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

Shi, Ruonan; Jung, Jae-Hun

doi:10.1007/s10915-017-0467-y

Cited by 3 publications

(9 citation statements)

References 18 publications

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“…In [3] the domain decomposition sparse PA method was proposed, with which the given domain is split into multiple subdomains and the sparse PA method is applied separately in each individual domain. To make the minimization of Eq (11) done locally, we used the Fourier continuation method.…”

Section: D Domain Decomposition Fourier Continuation Sparse Pa Methodsmentioning

confidence: 99%

“…However, most reconstruction methods, such as the Fourier reconstruction and filtered Fourier reconstruction, are carried out in a global manner [1, 2], so they are computationally expensive. And since global methods provide the reconstruction in one piece, so the quality of the reconstruction near different edges is not equal ([3] shows this limitation). In this work, we propose a local method that focuses on and yields the local reconstruction of the subdomain such that an accurate and non-oscillatory sharp image reconstruction is achieved in that subdomain.…”

Section: Introductionmentioning

confidence: 99%

“…As an example, we proposed a local method [3] based on the L 1 regularization method proposed in [4]. The method by [4], known as the sparse polynomial annihilation (PA) method, is an L 1 regularization method based on sparsity of edges for the Fourier reconstruction, for obtaining the sharp image profiles near edges.…”

Section: Introductionmentioning

confidence: 99%

“…If the L 1 regularization can be done locally, the reconstruction near weak edges can be improved. Such local operation could be done by our previous 1D method of domain decomposition Fourier L 2 and L 1 minimization based on the Fourier continuation [3]. The key element of our local method is to split the given domain into multiple subdomains first and then carry out the L 1 minimization individually in each subdomain.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is composed of the following sections. First, a brief explanation of the 1D domain decomposition Fourier continuation method based on the edge sparsity proposed in [3] is given. Then the extension of the 1D method to the 2D Fourier reconstruction will be introduced.…”

Section: Introductionmentioning

confidence: 99%

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

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Section: D Domain Decomposition Fourier Continuation Sparse Pa Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

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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

Cited by 3 publications

References 18 publications

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

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

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

Acoustic Tomography Temperature Reconstruction Based on Virtual Observation and Residual Network

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