A New Diffusion-Based Variational Model for Image Denoising and Segmentation

In this paper we propose a new variational model for image denoising and segmentation of both gray and color images. This method is inspired by the complex Ginzburg–Landau model and the weighted bounded variation model. Compared with active contour methods, our new algorithm can detect non-closed edges as well as quadruple junctions, and the initialization is completely automatic. The existence of the minimizer for our energy functional is proved. Numerical results show the effectiveness of our proposed model in image denoising and segmentation. Fang Li received the MSc degree in Mathematics from the South West China Normal University in 2004 and from then on she works in the South West University. Meanwhile, she studies mathematics at the East China Normal University as a doctoral student. Her research interests include anisotropic diffusion filtering, the variational methods and PDEs in image processing. Chaomin Shen received the MSc degree in Mathematics from the National University of Singapore (NUS) in 1998. He worked in the Centre for Remote Imaging, Sensing and Processing (CRISP), NUS as an associate scientist during 1998 to 2004. Currently he is a lecturer in Joint Laboratory for Imaging Science & Technology and Department of Computer Science, East China Normal University. His research interests include remote sensing applications and variational methods in image processing. Ling Pi received her MSc degree from the Department of Mathematics, East China Normal University in 2003. She is currently a lecturer in the Department of Applied Mathematics, Shanghai Jiaotong University. Her work involves the application of geometric and analytic methods to problems in image processing.     

Error Analysis for Image Inpainting

Image inpainting refers to restoring a damaged image with missing information. In recent years, there have been many developments on computational approaches to image inpainting problem [2, 4, 6, 9, 11–13, 27, 28]. While there are many effective algorithms available, there is still a lack of theoretical understanding on under what conditions these algorithms work well. In this paper, we take a step in this direction. We investigate an error bound for inpainting methods, by considering different image spaces such as smooth images, piecewise constant images and a particular kind of piecewise continuous images. Numerical results are presented to validate the theoretical error bounds. Tony F. Chan received the B.S. degree in engineering and the M.S. degree in aerospace engineering in 1973, from the California Institute of Technology, and the Ph.D. degree in computer science from Stanford University in 1978. He is Professor of Mathematics and currently also Dean of the division of Physical science at University of California, Los Angeles, where he has been a Professor since 1986. His research interests include mathematical and computational methods in image processing, multigrid, domain decomposition algorithms, iterative methods, Krylov subspace methods, and parallel algorithms. Sung Ha Kang received the Ph.D. degree in mathematics in 2002, from University of California, Los Angeles, and currently is Assistant Professor of Mathematics at University of Kentucky since 2002. Her research interests include mathematical and computational methods in image processing and computer vision.     

Centroidal Voronoi Tessellation Algorithms for Image Compression, Segmentation, and Multichannel Restoration

Centroidal Voronoi tessellations (CVT's) are special Voronoi tessellations for which the generators of the tessellation are also the centers of mass (or means) of the Voronoi cells or clusters. CVT's have been found to be useful in many disparate and diverse settings. In this paper, CVT-based algorithms are developed for image compression, image segmenation, and multichannel image restoration applications. In the image processing context and in its simplest form, the CVT-based methodology reduces to the well-known k-means clustering technique. However, by viewing the latter within the CVT context, very useful generalizations and improvements can be easily made. Several such generalizations are exploited in this paper including the incorporation of cluster dependent weights, the incorporation of averaging techniques to treat noisy images, extensions to treat multichannel data, and combinations of the aforementioned. In each case, examples are provided to illustrate the efficiency, flexibility, and effectiveness of CVT-based image processing methodologies. Qiang Du is a Professor of Mathematics at the Pennsylvania State University. He received his Ph.D. from the Carnegie Mellon University in 1988. Since then, he has held academic positions at several institutions such as the University of Chicago and the Hong Kong University of Science and Technology. He has published over 100 papers on numerical algorithms and their various applications. His recent research works include studies of bio-membranes, complex fluids, quantized vortices, micro-structure evolution, image and data analysis, mesh generation and optimization, and approximations of partial differential equations. Max Guzburger is the Frances Eppes Professor of Computational Science and Mathematics at Florida State University. He received his Ph.D. degree from New York University in 1969 and has held positions at the University of Tennessee, Carnegie Mellon University, Virginia Tech, and Iowa State University. He is the author of five books and over 225 papers. His research interest include computational methods for partial differential equations, control of complex systems, superconductivity, data mining, computational geometry, image processing, uncertainty quantification, and numerical analysis. Lili Ju is an Assistant Professor of Mathematics at the University of South Carolina, Columbia. He received a B.S. degree in Mathematics from Wuhan University in China in 1995, a M.S. degree in Computational Mathematics from the Chinese Academy of Sciences in 1998, and a Ph.D. in Applied Mathematics from Iowa State University in 2002. From 2002 to 2004, he was an Industrial Postdoctoral Researcher at the Institute of Mathematics and Its Applications at the University of Minnesota. His research interests include numerical analysis, scientific computation, parallel computing, and medical image processing. Xiaoqiang Wang is a graduate student in mathematics at the Pennsylvania State University, working under the supervision of Qiang Du. Starting in September 2005, he will be an Industrial Postdoctoral Researcher at the Institute of Mathematics and its Applications at the University of Minnesota. His research interests are in the fields of applied mathematics and scientific computation. His work involves numerical simulation and analysis, algorithms for image processing and data mining, parallel algorithms, and high-performance computing.     

相关扩散方程在图像修补中的应用

图像修复是图像复原研究中的一个重要内容,可以用于旧照片中丢失信息的恢复、视频文字去除以及视频错误隐藏等。目前有很多图像修补算法对于灰度图像的修补已经取得了很好的修补效果,但存在着时间消耗大和应用到彩色图像修补中时修补效果不理想的缺点。将相关扩散方程引入到图像修补中,并进行改进,使得图像的修补效率和修补效果都得到了有效提高,而且可以很自然地应用到彩色图像修补中。大量的实验证明了该方法的有效性。     

变分模型在矢量图像修复中的应用

图像修复是指恢复图像中破损区域的颜色信息或者去除图像中的多余物体。分析了基于整体变分法TV模型以及矢量图像耦合技术的原理,根据矢量图像耦合思想将整体变分法运用到矢量图像中并对矢量图像进行试验。实验结果表明：改进的矢量图像耦合修复模型能较好地修复大块彩色图像的缺失信息和移除多余物体,能保持彩色图像的边缘,且有较好的去噪功能。     

数字图像修复技术综述

图像修复是图像复原研究中的一个重要内容,它的目的是根据图像现有的信息来自动恢复丢失的信息,其可以用于旧照片中丢失信息的恢复、视频文字去除以及视频错误隐藏等。为了使人们对该技术有个概略了解,在对目前有关数字图像修复技术的文献进行理解和综合的基础上,首先通过对数字图像修复问题的描述,揭示了数字图像修复的数学背景;接着分别介绍了以下两类图像修复技术:一类是基于几何图像模型的图像修补(inpainting)技术,该技术特别适用于修补图像中的小尺度缺损;另一类是基于纹理合成的图像补全(comp letion)技术,该技术对于填充图像中大的丢失块有较好的效果;然后给出了这两类方法的应用实例;最后基于对数字图像修复问题的理解,提出了对数字图像修复技术的一些展望。     

Fast correction of bleed-through distortion in grayscale documents by a blind source separation technique

Ancient documents are usually degraded by the presence of strong background artifacts. These are often caused by the so-called bleed-through effect, a pattern that interferes with the main text due to seeping of ink from the reverse side. A similar effect, called show-through and due to the nonperfect opacity of the paper, may appear in scans of even modern, well-preserved documents. These degradations must be removed to improve human or automatic readability. For this purpose, when a color scan of the document is available, we have shown that a simplified linear pattern overlapping model allows us to use very fast blind source separation techniques. This approach, however, cannot be applied to grayscale scans. This is a serious limitation, since many collections in our libraries and archives are now only available as grayscale scans or microfilms. We propose here a new model for bleed-through in grayscale document images, based on the availability of the recto and verso pages, and show that blind source separation can be successfully applied in this case too. Some experiments with real-ancient documents arepresented and described. Anna Tonazzini graduated cum laude in Mathematics from the University of Pisa, Italy, in 1981. In 1984 she joined the Istituto di Scienza e Tecnologie dell'Informazione of the Italian National Research Council (CNR) in Pisa, where she is currently a researcher at the Signals and Images Laboratory. She cooperated in special programs for basic and applied research on image processing and computer vision, and is co-author of over 60 scientific papers. Her present interest is on inverse problems theory, image restoration and reconstruction, document analysis and recognition, independent component analysis, neural networks and learning. Emanuele Salerno graduated in Electronic Engineering from the University of Pisa, Italy, in 1985. In September 1987 he joined the Italian National Research Council (CNR) at the Department of Signal and Image Processing, Information Processing Institute (now Institute of Information Science and Technologies, ISTI, Signals and Images Laboratory), Pisa, Italy, where he has been working in applied inverse problems, image reconstruction and restoration, microwave nondestructive evaluation, and blind signal separation. He has been assuming different responsibilities in research programs in nondesctructive testing, robotics, numerical models for image reconstruction and computer vision, neural networks techniques in astrophysical imagery. At present, he is local scientific responsible in the framework of the European Space Agency's “Planck Surveyor Satellite” mission, and takes part in the European CRAFT project “ISyReADeT”, for document image restoration. Luigi Bedini graduated cum laude in Electronic Engineering from the University of Pisa, Italy, in 1968. Since 1970 he has been a Researcher of the Italian National Research Council, Istituto di Scienza e Tecnologie dell'Informazione, Pisa, Italy. His interests have been in modelling, identification, and parameter estimation of biological systems applied to non-invasive diagnostic techniques. At present, his research interest is in the field of digital signal processing, image reconstruction and neural networks applied to image processing. He is co-author of more than 80 scientific papers. From 1971 to 1989, he was Associate Professor of System Theory at the Computer Science Department, University of Pisa, Italy.     

DCT-IV computation

A new fast algorithm for computing a type of discrete cosine transform applied to compute modified cosine transform is described. The number of arithmetic operations of the algorithm is estimated and interpreted. The computational scheme with substitution of variables is presented, and methods for optimizing the code for digital signal processors are proposed. Vasilii S. Shaptala. Born 1978. Graduated from St. Petersburg Bonch-Bruevich University for Telecommunications in 2000. Received his candidate’s degree in 2003. At present, he is a software engineer at ARC International R&D Center. Scientific interests: digital signal processing in telecommunication systems. Author of 15 publications. Mikhail V. Korman. Born 1974. Graduated from the Department of Mathematics and Mechanics of St. Petersburg State University in 1997. Attended his postgraduate courses at the same university in 1997–2000. At present, he is a director at ARC International R&D Center. Scientific interests: fast orthogonal transformations, information encoding, rate-distortion optimization (encoding error optimization), numerical methods. Author of 5 articles on methods of solving parabolic and hyperbolic equations numerically.     

基于方向场的指纹图像偏微分方程修补模型

提出一个用于指纹图像修复的新的偏微分方程(PDE)模型,该模型对指纹图像的缺损区域能进行有效的修补。通过分析比较现有的技术方案对指纹图像修补的不足:一般常见的图像修复模型由于缺乏方向场的几何信息,对于指纹图像,这些模型不能给出满意的修补结果;或者虽然引入了方向场的几何信息,但在具体修补时会出现将不同的脊线连到一起不同的错误结果。该模型采用方向场作为扩散方向,在扩散过程中灰度信息沿着表征脊线方向的局部固定方向传播到待修复区域中, 改进了一般PDE模型不能用于修复指纹图像的不足。数值实验结果表明在修复指纹图像时提出的模型优于一般的模型     

一种基于整体变分的图象修补算法

图象修补是图象恢复研究中的一个重要内容，它的目的是根据图象现有的信息来自动恢复丢失的信息，可以用于旧照片中丢失信息的恢复。由于图象中的边缘代表了图象的重要信息，所以在设计修补算法时，必须着重考虑边缘的恢复，采用整体变分模型设计了一个图象修补算法，整体变分模型能够模拟人的低层视层，在修补图象时可以恢复图象中的边缘，数值实验表明，该模型能够较好地恢复待修补区域的信息，但是受修补区域大小的影响，同时又采用了一种向前传播操作来缩小修补区域。