• 한국수학교육학회지시리즈E:수학교육논문집
• /
• 제18권2호
• /
• pp.79-92
• /
• 2004

This paper offers an analysis of how students reasoned with the dynamic parameter time to support their mathematical activity and deepen their understandings of mathematical concepts. This mathematical thinking occurred as they participated in a differential equations class before, during, and instruction on solutions to linear systems of differential equations. Students participated in the following identified mathematical practices related to parametric reasoning during this time period: reasoning simultaneously in a qualitative and quantitative manner, reasoning by moving from discrete to continuous imaging of time, and reasoning by imagining the motion. Examples of this reasoning are provided in this report. Implications of this research include the possibility that instructional activities can build on this reasoning to help students learn about the mathematics of change at the middle school, high school, and the university.

• 대한수학회보
• /
• 제49권6호
• /
• pp.1291-1302
• /
• 2012

In this paper, we introduce a new elliptic PDE: $$\{{\nabla}{\cdot}\(\frac{|{\gamma}^{\omega}(r)|^2}{\sigma}{\nabla}v_{\omega}(r)\)=0,\;r{\in}{\Omega},\\v_{\omega}(r)=f(r),\;r{\in}{\partial}{\Omega},$$ where ${\gamma}^{\omega}={\sigma}+i{\omega}{\epsilon}$ is the admittivity distribution of the conducting material ${\Omega}$ and it is shown that the introduced elliptic PDE can replace the standard elliptic PDE with conductivity coefficient in EIT imaging. Indeed, letting $v_0$ be the solution to the standard elliptic PDE with conductivity coefficient, the solution $v_{\omega}$ is quite close to the solution $v_0$ and can show spectroscopic properties of the conducting object ${\Omega}$ unlike $v_0$. In particular, the potential $v_{\omega}$ can be used in detecting a thin low-conducting anomaly located in ${\Omega}$ since the spectroscopic change of the Neumann data of $v_{\omega}$ is inversely proportional to thickness of the thin anomaly.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제18권2호
• /
• pp.77-105
• /
• 2014

Medical imaging techniques have evolved to expand our ability to visualize new contrast information of electrical, optical, and mechanical properties of tissues in the human body using noninvasive measurement methods. In particular, electrical tissue property imaging techniques have received considerable attention for the last few decades since electrical properties of biological tissues and organs change with their physiological functions and pathological states. We can express the electrical tissue properties as the frequency-dependent admittivity, which can be measured in a macroscopic scale by assessing the relation between the time-harmonic electric field and current density. The main issue is to reconstruct spectroscopic admittivity images from 10 Hz to 1 MHz, for example, with reasonably high spatial and temporal resolutions. It requires a solution of a nonlinear inverse problem involving Maxwell's equations. To solve the inverse problem with practical significance, we need deep knowledge on its mathematical formulation of underlying physical phenomena, implementation of image reconstruction algorithms, and practical limitations associated with the measurement sensitivity, specificity, noise, and data acquisition time. This paper discusses a number of issues in electrical tissue property imaging modalities and their future directions.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제14권1호
• /
• pp.35-42
• /
• 2010

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].

• 대한의용생체공학회:의공학회지
• /
• 제29권2호
• /
• pp.89-98
• /
• 2008

NIRS (Near-infrared Spectroscopy) and DOI (Diffuse Optical Imaging) are relatively new, non-invasive, and non-ionizing methods that measure or image optical properties (Scattering and Absorption Coefficient) and physiological properties (Water Fraction, concentration of Oxy-, Deoxy-Hemoglobin, Cytochrome Oxidase, etc) of biological tissues. In this paper, three different types of NIRS systems, mathematical modeling, and reconstruction algorithms are described. Also, recent applications such as functional brain imaging, optical mammography, NIRS based BMI (Brain-Machine Interface), and small animal study are reviewed.

• Radiation Oncology Journal
• /
• 제35권2호
• /
• pp.101-111
• /
• 2017

The number of imaging data sets has significantly increased during radiation treatment after introducing a diverse range of advanced techniques into the field of radiation oncology. As a consequence, there have been many studies proposing meaningful applications of imaging data set use. These applications commonly require a method to align the data sets at a reference. Deformable image registration (DIR) is a process which satisfies this requirement by locally registering image data sets into a reference image set. DIR identifies the spatial correspondence in order to minimize the differences between two or among multiple sets of images. This article describes clinical applications, validation, and algorithms of DIR techniques. Applications of DIR in radiation treatment include dose accumulation, mathematical modeling, automatic segmentation, and functional imaging. Validation methods discussed are based on anatomical landmarks, physical phantoms, digital phantoms, and per application purpose. DIR algorithms are also briefly reviewed with respect to two algorithmic components: similarity index and deformation models.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제27권2호
• /
• pp.109-122
• /
• 2023

In this study, we consider the application of the bifocusing method (BFM) for identifying the locations and shapes of small anomalies from scattering parameter data when the exact values of background permittivity and conductivity are unknown. To this end, an imaging function using numerical focusing operator is introduced and its mathematical structure is revealed by establishing a relationship with an infinite series of Bessel functions, antenna arrangements, and anomaly properties. On the basis of the revealed structure, we demonstrate why inaccurate location and size of anomalies were retrieved via the BFM. Some simulation results are illustrated using synthetic data polluted by random noise to support the theoretical result.

• Current Optics and Photonics
• /
• 제6권1호
• /
• pp.69-78
• /
• 2022

A high-resolution camera is a precise optical system. Its vibrations during transportation and launch, together with changes in temperature and gravity field in orbit, lead to different degrees of defocus of the camera. Thermal refocusing is one of the solutions to the problems related to in-orbit defocusing, but there are few relevant thermal refocusing mathematical models for systematic analysis and research. Therefore, to further research thermal refocusing systems by using the development of a high-resolution micro-nano satellite (CX6-02) super-resolution camera as an example, we established a thermal refocusing mathematical model based on the thermal elasticity theory on the basis of the secondary mirror position. The detailed design of the thermal refocusing system was carried out under the guidance of the mathematical model. Through optical-mechanical-thermal integration analysis and Zernike polynomial calculation, we found that the data error obtained was about 1%, and deformation in the secondary mirror surface conformed to the optical index, indicating the accuracy and reliability of the thermal refocusing mathematical model. In the final ground test, the thermal vacuum experimental verification data and in-orbit imaging results showed that the thermal refocusing system is consistent with the experimental data, and the performance is stable, which provides theoretical and technical support for the future development of a thermal refocusing space camera.

• 한국정보통신학회논문지
• /
• 제19권5호
• /
• pp.1207-1212
• /
• 2015

본 논문에서는 광자 계수 집적 영상 현미경을 사용하여 광자가 희박한 조건에서 마이크로 물체의 3차원 시각화와 인식에 대한 기술을 제안한다. 제안하는 방법에서는 고해상도의 서로 다른 원근감을 가지는 2차원 영상을 획득하기 위해 합성조리개 집적 영상을 사용한다. 그리고 영상으로부터 광자를 추출하기 위해 광자계수 영상 시스템의 수학적 모델인 포아송 분포를 사용하며 통계적 추정법으로 부터 3차원 영상을 추정한다. 따라서, 광자가 희박한 조건에서 마이크로 물체가 손상되지 않으면서 그에 대한 3차원 영상을 획득하고 시각화할 수 있다. 추가적으로, 비선형 상관 필터를 사용하여 3차원 물체의 인식도 가능하다. 본 기술의 유용성을 증명하기 위해, 광학적 실험을 수행하였다.

• Journal of Biosystems Engineering
• /
• 제37권4호
• /
• pp.265-270
• /
• 2012

Purpose: An analytic method to design excitation and emission filters of a multispectral fluorescence imaging system is proposed and was demonstrated in an application to poultry fecal inspection Methods: A mathematical model of a multispectral imaging system is proposed and its system parameters, such as excitation and emission filters, were optimally determined by linear discriminant analysis (LDA). An alternating scheme was proposed for numerical implementation. Fluorescence characteristics of organic materials and feces of poultry carcasses are analyzed by LDA to design the optimal excitation and emission filters for poultry fecal inspection. Results: The most appropriate excitation filter was UV-A (about 360 nm) and blue light source (about 460 nm) and band-pass filter was 660-670 nm. The classification accuracy and false positive are 98.4% and 2.5%, respectively. Conclusions: The proposed method is applicable to other agricultural products which are distinguishable by their spectral properties.