Regularization and Scaling. In this chapter, we assume that the measured data obtained from the detector follow a Poisson distribution. To remove the noise and stabilize the solution, we need to introduce regularization terms. As the weights of each material correspond to an image, we can construct the regularization term with respect to the specified material. If we assume w is concatenated by w1, w2, · · · , wm where wi is the vectorization of the i-th material map, then we can build the regularization term as Σ Nm R (w) = αiR (wi) , (4.34) where R (wi) represents the regularization term for the i-th material and αi is the corresponding regularization parameter. Using this method, it gives us more freedom to choose the regularization term with respect to the material. In practice, we can usually find that one or several specified materials dominate the object and it is likely that the reconstructed material maps will contain many edges. So we select general- ized Tikhonov regularization for these materials to smooth the edges. The forward difference operator and zero boundary conditions are used to build the discrete dif- ferential operator. On the other hand, other materials might only occupy a small area so we think about restrict the sum of weights for these materials. To realize this idea, we introduce l1 regularization to penalize the sum of weights. With the generalized Tikhonov regularization and the l1 regularization, we need to select the corresponding regularization parameters. It is not obvious how we can use regular methods, such as L-curve and generalized cross-validation, to find proper parameters. In this case, we generate a log space for each regularization and use the grid search method to find the “best” regularization parameters. Another challenge associated with the regularization parameter is how to scale the problem. The point such that the objective function is zero might not be feasible and the residual corresponding to the global minimizer might still be a large number. In this case, choosing a proper regularization parameter is hard if the magnitude is large. In addition to selecting the regularization parameter, the infeasible step might result in the difficulty of meeting stopping criteria and thus increase the number of CG iterations. For this problem, we want to scale the objective function, gradient and Hessian with the spectrum radius of the Hessian based on the 2-norm. However, it is not necessary to compute the largest singular value of the Hessi...
