Visibility-Based Imaging

This section summarizes the elements of visibility-based imaging on which EPIC builds. Comprehensive treatments are given by Thompson, Moran & Swenson (2017) and Taylor, Carilli & Perley (1999).

The antenna measurement

The sky is modeled as a superposition of mutually incoherent emitters, with electric field $E (\hat{s}, f, t)$ arriving from direction $\hat{s}$ . Consider an antenna located at $r_{a}$ , with far-field voltage pattern $W_{a} (\hat{s})$ . After channelization, the complex voltage it measures at frequency $f$ and time $t$ is (Thyagarajan et al. 2017)

E_{a} (f, t) = \int W_{a} (\hat{s}) E (\hat{s}, f, t) e^{- i 2 π f r_{a} \cdot \hat{s} / c} d Ω + n_{a} (f, t),

where $n_{a}$ is the receiver noise. The geometric phase factor accounts for the excess path length to antenna $a$ , and is the origin of the angular resolution of the telescope.

The voltage pattern is the Fourier transform of the aperture illumination $W_{a} (r)$ . The same measurement may therefore be written in the aperture plane as a convolution of the ground-plane electric field with the aperture of the antenna:

E_{a} = \int W_{a} (r - r_{a}) E (r) d^{2} r + n_{a} .

EPIC is formulated in this aperture-plane representation.

Visibilities

The incident field is a stationary random process, so individual voltage samples carry no deterministic information about the sky. The measured quantity is the time-averaged correlation between antennas, the visibility:

V_{ab} \equiv ⟨ E_{a} E_{b}^{*} ⟩_{t} = \int B_{ab} (\hat{s}) I (\hat{s}) e^{- i 2 π f (r_{a} - r_{b}) \cdot \hat{s} / c} d Ω + n_{ab}, B_{ab} = W_{a} W_{b}^{*} .

This is the van Cittert–Zernike theorem. Each visibility samples the Fourier transform of the beam-weighted sky intensity at the spatial frequency $u_{ab} = (r_{a} - r_{b}) f / c$ . Cross terms between distinct directions vanish under time averaging because the emission is spatially incoherent.

For antennas lying in a common plane, in direction cosines $(l, m)$ ,

V (u, v) = \iint A (l, m) I (l, m) e^{- 2 π i (u l + v m)} \frac{d l d m}{1 - l ^{2} - m ^{2}} .

An image is obtained by gridding the visibility samples and applying an inverse two-dimensional FFT. The result, the dirty image, is the sky intensity convolved with the point-spread function determined by the $uv$ coverage.

The FX correlator

Real-time correlators for radio telescopes with many antennas are usually of the FX type. Processing proceeds in three steps:

F stage. The voltage stream from each antenna is channelized, typically with a polyphase filter bank.
X stage. For every channel, the product $E_{a} E_{b}^{*}$ is formed for all antenna pairs and accumulated over an integration time $Δ t$ .
Imaging. The visibilities are calibrated and imaged, usually offline.

The X stage dominates the computational cost. Per channel, per polarization product and per F-engine sample, it requires

C_{FX} = \frac{N _{a} ( N _{a} - 1 )}{2}

complex multiply–accumulate operations. With total bandwidth $Δ B$ and channel width $Δ f$ , the output data rate per polarization product is

r_{X} \sim \frac{N _{a} ( N _{a} - 1 )}{Δ t} \frac{Δ B}{Δ f}

real values per second. It scales as $N_{a}^{2}$ and inversely with the integration time.

For LWA-SV ( $N_{a} = 256$ ) there are 32 640 baselines per polarization product per channel. At 25 000 spectra per second this corresponds to approximately $8 \times 1 0^{8}$ complex multiply–accumulates per second per channel and product. A 3.2 MHz band with four polarization products requires approximately $4 \times 1 0^{11}$ . Reducing the integration time from several seconds to tens of milliseconds increases the visibility output rate by two orders of magnitude. For a telescope of this size, the principal cost of high time resolution under FX correlation therefore lies in the output data rate rather than in the arithmetic.

At larger scales the arithmetic itself becomes prohibitive. Morales (2011) quotes $1.55 \times 1 0^{13}$ complex multiply–accumulates per second for a 512-tile MWA, and Thyagarajan et al. (2017) quote 212 TMAC s⁻¹ for the full HERA telescope.

Relation to beamforming

A beamformer forms the weighted sum $\sum_{a} w_{a} (\hat{s}_{0}) E_{a}$ for a chosen direction $\hat{s}_{0}$ . Its properties are:

Cost. $O (N_{a})$ per beam.
Time resolution. It retains the full time resolution of the channelized data.
Field of view. Only the selected directions are observed.
Full-sky coverage. Covering the sky with $N_{beams}$ beams costs $O (N_{a} N_{beams})$ .

The first stage of EPIC may be viewed as an FFT beamformer that forms beams in all directions simultaneously. The subsequent squaring and averaging restore equivalence with visibility-based imaging.

Development History

The development of EPIC from the FFT telescope and the MOFF correlator to a GPU-based, commensal all-sky imager at LWA-SV.

The EPIC Algorithm

Derivation of the EPIC imaging equation, its equivalence to visibility-based imaging, gridding, autocorrelation removal, and computational cost.