<p>In this thesis, inverse problems related to microwave imaging and microwave component design are investigated. Our contribution in microwave imaging for breast tumor detection can be divided into two parts. In the first part, a vectorial 3D near-field microwave holography is proposed which is an improvement over the existing holography algorithms. In the second part, a simple and fast post-processing algorithm based on the principle of blind de-convolution is proposed for removing the integration effect of the antenna aperture. This allows for the data collected by the antennas to be used in 3D holography reconstruction. The blind deconvolution algorithm is a well-known algorithm in signal processing and our contribution here is its adaptation to microwave data processing.</p> <p>Second, a procedure for accelerating the space-mapping optimization process is presented. By exploiting both fine- and surrogate-model sensitivity information, a good mapping between the two model spaces is efficiently obtained. This results in a significant speed-up over direct gradient-based optimization of the original fine model and enhanced performance compared with other space-mapping approaches. Our approach utilizes commercially available software with adjoint-sensitivity analysis capabilities.</p> / Thesis / Master of Applied Science (MASc)

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The main contribution of this thesis is the concept of Fenchel duality with a focus on its application in the field of machine learning problems and image restoration tasks. We formulate a general optimization problem for modeling support vector machine tasks and assign a Fenchel dual problem to it, prove weak and strong duality statements as well as necessary and sufficient optimality conditions for that primal-dual pair. In addition, several special instances of the general optimization problem are derived for different choices of loss functions for both the regression and the classifification task. The convenience of these approaches is demonstrated by numerically solving several problems. We formulate a general nonsmooth optimization problem and assign a Fenchel dual problem to it. It is shown that the optimal objective values of the primal and the dual one coincide and that the primal problem has an optimal solution under certain assumptions. The dual problem turns out to be nonsmooth in general and therefore a regularization is performed twice to obtain an approximate dual problem that can be solved efficiently via a fast gradient algorithm. We show how an approximate optimal and feasible primal solution can be constructed by means of some sequences of proximal points closely related to the dual iterates. Furthermore, we show that the solution will indeed converge to the optimal solution of the primal for arbitrarily small accuracy. Finally, the support vector regression task is obtained to arise as a particular case of the general optimization problem and the theory is specialized to this problem. We calculate several proximal points occurring when using difffferent loss functions as well as for some regularization problems applied in image restoration tasks. Numerical experiments illustrate the applicability of our approach for these types of problems.

The main contribution of this thesis is the concept of Fenchel duality with a focus on its application in the field of machine learning problems and image restoration tasks. We formulate a general optimization problem for modeling support vector machine tasks and assign a Fenchel dual problem to it, prove weak and strong duality statements as well as necessary and sufficient optimality conditions for that primal-dual pair. In addition, several special instances of the general optimization problem are derived for different choices of loss functions for both the regression and the classifification task. The convenience of these approaches is demonstrated by numerically solving several problems. We formulate a general nonsmooth optimization problem and assign a Fenchel dual problem to it. It is shown that the optimal objective values of the primal and the dual one coincide and that the primal problem has an optimal solution under certain assumptions. The dual problem turns out to be nonsmooth in general and therefore a regularization is performed twice to obtain an approximate dual problem that can be solved efficiently via a fast gradient algorithm. We show how an approximate optimal and feasible primal solution can be constructed by means of some sequences of proximal points closely related to the dual iterates. Furthermore, we show that the solution will indeed converge to the optimal solution of the primal for arbitrarily small accuracy. Finally, the support vector regression task is obtained to arise as a particular case of the general optimization problem and the theory is specialized to this problem. We calculate several proximal points occurring when using difffferent loss functions as well as for some regularization problems applied in image restoration tasks. Numerical experiments illustrate the applicability of our approach for these types of problems.

Title: Image Deblurring in Demanding Conditions Author: Jan Kotera Department: Institute of Information Theory and Automation, Czech Academy of Sciences Supervisor: Doc. Ing. Filip Šroubek, Ph.D., DSc., Institute of Information Theory and Automation, Czech Academy of Sciences Abstract: Image deblurring is a computer vision task consisting of removing blur from image, the objective is to recover the sharp image corresponding to the blurred input. If the nature and shape of the blur is unknown and must be estimated from the input image, image deblurring is called blind and naturally presents a more difficult problem. This thesis focuses on two primary topics related to blind image deblurring. In the first part we work with the standard image deblurring based on the common convolution blur model and present a method of increasing robustness of the deblur- ring to phenomena violating the linear acquisition model, such as for example inten- sity clipping caused by sensor saturation in overexposed pixels. If not properly taken care of, these effects significantly decrease accuracy of the blur estimation and visual quality of the restored image. Rather than tailoring the deblurring method explicitly for each particular type of acquisition model violation we present a general approach based on flexible automatic...