Breast cancer is the most occurring cancer among women. Detecting it at an early stage enables to significantly reduce the mortality rate. Mammography is the golden standard technique used for screening but it has some disadvantages, among which the difficulty to scan women with dense breasts. Ultrasound is a promising technique enabling a cheap and fast diagnosis. More precisely, waveform inversion reconstructs the 3D speed of sound distribution within the breast, enabling an easier differentiation between healthy and non-healthy tissues.

In this work, two inversion methods are investigated, namely Algebraic Reconstruction Technique (ART) and Born inversion. Born inversion is a non-linear waveform inversion method aiming at solving an integral equation. To increase its performance while keeping the computation time constant, the frequencies used for the reconstruction are chosen randomly over a given bandwidth for each source receiver combination, at the expense of an additional noise in the reconstruction. The sparsity of the reconstruction in the wavelet
domain has been utilized to implement two new regularization methods aiming at reducing this noise. To test and validate the three algorithms (namely ART and Born Inversion with the two regularization methods), a phantom study has been performed with both synthetic and real data. The phantom, made out of agar, is
placed in a square water tank in the centre of a measurement setup designed during this study, allowing to scan the phantom from its four sides.

The results of the methods are presented and discussed. ART does not perform well with real data as the assumption on which it relies is not satisfied at the frequencies used in this study. Born inversion is presented with transmission and reflection data, as well as a combination of the two. With synthetic data, both reflections and transmissions give very satisfying results with the help of the regularization. With real data the phantom is correctly imaged by the reflections but the transmissions do not perform well enough to give an accurate speed of sound map.

The knowledge gathered in this work can help building a better measurement setup, with a phantom closer to the breast geometry and internal structure, with which the reconstruction techniques could be tested and further improved.