解线性不适定问题的一种方法及其应用
发布时间：2018-10-08 21:43
【摘要】：随着科技的发展,反问题理论的应用已经延伸到科学领域的各个方面,也成为了发展最快的数学研究领域之一。同时,推动了解决这类问题的正则化理论的发展。在解决不适定问题的一系列方法中,全变分(Total Voriation,TV)正则化方法由于能够较好地保持原问题的边缘信息而受到海内外学者的普遍关注。该方法经证明在目标边界不光滑的条件下,可以十分有效地将图像正则化。在图像去噪领域中,TV正则化也成为主要的方法之一。本文基于全变分(TV)模型,针对TV范数在零点的不可微性,引入参量?,结合同伦技术构造了同伦曲线??t?.得到了一种新的求解线性不适定问题的迭代格式,并对新的迭代格式进行了严格的收敛性证明。当数据为不存在扰动误差的真实数据时,本文结合Hilbert空间理论、不等式理论及Cauchy列原理等相关知识证明了迭代格式是收敛的。鉴于实际应用中,得到的测量数据都是具有一定扰动误差的,从而本文在数据带有扰动误差的情况下,利用不等式理论及Morozov偏差原则等相关知识证明了迭代格式是收敛的。在医学成像领域中,生物自发光层析成像(Bioluminescent Tomography,BLT)是一种新兴的分子成像技术,由于无创性、便捷性、成本低等优点而备受关注。BLT成像主要是通过荧光素标记目标基因的方式,诊断或预测组织体的病理情况。实质是通过组织体表面的可测信息及已知的光学知识确定组织体内部发光细胞的位置。这一过程是一个典型的数学物理反问题,并且求解组织体内部未知光源的问题是不适定的。常用的处理光在组织体内传播问题的数学模型为辐射传输方程(Radiative Transfer Equation,RTE)。然而大多数生物医学成像问题的研究都是针对RTE方程的扩散近似展开的。本文将直接从RTE方程入手,利用提出的新的迭代格式求解RTE方程的光源项。数值模拟的实验结果表明,新的迭代方法可以较好地还原生物组织体内的光源形状及位置信息,且光源的边界信息保留较好,即该方法用于处理线性不适定问题是有效的。从而,该迭代格式也可以应用于其它的线性反问题中,具有较高的应用前景。
[Abstract]:With the development of science and technology, the application of inverse problem theory has been extended to all aspects of science and has become one of the fastest growing fields of mathematical research. At the same time, it promotes the development of regularization theory for solving this kind of problems. Among a series of methods for solving ill-posed problems, total variation (Total Voriation,TV) regularization method has attracted widespread attention of scholars at home and abroad for its ability to maintain the edge information of the original problem. It is proved that the method can effectively regularize the image under the condition that the target boundary is not smooth. In the field of image denoising, TV regularization is also one of the main methods. In this paper, based on the total variational (TV) model, a parameter is introduced for the nondifferentiability of TV norm at zero point. Based on the homotopy technique, the homotopy curve is constructed. T? . In this paper, a new iterative scheme for solving linear ill-posed problems is obtained, and the convergence of the new iterative scheme is proved strictly. When the data is real data without perturbation error, this paper proves that the iterative scheme is convergent with the knowledge of Hilbert space theory, inequality theory and Cauchy sequence principle. In view of the fact that the measured data have some perturbation errors in practical application, this paper proves that the iterative scheme is convergent by using the theory of inequality and the Morozov deviation principle. In the field of medical imaging, bioluminescence tomography (Bioluminescent Tomography,BLT) is a new molecular imaging technology. Diagnosis or prediction of histopathology. The essence is to determine the location of the luminous cells in the tissue through measurable information on the tissue surface and known optical knowledge. This process is a typical inverse problem of mathematics and physics, and it is ill-posed to solve the problem of unknown light source in tissue. The commonly used mathematical model to deal with the propagation of light in tissues is the radiation transfer equation (Radiative Transfer Equation,RTE). However, most biomedical imaging problems are based on the diffusion approximation of RTE equation. In this paper, the light source term of RTE equation will be solved by using a new iterative scheme directly from the RTE equation. The experimental results of numerical simulation show that the new iterative method can effectively reduce the shape and position of light source in biological tissue, and the boundary information of light source is well preserved, that is to say, this method is effective in dealing with linear ill-posed problems. Therefore, the iterative scheme can also be applied to other linear inverse problems, and has a higher application prospect.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.6
[Abstract]:With the development of science and technology, the application of inverse problem theory has been extended to all aspects of science and has become one of the fastest growing fields of mathematical research. At the same time, it promotes the development of regularization theory for solving this kind of problems. Among a series of methods for solving ill-posed problems, total variation (Total Voriation,TV) regularization method has attracted widespread attention of scholars at home and abroad for its ability to maintain the edge information of the original problem. It is proved that the method can effectively regularize the image under the condition that the target boundary is not smooth. In the field of image denoising, TV regularization is also one of the main methods. In this paper, based on the total variational (TV) model, a parameter is introduced for the nondifferentiability of TV norm at zero point. Based on the homotopy technique, the homotopy curve is constructed. T? . In this paper, a new iterative scheme for solving linear ill-posed problems is obtained, and the convergence of the new iterative scheme is proved strictly. When the data is real data without perturbation error, this paper proves that the iterative scheme is convergent with the knowledge of Hilbert space theory, inequality theory and Cauchy sequence principle. In view of the fact that the measured data have some perturbation errors in practical application, this paper proves that the iterative scheme is convergent by using the theory of inequality and the Morozov deviation principle. In the field of medical imaging, bioluminescence tomography (Bioluminescent Tomography,BLT) is a new molecular imaging technology. Diagnosis or prediction of histopathology. The essence is to determine the location of the luminous cells in the tissue through measurable information on the tissue surface and known optical knowledge. This process is a typical inverse problem of mathematics and physics, and it is ill-posed to solve the problem of unknown light source in tissue. The commonly used mathematical model to deal with the propagation of light in tissues is the radiation transfer equation (Radiative Transfer Equation,RTE). However, most biomedical imaging problems are based on the diffusion approximation of RTE equation. In this paper, the light source term of RTE equation will be solved by using a new iterative scheme directly from the RTE equation. The experimental results of numerical simulation show that the new iterative method can effectively reduce the shape and position of light source in biological tissue, and the boundary information of light source is well preserved, that is to say, this method is effective in dealing with linear ill-posed problems. Therefore, the iterative scheme can also be applied to other linear inverse problems, and has a higher application prospect.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.6
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