Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography

We present a new approach to solve the inverse source problem arising in Fluorescence Tomography (FT). In general, the solution is non-unique and the problem is severely ill-posed. It poses tremendous challenges in image reconstructions. In practice, the most widely used methods are based on Tikhonov-type regularizations, which minimize a cost function consisting of a regularization term and a data fitting term. We propose an alternative method which overcomes the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, in two separate steps. First we find a particular solution called the orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills other regularity and physical requirements. The key ideas are that the correction function in the kernel has no impact on the data fitting, so that there is no parameter needed to balance the data fitting and additional constraints on the solution. Moreover, we use an efficient basis to represent the source function, and introduce a hybrid strategy combining spectral methods and finite element methods in the proposed algorithm. The resulting algorithm can dramatically increase the computation speed over the existing methods. Also the numerical evidence shows that it significantly improves the image resolution and robustness against noise.