Theory And Algorithm Research For Diffuse Optical Tomography Based On Radiative Transport Equation

As a new noninvasive medical imaging technology that can provide functional information of tissues,diffuse optical tomography(DOT)has become one of the focused topics.Due to the strong scattering and weak absorption nature of biological tissues,noise interference and the relatively limited number of available measurements,the inverse problems of DOT is severely ill-posed.In this paper,to overcome the ill-posedness of inverse problems and to improve the quality of reconstructed image and the imaging efficiency,we study the total variation related regularization methods for DOT based on radiative transport equation.For these regularization methods,both the well-posedness theory of solution and the feasibility and validity of application to DOT are researched.First,in a specific function space,we prove the Lipschitz continuity and differentiability of the forward operator based on the weak format of the boundary value problem of radiative transport equation.We strictly deduce the analytical form of adjoint derivative for forward operator and give the definition of adjoint equation.Second,considering that the objection of DOT reconstruction usually appears to be continuous or piecewise constant distribution,the total variation regularization that can keep the edge information is introduced in DOT for better dealing with the discontinuous situation.Total variation regularization which are suitable for DOT are developed.The existence,stability and convergence under Bregman distance of the minimal solution are proved.In the practical calculation,in view of the sparsity of gradient,we propose the split Bregman algorithm for solving total variation regularization and reweighted total variation regularization respectively.The two algorithms are compared in detail with numerical examples.The simulation results shows that the reweighted total variation regularization with split Bregman algorithm converges quickly.The reconstruction results are better on boundary and inside of inclusion.Moreover,the distribution of parameters can be recovered even with less measurement data.Next,to balance the discontinuity and smoothness of the solution,the total variation regularization method mixed with L~2 norm is proposed for parameters recovery.The existence,stability and convergence of the solution to the mixed regularization method are proved.The convergence rate is given as well.By adopting the lagged diffusivity fixed point method to this mixed regularization,the detailed comparison is made among this method,total variation regularization,and the H1 norm regularization.The results indicate the validity of total variation mixed with L~2 norm regularization and approve the convergence rate of the minimal solution.In the end,total variation mixed with L~1 norm regularization for optical parameters reconstruction is proposed for preserving the boundary and keeping the detailed information of the reconstructed image.In order to analyze the properties of the solution to the total variation mixed with L~1 norm regularization,we discuss the continuity and differentiability of the forward operator when the parameter space and solution space are chosen as general Banach space.Then the properties of the minimal solution to the mixed regularization is explained when the penalty term is Lpnorm,Hsnorm,BV norm and total variation mixed with L~1 norm.In order to improve the convergence rate and avoid the over sparse effect,we propose the split Bregman algorithm based on the lagged diffusivity fixed point method and compare the reconstruction results corresponds to this proposed algorithm,total variation method and L~1 method.The comparison shows that total variation mixed with L~1 norm regularization not only has a higher convergence rate and precision,but also can better identify the boundary.