  A tomographic technique called full-waveform inversion has recently shown
promise as a fast, affordable, and safe modality to image the brain using
ultrasound. However, its high computational cost and memory footprint currently
limit its clinical applicability. Here, we address these challenges through a
frequency-adaptive discretisation of the imaging domain and lossy compression
techniques. Because full-waveform inversion relies on the adjoint-state method,
every iteration involves solving the wave equation over a discretised
spatiotemporal grid and storing the numerical solution to calculate gradient
updates. The computational cost depends on the grid size, which is controlled
by the maximum frequency being modelled. Since the propagated frequency
typically varies during the reconstruction, we reduce reconstruction time and
memory use by allowing the grid size to change throughout the inversion.
Moreover, we combine this approach with multiple lossy compression techniques
that exploit the sparsity of the wavefield to further reduce its memory
footprint. We explore applying these techniques in the spatial, wavelet, and
wave atom domains. Numerical experiments using a human-head model show that our
methods lead to a 30% reduction in reconstruction time and up to three orders
of magnitude less memory, while negligibly affecting the accuracy of the
reconstructions.
