Browsing by Author "Tyrväinen, Sanna"

The push for cameras integrated into mobile devices has created strict size constraints for the hardware. This results in large compromises in lens quality, causing blur that can be modelled by using the convolution operator. We explore ways of reducing this blur via algorithms, a process that is also known as deconvolution. The deconvolution problem is considered ill-posed. According to Hadamard, an ill-posed problem is one that fails at least one of the following conditions: the solution is unique, the solution exist for any given data and the solution depends with continuity on the data. Traditional approaches to solve the deconvolution problem are Tikhonov and total variation regularization, but these methods have problems with fine structures and patterns in images, because they compare only points next to each other. Nonlocal methods were developed to alleviate these problems. They base on texture synthesis and neighbourhood filters. The main idea is that similar or repeating structures exist in the image. These structures can be used when reconstructing the unclear part of the image. The similarity between points is measured by comparing their environments with a weight function. This thesis presents the theory of total variation regularization, some properties of the total variation functional and the existence of the minimizer. Nonlocal operators and a weight function are then used to define nonlocal total variation regularization. We present an algorithm for solving the deconvolution problem by nonlocal total variation regularization using gradient descent. We compare our implementation to the code by Yifei, Zhang, Osher and Bertozzi (2010). Numerical examples using three different images with different added blur and noise show no significant difference between nonlocal methods, but they show great advance from local to nonlocal methods. Results are compared using the signal to noise ratio (SNR) and an ocular measuring. Tikhonov regularization turns out to be insufficient when measuring SNR.