多光源扩散光学层析成像的一类Kaczmarz型方法研究

Diffuse optical tomography with multi light sources is a problem with large scale and strong ill-posedness. Many deficiencies still exist in most imaging methods based on iteration algorithms on aspects of computational efficiency and precision, which impedes the clinical application of diffuse optical tomography. For the above issues, the study on the iterative regularization methods with high efficiency and accuracy for double-parameter identification in diffuse optical tomography will be carried out in this project. Our research work will mainly contains two parts. Firstly, we are devoted to the study of iterative regularization methods with fast convergence. A kind of stable iteration algorithm is constructed, and its acceleration problems will be studied by combining with the sequential subspace optimization method and Kaczmarz method, the regularity of the iterative algorithms are going to be analyzed. The obtained algorithms not only reduce the number of cycles for a satisfying solution, but also taper off the amount of sub-problems calculated in each cycle, which can significantly shorten the required time for parameters reconstruction, theoretically. Then a projection method is utilized for sparse processing in order to catch the detailed information of targets. The methods proposed in this project can be regarded as a profitable supplement to the study of diffuse optical tomography, which also have promotional value for other large-scale nonlinear inverse problems.

多光源扩散光学层析成像问题计算规模大、不适定性强，已有的基于迭代算法的成像方法在计算效率、计算精度等方面存在很大不足，阻碍了其在实际临床中的应用。针对上述问题，本项目拟开展扩散光学层析成像双参数重构的高效率、高精度迭代正则化方法研究。主要包含以下两个方面：1）快速的迭代正则化方法研究。构造一类稳定的迭代格式，并结合序列子空间优化方法以及Kaczmarz方法进一步研究算法的加速问题，分析算法的收敛性和稳定性。所得算法在减少迭代所需外部循环次数的同时可逐渐减少每次循环中子问题的计算个数，理论上可以显著缩短参数反演所需时间。2）利用投影方法进行稀疏处理，实现扩散光学层析成像对异常体细节信息的精细刻画。本项目所提出的方法是对扩散光学层析成像理论的一个有益补充，同时对其它大规模非线性反问题也具有推广价值。

多光源扩散光学层析成像因其能够进行无损检测而备受关注。由于问题的不适定性强，计算规模大，未被广泛进行临床应用。已有的迭代算法在计算效率和精度方面都存在一定的局限性。针对多光源扩散光学层析成像以及抽象的非线性不适定问题开展了算法研究，提出了基于同伦摄动的加速序列子空间迭代算法，投影平均Kaczmarz方法，邻近正则高斯牛顿Kaczmarz迭代算法，投影两点梯度Kaczmarz方法。针对非光滑非线性不适定问题基于Bouligand次梯度提出了加速同伦摄动迭代方法。对以上算法进行了收敛性、正则性的理论分析，给出了算法的数值模拟，验证了所提出方法能够减少计算时间，并且在一定程度上提高成像精度。本项目执行期间发表SCI论文5篇，授权发明专利1项，国内会议4次。