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A NOTE ON OPTIMAL RECONSTRUCTION OF MAGNETIC RESONANCE IMAGES FROM NON-UNIFORM SAMPLES IN k-SPACE
JUNE-YUB LEE 한국산업응용수학회 2010 Journal of the Korean Society for Industrial and A Vol.14 No.1
A goal of Magnetic Resonance Imaging is reproducing a spatial map of the effective spin density from the measured Fourier coefficients of a specimen. The imaging procedure can be done by inverse Fourier transformation or backward fast Fourier transformation if the data are sampled on a regular grid in frequency space; however, it is still a challenging question how to reconstruct an image from a finite set of Fourier data on irregular points in k-space. In this paper, we describe some mathematical and numerical properties of imaging techniques from non-uniform MR data using the pseudo-inverse or the diagonal-inverse weight matrix. This note is written as an easy guide to readers interested in the non-uniform MRI techniques and it basically follows the ideas given in the paper by Greengard-Lee-Inati [10, 11].
A NUMERICAL METHOD FOR CAUCHY PROBLEM USING SINGULAR VALUE DECOMPOSITION
Lee, June-Yub,Yoon, Jeong-Rock Korean Mathematical Society 2001 대한수학회논문집 Vol.16 No.3
We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of he domain, which provides the information on the choice of numerically meaningful modes. We suggest a pseudo-inverse regularization method based on singular value decomposition and present various numerical simulations.
A fast adaptive numerical solver for nonseparable elliptic partial differential equations
Lee, June-Yub 한국산업정보응용수학회 1998 한국산업정보응용수학회 Vol.2 No.1
We describe a fast numerical method for non-separable elliptic equations in self-adjoin form on irregular adaptive domains. One of the most successful results in numerical PDE is developing rapid elliptic solvers for separable EPDEs, for example, Fourier transformation methods for Poisson problem on a square, however, it is known that there is no rapid elliptic solvers capable of solving a general nonseparable problems [8]. It is the purpose of this paper to present an iterative solver for linear EPDEs in self-adjoint form. The scheme discussed in this paper solves a given non-separable equation using a sequence of solutions of Poisson equations, therefore, the most important key for such a method is having a good Poison solver. High performance is achieved by using a fast high-order adaptive Poisson solver which requires only about 500 floating point operations per gridpoint in order to obtain machine precision for both the computed solution and its partial derivatives. A few numerical examples have been presented.
GLOBAL SHAPE OF FREE BOUNDARY SATISFYING BERNOULLI TYPE BOUNDARY CONDITION
Lee, June-Yub,Seo, Jin-Keun Korean Mathematical Society 2000 대한수학회지 Vol.37 No.1
We study a free boundary problem satisfying Bernoulli type boundary condition along which the gradient of a piecewise harmonic solution jumps zero to a given constant value. In such problem, the free boundary splits the domain into two regions, the zero set and the harmonic region. Our main interest is to identify the global shape and the location of the zero set. In this paper, we find the lower and the upper bound of the zero set. In a convex domain, easier estimation of the upper bound and faster disk test technique are given to find a rough shape of the zero set. Also a simple proof on the convexity of zero set is given for a connected zero set in a convex domain.
김준혁 ( June Hyuk Kim ),유정현 ( Jeong Hyun Yoo ),곽호일 ( Ho Il Kwak ),송성훈 ( Seong Hun Song ),김준엽 ( Joon Yub Kim ) 대한스포츠의학회 2016 대한스포츠의학회지 Vol.34 No.2
The prevalence of metastatic tumorous lesion at medial clavicle or sternoclavicular joint is low. Herein, we presented a 45-year-old female patient with left sternoclavicular joint pain followed a weight-lifting exercise. The patient was initially diagnosed as traumatic sternoclavicular arthrosis, however, on later computed tomography scan demonstrated the bony destruction of medial clavicle which was diagnosed as metastasized non-small cell lung carcinoma after a fine needle biopsy.