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

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.