GoKawiilThe kine algorithm is a static or dynamic image reconstruction algorithm for interferometric data. It is a forward imaging method, based on a neural field representation43 of the brightness distribution of the source. In this section, we describe our methodology, starting with a general overview of VLBI measurements followed by an explanation of the kine imaging algorithm, which is shown in Fig. 4. Then, we describe the synthetic data used to test the performance of the algorithm and show the corresponding reconstructions. In the Supplementary Information, we present extensive tests validating the dynamic range, resolution, time interpolation and motion tracking of our method on synthetic datasets, of which we report the results here. We also present a performance comparison with the CLEAN method on real observations and the details of the optical flow velocity analysis.
VLBI measurements
In radio interferometric observations, each antenna in the array records a signal proportional to the received electromagnetic flux density. By the van Cittert–Zernike theorem, the time-averaged correlation product of the signals recorded by any pair of antennas (called visibility) is the Fourier transform of the flux density spatial distribution on the sky plane, evaluated at a frequency proportional to the distance of the antennas. For a pair of antennas A and B, with projected baseline vector \(\bar{b}=({b}_{x},{b}_{y})\), observing at wavelength λ, the ideal complex visibility V AB (u, v, t) is related to the flux density distribution \({\mathcal{I}}(x,y,t)\) by
$${V}_{\mathrm{AB}}^{({\mathcal{I}})}(u,v,t)=\int \int \,{{\rm{e}}}^{-2\mathrm{\pi i}(ux+vy)}\,{\mathcal{I}}(x,y,t){\rm{d}}x{\rm{d}}y,$$ (1)
where \((u,v)=\left(\frac{{b}_{x}}{\lambda },\frac{{b}_{y}}{\lambda }\right)\) are the x and y spatial frequencies (refer to ref. 44 for a detailed description of radio interferometry observations and imaging). The above equation, written for Stokes \({\mathcal{I}}\), holds for all Stokes visibilities \(({V}_{\mathrm{AB}}^{({\mathcal{I}})},{V}_{\mathrm{AB}}^{({\mathcal{Q}})},{V}_{\mathrm{AB}}^{({\mathcal{U}})},{V}_{\mathrm{AB}}^{({\mathcal{V}})})\).
In practice, different sources of noise corrupt the measurement of visibilities. They are classified as baseline-dependent errors and site-dependent errors, so the measured visibilities can be expressed as
$${V}_{{\rm{AB}}}^{{\prime} }={G}_{{\rm{A}}}{G}_{{\rm{B}}}{{\rm{e}}}^{{\rm{i}}({\phi }_{{\rm{A}}}-{\phi }_{{\rm{B}}})}({V}_{\mathrm{AB}}+{{\epsilon }}_{\mathrm{AB}}),$$ (2)
where \({G}_{{\rm{A}},{\rm{B}}}{{\rm{e}}}^{{\rm{i}}{\phi }_{{\rm{A}},{\rm{B}}}}\) are the site-dependent errors in amplitude and phase, referred to as complex gains, and ϵ AB is the thermal noise, which is Gaussian-distributed with baseline-dependent standard deviation44. The most problematic source of error is the complex gains, as they might be incorrectly estimated from the a priori calibration, whereas thermal noise can be fully characterized and incorporated in the loss function through the uncertainties σ AB of the visibilities.
Therefore, although complex visibilities are the fundamental data product resulting from VLBI observations, the imaging process can use different data products that are constructed to be unaffected by site-dependent amplitude or phase corruptions. kine supports the following data products: complex visibilities V AB , amplitudes of the visibilities |V AB |, closure phases \({\varPhi }_{\mathrm{ABC}}:= \arg ({V}_{\mathrm{AB}}{V}_{\mathrm{BC}}{V}_{\mathrm{CA}})\), closure amplitudes A ABCD ≔ |V AB V CD |/|V AC V BD | and the logarithm of closure amplitudes.
Model
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