
where
is some power of
.
On each pixel
, we
then center a scaling function of the form
Figure:
Object grid
(left hand)
and Fourier grid
(right hand) for
.
The object domain is characterized by its resolution scale
and the extension of its field
, where
is
some power of
(the larger is
, the more oversampled is the object
field). The basic Fourier sampling interval is
,
the extension of the Fourier domain is
.
![\begin{figure}\setlength{\unitlength}{1mm}\begin{picture}(110,60)(0,0)
%%
\put(2...
...vector(2,1){9.8}}
\put(96,41){$\mathbf{q}\delta u$}
%%
\end{picture}\end{figure}](img1637.png) |
It is easy to verify that these functions form an orthogonal set.
In this presentation of WIPE, the object space
is the Euclidian space generated by the basis vectors
,
spanning
(see Fig. 16.2). The dimension of
this space is equal to
: the number of pixels in the
grid
. The functions
lying in
can therefore
be expanded in the form
![$\displaystyle \phi(\mathbf{x}) = \sum_{\mathbf{p}\in\mathbb{G}} a_{\mathbf{p}} e_{\mathbf{p}}(\mathbf{x}),$](img1642.png) |
(16.3) |
where the
's are the components of
in the
interpolation basis of
.
The Fourier transform of
is defined by the relationship
e![$\displaystyle ^{\displaystyle -2\text{i}\pi \mathbf{u}\!\cdot\!\mathbf{x}} \text{d}\mathbf{x},$](img1645.png) |
|
where
is a two-dimensional angular spatial frequency:
. According to the expansion of
we therefore
have:
![$\displaystyle \widehat{\phi} (\mathbf{u}) = \sum_{\mathbf{p}\in\mathbb{G}} a_{\mathbf{p}} \widehat{e}_{\mathbf{p}}(\mathbf{u}),$](img1648.png) |
(16.4) |
where
e![$\displaystyle ^{\displaystyle -2\text{i}\pi \mathbf{p}\!\cdot\!\frac{\mathbf{...
...t{rect}\bigl(\frac{u}{\Delta u}\bigr) \text{rect}\bigl(\frac{v}{\Delta u}\bigr)$](img1650.png) |
(16.5) |
and
.
The dual space of the object space,
, is the
image of
by the Fourier transform operator:
is
the space of the Fourier transforms of the functions
lying
in
.
This space is characterized by two key parameters: its
extension
, and the basic Fourier sampling
interval
(see Fig. 16.2).
Next: 16.3 Experimental data space
Up: 16. Advanced Imaging Methods:
Previous: 16.1 Introduction
  Contents
Anne Dutrey