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 ($I_{\mathrm{i}\mathrm{n}}$) into $n$ continuous scale-space components, denoted as $I_l={\mathrm{C}\mathrm{D}\mathrm{D}}_l(I_{\mathrm{i}\mathrm{n}})$ 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 ($\mathtt{\text{erf}}$), defined as $R(l)=\frac{B}{2}[1-\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{l-c_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}}{w}\right)]$, where compact structures are effectively recovered, while extended emission decays monotonically as scales approach the maximum recoverable scale. The proposed CDD–$\mathtt{\text{erf}}$ framework predicts the spatially filtered interferometric image $I_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ directly in the image domain, bypassing visibility simulations, mapping the true sky brightness distribution via the equation $I_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={\sum }_{l=1}^n[{\mathrm{C}\mathrm{D}\mathrm{D}}_l(I_{\mathrm{i}\mathrm{n}})\times R(l)]$. This provides a quantitative bridge between model and interferometric observations.