16.2 Object space

In the problems of Fourier synthesis encountered in astronomy, the object function of interest, $\Phi_{\! o}$, is a real-valued function of an angular position variable $\mathbf{\sigma} \equiv \mathbf{x} = (x,y)$. 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 $A_j$ and $A_k$ point toward a distant radio source in a direction indicated by the unit vector  $\mathbf {s}$, and  $\mathbf {b}$ is the interferometer baseline vector. The position pointed by the unit vector  $\mathbf {s}_o$ is commonly referred to as the phase tracking center or phase reference position: $\mathbf {s} - \mathbf {s}_o = \mathbf {\sigma}$.

The object model variable $\phi$ lies in some object space $H_o$ whose vectors, the functions $\phi$, are defined at a high level of resolution. This space is characterized by two key parameters: the extension $\Delta x$ of its field, and its resolution scale $\delta x$. To define this object space more explicitly, we first introduce the finite grid (see Fig. 16.2):

$$\mathbb{G}= \mathbb{L}\times \mathbb{L}, \quad \mathbb{L}= \left\{ p\in\mathbb{Z}: -\frac{N}{2} \leq p \leq \frac{N}{2}-1 \right\},$$ (16.1)

where $N$ is some power of $2$.

On each pixel $\mathbf{p} \delta x (\mathbf{p}\in\mathbb{G})$, we then center a scaling function of the form

$$e_{\mathbf{p}}(\mathbf{x}) = e_{\mathbf{0}}(\mathbf{x}-\mathbf{p} \delta x) \quad e_{\mathbf{0}}(\mathbf{x}) = \bigl( \frac{x}{\delta x}\bigr) \bigl( \frac{y}{\delta x}\bigr).$$ (16.2)

Figure: Object grid $\mathbb{G} \delta x$ (left hand) and Fourier grid $\mathbb{G} \delta u$ (right hand) for $N=8$. The object domain is characterized by its resolution scale $\delta x$ and the extension of its field  $\Delta x = N\delta x$, where $N$ is some power of $2$ (the larger is $N$, the more oversampled is the object field). The basic Fourier sampling interval is  $\delta u = 1 / \Delta x$, the extension of the Fourier domain is  $\Delta u = 1/\delta x$.

It is easy to verify that these functions form an orthogonal set. In this presentation of WIPE, the object space $H_o$ is the Euclidian space generated by the basis vectors $e_{\mathbf{p}}$, $\mathbf{p}$ spanning $\mathbb{G}$ (see Fig. 16.2). The dimension of this space is equal to $N^2$: the number of pixels in the grid  $\mathbb{G}$. The functions $\phi$ lying in $H_o$ can therefore be expanded in the form

$$\phi(\mathbf{x}) = \sum_{\mathbf{p}\in\mathbb{G}} a_{\mathbf{p}} e_{\mathbf{p}}(\mathbf{x}),$$ (16.3)

where the  $a_{\mathbf{p}}$'s are the components of $\phi$ in the interpolation basis of $H_o$.

The Fourier transform of $\phi$ is defined by the relationship

$$\widehat{\phi} (\mathbf{u}) = \int \phi(\mathbf{x}) ^{\displaystyle -2\text{i}\pi \mathbf{u}\!\cdot\!\mathbf{x}} \text{d}\mathbf{x},$$

where $\mathbf{u}$ is a two-dimensional angular spatial frequency: $\mathbf{u} = (u,v)$. According to the expansion of $\phi$ we therefore have:

$$\widehat{\phi} (\mathbf{u}) = \sum_{\mathbf{p}\in\mathbb{G}} a_{\mathbf{p}} \widehat{e}_{\mathbf{p}}(\mathbf{u}),$$ (16.4)

where

$$\widehat{e}_{\mathbf{p}}(\mathbf{u}) = \widehat{e}_{\mathbf{0}}(\mathbf{u}) ^{\displaystyle -2\text{i}\pi \mathbf{p}\!\cdot\!\frac{\mathbf{... ...t{rect}\bigl(\frac{u}{\Delta u}\bigr) \text{rect}\bigl(\frac{v}{\Delta u}\bigr)$$ (16.5)

and $\Delta u = 1/\delta x$.

The dual space of the object space, $\widehat{H}_o$, is the image of $H_o$ by the Fourier transform operator: $\widehat{H}_o$ is the space of the Fourier transforms of the functions $\phi$ lying in $H_o$. This space is characterized by two key parameters: its extension  $\Delta u = 1/\delta x$, and the basic Fourier sampling interval  $\delta u = 1 / \Delta x$ (see Fig. 16.2).