New contributions. In this paper we prove new lower and upper bounds for both the source model and the channel model. In each case, an example is provided to show that the new bound represents an strict improvement over the corresponding previously known bound. Roughly speaking our new lower bound in the source model is proved by following the interactive communication stage by stage, however we have to do some careful bookkeeping of the buildup of the secret-key rate by controlling the amount of reduction of secret key rate built up in earlier stages due to the communication in later stages. The lower bound in the source model is exploited for deriving a new lower bound on the secret key capacity in the channel model. 1Maurer provided a different technique for deriving lower bounds on the secret key capacity in [10]. He proved, for instance, that even when the maximum of the two one-way secret key capacities vanishes, the secret key capacity may still be positive. This technique however seems to give us a rather low secret key rate in this case. A generally applicable single letter form of a lower bound based on the ideas in [10] is not known. The technique used for deriving the upper bounds is to consider functions of joint distributions which satisfy specific properties that eventually lead to their dominating the secret key capacity. More specifically, in the source model, we consider a specific class of functions of joint distributions, called potential functions, and show that they satisfy the following property: for any secret key generating protocol, the potential function starts from the upper bound and decreases as we move along the protocol, and eventually becomes equal to the secret key rate of the protocol. See section III for more details. The technique takes the following form in the case of the channel model. Take an arbitrary secret key generation scheme that uses the DMBC for say n times. During the simulation of the protocol, the “secret key reservoir” (representing the amount of secret key bits built up so far)2 of the legitimate terminals gradually increases until it reaches its final state where the legitimate terminals create the common secret key. The idea is to quantify this gradual evolution, bound the derivative of its increase at each stage from above by showing that one use of the DMBC can buy us at most a certain amount of secret bits; and that the use of the public channel does not increase the “secret key reservoir”. See section II of the seco...
New contributions. This thesis makes contributions on several aspects of the image deblurring problem, including modeling, algorithms, and software. New synthetic boundary conditions are devised, including development of an efficient implementation. In addition, a new regularized DCT pre- conditioner is used for iterative deblurring algorithms when using synthetic boundary conditions. Extensive experiments presented in this thesis illus- trate the effectiveness of synthetic boundary conditions and the regularized DCT preconditioner. To facilitate research in image deblurring, two software packages, PYRET and XXXXXX, were developed. PYRET (Python RestoreTools), which uses object oriented programming in Python, is a serial implementation on CPUs; XXXXXX (Parallel RestoreTools), which makes use of the computing power of GPUs, is a parallel implementation. A Web user interface has also been developed for PYRET. In the course of writing software for these packages, it was necessary to contribute a new complex branch to the open-source soft- xxxx PyCUDA, and to create Python wrappers so that CUBLAS and CUFFT libraries will work with PyCUDA. Benchmark results presented in this thesis show a significant speedup of the GPU implementation (XXXXXX) over the CPU implementation (PYRET). For blind deconvolution, the variable projection technique is used to simplify the problem. The formulas involved are carefully derived using the spectral decomposition and two lemmas on conjugate symmetric vectors. Specific details are provided when tackling pupil phase blurs, especially on how to decompose the Jacobian matrix for fast multiplications. In addition, a new approach is proposed to provide a mathematical decoupling of the optimization problem when multiple frames from the same object are used. This approach leads to a block structure of the Jacobian matrix, which allows efficient multiplications. Numerical experiments show the benefits gained by using more than one frame.
