Quantitative study of undersampled recoverability for sparse images in computed tomography

Jakob Heide Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan

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    Image reconstruction methods based on exploiting image sparsity, motivated by compressed sensing (CS), allow reconstruction from a significantly reduced number of projections in X-ray computed tomography (CT). However, CS provides neither theoretical guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery. In this paper, we demonstrate empirically through computer simulations that minimization of the image 1-norm allows for recovery of sparse images from fewer measurements than unknown pixels, without relying on artificial random sampling patterns. We establish quantitatively an average-case relation between image sparsity and sufficient number of measurements for recovery, and we show that the transition from non-recovery to recovery is sharp within well-defined classes of simple and semi-realistic test images. The specific behavior depends on the type of image, but the same quantitative relation holds independently of image size.
    Original languageEnglish
    Pages (from-to)1-20
    Publication statusPublished - 2012


    • Computed Tomography
    • Image Reconstruction
    • Sparse approximation
    • Compressed Sensing
    • Recoverability
    • Inverse Problems


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