Ultrafast ultrasound imaging as an inverse problem : Fast and matrix-free measurement model for sparsity-driven image reconstruction methods

Pulse-echo ultrasonography aims at imaging tissues using an array of piezoelectric elements by transmitting ultrasound (US) pulses and receiving backscattered echoes. Retrieving the image from the received signals can be mathematically formulated as an ill-posed inverse problem. Conventional realtime image reconstruction methods rely on delay-and-sum (DAS) beamforming, which recovers a relatively poor solution of the problem. An alternative to DAS consists in using iterative techniques. Such methods require both an accurate and fast measurement model as well as a strong prior on the image under scrutiny. Towards this goal, much effort has been deployed in formulating models for US imaging. However, current measurement models require a tremendous amount of memory to store the matrix coefficients. We propose fast and matrix-free formulations of the measurement model and its adjoint for real-time US image reconstruction. We use sparse regularization methods to inject additional prior information of the signal with the aim to improve upon traditional methods. We present two different techniques which take advantage of fast operators developed for the forward model and its adjoint and rely on sparsity of US images in well-chosen models. Sparse regularization is used for enhanced image reconstruction. Compressed beamforming exploits the compressed sensing framework to restore high quality images from fewer raw-data than state-of-the-art approaches. Using simulated and experimental data, on in vivo examples, we explore the potential of the proposed methods in comparison with state-of-the-art approaches, for both planeand diverging-wave imaging.