基于偏微分方程的图像去噪和增强研究
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摘要
随着计算机处理能力的不断增强和信息社会对多媒体信息处理要求的不断增加，图像处理已成为一个相当活跃的研究领域。图像的去噪和增强就是要改善图像的质量，使之更适合于实际的应用需求，因此具有较高的研究价值。本论文主要研究了在偏微分方程理论框架下进行图像去噪和增强的方法，特别是该类方法在理论和实际应用中遇到的难题。本论文围绕图像的去噪和增强，分析了公理性的偏微分方程模型、基于变分的偏微分方程模型、几何偏微分方程模型三个方面，对基于偏微分方程的图像处理方法中的一系列难点问题进行了讨论，并提出了解决方法。本论文的工作主要包括以下的内容：
1)研究并提出了在非线性扩散方程中引入自适应数值保真项的方法。论文首先根据非线性扩散滤波模型很难控制对目标结构的破坏和总变差去噪方法易保留过多的噪声等小尺度信息的情况，分析了数值保真项在图像去噪中的作用。并针对已有方法在利用数值保真项时的不足，提出了利用图像中目标的局部信息构造自适应数值保真项引入到非线性扩散模型中，克服经典方法在利用数值保真项方面的不足，使新的非线性扩散去噪模型能够在有效地去除噪声的同时很好地保持目标尖角、边缘等重要的几何结构。在数值求解方法方面，本文根据差分格式优化的思想提出了一种稳定、可靠的数值求解方法。
2)研究了利用非线性偏微分方程进行图像去噪易产生阶越效应的原因，并提出了利用梯度保真项来消除这种现象。为了能够在去噪的同时保持目标的边缘信息，需要采用高度非线性的偏微分方程模型，而这将直接导致阶越效应的产生。文中首先回顾了消除阶越效应的已有方法，接着从变分的角度分析了防止阶越效应的策略，提出在非线性方程中引入对图像梯度的约束保真项。该模型能使得结果图像的灰度变化和原噪声图像保持一致，从而防止阶越效应的产生。由于改进的去噪模型保持了低阶的非线性扩散方程的形式，相对于高阶偏微分方程去噪模型具有数值求解简单、稳定的优点。同时，新模型在有界变差函数空间中有解，使得该方法能够去噪的同时很好的保留了图像的边缘信息，避免了运用高阶非线性方程去噪时易破坏图像高频信息的缺点。
3)研究了几何图像模型的已有理论框架，包括曲线演化理论和曲面演化理论，提出了利用图像水平线等分布约束消除阶越效应的模型。文中首先分析了阶越效应的产生会使得图像水平集出现非均匀分布的现象，据此提出了对图像进行水平线等分布约束方法用于防止阶越效应的产生，并推导出约束方法对应的三阶线性偏微分方程。文中证明了三阶约束偏微分方程不改变图像中零交叉点处的梯度值。在改进已有的基于曲面演化的去噪模型方面，本论文提出了一种能够保持图像纹理的法曲率驱动的曲面演化方程，由于采用了沿纹理方向的曲率来控制曲面演化速度，使得图像的纹理成分具有相对缓慢的演变，能够在有效去除噪声的同时保持图像中纹理的成分。
4)根据指纹图像上脊线的方向特性，提出一种基于伪线性方向相关扩散方程的指纹增强方法。指纹图像是一种具有特殊的方向信息和结构信息的图像，指纹增强就是要保持和增强它的这种特有结构。文中首先指出了散度形式的相关扩散方程的缺点，并针对该方程在局部方向信息较复杂情况下不能够沿着指纹脊线方向扩散提出了改进方法。改进后的方法具有更好的方向扩散特性，在去噪和保持指纹图像脊线信息方面表现更优。另外，本文针对改进的方法计算复杂度较高的不足，提出伪线性形式的方向相关扩散方程。该方法具有线性算子的优点、计算量小，有利于指纹识别实时系统应用。
5)研究了光照对图像对比度的影响及已有的处理方法，并在偏微分方程理论框架下提出了一种能够有效增强由光照不均匀而造成的低对比度图像的方法。图像中阴影部分中包含丰富的细节信息，但是这些信息的梯度变化相对较小。本文通过在梯度域进行直方图均衡化调整图像梯度的非均匀分布，使图像阴影中的细节能够得到增强，最后利用最小二乘原理从校正的梯度场中重建出增强的图像。在结果图像中，阴影区域的细节信息被很清晰地表达出来。本文通过引入Lab彩色空间将增强方法推广到了对彩色图像的处理中。同时，本文根据Laplacian算子的特点，改进了原有的Poisson方程数值求解方法。
Image processing has been an active research field resulting from the development ofcomputer technology and increasing requirements of multimedia data processing. Sinceimage denoising and enhancement are often applied to improve the image quality and tomake them fulfill the specific application requirements, both of them are importantresearch subjects in image processing. In this paper, more attention is paid to the partialdifferential equation based methods, especially to the theoretical and applicationaldifficulties in these methods. With the topics of image denoising and enhancement, wehave reviewed and studied the following aspects in this thesis: the axiomal PDEs based onscale space theory, variational PDEs and geometrical PDEs based on curve or surfaceevolution. The difficulties in them are analyzed and solutions to them are proposed.
Some novel models and algorithms have been proposed as following:
(1) Anisotropic diffusion based image denoising couplying adaptive data fidelity termis proposed in this paper. At first, three requirements of image denoising are proposed,based on which the defects of using the fidelity term in classical image denoising modelsare investigated. After that, an adaptive data fidelity term based on local image structuralinformation is brought up to preserve the objects in the images. The new adaptive datafidelity term will make nonlinear PDE based denoising methods capable of preservingsufficiently the geometric structures such as edges and comers besides its effectiveness inimage denoising. To the numerical schemes, a more stable and reliable implementation ofproposed denoising model is introduced.
(2) Image denoising with second order non-linear diffusion PDEs often leads to anundesirable staircase effect, namely, the transformation of smooth regions into piecewiseconstant regions. In this paper, the gradient fidelity term is introduced which describles thesimilarity between the restored images and noise ones in gray variation in order to alleviatethe staircase effect. Anisotropic diffusion models are improved by adding the Eulerequation derived from the gradient fidelity term. After coupling the new restriction derivedfrom the gradient fidelity term, the classical second order PDE-based denoising modelswill produce piecewise smooth results, while preserving sharp jump discontinuities inimages. In addition, the gradient fidelity term is integrabel in bounded variation functionspace which makes our model outperform fourth order nonlinear PDE based denoisingmethods suffering from leakage problems and sensitivity to texture components. Experimental results show that our new model alleviates the staircase effect to some extentand preserves the image features, such as textures and edges.
(3) The research on the geometrical image modeling has been made and we haveproposed to alleviate staircase effect with the uniform distribution restriction of imagelevel sets. In order to inhibit the staircase effect, a new third order PDE model is derived toimplement the uniform distribution restriction of image level sets. Coupling new modelinto classical PDE-based denoising models will make them produce piecewise smoothresults. Moreover, we prove that the proposed third order PDE model does not change theposition of edges in the noisy images and preserves the image edges well. On the otherhand, the surface evolution equation driven by normal curvature is also proposed in thispaper after having reviewed the existing surface evolution equation based image denoisingmethods. To normal curvature driven diffusion PDE model, we find that it will preservetexture components in images better than other geometrical diffusion model because thenormal curvature along the texture direction is smaller.
(4) A pseudo-linear diffusion filter was proposed to enhance fingerprint imagesaccording to the directional infomation of them. In this part, the defects of coherentdiffusion in divergence form are discussed which produce artificial structure and a newnonlinear directional diffusion PDE is proposed in order to improve its inability. The newalgorithm outperforms the coherent diffusion based image enhancement in denoisingability and structure preservation. At last, a pseudo-linear diffusion equation derived fromnew method is proposed in order to decreasing the computational burden. Thepseudo-linear method is more suitable to real-time fingerprint recognition system becauseof its efficiency.
(5) The contrast enhancement models to images with uneven illumination distributionare reviewed and an efficient image enhancement method based on the PDE is proposed toadjust the uneven distributed illumination in the image. This model improves the imagedetails in shadows by adjusting the distribution of image gradient firstly and reconstructsthe result image from new gradient field in least square sense finally. With the introductionof Lab color space, the new method is extended to color image applications. The efficientnumerical scheme for Poisson equation is very important in real application and a moreefficient scheme is proposed according to the features of Laplacian operator.
1)研究并提出了在非线性扩散方程中引入自适应数值保真项的方法。论文首先根据非线性扩散滤波模型很难控制对目标结构的破坏和总变差去噪方法易保留过多的噪声等小尺度信息的情况，分析了数值保真项在图像去噪中的作用。并针对已有方法在利用数值保真项时的不足，提出了利用图像中目标的局部信息构造自适应数值保真项引入到非线性扩散模型中，克服经典方法在利用数值保真项方面的不足，使新的非线性扩散去噪模型能够在有效地去除噪声的同时很好地保持目标尖角、边缘等重要的几何结构。在数值求解方法方面，本文根据差分格式优化的思想提出了一种稳定、可靠的数值求解方法。
2)研究了利用非线性偏微分方程进行图像去噪易产生阶越效应的原因，并提出了利用梯度保真项来消除这种现象。为了能够在去噪的同时保持目标的边缘信息，需要采用高度非线性的偏微分方程模型，而这将直接导致阶越效应的产生。文中首先回顾了消除阶越效应的已有方法，接着从变分的角度分析了防止阶越效应的策略，提出在非线性方程中引入对图像梯度的约束保真项。该模型能使得结果图像的灰度变化和原噪声图像保持一致，从而防止阶越效应的产生。由于改进的去噪模型保持了低阶的非线性扩散方程的形式，相对于高阶偏微分方程去噪模型具有数值求解简单、稳定的优点。同时，新模型在有界变差函数空间中有解，使得该方法能够去噪的同时很好的保留了图像的边缘信息，避免了运用高阶非线性方程去噪时易破坏图像高频信息的缺点。
3)研究了几何图像模型的已有理论框架，包括曲线演化理论和曲面演化理论，提出了利用图像水平线等分布约束消除阶越效应的模型。文中首先分析了阶越效应的产生会使得图像水平集出现非均匀分布的现象，据此提出了对图像进行水平线等分布约束方法用于防止阶越效应的产生，并推导出约束方法对应的三阶线性偏微分方程。文中证明了三阶约束偏微分方程不改变图像中零交叉点处的梯度值。在改进已有的基于曲面演化的去噪模型方面，本论文提出了一种能够保持图像纹理的法曲率驱动的曲面演化方程，由于采用了沿纹理方向的曲率来控制曲面演化速度，使得图像的纹理成分具有相对缓慢的演变，能够在有效去除噪声的同时保持图像中纹理的成分。
4)根据指纹图像上脊线的方向特性，提出一种基于伪线性方向相关扩散方程的指纹增强方法。指纹图像是一种具有特殊的方向信息和结构信息的图像，指纹增强就是要保持和增强它的这种特有结构。文中首先指出了散度形式的相关扩散方程的缺点，并针对该方程在局部方向信息较复杂情况下不能够沿着指纹脊线方向扩散提出了改进方法。改进后的方法具有更好的方向扩散特性，在去噪和保持指纹图像脊线信息方面表现更优。另外，本文针对改进的方法计算复杂度较高的不足，提出伪线性形式的方向相关扩散方程。该方法具有线性算子的优点、计算量小，有利于指纹识别实时系统应用。
5)研究了光照对图像对比度的影响及已有的处理方法，并在偏微分方程理论框架下提出了一种能够有效增强由光照不均匀而造成的低对比度图像的方法。图像中阴影部分中包含丰富的细节信息，但是这些信息的梯度变化相对较小。本文通过在梯度域进行直方图均衡化调整图像梯度的非均匀分布，使图像阴影中的细节能够得到增强，最后利用最小二乘原理从校正的梯度场中重建出增强的图像。在结果图像中，阴影区域的细节信息被很清晰地表达出来。本文通过引入Lab彩色空间将增强方法推广到了对彩色图像的处理中。同时，本文根据Laplacian算子的特点，改进了原有的Poisson方程数值求解方法。
Image processing has been an active research field resulting from the development ofcomputer technology and increasing requirements of multimedia data processing. Sinceimage denoising and enhancement are often applied to improve the image quality and tomake them fulfill the specific application requirements, both of them are importantresearch subjects in image processing. In this paper, more attention is paid to the partialdifferential equation based methods, especially to the theoretical and applicationaldifficulties in these methods. With the topics of image denoising and enhancement, wehave reviewed and studied the following aspects in this thesis: the axiomal PDEs based onscale space theory, variational PDEs and geometrical PDEs based on curve or surfaceevolution. The difficulties in them are analyzed and solutions to them are proposed.
Some novel models and algorithms have been proposed as following:
(1) Anisotropic diffusion based image denoising couplying adaptive data fidelity termis proposed in this paper. At first, three requirements of image denoising are proposed,based on which the defects of using the fidelity term in classical image denoising modelsare investigated. After that, an adaptive data fidelity term based on local image structuralinformation is brought up to preserve the objects in the images. The new adaptive datafidelity term will make nonlinear PDE based denoising methods capable of preservingsufficiently the geometric structures such as edges and comers besides its effectiveness inimage denoising. To the numerical schemes, a more stable and reliable implementation ofproposed denoising model is introduced.
(2) Image denoising with second order non-linear diffusion PDEs often leads to anundesirable staircase effect, namely, the transformation of smooth regions into piecewiseconstant regions. In this paper, the gradient fidelity term is introduced which describles thesimilarity between the restored images and noise ones in gray variation in order to alleviatethe staircase effect. Anisotropic diffusion models are improved by adding the Eulerequation derived from the gradient fidelity term. After coupling the new restriction derivedfrom the gradient fidelity term, the classical second order PDE-based denoising modelswill produce piecewise smooth results, while preserving sharp jump discontinuities inimages. In addition, the gradient fidelity term is integrabel in bounded variation functionspace which makes our model outperform fourth order nonlinear PDE based denoisingmethods suffering from leakage problems and sensitivity to texture components. Experimental results show that our new model alleviates the staircase effect to some extentand preserves the image features, such as textures and edges.
(3) The research on the geometrical image modeling has been made and we haveproposed to alleviate staircase effect with the uniform distribution restriction of imagelevel sets. In order to inhibit the staircase effect, a new third order PDE model is derived toimplement the uniform distribution restriction of image level sets. Coupling new modelinto classical PDE-based denoising models will make them produce piecewise smoothresults. Moreover, we prove that the proposed third order PDE model does not change theposition of edges in the noisy images and preserves the image edges well. On the otherhand, the surface evolution equation driven by normal curvature is also proposed in thispaper after having reviewed the existing surface evolution equation based image denoisingmethods. To normal curvature driven diffusion PDE model, we find that it will preservetexture components in images better than other geometrical diffusion model because thenormal curvature along the texture direction is smaller.
(4) A pseudo-linear diffusion filter was proposed to enhance fingerprint imagesaccording to the directional infomation of them. In this part, the defects of coherentdiffusion in divergence form are discussed which produce artificial structure and a newnonlinear directional diffusion PDE is proposed in order to improve its inability. The newalgorithm outperforms the coherent diffusion based image enhancement in denoisingability and structure preservation. At last, a pseudo-linear diffusion equation derived fromnew method is proposed in order to decreasing the computational burden. Thepseudo-linear method is more suitable to real-time fingerprint recognition system becauseof its efficiency.
(5) The contrast enhancement models to images with uneven illumination distributionare reviewed and an efficient image enhancement method based on the PDE is proposed toadjust the uneven distributed illumination in the image. This model improves the imagedetails in shadows by adjusting the distribution of image gradient firstly and reconstructsthe result image from new gradient field in least square sense finally. With the introductionof Lab color space, the new method is extended to color image applications. The efficientnumerical scheme for Poisson equation is very important in real application and a moreefficient scheme is proposed according to the features of Laplacian operator.
引文
[1] A. K. Jain. Fundamentals of Digital Image Processing. Journal of Magnetic Resonance Imaging. Prentice-Hall, Englewood Cliffs, NJ, 1989.
[2] M. A. Sid-Ahmed. Image Processing. McGraw-Hill, New York, 1995.
[3] O. Salvado, C. Hillenbrand, et al. Method to Correct Intensity Inhomogeneity in MR Images for Atherosclerosis Characterization. IEEE Transactions on Medical Imaging, 2006, 25(5):539-552.
[4] L. Meylan, S. Susstrunk. High Dynamic Range Image Rendering with a Retinex-based Adaptive Filter. IEEE Transactions on Image Processing, 2006,15(9):2820-2830.
[5] Milan Sonka, Vaclav Hlavac, Roger Botle. Image Processing, Analysis, and Machine Vision. International Thomson Publishing, 1998.
[6] S. Mallat. A wavelet tour of signal processing. Academic Press, 1997.
[7] K. R. Castleman. Digital Image Processing. Prentice-Hall, Englewood Cliffs, NJ, 1996.
[8] A. Wrangsjo, M. Borga, H. Knutsson. A Bayesian Approach to Image Restoration. IEEE International Symposium on Biomedical Imaging, 2004.
[9] Leipo Yan, Lipo Wang, Kim-Hui Yap. A Noisy Chaotic Neural Net work Approach to Image Denoising. International Conference on Image Processing. 2004.
[10] P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. Proc. IEEE Comp. Soc. Workshop on Computer Vision, 1987: 16-22.
[11] A. Buades, B. Coll, J. M. Morel. A Review of image denoising algorithms, with a new one. Multiscale Model. Simul.. 2005,4(2):490-530.
[12] Jiangwen D., H. T. Tsui. A fast level set method for segmentation of low contrast noisy biomedical images. Pattern Recognition Letter, 2002, 23:161-169.
[13] A. Bruhn, J. Weickert, C. Fedderm, et al. Variational Optical Flow Computation in Real Time. IEEE Transactions on Image Processing, 2005, 14(5):608-615.
[14] O. Faugeras, R. Keriven. Variational principles, surface evolution, PDE's level set methods and the Stereo Problem. INRIA research report, 1996, 3021.
[15]D. P. Mukherjee, N. Ray, S. T. Acton, et al. Level set analysis for Leukoyte Detection and tracking. IEEE Transactions on Image Processing, 2004, 13(4):562-572.
[16]J. J. Koenderink. The structures of images. Biol. Cybernet, 1984, 50: 363-370.
[17]A. P. Witkin. Scale-space filtering. In Proc, of IJCAI, 1983, vol. 2,1019-1022.
[18]L. Alvarez, F. Guichard, P. L. Lions, J. M. Morel. Axioms and Fundamental Equations of Image Processing. Arch. Rational Mech. Anal. 1993, 123:199-257.
[19] P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE transactions on pattern recognition and machine intelligence, 1990,12(7): 629-639.
[20] F. Carte, T. Coll, P. Lions, J. M. Morel. Image Selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. ANal.. 1992,29: 182-193.
[21]M. J. Black, G Sapiro, D. H. Marimont, et al. Robust anisotropic diffusion. IEEE Transactions on Image Processing, 1998, 7(3):421-432.
[22] G Gilboa, Y. Y. Zeevi, N. Sochen. Complex Diffusion Processes for Image Filtering. Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision, 2001,299-307.
[23] J. Weickert. A Review of Nonlinear Diffusion Filtering. Proceedings of the First International Conference on Scale-Space Theory in Computer Vision, 1997, 1252: 3-28.
[24] G W. Wei. Generalized Perona-Malik Equation for Image Restoration. IEEE Signal Processing Letters, 1999, 6(7): 165-167.
[25] S. Esedoglu. Stability Properties of Perona-Malik Scheme. SIAM Journal on Numerical Analysis, 2006,44(3): 1297 - 1313.
[26] J. Weickert. Multiscale Texture Enhancement. Proceedings of the 6th International Conference on Computer Analysis of Images and Patterns. 1995, 970: 230-237.
[27] J. Weickert, C. Schnorr. A theoretical framework for convex regularizers in PDE-based computation of image motion. International Journal of Computer Vision, 2001, 45(3): 245-264.
[28] J. Weickert. Theoretical Foundations of Anisotropic Diffusion in Image Processing. Proceedings of the 7th TFCV on Theoretical Foundations of Computer Vision, 1994, 221-236.
[29] David Tschumperle, R. Deriche. PDE's Based Regularization of Multivalued Images and Applications. IEEE Trans. Pattern Ana. Mach. Intell. 2005,27(4): 506-517.
[30] David Tschumperle. PDE's Based Regularization of Multivalued Images and Applications. PhD These, France: University of Nice-Sophia Antipolis, 2002.
[31]L. Rudin. Images, numerical analysis of singularities, and Shock Filter. PhD. These, California Institute of Technology, Pasadena, CA, 1987.
[32] L. Alvarez, L. Mazorra. Signal and Image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 1994, 31(2):590-605.
[33] L. Remaki, M. Cheriet. Numerical Schemes of shock filters models for image enhancement and restoration. Journal of Mathematical Imaging and Vision, 2003, 18:129-143.
[34] L. Remaki, M. Cheriet. Enhanced and restored signals as generalized solution for shock filter models. Part I-existence and uniqueness result of the Cauchy problem. J. Math. Anal. Appl. 2003, 279:189-209.
[35] L. Remaki, M. Cheriet. Enhanced and restored signals as generalized solution for shock filter models. Part Ⅰ-numerical study. J. Math. Anal. Appl. 2003, 279:398-417.
[36] Guy Gilboa, Nir Sochen, Yehoshua Y. Zeevi. Anisotropic selective inverse diffusion for signal enhancement inthe presence of noise. Proceedings. 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000, Vol. 1:221-224
[37] Guy Gilboa, Nir Sochen, Yehoshua Y. Zeevi. Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Transactions Image Processing, 2002, 11 (7): 689-703.
[38] I. Pollak, A. S. Willsky, H. Krim. Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Transactions on image processing, 2000, 9(2):256-266.
[39] 石澄贤．几何图像模型及其在医学图像处理中的应用研究．南京理工大学博士论文，2005．
[40] 肖亮．基于变分PDE和多重分形的图像建模理论、算法与应用．南京理工大学博士论文，2003．
[41] M. G. Crandall, H. Ishii, P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull Amer. Math. Sot., 1992, 33:1～67.
[42] T. F. Chan and J. Shen. On the role of the BV image model in image restoration. Amer. Math. Soc. Contemp. Math., 2003, vol. 330, pp. 25-41.
[43] L. Rudin, S. Osher, E. Fatemi. Nonlinear Total Variation based noise removal algorithms. Physica D. 1992(60): 259-268.
[44] A. Chambolle, P.-L. Lions. Image Recovery via total variation minimization and related problems. Numer. Math., 1997, 76(2): 167-188.
[45] Chan T F, Golub G H and P.Mulet. A nonlinear prime dual method for total variation-based image restoration. Technique Report, Univ. California,, Los Angeles, 1995.
[46] Strong D M. Adaptive total variation minimizing image restoration. Ph.D Dissertation, University of California, Los Angeles, 1997.
[47] Blomgren P, Chan T F and Mullet P. Extensions to total variation denoising. Technique report, University of California, Los Angeles, 1995.
[48] T. F. Chan, S. Esedoglu, F. Park, et al. Recent Developments in Total Variation Image Restoration. Technique Report, University of California, Los Angeles, 2005.
[49] T. Chan, A. Marquina, P. Mulet. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 2000, Vol. 22, no. 2, pp.503-516.
[50] Tony Chan and Shen Jianhong. Mathematical models for local non-texture inpalntings SIAM J.Appl.Math., 2001, 62(3): 1019-1043.
[51] G. Aubert, Luminita Vese. A variational method in image recovery. SIAM Journal on Numerical Analysis, 1997, 34(5): 1948-1979.
[52] Yu-Li You, Wenyuan Xu, Allen Tannenbaum, Mostafa Kaveh. Behavioral Analysis of anisotropic diffusion in Image Processing. IEEE Transactions on Image Processing. 1996, 11 (5): 1539-1553.
[53] Hamzn A. B., Krim Hamid. A variational approach to maximum a posteriori estimation for image denoising. EMMCVPR, Figueiredo MAT, 2001, 19-34.
[54] German S amd German D. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. on PAMI. 1984, 6: 721-741.
[55] Geman D,Yang C. Nonlinear image recovery with half-quadratic regularizetion. IEEE Transaction on Image Processing. 1995, 4(7): 932-945.
[56] Charbonnier P, Feraud L B, et al. Deterministic edge preserving regularizetion in computed imaging. IEEE Transaction on Image Processing, 1997, 6(2): 298-311.
[57] Mumford D, Shah J.. Optimal approximation by piecewise smooth functions and associated variational problems. Communication on Pure and Applied Mathematics. 1989, 42(5): 577-685.
[58] Giorgi De, Carriero M, Leac A.. Existence theorem for a minimum problem with free discontinuity set. Archive for Rational Mechanics and Analysis. 1989, 108(3): 195-218.
[59] Chambolle A.. Finite differences discretizations of the Mumford-Shah functional. Mathematics modeling and numerical analysis. 1999, 23(2): 261-288.
[60] Black A and Zisserman A. Visual reconstruction. MIT Press. 1987.
[61] Ambrosio L, Tortorelli V.. Approximation of functionals depending on jumps by elliptic functionals via-convergence. Communication on Pure and Applied Mathematics, 1990, 43(8): 999-1036.
[62] Richardson T J, Mitter S K.. A variational formulation based edge focusing algorithm. Sadhana Acad. P. Eng. S.. 1999, 22: 553-574.
[63]Yunmei Chen, B. C. Vemuri, L. Wang. Image Denoising and segmentation via nonlinear diffusion. Computers and Mathematics with applications, 2000, 39:131-149.
[64] A. Tsai, J. Yezzi, A. Willsky. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation and magnification, IEEE Transactions on image processing, 2001,10(8): 1169-1186.
[65] Patrick Perez, M. Gangnet, A. Blake. Poisson image editing. Proceedings of ACM SIGGRAPH 2003,2003, 313-318.
[66] Luis Alvarez, Pierre Louis Lions., Jean Michel Morel. Image Selective Smoothing and Edge Detection by Nonlinear Diffusion. SIAM Journal of Numerical Analysis, 1992, 29(3): 845-866.
[67] R. Malladi, J. A. Sethian. Image Processing: Flows under Min/Max Curvatureand Mean Curvature. Graphical models and image processing, 1996, 58(2): 127-141.
[68]Nir Sochen, Ron Kimmel, Ravikanth Malladi. A general framework for low level vision. IEEE Transactions on Image Processing, 1998, 7(3):310—318.
[69]Suk-Ho Lee, Jin Keun Seo. Noise Removal with Gauss Curvature-Driven Diffusion. IEEE Transactions on Image Processing. 2005, 14(7): 904-909.
[70] F. Shih, O. R. Mitchell. Threshold Decomposition of Gray Scale Morphology into Binary Morphology. IEEE Transactions on PAMI, 1989, 11(1):31-42.
[71 ]G Sapiro. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press. 2001.
[72] Pierre Kornprobst, Rachid Deriche, Gilles Aubert. Nonlinear operators in image restoration. CVPR, Porto Rico. 1997.
[73]Manfredo Do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall. 1976.
[74] Anthony Yezzi. Modified Curvature Motion for Image Smoothing and Enhancement. IEEE Transactions on Image Processing. 1998, 7(3): 345-352.
[75] D. Barash. A Fundamental Relationship between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Diffusion Eqaution. IEEE transactions PAMI, 2002, 24(6): 844-847.
[76] D. Barash, D. Comaniciu. A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift. Image and Vision Computing, 2004, 22: 73-81.
[77] H. R. Sheikh, M. F. Sabir, A. C. Bovik. A Statistical Evaluation of Recent Full Reference Image Quality Assessment Algorithms. IEEE Transactions on image processing, 2006, 15(11):3441-3452.
[78] 石澄贤，王洪元，王元全，夏德深．小波域上的图像非线性扩散滤波．中国图象图形学报，2004，9(12)，1449-1453．
[79] G. Gilboa, N. Sochen, Y. Y. Zeevi. Variational Denoising of Partly Textured Images by Spatially Varying Constraints. IEEE Transactions on image processing, 2006, 15(8):2281-2289.
[80] G. Gilboa, N. Sochen, Y. Y. Zeevi. Esitmation of optimal PDE-based Denoising in the SNR sense. IEEE Transactions on image processing, 2006, 15(8):2269-2280.
[81] P. Mrazek, M. Navara. Selection of Optimal stopping time for nonlinear diffusion filtering. International Journal of Computer Vision, 2003, 52(2/3): 189-203.
[82] Y. Meyer. Osillating patterns in image processing and nonlinear evolution equation. Vol. 22 of University Lecture Series, AMS, Providence, 2001.
[83] S. Osher, A. Sole, L. Vese. Image Decomposition and Restoration Using Total Variation Minimization and the H1 Norm. SIAM J. Multiscale Model. Simul. 2003, 1(3):349-370.
[84] T. F. Chan, S. Esedoglu. Aspects of total variation regularized L1 function approximation. SIAM J. Multiscale Model. Simul., 2005,65(5): 1817-1837.
[85] S. Osher, M. Burger, D. Goldfarb, et al. Iterative Regularization Method for Total Varitaion-Based Image Restoration. SIAM J. Multiscale Modeling and Simulation 2005, 4(2), 460-489.
[86] L. Lucido, R. Deriche, L. Alvarez, V.Rigaud. Sur quelques schemas numeriques de resolution d'equations aux derivees partielles pour le traitement d'images. France: The French National Institute for Research in Computer and Control, 1997, No. 3132.
[87] S. Kim. Numerical modeling for the recovery of fine structures in PDE-based image denoising, http://www2.msstate.edu/～sk38/publication.html
[88] You Y. L., Kaveh M.. Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 2000, 9(10): 1723-1730
[89] Marius Lysaker, Arvid Lundervold, Xue-Cheng Tai. Noise Removal Using Fourth-Order Partial Differential Equation with Applications to Medical Magnetic Resonance Images in Space and Time. IEEE Transactions on Image Processing. 2003, 12(12): 1579-1589.
[90] 贾迪野，黄凤岗，苏菡．一种新的基于高阶非线性扩散的图象平滑方法[J]．计算机学报，2005，28(5)，882-891．
[91] Kass M., Witldn A., TERZOPOULOS D.. Snakes: Active Contour models, International Journal of Computer Vision, 1988, 1:321-331.
[92] J. Tumblin and G Turk. LCIS: A boundary hierarchy for detail preserving contrast reduction. In Proceedings of the SIGGRAPH 1999 annual conference on computer graphics, August 8-13,1999, Los Angeles, CA US, pp. 83-90.
[93]Antoni Buades, Bartomeu Coll, Jean-Michel Morel. The Staircasing Effect in Neighborhood Filters and Its Solution. IEEE Transactions on Image Processing, 2006, 15(6): 1499-1505.
[94] G Sapiro. Inpainting the colors. IEEE International Conference on Image Processing, 2005. Vol. 2, pp. 11-14.
[95] R. Fattal, D. Lischinski, M. Werman. Gradient Domain High Dynamic Range Compression, Proceedings of the 29th annual conference on Computer graphics and interactive techniques, San Antonio, Texas, 2002, pp. 249 - 256.
[96] S. Didas, S. Setzer, G Steidl. Combined l_2 data and gradient fitting in conjunction with l_1 regularization. Preprint No. 143, DFG Priority Programme 1114, Department of Mathematics, University of Bremen, Germany, June 2006.
[97] J. Weickert, B. M. Romeny, M. Viergever. Efficient and reliable schemes for nonlinear diffusion filtering. IEEE transactions on image processing, 1998, 7(3):398-410.
[98] T. Young, L. J. Vliet. Recursive implementation of the Gaussian filters. Signal Processing, 1995,44:139-151.
[99] R. Malladi, J. A. Sethian. B. C. Vemuri. Shape Modeling with front propagation: A level set approach. IEEE transactions on PAMI, 1995, 17(2):158-175.
[100] S. J. Osher, J. A. Sethian. Fronts propagation with curvature dependent speed: algorithms based on Hamilton Jacobi formulations, J. Computational Phys. 79, 12-49, 1988.
[101] Marcelo Bertalmio. Strong-Continuation, Contrast-Invariant Inpainting with a Third-Order Optimal PDE, IEEE transactions on Image Processing, 2006, Vol. 15(7):1934-1938.
[102] V. Torre, T.A. Poggio. On edge detection, IEEE Trans. Pattern Anal. Machine Intell., 1986, vol. PAMI-8, pp. 147-163.
[103] A. I. El-Fallah, G. E. Ford. Mean curvature evolution and surface area scaling in image filtering. IEEE transactions on image proessing, 1997, 6(5):750-753.
[104] Dario Maio, Davide Maltoni, Raffaele Cappelli, etc. FVC2000: Fingerprint Verfication Competition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(3):402-412.
[105] Lin Hong, Yifei Wan, Anil Jain. Fingerprint Image Enhancement Algorithm and Performance Evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1998, 20(8):777-789.
[106] L. O'Gorman, J.V. Nickerson. Anapproach to fingerprint filter design. Pattern Recognition, 1989, 22(1):29-38.
[107] 尹义龙，詹小四，谭台哲，宁新宝．基于加搏函数的指纹增强算法及其应用．软件学报，2003，14(3)：484-489．
[108] Joachim Weickert. A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance, Journal of Visual Communication and Image Representation, 2002, Vol. 13: 103-118.
[109] 程建刚，田捷，何余良，杨鑫．基于非线性扩散滤波的指纹增强算法，自动化学报，2004，30(6)：854-862．
[110] Z. Rahman, D. J. Jobson, G. Woodell. Retinex Processing for Automatic Image Enhancement. Journal of Electionie Imaging, 2004, 13 (1):78-94.
[111] M. Hanmandu, D. Jha. An optimal fuzzy system for color image enhancement. IEEE transactions on image proessing, 2006, 15(10):2956-2966.
[112] S. Kartic, M. Aditi, I. Sandy, Roli Fabio, et al. Greedy algorithm for local contrast enhancement of images. International conference on image analysis and processing, Cagliari, ITALIE, 2005, Vol. 3617, pp.171-179.
[113] E. Land. An alternative technique for the computation of the designator in the retinex theory of color vision. Proe. Nat. Acad. Sci., 1986, 83:3078-3080.
[114] D. J. Jobson, Z. Rahman, G. Woodell, Properties and performance of a center/surround retinex. IEEE transactions on Image Processing, 1997, 6:451-462.
[115] D.J. Jobson, Z. Rahman, G. Woodell, A multiscale retinex for bridging the gap between color images and the human observation of scenes. IEEE transactions on Image Processing, 1997, 16:965-976.
[116] R. Kimmel, M. Elad, I. Sobel. A variational framework for retinex. International Journal of Computer vision, 2003, 52(1):7-23.
[117] R. Kimmel, D. Shaked, M. Elad, I. Sobel. Space-depedent color gamut mapping: a variational approach. IEEE transactions on Image Processing, 2005, 14(6):796-803.
[118] O. Scherzer, J. Weichert. Relations Between Regularization and Diffusion Filtering. Journal of Mathematical Imaging and Vision, 2000, 12(1):43-63.
[119] R. W. Brockett, P. Maragos. Evolution Equations for Continous-Scale Morphological Filtering, IEEE Transactions on Signal Processing, 1994, 42(12):3377-3356.
[120] P. Maragos. Differential Morphology and Image Processing. IEEE Transactions on Image Processing, 1996, 5(6):922-937.
[121] G. Gilboa, N. Sochen, Y. Y. Zeevi. Image Sharpening by Flows based on Triple Well Potentials, Journal of Mathematical and Vision, 2004, 20:121-131.
[2] M. A. Sid-Ahmed. Image Processing. McGraw-Hill, New York, 1995.
[3] O. Salvado, C. Hillenbrand, et al. Method to Correct Intensity Inhomogeneity in MR Images for Atherosclerosis Characterization. IEEE Transactions on Medical Imaging, 2006, 25(5):539-552.
[4] L. Meylan, S. Susstrunk. High Dynamic Range Image Rendering with a Retinex-based Adaptive Filter. IEEE Transactions on Image Processing, 2006,15(9):2820-2830.
[5] Milan Sonka, Vaclav Hlavac, Roger Botle. Image Processing, Analysis, and Machine Vision. International Thomson Publishing, 1998.
[6] S. Mallat. A wavelet tour of signal processing. Academic Press, 1997.
[7] K. R. Castleman. Digital Image Processing. Prentice-Hall, Englewood Cliffs, NJ, 1996.
[8] A. Wrangsjo, M. Borga, H. Knutsson. A Bayesian Approach to Image Restoration. IEEE International Symposium on Biomedical Imaging, 2004.
[9] Leipo Yan, Lipo Wang, Kim-Hui Yap. A Noisy Chaotic Neural Net work Approach to Image Denoising. International Conference on Image Processing. 2004.
[10] P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. Proc. IEEE Comp. Soc. Workshop on Computer Vision, 1987: 16-22.
[11] A. Buades, B. Coll, J. M. Morel. A Review of image denoising algorithms, with a new one. Multiscale Model. Simul.. 2005,4(2):490-530.
[12] Jiangwen D., H. T. Tsui. A fast level set method for segmentation of low contrast noisy biomedical images. Pattern Recognition Letter, 2002, 23:161-169.
[13] A. Bruhn, J. Weickert, C. Fedderm, et al. Variational Optical Flow Computation in Real Time. IEEE Transactions on Image Processing, 2005, 14(5):608-615.
[14] O. Faugeras, R. Keriven. Variational principles, surface evolution, PDE's level set methods and the Stereo Problem. INRIA research report, 1996, 3021.
[15]D. P. Mukherjee, N. Ray, S. T. Acton, et al. Level set analysis for Leukoyte Detection and tracking. IEEE Transactions on Image Processing, 2004, 13(4):562-572.
[16]J. J. Koenderink. The structures of images. Biol. Cybernet, 1984, 50: 363-370.
[17]A. P. Witkin. Scale-space filtering. In Proc, of IJCAI, 1983, vol. 2,1019-1022.
[18]L. Alvarez, F. Guichard, P. L. Lions, J. M. Morel. Axioms and Fundamental Equations of Image Processing. Arch. Rational Mech. Anal. 1993, 123:199-257.
[19] P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE transactions on pattern recognition and machine intelligence, 1990,12(7): 629-639.
[20] F. Carte, T. Coll, P. Lions, J. M. Morel. Image Selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. ANal.. 1992,29: 182-193.
[21]M. J. Black, G Sapiro, D. H. Marimont, et al. Robust anisotropic diffusion. IEEE Transactions on Image Processing, 1998, 7(3):421-432.
[22] G Gilboa, Y. Y. Zeevi, N. Sochen. Complex Diffusion Processes for Image Filtering. Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision, 2001,299-307.
[23] J. Weickert. A Review of Nonlinear Diffusion Filtering. Proceedings of the First International Conference on Scale-Space Theory in Computer Vision, 1997, 1252: 3-28.
[24] G W. Wei. Generalized Perona-Malik Equation for Image Restoration. IEEE Signal Processing Letters, 1999, 6(7): 165-167.
[25] S. Esedoglu. Stability Properties of Perona-Malik Scheme. SIAM Journal on Numerical Analysis, 2006,44(3): 1297 - 1313.
[26] J. Weickert. Multiscale Texture Enhancement. Proceedings of the 6th International Conference on Computer Analysis of Images and Patterns. 1995, 970: 230-237.
[27] J. Weickert, C. Schnorr. A theoretical framework for convex regularizers in PDE-based computation of image motion. International Journal of Computer Vision, 2001, 45(3): 245-264.
[28] J. Weickert. Theoretical Foundations of Anisotropic Diffusion in Image Processing. Proceedings of the 7th TFCV on Theoretical Foundations of Computer Vision, 1994, 221-236.
[29] David Tschumperle, R. Deriche. PDE's Based Regularization of Multivalued Images and Applications. IEEE Trans. Pattern Ana. Mach. Intell. 2005,27(4): 506-517.
[30] David Tschumperle. PDE's Based Regularization of Multivalued Images and Applications. PhD These, France: University of Nice-Sophia Antipolis, 2002.
[31]L. Rudin. Images, numerical analysis of singularities, and Shock Filter. PhD. These, California Institute of Technology, Pasadena, CA, 1987.
[32] L. Alvarez, L. Mazorra. Signal and Image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 1994, 31(2):590-605.
[33] L. Remaki, M. Cheriet. Numerical Schemes of shock filters models for image enhancement and restoration. Journal of Mathematical Imaging and Vision, 2003, 18:129-143.
[34] L. Remaki, M. Cheriet. Enhanced and restored signals as generalized solution for shock filter models. Part I-existence and uniqueness result of the Cauchy problem. J. Math. Anal. Appl. 2003, 279:189-209.
[35] L. Remaki, M. Cheriet. Enhanced and restored signals as generalized solution for shock filter models. Part Ⅰ-numerical study. J. Math. Anal. Appl. 2003, 279:398-417.
[36] Guy Gilboa, Nir Sochen, Yehoshua Y. Zeevi. Anisotropic selective inverse diffusion for signal enhancement inthe presence of noise. Proceedings. 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000, Vol. 1:221-224
[37] Guy Gilboa, Nir Sochen, Yehoshua Y. Zeevi. Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Transactions Image Processing, 2002, 11 (7): 689-703.
[38] I. Pollak, A. S. Willsky, H. Krim. Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Transactions on image processing, 2000, 9(2):256-266.
[39] 石澄贤．几何图像模型及其在医学图像处理中的应用研究．南京理工大学博士论文，2005．
[40] 肖亮．基于变分PDE和多重分形的图像建模理论、算法与应用．南京理工大学博士论文，2003．
[41] M. G. Crandall, H. Ishii, P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull Amer. Math. Sot., 1992, 33:1～67.
[42] T. F. Chan and J. Shen. On the role of the BV image model in image restoration. Amer. Math. Soc. Contemp. Math., 2003, vol. 330, pp. 25-41.
[43] L. Rudin, S. Osher, E. Fatemi. Nonlinear Total Variation based noise removal algorithms. Physica D. 1992(60): 259-268.
[44] A. Chambolle, P.-L. Lions. Image Recovery via total variation minimization and related problems. Numer. Math., 1997, 76(2): 167-188.
[45] Chan T F, Golub G H and P.Mulet. A nonlinear prime dual method for total variation-based image restoration. Technique Report, Univ. California,, Los Angeles, 1995.
[46] Strong D M. Adaptive total variation minimizing image restoration. Ph.D Dissertation, University of California, Los Angeles, 1997.
[47] Blomgren P, Chan T F and Mullet P. Extensions to total variation denoising. Technique report, University of California, Los Angeles, 1995.
[48] T. F. Chan, S. Esedoglu, F. Park, et al. Recent Developments in Total Variation Image Restoration. Technique Report, University of California, Los Angeles, 2005.
[49] T. Chan, A. Marquina, P. Mulet. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 2000, Vol. 22, no. 2, pp.503-516.
[50] Tony Chan and Shen Jianhong. Mathematical models for local non-texture inpalntings SIAM J.Appl.Math., 2001, 62(3): 1019-1043.
[51] G. Aubert, Luminita Vese. A variational method in image recovery. SIAM Journal on Numerical Analysis, 1997, 34(5): 1948-1979.
[52] Yu-Li You, Wenyuan Xu, Allen Tannenbaum, Mostafa Kaveh. Behavioral Analysis of anisotropic diffusion in Image Processing. IEEE Transactions on Image Processing. 1996, 11 (5): 1539-1553.
[53] Hamzn A. B., Krim Hamid. A variational approach to maximum a posteriori estimation for image denoising. EMMCVPR, Figueiredo MAT, 2001, 19-34.
[54] German S amd German D. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. on PAMI. 1984, 6: 721-741.
[55] Geman D,Yang C. Nonlinear image recovery with half-quadratic regularizetion. IEEE Transaction on Image Processing. 1995, 4(7): 932-945.
[56] Charbonnier P, Feraud L B, et al. Deterministic edge preserving regularizetion in computed imaging. IEEE Transaction on Image Processing, 1997, 6(2): 298-311.
[57] Mumford D, Shah J.. Optimal approximation by piecewise smooth functions and associated variational problems. Communication on Pure and Applied Mathematics. 1989, 42(5): 577-685.
[58] Giorgi De, Carriero M, Leac A.. Existence theorem for a minimum problem with free discontinuity set. Archive for Rational Mechanics and Analysis. 1989, 108(3): 195-218.
[59] Chambolle A.. Finite differences discretizations of the Mumford-Shah functional. Mathematics modeling and numerical analysis. 1999, 23(2): 261-288.
[60] Black A and Zisserman A. Visual reconstruction. MIT Press. 1987.
[61] Ambrosio L, Tortorelli V.. Approximation of functionals depending on jumps by elliptic functionals via-convergence. Communication on Pure and Applied Mathematics, 1990, 43(8): 999-1036.
[62] Richardson T J, Mitter S K.. A variational formulation based edge focusing algorithm. Sadhana Acad. P. Eng. S.. 1999, 22: 553-574.
[63]Yunmei Chen, B. C. Vemuri, L. Wang. Image Denoising and segmentation via nonlinear diffusion. Computers and Mathematics with applications, 2000, 39:131-149.
[64] A. Tsai, J. Yezzi, A. Willsky. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation and magnification, IEEE Transactions on image processing, 2001,10(8): 1169-1186.
[65] Patrick Perez, M. Gangnet, A. Blake. Poisson image editing. Proceedings of ACM SIGGRAPH 2003,2003, 313-318.
[66] Luis Alvarez, Pierre Louis Lions., Jean Michel Morel. Image Selective Smoothing and Edge Detection by Nonlinear Diffusion. SIAM Journal of Numerical Analysis, 1992, 29(3): 845-866.
[67] R. Malladi, J. A. Sethian. Image Processing: Flows under Min/Max Curvatureand Mean Curvature. Graphical models and image processing, 1996, 58(2): 127-141.
[68]Nir Sochen, Ron Kimmel, Ravikanth Malladi. A general framework for low level vision. IEEE Transactions on Image Processing, 1998, 7(3):310—318.
[69]Suk-Ho Lee, Jin Keun Seo. Noise Removal with Gauss Curvature-Driven Diffusion. IEEE Transactions on Image Processing. 2005, 14(7): 904-909.
[70] F. Shih, O. R. Mitchell. Threshold Decomposition of Gray Scale Morphology into Binary Morphology. IEEE Transactions on PAMI, 1989, 11(1):31-42.
[71 ]G Sapiro. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press. 2001.
[72] Pierre Kornprobst, Rachid Deriche, Gilles Aubert. Nonlinear operators in image restoration. CVPR, Porto Rico. 1997.
[73]Manfredo Do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall. 1976.
[74] Anthony Yezzi. Modified Curvature Motion for Image Smoothing and Enhancement. IEEE Transactions on Image Processing. 1998, 7(3): 345-352.
[75] D. Barash. A Fundamental Relationship between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Diffusion Eqaution. IEEE transactions PAMI, 2002, 24(6): 844-847.
[76] D. Barash, D. Comaniciu. A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift. Image and Vision Computing, 2004, 22: 73-81.
[77] H. R. Sheikh, M. F. Sabir, A. C. Bovik. A Statistical Evaluation of Recent Full Reference Image Quality Assessment Algorithms. IEEE Transactions on image processing, 2006, 15(11):3441-3452.
[78] 石澄贤，王洪元，王元全，夏德深．小波域上的图像非线性扩散滤波．中国图象图形学报，2004，9(12)，1449-1453．
[79] G. Gilboa, N. Sochen, Y. Y. Zeevi. Variational Denoising of Partly Textured Images by Spatially Varying Constraints. IEEE Transactions on image processing, 2006, 15(8):2281-2289.
[80] G. Gilboa, N. Sochen, Y. Y. Zeevi. Esitmation of optimal PDE-based Denoising in the SNR sense. IEEE Transactions on image processing, 2006, 15(8):2269-2280.
[81] P. Mrazek, M. Navara. Selection of Optimal stopping time for nonlinear diffusion filtering. International Journal of Computer Vision, 2003, 52(2/3): 189-203.
[82] Y. Meyer. Osillating patterns in image processing and nonlinear evolution equation. Vol. 22 of University Lecture Series, AMS, Providence, 2001.
[83] S. Osher, A. Sole, L. Vese. Image Decomposition and Restoration Using Total Variation Minimization and the H1 Norm. SIAM J. Multiscale Model. Simul. 2003, 1(3):349-370.
[84] T. F. Chan, S. Esedoglu. Aspects of total variation regularized L1 function approximation. SIAM J. Multiscale Model. Simul., 2005,65(5): 1817-1837.
[85] S. Osher, M. Burger, D. Goldfarb, et al. Iterative Regularization Method for Total Varitaion-Based Image Restoration. SIAM J. Multiscale Modeling and Simulation 2005, 4(2), 460-489.
[86] L. Lucido, R. Deriche, L. Alvarez, V.Rigaud. Sur quelques schemas numeriques de resolution d'equations aux derivees partielles pour le traitement d'images. France: The French National Institute for Research in Computer and Control, 1997, No. 3132.
[87] S. Kim. Numerical modeling for the recovery of fine structures in PDE-based image denoising, http://www2.msstate.edu/～sk38/publication.html
[88] You Y. L., Kaveh M.. Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 2000, 9(10): 1723-1730
[89] Marius Lysaker, Arvid Lundervold, Xue-Cheng Tai. Noise Removal Using Fourth-Order Partial Differential Equation with Applications to Medical Magnetic Resonance Images in Space and Time. IEEE Transactions on Image Processing. 2003, 12(12): 1579-1589.
[90] 贾迪野，黄凤岗，苏菡．一种新的基于高阶非线性扩散的图象平滑方法[J]．计算机学报，2005，28(5)，882-891．
[91] Kass M., Witldn A., TERZOPOULOS D.. Snakes: Active Contour models, International Journal of Computer Vision, 1988, 1:321-331.
[92] J. Tumblin and G Turk. LCIS: A boundary hierarchy for detail preserving contrast reduction. In Proceedings of the SIGGRAPH 1999 annual conference on computer graphics, August 8-13,1999, Los Angeles, CA US, pp. 83-90.
[93]Antoni Buades, Bartomeu Coll, Jean-Michel Morel. The Staircasing Effect in Neighborhood Filters and Its Solution. IEEE Transactions on Image Processing, 2006, 15(6): 1499-1505.
[94] G Sapiro. Inpainting the colors. IEEE International Conference on Image Processing, 2005. Vol. 2, pp. 11-14.
[95] R. Fattal, D. Lischinski, M. Werman. Gradient Domain High Dynamic Range Compression, Proceedings of the 29th annual conference on Computer graphics and interactive techniques, San Antonio, Texas, 2002, pp. 249 - 256.
[96] S. Didas, S. Setzer, G Steidl. Combined l_2 data and gradient fitting in conjunction with l_1 regularization. Preprint No. 143, DFG Priority Programme 1114, Department of Mathematics, University of Bremen, Germany, June 2006.
[97] J. Weickert, B. M. Romeny, M. Viergever. Efficient and reliable schemes for nonlinear diffusion filtering. IEEE transactions on image processing, 1998, 7(3):398-410.
[98] T. Young, L. J. Vliet. Recursive implementation of the Gaussian filters. Signal Processing, 1995,44:139-151.
[99] R. Malladi, J. A. Sethian. B. C. Vemuri. Shape Modeling with front propagation: A level set approach. IEEE transactions on PAMI, 1995, 17(2):158-175.
[100] S. J. Osher, J. A. Sethian. Fronts propagation with curvature dependent speed: algorithms based on Hamilton Jacobi formulations, J. Computational Phys. 79, 12-49, 1988.
[101] Marcelo Bertalmio. Strong-Continuation, Contrast-Invariant Inpainting with a Third-Order Optimal PDE, IEEE transactions on Image Processing, 2006, Vol. 15(7):1934-1938.
[102] V. Torre, T.A. Poggio. On edge detection, IEEE Trans. Pattern Anal. Machine Intell., 1986, vol. PAMI-8, pp. 147-163.
[103] A. I. El-Fallah, G. E. Ford. Mean curvature evolution and surface area scaling in image filtering. IEEE transactions on image proessing, 1997, 6(5):750-753.
[104] Dario Maio, Davide Maltoni, Raffaele Cappelli, etc. FVC2000: Fingerprint Verfication Competition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(3):402-412.
[105] Lin Hong, Yifei Wan, Anil Jain. Fingerprint Image Enhancement Algorithm and Performance Evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1998, 20(8):777-789.
[106] L. O'Gorman, J.V. Nickerson. Anapproach to fingerprint filter design. Pattern Recognition, 1989, 22(1):29-38.
[107] 尹义龙，詹小四，谭台哲，宁新宝．基于加搏函数的指纹增强算法及其应用．软件学报，2003，14(3)：484-489．
[108] Joachim Weickert. A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance, Journal of Visual Communication and Image Representation, 2002, Vol. 13: 103-118.
[109] 程建刚，田捷，何余良，杨鑫．基于非线性扩散滤波的指纹增强算法，自动化学报，2004，30(6)：854-862．
[110] Z. Rahman, D. J. Jobson, G. Woodell. Retinex Processing for Automatic Image Enhancement. Journal of Electionie Imaging, 2004, 13 (1):78-94.
[111] M. Hanmandu, D. Jha. An optimal fuzzy system for color image enhancement. IEEE transactions on image proessing, 2006, 15(10):2956-2966.
[112] S. Kartic, M. Aditi, I. Sandy, Roli Fabio, et al. Greedy algorithm for local contrast enhancement of images. International conference on image analysis and processing, Cagliari, ITALIE, 2005, Vol. 3617, pp.171-179.
[113] E. Land. An alternative technique for the computation of the designator in the retinex theory of color vision. Proe. Nat. Acad. Sci., 1986, 83:3078-3080.
[114] D. J. Jobson, Z. Rahman, G. Woodell, Properties and performance of a center/surround retinex. IEEE transactions on Image Processing, 1997, 6:451-462.
[115] D.J. Jobson, Z. Rahman, G. Woodell, A multiscale retinex for bridging the gap between color images and the human observation of scenes. IEEE transactions on Image Processing, 1997, 16:965-976.
[116] R. Kimmel, M. Elad, I. Sobel. A variational framework for retinex. International Journal of Computer vision, 2003, 52(1):7-23.
[117] R. Kimmel, D. Shaked, M. Elad, I. Sobel. Space-depedent color gamut mapping: a variational approach. IEEE transactions on Image Processing, 2005, 14(6):796-803.
[118] O. Scherzer, J. Weichert. Relations Between Regularization and Diffusion Filtering. Journal of Mathematical Imaging and Vision, 2000, 12(1):43-63.
[119] R. W. Brockett, P. Maragos. Evolution Equations for Continous-Scale Morphological Filtering, IEEE Transactions on Signal Processing, 1994, 42(12):3377-3356.
[120] P. Maragos. Differential Morphology and Image Processing. IEEE Transactions on Image Processing, 1996, 5(6):922-937.
[121] G. Gilboa, N. Sochen, Y. Y. Zeevi. Image Sharpening by Flows based on Triple Well Potentials, Journal of Mathematical and Vision, 2004, 20:121-131.