Browsing by Subject "Compressed Sensing"

Now showing 1 - 4 of 4

Open Access
Fast calibration and image reconstruction for magnetic particle imaging
(2018-05) İlbey, Serhat
Magnetic particle imaging (MPI) is a relatively new medical imaging modality that images the spatial distribution of magnetic nanoparticles (MNPs) administered to the body. For image reconstruction with the system matrix (SM) approach, a time-consuming calibration scan is necessary, in which a single MNP sample is placed and scanned inside the full eld-of-view (FOV). Moreover, for a relatively large 3D high-resolution FOV, the reconstructed SM is too large to get high quality images in real-time using the standard state-of-the-art techniques. In this thesis, for the calibration scan, the use of coded calibration scenes (CCSs) is proposed, which utilizes MNP samples at multiple positions inside the FOV. The SM, which is sparse in the discrete cosine transform domain, is reconstructed using the Alternating Direction Method of Multipliers (ADMM) with l1-norm minimization. The e ectiveness of the CCSs for di erent parameter sets is analyzed via simulations, and the results are compared with the standard sparse reconstruction technique. As the MPI images are naturally sparse, ADMM is also proposed for image reconstruction, minimizing the total variation and l1-norm. Image quality is compared with the images obtained by widely used MPI image reconstruction algorithms: Algebraic Reconstruction Technique, Nonnegative Fused LASSO, and X-space-based projection reconstruction. Moreover, ADMM is parallelized on a GPU for real-time image reconstruction. The results show that the required number of measurements for system calibration is substantially reduced with the proposed methods, and the reconstruction performance is signi cantly improved in terms of both image quality and speed.
Open Access
Novel methods for SAR imaging problems
(2013) Uğur, Salih
Synthetic Aperture Radar (SAR) provides high resolution images of terrain reflectivity. SAR systems are indispensable in many remote sensing applications. High resolution imaging of terrain requires precise position information of the radar platform on its flight path. In target detection and identification applications, imaging of sparse reflectivity scenes is a requirement. In this thesis, novel SAR image reconstruction techniques for sparse target scenes are developed. These techniques differ from earlier approaches in their ability of simultaneous image reconstruction and motion compensation. It is shown that if the residual phase error after INS/GPS corrected platform motion is captured in the signal model, then the optimal autofocused image formation can be formulated as a sparse reconstruction problem. In the first proposed technique, Non-Linear Conjugate Gradient Descent algorithm is used to obtain the optimum reconstruction. To increase robustness in the reconstruction, Total Variation penalty is introduced into the cost function of the optimization. To reduce the rate of A/D conversion and memory requirements, a specific under sampling pattern is introduced. In the second proposed technique, Expectation Maximization Based Matching Pursuit (EMMP) algorithm is utilized to obtain the optimum sparse SAR reconstruction. EMMP algorithm is greedy and computationally less complex resulting in fast SAR image reconstructions. Based on a variety of metrics, performances of the proposed techniques are compared. It is observed that the EMMP algorithm has an additional advantage of reconstructing off-grid targets by perturbing on-grid basis vectors on a finer grid.
Open Access
Recovery of sparse perturbations in Least Squares problems
(IEEE, 2011) Pilanci, M.; Arıkan, Orhan
We show that the exact recovery of sparse perturbations on the coefficient matrix in overdetermined Least Squares problems is possible for a large class of perturbation structures. The well established theory of Compressed Sensing enables us to prove that if the perturbation structure is sufficiently incoherent, then exact or stable recovery can be achieved using linear programming. We derive sufficiency conditions for both exact and stable recovery using known results of ℓ 0/ℓ 1 equivalence. However the problem turns out to be more complicated than the usual setting used in various sparse reconstruction problems. We propose and solve an optimization criterion and its convex relaxation to recover the perturbation and the solution to the Least Squares problem simultaneously. Then we demonstrate with numerical examples that the proposed method is able to recover the perturbation and the unknown exactly with high probability. The performance of the proposed technique is compared in blind identification of sparse multipath channels. © 2011 IEEE.
Open Access
Uncertain linear equations
(2010) Pilancı, Mert
In this thesis, new theoretical and practical results on linear equations with various types of uncertainties and their applications are presented. In the first part, the case in which there are more equations than unknowns (overdetermined case) is considered. A novel approach is proposed to provide robust and accurate estimates of the solution of the linear equations when both the measurement vector and the coefficient matrix are subject to uncertainty. A new analytic formulation is developed in terms of the gradient flow to analyze and provide estimates to the solution. The presented analysis enables us to study and compare existing methods in literature. We derive theoretical bounds for the performance of our estimator and show that if the signal-to-noise ratio is low than a treshold, a significant improvement is made compared to the conventional estimator. Numerical results in applications such as blind identification, multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean-squared error for a significant range of signal-to-noise ratio values. The second type of uncertainty analyzed in the overdetermined case is where uncertainty is sparse in some basis. We show that this type of uncertainty on the coefficient matrix can be recovered exactly for a large class of structures, if we have sufficiently many equations. We propose and solve an optimization criterion and its convex relaxation to recover the uncertainty and the solution to the linear system. We derive sufficiency conditions for exact and stable recovery. Then we demonstrate with numerical examples that the proposed method is able to recover unknowns exactly with high probability. The performance of the proposed technique is compared in estimation and tracking of sparse multipath wireless channels. The second part of the thesis deals with the case where there are more unknowns than equations (underdetermined case). We extend the theory of polarization of Arikan for random variables with continuous distributions. We show that the Hadamard Transform and the Discrete Fourier Transform, polarizes the information content of independent identically distributed copies of compressible random variables, where compressibility is measured by Shannon’s differential entropy. Using these results we show that, the solution of the linear system can be recovered even if there are more unknowns than equations if the number of equations is sufficient to capture the entropy of the uncertainty. This approach is applied to sampling compressible signals below the Nyquist rate and coined ”Polar Sampling”. This result generalizes and unifies the sparse recovery theory of Compressed Sensing by extending it to general low entropy signals with an information theoretical analysis. We demonstrate the effectiveness of Polar Sampling approach on a numerical sub-Nyquist sampling example.