Conclusions and Remarks. By taking multiple materials into consideration, the reconstructed images can reveal the weights of materials that compose the object, providing substantially more useful information for the clinicians. Furthermore, the objective function and the gradient are uncomplicated to implement and the modified Hessian is a sufficient and stable estimate to the true Hessian. In addition, the merits of using a nonlinear interior point method are easy to identify. It is a globally convergent method with superlinear rate of convergence. It is also a stable and robust algorithm that can handle large-scale problems. Furthermore, there is substantial flexibility in choosing the initial guess because it does not need to satisfy the constraints. Although this method has advantages such as faster convergence, robust compu- tation and flexibility, it still has a few limitations. For example, implementation of nonlinear interior point method is not straightforward for large-scale problems. It re- quires solving a normal subproblem as well as a tangential subproblem. Furthermore, we need to decide the size of trust region in each iteration. Meanwhile, this method involves many parameters that we need to choose manually. So far, we have only tested 2D images rather than 3D images. For 3D images, the evaluation of each part might be more complicated, which is likely to increase the expense for solving this problem. For further research, we might consider the gradient-based methods such as the scaled gradient descent method or splitting methods such as the alternating direction method of multipliers (ADMM) [21]. Chapter 4 Nonlinear Optimization for Energy-windowed Spectral Computed Tomography The development of new energy-windowed spectral computed tomography (CT) ma- chines have received a great deal of interest in recent years; see, e.g. [2, 57]. These detectors assume that x-rays emitted by the x-ray source are composed of a spectrum of different energies, and in each energy window, the detector can detect a specific range of energy. Moreover, it assumes that the detector can perform photon count- ing and the data collected by the detector are nonnegative integers. Compared with traditional CT machines, we can avoid introducing beam-hardening artifacts [45] and improve quality of reconstructed images. To reconstruct the material maps of an object, we need to solve a nonlinear equation of the form Y = exp .−AWCT Σ S + E, (4.1) where Y is a matrix that gathers the projected d...
Conclusions and Remarks. With the energy-windowed spectral CT model, we set up a nonlinear least squares problem under bound constraints. To solve this optimization problem, we propose a new preconditioner and then implement it into a two-step method. The new pre- conditioner can transform the eigenvalues of the original Xxxxx-Xxxxxx system into more clustered ones, which will lead to faster convergence and higher accuracy. With the introduction of the two-step method, we can guarantee that the obtained step is at least better than the approximate Cauchy point. By solving the Xxxxx-Xxxxxx system in the second step, we expect further reduction from the solution. Therefore, the convergence rate should be better than linearity. Moreover, we further restrict the size of each step with the help of trust regions. In addition, we can remove parts of noise and speed up the convergence rate with the scaling parameter and the regularization terms. On the other hand, it still has several limitations. Because of the nonlinearity of the objective function, it is hard to decide if we should accept the new step or not under certain circumstances. Furthermore, with multiple materials, we cannot use regular methods to choose regularization parameters. In each iteration, we need to solve a NKP problem to obtain the preconditioner. Even if the computational cost is cheap, it might be better if we can find a preconditioner that is feasible for all steps. We might also think about how to implement the preconditioner into first order methods such as FISTA [3]. Chapter 5 Preconditioning and Optimization for Energy-windowed Spectral Computed Tomography In Chapter 4, we discussed the energy-windowed spectral CT model and presented a preconditioning framework and a nonlinear optimization approach to compute the solution. In this chapter, we still focus on the energy-windowed spectral CT model Y = exp .−AWCT Σ S + E, (5.1) where Y is a matrix that gathers the projected data of each energy window in the corresponding column and the exponential operator is applied element-wise (i.e., it is not a matrix function). A is a matrix that is related to the quantitative information of ray trace and C is a matrix that contains linear attenuation coefficients for par- ticular (known) materials at specified energies. S is the matrix that accumulates the spectrum energies for each energy window in the corresponding column. We assume that S is square and invertible. Moreover, E represents the noise term and we assume t...
Conclusions and Remarks. In this paper, we introduced a certificateless two- party authenticated key agreement protocol; The KGC publishes the system parameters and public keys in a public directory. The public directory (LDAP server) possesses a certificate to prove its identity to both communicating parties. The communications between the KGC and the LDAP server is offline. Both the two communicating parties are able to compute the same secret symmetric key without messages exchanges, except the first hello message, which is encrypted. This makes it impossible to carry out a man in the middle attack to obtain information about the encryption/decryption key. Moreover, the scheme is secure against known key attack, resilient against unknown key share and key-compromise impersonation, and secure against weak perfect forward secrecy. Compared to the CTAKA protocol presented by Yang et. al (Yang et al., 2011) and the MHM protocol (Mokhtarnameh et al., 2011), the proposed scheme in this paper is resistant to the key escrow problem through using the second secret value x'A. This comes in contrast to the scheme proposed by Yang et. al in(Yang et al., 2011) where the KGC can easily compute the full private key for user A by multiplying the public term YA by its master secret value s and also in contrast to the scheme proposed by Mokhtarnameh et al. (Mokhtarnameh et al., 2011), in which the KGC can easily compute the full private key for client A by multiplying the public term PA by its master secret value s. The proposed protocol is immune against any inside (type 2) or outside (type 1) attacks. Therefore, the CTAKA protocol introduced in this paper is fully secure, if we assume that the KGC is honest and each party within the network has the full capability to fully protect its secret values. On the other hand, The results explained in Table 1 show that all the pairing free protocols stated in the table have costs of at least five points multiplications, two hash function evaluations and two message exchanges, except Mohamed et al. [18] and the improved CTAKA protocol. The proposed protocol requires more point multiplications and hash function evaluations than the protocol of Mohamed et al. But it is much practical as it does not require the involvement of the KGC at each session initiation.
