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 in ) into n continuous scale-space components, denoted as I l = CDD l ( I in ) for l ∈ [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 ( erf ), defined as R ( l ) = B 2 1 − erf l − c recover w , where compact structures are effectively recovered, while extended emission decays monotonically as scales approach the maximum recoverable scale. The proposed CDD– erf framework predicts the spatially filtered interferometric image I pred directly in the image domain, bypassing visibility simulations, mapping the true sky brightness distribution via the equation I pred = ∑ l = 1 n [ CDD l ( I in ) × R ( l ) ] . This provides a quantitative bridge between theoretical hydrodynamical simulations and interferometric observations.