Demystifying image-recovery from radio interferometers: toward a multiscale predictive model

Abstract

Radio interferometers suffer from the missing short-spacing problem, losing large-scale diffuse emission. This missing flux underestimates gas mass and biases key metrics like star formation efficiency. Quantifying this scale-dependent loss currently relies on computationally intensive mock observations, lacking an analytical image-domain framework. We introduce the Constrained Diffusion Decomposition (CDD) method to decompose an input image ($I_{\mathrm{in}}$) into $n$ continuous scale-space components, denoted as $I_l={\mathrm{CDD}}_l(I_{\mathrm{in}})$ for $l\in [1,n]$, and apply it to simulated Atacama Large Millimeter/submillimeter Array (ALMA) observations of the Perseus molecular cloud across multiple array configurations. We find that the interferometric spatial filtering response can be mathematically decoupled: the scale-dependent flux recovery fraction follows a one-dimensional error function (\texttt{erf}), defined as $R(l)=\frac{B}{2}\left[1-\mathrm{erf}\left(\frac{l-c_{\mathrm{recover}}}{w}\right)\right]$, where compact structures are effectively recovered, while extended emission decays monotonically as scales approach the maximum recoverable scale. The proposed CDD--\texttt{erf} framework predicts the spatially filtered interferometric image $I_{\mathrm{pred}}$ directly in the image domain, bypassing visibility simulations, mapping the true sky brightness distribution via the equation $I_{\mathrm{pred}}={\sum }_{l=1}^n[{\mathrm{CDD}}_l(I_{\mathrm{in}})\times R(l)]$. This provides a quantitative bridge between model and interferometric observations.

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