Diffuse optical tomography with near-infrared light is a promising technique for noninvasive study of the functional characters of human tissues. Mathematically, it is a seriously ill-posed parameter identification problem. For the purpose of better providing both segmentation and piecewise constant approximation of the underlying solution, nonconvex nonsmooth total variation based regularization functional is considered in this paper. We first give a theoretical study on the well-posedness of solutions corresponding to this minimization problem in the Banach space of piecewise constant functions. Moreover, our theoretical results show that the minimizers corresponding to a sequence nonconvex nonsmooth potential functions which converge to the 0–1 functions, can be used to approximate the solution to the weak Mumford–Shah regularization. Then from the numerical side, we propose a double graduated nonconvex Gauss–Newton algorithm to solve this nonconvex nonsmooth regularization. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of the proposed algorithm.

具有近红外光的漫射光学断层扫描是一种用于无创研究人体组织功能特征的有前途的技术。从数学上讲，这是一个严重不适定的参数识别问题。为了更好地提供底层解的​​分割和分段常数近似，本文考虑了基于非凸非光滑总变差的正则化函数。我们首先对分段常数函数的Banach空间中对应于这个最小化问题的解的适定性进行了理论研究。此外，我们的理论结果表明，对应于收敛到 0-1 函数的序列非凸非光滑势函数的最小化器可用于逼近弱 Mumford-Shah 正则化的解。然后从数值方面，我们提出了一种双梯度非凸高斯-牛顿算法来解决这种非凸非光滑正则化问题。所有的插图和数值实验都给出了所提出算法的最小化器提供的可能性的味道。