  Sonography techniques use multiple transducer elements for tissue
visualization. Signals detected at each element are sampled prior to digital
beamforming. The required sampling rates are up to 4 times the Nyquist rate of
the signal and result in considerable amount of data, that needs to be stored
and processed. A developed technique, based on the finite rate of innovation
model, compressed sensing (CS) and Xampling ideas, allows to reduce the number
of samples needed to reconstruct an image comprised of strong reflectors. A
significant drawback of this method is its inability to treat speckle, which is
of significant importance in medical imaging. Here we build on previous work
and show explicitly how to perform beamforming in the Fourier domain.
Beamforming in frequency exploits the low bandwidth of the beamformed signal
and allows to bypass the oversampling dictated by digital implementation of
beamforming in time. We show that this allows to obtain the same beamformed
image as in standard beamforming but from far fewer samples. Finally, we
present an analysis based CS-technique that allows for further reduction in
sampling rate, using only a portion of the beamformed signal's bandwidth,
namely, sampling the signal at sub-Nyquist rates. We demonstrate our methods on
in vivo cardiac ultrasound data and show that reductions up to 1/25 over
standard beamforming rates are possible.
