Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform

Archibald, Rick; Gelb, Anne; Platte, Rodrigo B.

doi:10.1007/s10915-015-0088-2

Cited by 47 publications

(93 citation statements)

References 25 publications

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“…where µ = 2 λ is referred to as the data fidelity parameter. Note that (26) and (27) are equivalent. In the later numerical tests, the data fidelity parameter µ and the regularization parameter λ will be steered by a discontinuity sensor proposed in §3.3.…”

Section: Methodsmentioning

confidence: 99%

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Glaubitz

Gelb

2019

SIAM J. Sci. Comput.

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This paper investigates the use of 1 regularization for solving hyperbolic conservation laws based on high order discontinuous Galerkin (DG) approximations. We first use the polynomial annihilation method to construct a high order edge sensor which enables us to flag "troubled" elements. The DG approximation is enhanced in these troubled regions by activating 1 regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting 1 optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting 1 regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers' equation as well as for a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.

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Section: Methodsmentioning

confidence: 99%

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Glaubitz

Gelb

2019

SIAM J. Sci. Comput.

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“…Magnetic resonance imaging (MRI) is a commonly used medical imaging technique. 2 MRI image reconstruction is based on the inverse Fourier transform of a 3 frequency-limited acquired Fourier spectrum of the object. In this work, we are 4 2018 1/21 interested in the 2D Fourier reconstruction of given MRI data.…”

Section: Introductionmentioning

confidence: 99%

“…f R is the reconstruction we want to find andf is the given Fourier 119 coefficient vector. The edge operator E p is the sparse polynomial annihilation transform 120 and the superscript p denotes the order of the derivative of the interpolation [2,3]. The 121 polynomial annihilation (PA) is basically higher order derivative of the interpolation.…”

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confidence: 99%

“…γ is a positive integer), with periodicity of 185 L + d on x and y directions. The periodic extension g(z (1) , z (2) ) can be obtained as…”

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confidence: 99%

“…where i = i 1 − n m , i 1 − n m + 1, · · · , i 2 + n m and j = j 1 − n m , j 1 − n m + 1, · · · , j 2 + n m . 190 The reconstruction of f (x, y) within J 1 based on {f (x i , y j )} is given by g(z (1) , z (2) ) for 191…”

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Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method

Shi

Jung

Schweser

2018

Preprint

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The MRI image is obtained in the spatial domain from the given Fourier coefficients in the frequency domain. It is costly to obtain the high resolution image because it requires higher frequency Fourier data while the lower frequency Fourier data is less costly and effective if the image is smooth. However, the Gibbs ringing, if existent, prevails with the lower frequency Fourier data. We propose an efficient and accurate local reconstruction method with the lower frequency Fourier data that yields sharp image profile near the local edge. The proposed method utilizes only the small number of image data in the local area. Thus the method is efficient. Furthermore the method is accurate because it minimizes the global effects on the reconstruction near the weak edges shown in many other global methods for which all the image data is used for the reconstruction. To utilize the Fourier method locally based on the local non-periodic data, the proposed method is based on the Fourier continuation method. This work is an extension of our previous 1D Fourier domain decomposition method to 2D Fourier data. The proposed method first divides the MRI image in the spatial domain into many subdomains and applies the Fourier continuation method for the smooth periodic extension of the subdomain of interest. Then the proposed method reconstructs the local image based on L 2 minimization regularized by the L 1 norm of edge sparsity to sharpen the image near edges. Our numerical results suggest that the proposed method should be utilized in dimension-by-dimension manner instead of in a global manner for both the quality of the reconstruction and computational efficiency. The numerical results show that the proposed method is effective when the local reconstruction is sought and that the solution is free of Gibbs oscillations.Since the function that we want to find, f (x), is a piecewise smooth function, the 13 approximation by Eq (2) may yield the Gibbs phenomenon.14 When the patient's MRI image is obtained, usually we want to reconstruct the 15 image so as to reveal the detailed structure of a particular region with the given Fourier 16 coefficients. However, most reconstruction methods are carried out in a global manner, 17 so they are computationally expensive. And since global methods provide the 18 reconstruction in one piece, so the quality of the reconstruction near different edges is 19 not equal ( [12] shows this limitation). In this work, we propose a local method that 20 focuses on and yields the local reconstruction of the subdomain such that an accurate 21 and non-oscillatory sharp image reconstruction is achieved in that subdomain. If we 22 patch all these local reconstructions together to constitute the whole image, the result 23 will be more accurate than the one obtained by the global method while the subdomain 24 that we are interested in is much enhanced. 25As an example, we propose a local method base on the L 1 regularization method 26 proposed in [13]. The method by [13], known as the sparse polynomial annihi...

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A Domain Decomposition Fourier Continuation Method for Enhanced $$L_1$$ L 1 Regularization Using Sparsity of Edges in Reconstructing Fourier Data

Shi

Jung

2017

J Sci Comput

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Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform

Cited by 47 publications

References 25 publications

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method

A Domain Decomposition Fourier Continuation Method for Enhanced $$L_1$$ L 1 Regularization Using Sparsity of Edges in Reconstructing Fourier Data

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