Abstract
Data of piecewise smooth images are sometimes acquired as Fourier samples. Standard reconstruction techniques yield the Gibbs phenomenon, causing spurious oscillations at jump discontinuities and an overall reduced rate of convergence to first order away from the jumps. Filtering is an inexpensive way to improve the rate of convergence away from the discontinuities, but it has the adverse side effect of blurring the approximation at the jump locations. On the flip side, high resolution post processing algorithms are often computationally cost prohibitive and also require explicit knowledge of all jump locations. Recent convex optimization algorithms using \(l^1\) regularization exploit the expected sparsity of some features of the image. Wavelets or finite differences are often used to generate the corresponding sparsifying transform and work well for piecewise constant images. They are less useful when there is more variation in the image, however. In this paper we develop a convex optimization algorithm that exploits the sparsity in the edges of the underlying image. We use the polynomial annihilation edge detection method to generate the corresponding sparsifying transform. Our method successfully reduces the Gibbs phenomenon with only minimal blurring at the discontinuities while retaining a high rate of convergence in smooth regions.
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Notes
Technically, the \(l^0\) norm of an expression is a better measure of sparsity. However, the \(l^0\) norm does not meet the convexity requirements and is very slow to compute. Additional detail on using the \(l^1\) norm in place of the \(l^0\) in order to measure sparsity can be found in [8].
The images were acquired on a 3T Siemens Skyra MRI scanner at University of Arizona, courtesy of Professor Ali Bilgin, in accordance with Institutional Review Board policies. A Cartesian Turbo Spin-Echo sequence was used with the following parameters: Effective TE = 147 ms, TR = 1,500 ms, Echo train length = 16, Field-of-View = \(350 \times 350\) mm, Slice thickness=5mm, Bandwidth = 130 Hz/pixel, Image matrix = \(1{,}024\times 1{,}024\) pixels.
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The submitted manuscript has been authored in part by contractors [UT-Battelle LLC, manager of Oak Ridge National Laboratory (ORNL)] of the U.S. Government under Contract No. DE-AC05-00OR22725. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.
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This work is supported in part by Grants NSF-DMS 1216559 and AFOSR FA9550-12-1-0393.
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Wasserman, G., Archibald, R. & Gelb, A. Image Reconstruction from Fourier Data Using Sparsity of Edges. J Sci Comput 65, 533–552 (2015). https://doi.org/10.1007/s10915-014-9973-3
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DOI: https://doi.org/10.1007/s10915-014-9973-3