  This paper presents an innovative approach to computational acoustic imaging
of biperiodic surfaces, exploiting the capabilities of an acoustic superlens to
overcome the diffraction limit. We address the challenge of imaging physical
entities in complex environments by considering the partial differential
equations that govern the physics and solving the corresponding inverse
problem. We focus on imaging infinite rough surfaces, specifically 2D
diffraction gratings, and propose a method that leverages the transformed field
expansion (TFE). We derive a reconstruction formula connecting the Fourier
coefficients of the surface and the measured field, demonstrating the potential
for unlimited resolution under ideal conditions. We also introduce an
approximate discrepancy principle to determine the cut-off frequency for the
truncated Fourier series expansion in surface profile reconstruction.
Furthermore, we elucidate the resolution enhancement effect of the superlens by
deriving the discrete Fourier transform of white Gaussian noise. Our numerical
experiments confirm the effectiveness of the proposed method, demonstrating
high subwavelength resolution even under slightly non-ideal conditions. This
study extends the current understanding of superlens-based imaging and provides
a robust framework for future research.
