Application of compress sensing in M.R.I. imaging

The sparsity of signals in a certain transform domain or dictionary has been extended in different applications in signal processing, image processing, and medical imaging. Wavelets and DCT have been widely used for compression. \ud Recently, new application of the data-driven learning of sparsifying dictionaries has discovered in denoising, inpainting, and compressed sensing. Here We study the sparsifying transform model related to its prior linear \ud sparse models. Then, we formulate the problem for learning square sparsifying transforms from data. Here algorithm alternate between a sparse coding step and a transform update step. Compressed sensing (CS) utilizes the sparsity of MR images to enable reconstruction from \ud undersampled k-space data. Recent CS methods have employed analytical sparsifying transforms such as wavelets, curvelets, and finite differences. This transform is sele\ud cted on the basis of image type. In this thesis, we propose a framework for adaptively learning the sparsifying transform (dictionary), and reconstructing the image simultaneously from highly under sampled k-space \ud data. This framework is enforced on overlapping image patches emphasizing local structure. Reconstruction algorithm learns the sparsifying dictionary, \ud and uses it to remove aliasing and noise in one step, and restores . Then it fills in the k-space data in the other step