In synthetic aperture (SA) imaging reported in the ultrasound imaging literature, typically, the delay and sum (DAS) beamformer is used; however, it is computationally expensive due to the pixel-by-pixel processing performed in the time domain. Recently, the adaptation of frequency-domain beamformers for medical ultrasound SA imaging, particularly to single-element/multielement synthetic transmit aperture (STA/MSTA) schemes, has been reported. In such reports, usually, less attention is paid to reducing system complexity. Recently, a sparse-transmit sparse-receive version of diverging beam-based synthetic aperture technique (DBSAT) was shown to achieve a reduction in system complexity by using fewer parallel receive channels, yet it achieves better quality and higher frame rate than conventional focused beamforming. However, this was also demonstrated using the DAS beamformer. In this work, we aim at achieving a reduction in computational cost, in addition to a reduction in system complexity, by implementing a fast and efficient frequency-wavenumber (ω-κ) algorithm for the sparse DBSAT scheme. In doing so, an additional novel step of recovering missing frame data due to sparse transmit is introduced, namely, projection onto elliptical sets (POES). The results from this novel combination of ω-κ with POES recovery showed that it is feasible to achieve several orders of magnitude faster reconstruction compared with the standard DAS beamforming, without any compromise in the image quality and, in some cases, with improved image quality. The average value of the contrast-to-noise ratio (CNR) and signal-to-noise ratio (SNR) calculated from cyst at 15-mm depth obtained using the different schemes was 4.94 and 5.73 dB better when ω-κ was employed instead of DAS, respectively. In addition, for the sparse data set acquired with a 50% overlap during transmit and 64 active receive elements, DAS reconstruction takes as long as 647 s, whereas the ω-κ algorithm takes only 2 s when programmed and executed in MATLAB. © 1986-2012 IEEE.