解線性不適定問題的一種方法及其應(yīng)用
發(fā)布時(shí)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(xué)領(lǐng)域的各個(gè)方面,也成為了發(fā)展最快的數(shù)學(xué)研究領(lǐng)域之一。同時(shí),推動(dòng)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(xué)者的普遍關(guān)注。該方法經(jīng)證明在目標(biāo)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(diǎn)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(jìn)行了嚴(yán)格的收斂性證明。當(dāng)數(shù)據(jù)為不存在擾動(dòng)誤差的真實(shí)數(shù)據(jù)時(shí),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(shí)證明了迭代格式是收斂的。鑒于實(shí)際應(yīng)用中,得到的測量數(shù)據(jù)都是具有一定擾動(dòng)誤差的,從而本文在數(shù)據(jù)帶有擾動(dòng)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(shí)證明了迭代格式是收斂的。在醫(yī)學(xué)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(diǎn)而備受關(guān)注。BLT成像主要是通過熒光素標(biāo)記目標(biāo)基因的方式,診斷或預(yù)測組織體的病理情況。實(shí)質(zhì)是通過組織體表面的可測信息及已知的光學(xué)知識(shí)確定組織體內(nèi)部發(fā)光細(xì)胞的位置。這一過程是一個(gè)典型的數(shù)學(xué)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(xué)模型為輻射傳輸方程(Radiative Transfer Equation,RTE)。然而大多數(shù)生物醫(yī)學(xué)成像問題的研究都是針對RTE方程的擴(kuò)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(xiàng)。數(shù)值模擬的實(shí)驗(yàn)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(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：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(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.6
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1 石宗寶;;球面上渦度方程的一個(gè)不適定問題[J];湖南師范大學(xué)自然科學(xué)學(xué)報(bào);1984年01期
2 郭慶平,王偉滄,向平波,童仕寬;不適定問題研究的若干進(jìn)展[J];武漢理工大學(xué)學(xué)報(bào)(交通科學(xué)與工程版);2001年01期
3 欒文貴;地球物理中的反問題與不適定問題[J];地球物理學(xué)報(bào);1988年01期
4 張改榮;不適定問題的Tikhonov正則化方法[J];山東科學(xué);1995年03期
5 凌捷,曾文曲,盧建珠,溫為民;近似數(shù)據(jù)的不適定問題正則參數(shù)的后驗(yàn)選擇[J];廣東工業(yè)大學(xué)學(xué)報(bào);1999年04期
6 金其年,侯宗義;非線性不適定問題的最大熵方法Ⅱ[J];復(fù)旦學(xué)報(bào)(自然科學(xué)版);1997年06期
7 傅初黎,傅鵬;小波分析及其在不適定問題研究中的應(yīng)用[J];高等理科教育;2003年03期
8 傅初黎,朱佑彬,陶建紅,邱春雨;一個(gè)不適定問題的頻域?qū)ΨQ截?cái)嗾齽t化方法[J];甘肅科學(xué)學(xué)報(bào);2001年04期
9 李招文;李景;劉振海;;非線性不適定問題的雙參數(shù)正則化[J];中國科學(xué)(A輯:數(shù)學(xué));2007年09期
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