Melinda M. Au

ORCID Identifier(s)


Graduation Semester and Year




Document Type


Degree Name

Doctor of Philosophy in Mathematics



First Advisor

Ren-Cang Li


Sparse signal reconstruction has been steadily gaining tremendous attention, specifically in applications of compressed sensing as well as feature selection in signal processing methods [IEEE Signal Processing, Vol. 25 (2008) pp.21-30]. Standard techniques to solve these problems such as the nonlinear Conjugate Gradient method have been used successfully on small and medium sized problems. However, as the desire to collect more data becomes the trend, and the need to process information more quickly and reliably becomes more prevalent than before, these standard meth- ods become inadequate. In this thesis, we present a new approach for sparse signal reconstruction called the Three Dimensional Image Reconstruction (3DIRECT) method. This method minimizes total variation using the interior point method via the log-barrier function to recover an image sparsely sampled in the frequency domain. Furthermore, this method is able to recover all of the slices of 3D images simultaneously, as oppose to the current state-of-the-art methods which traditionally recover 3D images one slice at a time. We applied our method to MRI data and observed the following results. First, our method improves upon the speed of existing interior point methods by leveraging object oriented coding, as well as analyzing and improving the stopping criteria in the Conjugate Gradient (CG) method. The 3DIRECT method also benefits from speed improvements due to the structure of the optimization model. The 3D model requires less calls to iterative methods compared to the 2D models. Thus, our method achieves similar or better results than current state-of-the-art methods in less time, e.g. for a 128×128×128 image using 17% sampling, 3DIRECT is 22% faster in mean and 33% faster in minimum observed run times. Finally, our method is able to achieve a cleaner recovered MRI image under certain conditions, due to the ability to exploit physical information in the z-direction. We provide an algorithm which is able to detect this superior region of performance, and provide an indication of when to use the 3DIRECT method versus traditional 2D methods. In cases, where the image has homogeneity in the diameter of the cross- section from slice to slice, the 3DIRECT method provides the best results.


Compressed sensing, Image recovery, MRI, Log-barrier, Interior point method, 3D image recovery


Mathematics | Physical Sciences and Mathematics


Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons