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  • 1.
    Dadras, Ali
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Leffler, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A ridgelet approach to poisson denoising2024Manuscript (preprint) (Other academic)
    Abstract [en]

    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.

  • 2.
    Leffler, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The PET sampling puzzle: intelligent data sampling methods for positron emission tomography2024Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    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.

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  • 3.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Axelsson, Jan
    Umeå University, Faculty of Medicine, Department of Radiation Sciences, Radiation Physics.
    Larsson, Anne
    Umeå University, Faculty of Medicine, Department of Radiation Sciences, Radiation Physics.
    Häggström, Ida
    Umeå University, Faculty of Medicine, Department of Radiation Sciences, Radiation Physics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sharper Positron Emission Tomography: Intelligent Data Sampling to Promote Accelerated Medical Imaging2019Conference paper (Other academic)
  • 4.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Häggström, Ida
    Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Compressed sensing for low-count PET denoising in measurement space2023In: NORDSTAT 2023 Gothenburg, Göteborgs universitet, 2023Conference paper (Refereed)
    Abstract [en]

    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.

  • 5.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Häggström, Ida
    Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Compressed sensing for low-count positron emission tomography denoising in measurement spaceManuscript (preprint) (Other academic)
  • 6.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Häggström, Ida
    Umeå University, Faculty of Medicine, Department of Radiation Sciences, Radiation Physics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Intelligent data sampling promotes accelerated medical imaging: sharper positron emission tomography2018Conference paper (Refereed)
  • 7.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tommaso Luppino, Luigi
    Department Physics and Technology, UiT The Arctic University of Norway, Tromsø, Norway.
    Kuttner, Samuel
    University Hospital of North Norway, Tromsø, Norway; Physics and Technology, UiT The Arctic University of Norway, Tromsø, Norway; Department of Clinical Medicine, UiT The Arctic University of Norway, Tromsø, Norway..
    Axelsson, Jan
    Umeå University, Faculty of Medicine, Department of Diagnostics and Intervention.
    Deep learning-based filling of incomplete sinograms from low-cost, long axial field-of-view PET scanners with inter-detector gaps2023In: The international networking symposiumon artificial intelligence and informatics in nuclear medicine: Program book, University Medical Center Groningen , 2023, p. 59-59Conference paper (Refereed)
  • 8.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tommaso Luppino, Luigi
    Department Physics and Technology, UiT The Arctic University of Norway, Tromsø, Norway.
    Kuttner, Samuel
    University Hospital of North Norway, Tromsø, Norway; Physics and Technology, UiT The Arctic University of Norway, Tromsø, Norway; Department of Clinical Medicine, UiT The Arctic University of Norway, Tromsø, Norway.
    Söderkvist, Karin
    Umeå University, Faculty of Medicine, Department of Diagnostics and Intervention.
    Axelsson, Jan
    Umeå University, Faculty of Medicine, Department of Diagnostics and Intervention.
    Filling of incomplete sinograms from sparse PET detector configurations using a residual U-NetManuscript (preprint) (Other academic)
  • 9.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Department of Statistics, Zhejiang University City College, China.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    An extended block restricted isometry property for sparse recovery with non-Gaussian noise2020In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 38, no 6, p. 827-838Article in journal (Refereed)
    Abstract [en]

    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.

  • 10.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    An Extended Block Restricted Isometry Property for Sparse Recovery with Non-Gaussian Noise2018Conference paper (Refereed)
    Abstract [en]

    Recovering an unknown signal from significantly fewer measurements is a fundamental aspect in computational sciences today. The key ingredient is the sparsity of the unknown signal, a realisation that that has led to the theory of compressed censing, through which successful recovery of high dimensional (approximately) sparse signals is now possible at a rate significantly lower than the Nyquist sampling rate. Today, an interesting challenge lies in customizing the recovery process to take into account prior knowledge about e.g. signal structure and properties of present noise. We study recovery conditions for block sparse signal reconstruction from compressed measurements when partial support information is available via weighted mixed l2/lp minimization. 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. Thereby, we also establish a setting wherein we are not restricted to a Gaussian measurement noise. The results are illustrated with a series of numerical experiments.

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