Browsing by Subject "inversio-ongelmat"

The motivation for the methods developed in this thesis rises from solving the severely ill-posed inverse problem of limited angle computed tomography. Breast tomosynthesis provides an example where the inner structure of the breast should be reconstructed from a very limited measurement angle. Some parts of the boundaries of the structure can be recovered from the X-ray measurements and others can not. These are referred to as visible and invisible boundaries. For parallel beam measurement geometry directions of visible and invisible boundaries can be deduced from the measurement angles. This motivates the usage of the concept of wavefront set. Roughly speaking, a wavefront set contains boundary points and their directions. The definition of wavefront set is based on Fourier analysis, but its characterization with the decay properties of functions called shearlets is used in this thesis. Shearlets are functions based on changing resolution, orientation, and position of certain generating functions. The theoretical part of this thesis focuses on studying this connection between shearlets and wavefront sets. This thesis applies neural networks to the limited angle CT problem since neural networks have become state-of-the-art in many computer vision tasks and achieved impressive performance in inverse problems related to imaging. Neural networks are compositions of multiple simple functions, typically alternating linear functions and some element-wise non-linearities. They are trained to learn values for a huge amount of parameters to approximate the desired relation between input and output spaces. Neural networks are very flexible function approximators, but high dimensional optimization of parameters from data makes them hard to interpret. Convolutional neural networks (CNN) are the ones that succeed in tasks with image-like inputs. U-Net is a CNN architecture with very good properties, like learning useful parameters form considerably small data sets. This thesis provides two U-Net based CNN methods for solving limited angle CT problems. The main focus is on method projecting model-based reconstructions such that the projections have the desired wavefront sets. The guiding principle of this projector network is that it should not change reconstruction already projected to the given wavefront set. Another network estimates the invisible part of the wavefront set from the visible one. Few different data sets are simulated to train and evaluate these methods and performance on real data is also tested. A combination of the wavefront set estimator and the projector networks were used to postprocess model-based reconstructions. The fact this postprocessing has two steps increases the interpretability and the control over the processes performed by neural networks. This postprocessing increased the quality of reconstructions significantly and quality was even better when the true wavefront set was given for the projector as a prior.