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

$$V(u,v) = \ensuremath{\int\!\!\int}B(x,y) I(x,y) \ensuremath{e^{-2i\pi(ux+vy)}} dx dy$$ (15.1)

over an ensemble of points $(u_i,v_i), i = 1,n$, 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.15.1 is a convolution, the imaging process will involve deconvolution techniques.

Let $S(u,v)$ be the sampling (or spectral sensitivity) function

$$S(u,v) \neq 0 \Longleftrightarrow \exists i \in 1,n (u_i,v_i) = (u,v)$$

$$S(u,v) = 0 \Longleftrightarrow \forall i \in 1,n (u_i,v_i) \neq (u,v)$$ (15.2)

The spectral sensitivity function $S$ 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

$$I_w(x,y) = \ensuremath{\int\!\!\int}S(u,v) W(u,v) V(u,v) \ensuremath{e^{2i\pi(ux+vy)}} du dv$$ (15.3)

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, $I_w(x,y)$ can be identified with

$$I_w(x,y) = (B(x,y) I(x,y)) \ensuremath{\ast\!\ast}(D_w(x,y))$$ (15.4)

where

$$D_w(x,y) = \ensuremath{\int\!\!\int}S(u,v) W(u,v) \ensuremath{e^{2i\pi(ux+vy)}} du dv = \widehat{S W}$$ (15.5)

$D_w(x,y)$ 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 $S$. $I_w(x,y)$ is usually called the dirty image.

Fourier Transform, which allows to directly derive $I_w$ from the measured visibilities $V$ and spectral sensitivity function $S$, and Deconvolution, which allows to derive the sky brightness $I$ from $I_w$, are thus two key issues in imaging (see Eq.15.4).