Sparsity constrained born inversion for breast cancer detection

Ultrasound has been successfully used for detecting and characterizing small breast lesions. To improve the image quality, an imaging method known as Born inversion (BI) in combination with compressive sensing (CS) theory has been studied. BI is an iterative method which is simple and computationally efficient. However, due to the ill-posedness of the inverse problem, the resulting image will diverge from the correct solution after several iterations. In this work, we use CS ideas to (1) regularize the inversion process and (2) reduce the number of unknowns by restricting the solution of the BI method to be sparse in a wavelet domain. The sparse BI method estimates a contrast function by minimizing a cost functional given by the mean square error between the measured and modeled data. To regularize the inverse problem and to include sparseness, an extra penalty term is added to this cost functional. This penalty term defines the set of coefficients of the estimated image in the wavelet transform domain whose L₀-norm is minimal. A solution for the sparse BI method is found by using an iterative algorithm, which at each iteration first estimates a contrast function image using the conjugate gradient solution, then computes the wavelet transform of the estimated image and finally selects the wavelets with the largest coefficients. The sparse BI method has been tested successfully on noise-free and noisy synthetic data set representing a tomographic scan of a cancerous breast. Experimental results show that the sparse Born inversion method remains convergent as the number of iterations increases, in comparison to conventional BI method.