Inverse Problems With Multiple Plane Waves: The Angular Simplification

In unfocused ultrasound imaging, a delay-and-sum algorithm is commonly used to reconstruct one image per emission. When multiple emissions are performed, individual images can be combined by coherent compounding to improve image quality. Alternative methods based on tomographic inverse problems have been recently introduced and prove a superior image quality. However, the high dimensionality of the operators involved in such tomographic problems –especially in the case of multiple emissions– leads to prohibitive computation times and memory requirements, preventing their use in practice. We propose to use an angular framework in which plane waves are considered both in emission and reception. In this new framework, we show that the delay-an-sum and the compounding operators are commutative. Using this property, we formulate a low-dimensional tomographic inverse problem and describe a matrix-free method able to reconstruct high-quality images with a computation time independent of the number of emissions.