@phdthesis{Fischer2011,
  author    = {Fischer, Andr{\´e}},
  title     = {On the Application of Compressed Sensing to Magnetic Resonance Imaging},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-72496},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2011},
  abstract  = {This thesis investigated the potential of Compressed Sensing (CS) applied to Magnetic Resonance Imaging (MRI). CS is a novel image reconstruction method that emerged from the field of information theory. The framework of CS was first published in technical reports in 2004 by Cand{\`e}s and Donoho. Two years later, the theory of CS was published in a conference abstract and two papers. Cand{\`e}s and Donoho proved that it is possible, with overwhelming probability, to reconstruct a noise-free sparse signal from incomplete frequency samples (e.g., Fourier coefficients). Hereby, it is assumed a priori that the desired signal for reconstruction is sparse. A signal is considered "sparse" when the number of non-zero elements is significantly smaller than the number of all elements. Sparsity is the most important foundation of CS. When an ideal noise-free signal with few non-zero elements is given, it should be understandably possible to obtain the relevant information from fewer Fourier coefficients than dictated by the Nyquist-Shannon criterion. The theory of CS is based on noise-free sparse signals. As soon as noise is introduced, no exact sparsity can be specified since all elements have signal intensities that are non-zero. However, with the addition of little or moderate noise, an approximate sparsity that can be exploited using the CS framework will still be given. The ability to reconstruct noisy undersampled sparse MRI data using CS has been extensively demonstrated. Although most MR datasets are not sparse in image space, they can be efficiently sparsified by a sparsifying transform. In this thesis, the data are either sparse in the image domain, after Discrete Gradient transformation, or after subtraction of a temporally averaged dataset from the data to be reconstructed (dynamic imaging). The aim of this thesis was to identify possible applications of CS to MRI. Two different algorithms were considered for reconstructing the undersampled sparse data with the CS concept. The Nonlinear Conjugate Gradient based technique with a relaxed data consistency constraint as suggested by Lustig et al. is termed Relaxed DC method. An alternative represents the Gradient or Steepest Descent algorithm with strict data consistency and is, therefore, termed the Strict DC method. Chapter 3 presents simulations illustrating which of these two reconstruction algorithms is best suited to recover undersampled sparse MR datasets. The results lead to the decision for the Strict DC method as reconstruction technique in this thesis. After these simulations, different applications and extensions of CS are demonstrated. Chapter 4 shows how CS benefits spectroscopic 19F imaging at 7 T, allowing a significant reduction of measurement times during in vivo experiments. Furthermore, it allows highly resolved spectroscopic 3D imaging in acceptable measurement times for in vivo applications. Chapter 5 introduces an extension of the Strict DC method called CS-CC (CS on Combined Coils), which allows efficient processing of sparse undersampled multi-coil data. It takes advantage of a concept named "Joint Sparsity", which exploits the fact that all channels of a coil array detect the same sparse object weighted with the coil sensitivity profiles. The practical use of this new algorithm is demonstrated in dynamic radial cardiac imaging. Accurate reconstructions of cardiac motion in free breathing without ECG triggering were obtained for high undersampling factors. An Iterative GRAPPA algorithm is introduced in Chapter 6 that can recover undersampled data from arbitrary (Non-Cartesian) trajectories and works solely in the Cartesian plane. This characteristic makes the proposed Iterative GRAPPA computationally more efficient than SPIRiT. Iterative GRAPPA was developed in a preceding step to combine parallel imaging with CS. Optimal parameters for Iterative GRAPPA (e.g. number of iterations, GRAPPA kernel size) were determined in phantom experiments and verified by retrospectively undersampling and reconstructing a radial cardiac cine dataset. The synergistic combination of the coil-by-coil Strict DC CS method and Iterative GRAPPA called CS-GRAPPA is presented in Chapter 7. CS-GRAPPA allows accurate reconstruction of undersampled data from even higher acceleration factors than each individual method. It is a formulation equivalent to L1-SPIRiT but computationally more efficient. Additionally, a comparison with CS-CC is given. Interestingly, exploiting joint sparsity in CS-CC is slightly more efficient than the proposed CS-GRAPPA, a hybrid of parallel imaging and CS. The last chapter of this thesis concludes the findings presented in this dissertation. Future applications expected to benefit from CS are discussed and possible synergistic combinations with other existing MR methodologies for accelerated imaging are also contemplated.},
  subject      = {NMR-Tomographie},
  language  = {en}
}