Solutions to Space-Time Inverse Problems

Abstract

Two inverse problems are investigated in this dissertation, taking into account both the spatial and temporal aspects. The first problem addresses the under determined image reconstruction problem for dynamic SPECT. The quality of the reconstructed image is often limited due to having fewer observations than the number of voxels. The proposed algorithms make use of the generalized α-divergence function to improve the estimation performance. The first algorithm is based on an alternating minimization framework to minimize a regularized α-divergence objective function. We demonstrate that selecting an adaptive α policy depending on the time evolution of the voxels gives better performance than a fixed α assignment. The second algorithm is based on Newton's method. A regularized approach has been taken to avoid stability issues. Newton's method is generally computationally demanding due to the complexity associated with inverting the Hessian matrix. A fast Newton-based method is proposed using majorization-minimization techniques that diagonalize the Hessian matrix. In dynamically evolving systems, the prediction matrix plays an important role in the estimation process. An estimation technique is proposed to estimate the prediction matrix using the α-divergence function. The simulation results show that our algorithms provide better performance than the techniques based on the Kullback-Leibler distance. The second problem is the recovery of data transmitted over free-space optical communication channels using orbital angular momentum (OAM). In the presence of atmospheric turbulence, crosstalk occurs among OAM optical modes resulting in an error floor at a relatively high bit error rate. The modulation format considered for the underlying problem is Q-ary pulse position modulation (PPM). We propose and evaluate three joint detection strategies to overcome the OAM crosstalk problem: i) maximum likelihood sequence estimation (MLSE). ii) Q-PPM factor graph detection. iii) branch-and-bound detection. We compare the complexity and the bit-error-rate performance of these strategies in realistic scenarios.