Ultrasound tomography for breast screening
Ultrasound Tomography Setup
Ultrasound tomography aims at recovering the parameters of an unknown medium by studying the characteristics of sound propagated through the medium. Instead of modeling the forward problem using the wave equation, travel-time tomography employs the principles of geometrical acoustics to estimate the sound speed distribution. It is based on the fact that acoustic energy travels along the lines perpendicular to the equal-phase wavefronts which is a valid assumption at high frequencies. In contrast to the problems with straight-line propagations such as X-ray tomography, the ultrasound propagation paths are not straight in an inhomogeneous medium and depend on the sound speed distribution. Therefore, travel-times are a nonlinear function of the unknown sound speed values.
The scanner consists of a circular array of transmitters and receivers which encloses the object to be imaged. By solving a nonlinear system of equations, the reconstruction algorithm estimates the sound speed of the object using the set of travel-time measurements. The main difficulty in this inverse problem is to ensure the convergence and robustness to noise. We used a gradient method to find a solution for which the corresponding travel-times are closest to the measured travel-times in the least squares sense. To this end, first the gradient of the cost function is derived using Fermat’s Principle. Then, the iterative nonlinear conjugate gradient algorithm solves the minimization problem. This is combined with the backtracking line search method to efficiently find the step size in each iteration. This approach is guaranteed to converge to a local minimum of the cost function where the convergence point depends on the initial guess. Moreover, the method has the potential to easily incorporate regularity constraints such as sparsity as a priori information on the model. The method is tested both numerically and using in vivo data obtained from a UT scanner. The results confirm the stability and robustness of our approach for breast screening applications.