UCLA

Most of the studies of noise-induced phenomena assume that the noise source is Gaussian because of the possibility of obtaining some analytical results when working with Gaussian noises. The use of non-Gaussian noises is rare, mainly because of the difficulties in handling them. However, there is experimental evidence indicating that in many phenomena, the noise sources could be non-Gaussian, for example Poisson data and sparsely corrupted data. This thesis provides two classes of algorithms for dealing with some special types of non-Gaussian noise.

Obtaining high quality images is very important in many areas of applied sciences, and the first part of this thesis is on expectation maximization (EM)-Type algorithms for image reconstruction with Poisson noise and weighted Gaussian noise. In these two chapters, we proposed general robust expectation maximization (EM)-Type algorithms for image reconstruction when the measured data is corrupted by Poisson noise and weighted Gaussian noise, without and with background emission. This method is separated into two steps: EM step and regularization step. In order to overcome the contrast reduction introduced by some regularizations, we suggested EM-Type algorithms with Bregman iteration by applying a sequence of modified EM-Type algorithms. One algorithm with total variation being the regularization is used for image reconstruction in computed tomography application.

The second part of this thesis is on adaptive outlier pursuit method for sparsely corrupted data. In many real world applications, there are all kinds of errors in the measurements during data acquisition and transmission. Some errors will damage the data seriously and make the obtained data containing no information about the true signal, for example, sign flips in measurements for 1-bit compressive sensing and impulse noise in images. Adaptive outlier pursuit is used to detect the outlier and reconstruct the image or signal by iteratively reconstructing the image or signal and adaptively pursuing the outlier. Adaptive outlier pursuit method is used for robust 1-bit compressive sensing and impulse noise removal in chapters 4 and 5 respectively.