非凸函數(shù)在圖像復(fù)原中的應(yīng)用
發(fā)布時(shí)間：2019-03-20 19:24
【摘要】：圖像信息是人類認(rèn)識(shí)世界的重要信息來(lái)源,然而由于在圖像成像條件和圖像傳輸過(guò)程中存在各種不利因素致使圖像質(zhì)量下降,從而影響圖像的使用及其后續(xù)處理.如何從退化圖像復(fù)原出清晰的、內(nèi)容豐富的圖像是人們所普遍關(guān)注的問(wèn)題,這正是圖像復(fù)原要解決的問(wèn)題.圖像復(fù)原是圖像處理領(lǐng)域中重要的研究?jī)?nèi)容之一.通常情況下,由于圖像復(fù)原問(wèn)題是一個(gè)不適定的反問(wèn)題,這就需要利用先驗(yàn)信息將不適定問(wèn)題正則化處理轉(zhuǎn)化為適定模型.同時(shí),自然圖像統(tǒng)計(jì)學(xué)顯示圖像邊緣分布既不全是Gaussian分布也不全是Laplacian分布,而是類似于Hyper-Laplacian分布,即先驗(yàn)信息是非凸的.本論文正是基于各種非凸勢(shì)函數(shù),包括Lipschitz非凸函數(shù)與non-Lipschitz非凸函數(shù),建立相應(yīng)的非凸非光滑優(yōu)化模型,采用交替最小化算法求解,分析其收斂性.本論文的主要工作與取得的創(chuàng)新性成果主要有:針對(duì)Lipschitz非凸正則函數(shù)與加性噪聲,建立L2+Lipschitz正則函數(shù)的非凸能量函數(shù).先采用非凸累進(jìn)算法處理,相應(yīng)的非凸累進(jìn)能量函數(shù)隨著變系數(shù)增大而由凸能量函數(shù)趨近于原目標(biāo)非凸能量函數(shù).對(duì)每一個(gè)固定的系數(shù),代理能量函數(shù)分別采用四種交替最小化算法求解,在求解過(guò)程中,為了保證海森陣正定性,僅考慮能量函數(shù)海森陣中的正定部分.同時(shí),將四種交替最小化算法歸結(jié)為一種模式處理,且選擇其中一個(gè)算法,利用Kurdyka-Lojasiewicz不等式分析該算法的收斂性,且成功分析了代理能量函數(shù)隨著變系數(shù)改變而趨于原目標(biāo)非凸能量函數(shù)時(shí)的收斂性;針對(duì)non-Lipschitz擬范數(shù)?p(0p1)正則函數(shù)與乘性噪聲,建立L1+TVp的非凸能量函數(shù),采用變量分離與鄰近點(diǎn)交替最小化算法處理.而對(duì)含TVp項(xiàng)的子問(wèn)題(去噪模型),先采用Huber函數(shù)處理?p范數(shù),再對(duì)相應(yīng)的歐拉方程采用原始對(duì)偶牛頓法求解對(duì)偶向量,進(jìn)而得到廣義正定海森陣,接下來(lái)用信賴域法求該子問(wèn)題的最優(yōu)解,且分析了處理該子問(wèn)題的算法具有超線性收斂性.之后,利用KurdykaLojasiewicz不等式對(duì)整個(gè)算法,進(jìn)行了收斂性分析;給出了三種近年來(lái)出現(xiàn)的非凸函數(shù)模型應(yīng)用于圖像復(fù)原,如箱式約束非凸最小化模型,p-壓縮算子(0p1)與修正的非凸最小化模型,且針對(duì)不同噪聲分別給出了相應(yīng)的算法與相關(guān)收斂性分析.在數(shù)值試驗(yàn)中,分別驗(yàn)證了各種算法的有效性.特別的,在第三章中由數(shù)值試驗(yàn)分析了變系數(shù)取值區(qū)間,在第四章中分析了擬范數(shù)?p作為正則函數(shù)處理乘性噪聲時(shí),發(fā)現(xiàn)當(dāng)p=12時(shí),圖像復(fù)原效果最優(yōu),以及其他一些非凸模型的特殊效果,如箱式約束非凸模型在特定的圖像區(qū)域能改善復(fù)原效果,修正非凸模型能加速處理速度.
[Abstract]:Image information is an important source of information for human beings to understand the world. However, due to various unfavorable factors in image imaging conditions and image transmission, the image quality is degraded, thus affecting the use of images and their subsequent processing. How to recover clear and rich images from degraded images is a common concern, which is exactly the problem to be solved in image restoration. Image restoration is one of the important research contents in the field of image processing. Usually, since the image restoration problem is an ill-posed inverse problem, it is necessary to transform the regularization of the ill-posed problem into a well-posed model by using prior information. At the same time, natural image statistics show that image edge distribution is not all Gaussian distribution or Laplacian distribution, but similar to Hyper-Laplacian distribution, that is, prior information is non-convex. Based on various non-convex potential functions, including Lipschitz nonconvex function and non-Lipschitz nonconvex function, the corresponding non-convex and non-smooth optimization model is established and solved by alternating minimization algorithm, and its convergence is analyzed. The main work and innovative results of this thesis are as follows: for Lipschitz nonconvex regularization function and additive noise, the non-convex energy function of L2-Lipschitz regular function is established. First, the non-convex cumulant algorithm is used to deal with it. The corresponding nonconvex cumulant energy function approaches to the non-convex energy function of the original target with the increase of the variable coefficient. For each fixed coefficient, the proxy energy function is solved by four alternating minimization algorithms. In order to ensure the positive definiteness of the Haysen matrix, only the positive definite part of the energy function is considered. At the same time, the four alternative minimization algorithms are reduced to a kind of pattern processing, and one of them is chosen to analyze the convergence of the algorithm by using Kurdyka-Lojasiewicz inequality. The convergence of the proxy energy function tends to the non-convex energy function of the original target with the change of the variable coefficient, and the convergence of the proxy energy function is analyzed successfully. For the non-Lipschitz quasi-norm p (0p1) regularization function and multiplicative noise, the non-convex energy function of L1-TVp is established, and the algorithm of variable separation and alternating minimization of adjacent points is used to deal with it. For the sub-problem (de-noising model) with TVp term, the Huber function is first used to deal with the p-norm, then the primal dual Newton method is used to solve the dual vector for the corresponding Euler equation, and then the generalized positive definite Hessen matrix is obtained. Next, the trust region method is used to find the optimal solution of the sub-problem, and the superlinear convergence of the algorithm for dealing with the sub-problem is analyzed. Then, the convergence of the whole algorithm is analyzed by using KurdykaLojasiewicz inequality. Three nonconvex function models, such as box constrained nonconvex minimization model, p-contractive operator (0p1) and modified nonconvex minimization model, are presented in this paper. The corresponding algorithm and related convergence analysis are given for different noises. In numerical experiments, the validity of each algorithm is verified. In the third chapter, the range of variable coefficients is analyzed by numerical experiments. In the fourth chapter, when the quasi-norm? p is used as a regular function to deal with multiplicative noise, it is found that the image restoration effect is the best when p is 12. And some other special effects of non-convex model, such as box constrained non-convex model can improve the restoration effect in a specific image area, and the modified non-convex model can accelerate the processing speed.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：TP391.41;O174.13
[Abstract]:Image information is an important source of information for human beings to understand the world. However, due to various unfavorable factors in image imaging conditions and image transmission, the image quality is degraded, thus affecting the use of images and their subsequent processing. How to recover clear and rich images from degraded images is a common concern, which is exactly the problem to be solved in image restoration. Image restoration is one of the important research contents in the field of image processing. Usually, since the image restoration problem is an ill-posed inverse problem, it is necessary to transform the regularization of the ill-posed problem into a well-posed model by using prior information. At the same time, natural image statistics show that image edge distribution is not all Gaussian distribution or Laplacian distribution, but similar to Hyper-Laplacian distribution, that is, prior information is non-convex. Based on various non-convex potential functions, including Lipschitz nonconvex function and non-Lipschitz nonconvex function, the corresponding non-convex and non-smooth optimization model is established and solved by alternating minimization algorithm, and its convergence is analyzed. The main work and innovative results of this thesis are as follows: for Lipschitz nonconvex regularization function and additive noise, the non-convex energy function of L2-Lipschitz regular function is established. First, the non-convex cumulant algorithm is used to deal with it. The corresponding nonconvex cumulant energy function approaches to the non-convex energy function of the original target with the increase of the variable coefficient. For each fixed coefficient, the proxy energy function is solved by four alternating minimization algorithms. In order to ensure the positive definiteness of the Haysen matrix, only the positive definite part of the energy function is considered. At the same time, the four alternative minimization algorithms are reduced to a kind of pattern processing, and one of them is chosen to analyze the convergence of the algorithm by using Kurdyka-Lojasiewicz inequality. The convergence of the proxy energy function tends to the non-convex energy function of the original target with the change of the variable coefficient, and the convergence of the proxy energy function is analyzed successfully. For the non-Lipschitz quasi-norm p (0p1) regularization function and multiplicative noise, the non-convex energy function of L1-TVp is established, and the algorithm of variable separation and alternating minimization of adjacent points is used to deal with it. For the sub-problem (de-noising model) with TVp term, the Huber function is first used to deal with the p-norm, then the primal dual Newton method is used to solve the dual vector for the corresponding Euler equation, and then the generalized positive definite Hessen matrix is obtained. Next, the trust region method is used to find the optimal solution of the sub-problem, and the superlinear convergence of the algorithm for dealing with the sub-problem is analyzed. Then, the convergence of the whole algorithm is analyzed by using KurdykaLojasiewicz inequality. Three nonconvex function models, such as box constrained nonconvex minimization model, p-contractive operator (0p1) and modified nonconvex minimization model, are presented in this paper. The corresponding algorithm and related convergence analysis are given for different noises. In numerical experiments, the validity of each algorithm is verified. In the third chapter, the range of variable coefficients is analyzed by numerical experiments. In the fourth chapter, when the quasi-norm? p is used as a regular function to deal with multiplicative noise, it is found that the image restoration effect is the best when p is 12. And some other special effects of non-convex model, such as box constrained non-convex model can improve the restoration effect in a specific image area, and the modified non-convex model can accelerate the processing speed.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：TP391.41;O174.13
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