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This paper introduces a novel ridgelet transform-based method for Poisson image denoising. Our work focuses on harnessing the Poisson noise's unique non-additive and signal-dependent properties, distinguishing it from Gaussian noise. The core of our approach is a new thresholding scheme informed by theoretical insights into the ridgelet coefficients of Poisson-distributed images and adaptive thresholding guided by Stein's method. We verify our theoretical model through numerical experiments and demonstrate the potential of ridgelet thresholding across assorted scenarios. Our findings represent a significant step in enhancing the understanding of Poisson noise and offer an effective denoising method for images corrupted with it.

Much like a backwards computed Sudoku puzzle, starting from the completed number grid and working ones way down to a partially completed grid without damaging the route back to the full unique solution, this thesis tackles the challenges behind setting up a number puzzle in the context of biomedical imaging. By leveraging sparse signal processing theory, we study the means of practical undersampling of positron emission tomography (PET) measurements, an imaging modality in nuclear medicine that visualises functional processes within the body using radioactive tracers. What are the rules for measurement removal? How many measurements can be removed without damaging the route back to the full solution? Moreover, how is the original solution retained once the data has been altered? This thesis aims to investigate and answer such questions in relation to PET data sampling, thereby creating a foundation for a PET Sampling Puzzle.

The objective is to develop intelligent data sampling strategies that allow for practical undersampling of PET measurements combined with sophisticated computational compensations to address the resulting data distortions. We focus on two main challenges in PET undersampling: low-count measurements due to reduced radioactive dose or reduced scan times and incomplete measurements from sparse PET detector configurations. The methodological framework is based on key aspects of sparse signal processing: sparse representations, sparsity patterns and sparse signal recovery, encompassing denoising and inpainting. Following the characteristics of PET measurements, all elements are considered with an underlying assumption of signal-dependent Poisson distributed noise.

The results demonstrate the potential of noise awareness, sparsity, and deep learning to enhance and restore measurements corrupted with signal-dependent Poisson distributed noise, such as those in PET imaging, thereby marking a notable step towards unravelling the PET Sampling Puzzle.

Low-count positron emission tomography (PET) data suffer from high noise levels, leading topoor image quality and reduced diagnostic accuracy. Compressed sensing (CS) based denoisingmethods have shown potential in medical imaging. This study investigates the performance ofCS-based denoising methods on PET sinograms.Three simulated datasets were used in this study, including circular phantom, patient pelvisphantom, and patient brain phantom. Ten sampling levels were employed to investigate the effect of data reduction on diagnostic accuracy. CS-based denoising methods were applied prereconstruction, and a conventional Gaussian post-filter was used for comparison. Performancemeasures included rRMSE, SSIM, SNR, line profiles, and FWHM.Overall, the proposed CS-based denoising methods performed similarly to the benchmark interms of lesion contrast, spatial resolution, and noise texture. The proposed methods outperformed the benchmark in low-count situations by suppressing background noise and preservingcontrast better.The results of this study demonstrate that CS-based denoising methods in the sinogram domain can improve the quality of low-count PET images, particularly in suppressing backgroundnoise and preserving contrast. These findings suggest that CS-based denoising could be apromising solution for improving the diagnostic accuracy of low-count PET data.

Explicitly using the block structure of the unknown signal can achieve better reconstruction performance in compressive sensing. An unknown signal with block structure can be accurately recovered from under-determined linear measurements provided that it is sufficiently block sparse. However, in practice, the block sparsity level is typically unknown. In this paper, we propose a soft measure of block sparsity k_α(x) = (||x||_2,α/||x||_2,1) ^α/(1−α) with α ∈ [0,∞], and present a procedure to estimate it by using multivariate centered isotropic symmetric α-stable random projections. The limiting distribution of the estimator is given. Simulations are conducted to illustrate our theoretical results.

The purpose of this work is to investigate spatial statistical modelling approaches to improve contrast agent quantification in dynamic contrast enhanced MRI, by utilising the spatial dependence among image voxels. Bayesian hierarchical models (BHMs), such as Besag model and Leroux model, were studied using simulated MRI data. The models were built on smaller images where spatial dependence can be incorporated, and then extended to larger images using the maximum a posteriori (MAP) method. Notable improvements on contrast agent concentration estimation were obtained for both smaller and larger images. For smaller images: the BHMs provided substantial improved estimates in terms of the root mean squared error (rMSE), compared to the estimates from the existing method for a noise level equivalent of a 12-channel head coil at 3T. Moreover, Leroux model outperformed Besag models with two different dependence structures. Specifically, the Besag models increased the estimation precision by 27% around the peak of the dynamic curve, while the Leroux model improved the estimation by 40% at the peak, compared with the existing estimation method. For larger images: the proposed MAP estimators showed clear improvements on rMSE for vessels, tumor rim and white matter.

In this paper, we propose a general scale invariant approach for sparse signal recovery via the minimization of the q-ratio sparsity s_q(z) = (||z||₁ / ||z||_q )^q/(q-1) with q ∈ [0 , ∞]. The properties of the q-ratio sparsity measure are studied and illustrated with examples. For the proposed q-ratio sparsity minimization problem with 1 < q ≤∞ , we establish a verifiable exact reconstruction condition and derive its concise error bounds in terms of q-ratio constrained minimal singular values (CMSV). From an algorithmic point of view, we recognize that the proposed problem belongs to the nonlinear fractional programming and investigate two kinds of methods for solving it including the parametric methods and the change of variable method. Numerical experiments are conducted to demonstrate the advantageous performance of the proposed approaches over the state-of-the-art sparse recovery methods. 

We study the recovery conditions of weighted mixed ℓ2/ℓp minimization for block sparse signal reconstruction from compressed measurements when partial block supportinformation is available. We show theoretically that the extended block restricted isometry property can ensure robust recovery when the data fidelity constraint is expressed in terms of an ℓq norm of the residual error, thus establishing a setting wherein we arenot restricted to Gaussian measurement noise. We illustrate the results with a series of numerical experiments.

In this paper, we propose a general scale invariant approach for sparse signal recovery via the minimization of the q-ratio sparsity. When 1<q≤∞, both the theoretical analysis based on q-ratio constrained minimal singular values (CMSV) and the practical algorithms via nonlinear fractional programming are presented. Numerical experiments are conducted to demonstrate the advantageous performance of the proposed approaches over the state-of-the-art sparse recovery methods.

We study the recovery conditions of weighted ℓ₁ minimization for real-valued signal reconstruction from phaseless compressive sensing measurements when partial support information is available. A strong restricted isometry property condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success of k-sparse phaseless recovery via weighted ℓ₁ minimization. Numerical experiments are conducted to illustrate our results.