In recent years, dynamic metasurface antennas (DMAs) have been proposed as an efficient alternative platform for computational imaging, which can drastically simplify the hardware architecture. In this paper, we first mathematically describe the existing solution to be able to convert raw measurements obtained by a DMA in the frequency-space domain into raw data on Fourier bases. Next, an optimization problem based on compressive sensing theory is defined, through which only a limited share of the total frequency/spatial data will be needed. The converted/retrieved data are used to reconstruct the image in the Fourier domain. The performance of the corresponding image reconstruction techniques (with/without Stolt interpolation operation) is evaluated in terms of the quality of the reconstructed image (both visually and quantitatively) and computational time with computer simulations.