Computed Tomography for RegionofInterest Problems with Limited Data 
 Abstract  In this thesis the mathematical model for reconstructing crosssectional images of deep sea oil pipes is studied for limited regionofinterest Xray measurement data. This is motivated by a research and development project in collaboration with the Digital Xray inspection division within the company FORCE Technology. The images are used to detect defects in the pipes such as cracks and corrosion.
The first part of the thesis provides insight into a mathematical model describing regionofinterest Xray tomography. For this model it is shown exactly which singularities of a measured object are (or are not) visible in the data using a framework derived from microlocal analysis. This provides an expectation of the challenges in reconstructions from limited data.
The second part of the thesis studies reconstruction algorithms for the regionofinterest model. Firstly, the expected challenges of reconstructing using this model are verified numerically and additional challenges, when using standard algorithms, are shown. With the aim of overcoming these challenges a weighted framebased sparsity penalty in a variational formulation is used to incorporate prior knowledge of the measurement geometry and object. This method is shown to include only significant details of the object that are visible in the data and is wellrepresented by the frame.
In the third and last part of the thesis this insight is used on real measurement data provided by FORCE from a prototype setup. The expected challenges of ROI are shown to hold for real data. Hence, an exterior measurement geometry is proposed as an alternative to ROI yielding more singularities of the object in the data. The weighted framebased methods are shown to provide reliable reconstructions on this type of data.  Type  Master's thesis [Industrial collaboration]  Year  2017  Publisher  Technical University of Denmark, Department of Applied Mathematics and Computer Science  Address  Richard Petersens Plads, Building 324, DK2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk  Series  DTU Compute M.Sc.2017  Note  DTU supervisors: Per Christian Hansen, pcha@dtu.dk, DTU Compute, and Yiqiu Dong, yido@dtu.dk, DTU Compute  Electronic version(s)  [pdf]  Publication link  http://www.compute.dtu.dk/English.aspx  BibTeX data  [bibtex]  IMM Group(s)  Scientific Computing 
