解線性不適定問題的一種方法及其應(yīng)用
發(fā)布時間：2018-10-08 21:43
【摘要】：隨著科技的發(fā)展,反問題理論的應(yīng)用已經(jīng)延伸到科學領(lǐng)域的各個方面,也成為了發(fā)展最快的數(shù)學研究領(lǐng)域之一。同時,推動了解決這類問題的正則化理論的發(fā)展。在解決不適定問題的一系列方法中,全變分(Total Voriation,TV)正則化方法由于能夠較好地保持原問題的邊緣信息而受到海內(nèi)外學者的普遍關(guān)注。該方法經(jīng)證明在目標邊界不光滑的條件下,可以十分有效地將圖像正則化。在圖像去噪領(lǐng)域中,TV正則化也成為主要的方法之一。本文基于全變分(TV)模型,針對TV范數(shù)在零點的不可微性,引入?yún)⒘?,結(jié)合同倫技術(shù)構(gòu)造了同倫曲線??t?.得到了一種新的求解線性不適定問題的迭代格式,并對新的迭代格式進行了嚴格的收斂性證明。當數(shù)據(jù)為不存在擾動誤差的真實數(shù)據(jù)時,本文結(jié)合Hilbert空間理論、不等式理論及Cauchy列原理等相關(guān)知識證明了迭代格式是收斂的。鑒于實際應(yīng)用中,得到的測量數(shù)據(jù)都是具有一定擾動誤差的,從而本文在數(shù)據(jù)帶有擾動誤差的情況下,利用不等式理論及Morozov偏差原則等相關(guān)知識證明了迭代格式是收斂的。在醫(yī)學成像領(lǐng)域中,生物自發(fā)光層析成像(Bioluminescent Tomography,BLT)是一種新興的分子成像技術(shù),由于無創(chuàng)性、便捷性、成本低等優(yōu)點而備受關(guān)注。BLT成像主要是通過熒光素標記目標基因的方式,診斷或預(yù)測組織體的病理情況。實質(zhì)是通過組織體表面的可測信息及已知的光學知識確定組織體內(nèi)部發(fā)光細胞的位置。這一過程是一個典型的數(shù)學物理反問題,并且求解組織體內(nèi)部未知光源的問題是不適定的。常用的處理光在組織體內(nèi)傳播問題的數(shù)學模型為輻射傳輸方程(Radiative Transfer Equation,RTE)。然而大多數(shù)生物醫(yī)學成像問題的研究都是針對RTE方程的擴散近似展開的。本文將直接從RTE方程入手,利用提出的新的迭代格式求解RTE方程的光源項。數(shù)值模擬的實驗結(jié)果表明,新的迭代方法可以較好地還原生物組織體內(nèi)的光源形狀及位置信息,且光源的邊界信息保留較好,即該方法用于處理線性不適定問題是有效的。從而,該迭代格式也可以應(yīng)用于其它的線性反問題中,具有較高的應(yīng)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.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：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.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O241.6
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