Browsing by Subject "Haar wavelet"

In today's world, we often wish to see the inner structure of a variety of objects without actually having to cut open the object in question. This can be accomplished for example through X-ray computed tomography (CT), where we measure the attenuated X-rays going through the object around the whole object (full view) or around a part of it, and we try to recover the inner structure from the noisy measurement with an algorithm. This problem is usually referred to as an ill-posed inverse problem, since the data is often incomplete and the noise makes it unstable. To overcome these problems, we will use a sparsity promoting regularization method using a dictionary. In this thesis the tilted Haar wavelet transform is introduced. It is shown how it can be formulated as a matrix, and used as a part of a larger dictionary to solve a computed tomography regularization problem using the primal-dual fixed point (PDFP) algorithm. Three different X-ray tomographic situations are studied, namely a full-view, a sparse-angle view and a limited-angle view with two different noise levels. First, we use a dictionary with the Haar wavelet transform and the tilted wavelet transform. For comparison, we also present the results when the dictionary has either only the Haar wavelet transform or only the tilted Haar wavelet transform. A Tikhonov regularization reconstruction is also presented to give a wider picture of the suitability of the proposed algorithm. The reconstructions were done for simulated data. The results show the larger dictionary, with both of the wavelet transforms, reduced the noise in the reconstruction more than just using one of them as the dictionary or by using the Tikhonov regularization. In conclusion, this thesis introduces the tilted Haar wavelets, builds an algorithm to do the reconstructions and presents the results when using simulated data.