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.     

Fuzzy approach for image retrieval

Current technology allows the acquisition, transmission, storing, and manipulation of large collections of images. Content-based information retrieval is now a widely investigated issue that aims at allowing users of multimedia information systems to retrieve images coherent with a sample image. A way to achieve this goal is the automatic computation of features such as color, texture, and shape and the use of these features as query terms. Feature extraction is a crucial part of any such system. Current methods for feature extraction suffer from two main problems: firstly, many methods do not retain any spatial information, and secondly, the problem of invariance with respect to standard transformation is still unsolved. In this paper, we describe some results of a study on similarity evaluation in image retrieval using shape, texture, and color as content features. Images are retrieved based on similarity of features, where features of the query specification are compared with features of the image database to determine which images match similarly with given features. In this paper, we propose an effective method for image representation which utilizes fuzzy features. The text was submitted by the author in English. Ryszard S. Choraś is Professor of Computer Science in the Department of Telecommunications and EE of University of Technology and Agriculture, Bydgoszcz, Poland. He also holds a courtesy appointment with the Faculty of Mathematics, Technology, and Natural Sciences of Kazimierz Wielki University, Bydgoszcz and the College of Computer Science, Lódz, Poland. His research interests include image signal compression and coding, computer vision, and multimedia data transmission. He received his M.S. degree in Electrical Engineering from Electronics from the Technical University of Wroclaw, Poland in 1973, and his Ph.D. degree in Electronics from Technical University of Wroclaw, Poland, in 1980, and D.Sc. (Habilitation degree) in Computer Science from Warsaw Technical University, Poland, in 1993. Until 1973–1976 he was a member of the research staff at the Institute of Mathematical Machines Silesian Division, Gliwice, working on graphics hardware and human visual perception. In 1976, he joined University of Technology and Agriculture, Bydgoszcz, Poland, first as an Assistant, then as a Professor of Computer Science at the Department of Telecommunications and EE. From 1994 to 1996, he was also Professor of Computer Sciences of the Zielona Góra University, Poland. He has served as the Chairman of the Communication Switching Division and as Chief of the Image Processing and Recognition Group. Until 1996–2002 he was the Vice Rector of University of Technology and Agriculture, Bydgoszcz. Prof. Choraś has an expertise in EU Programs and National Programs, e.g., he was coordinator of EU Program CME-02060, EU Program on Continuous Education and Technology Transfer, and coordinator of national programs in IST and multimedia in e-learning. Prof. Choraś has authored two monographs, and over 130 book chapters, journal articles, and conference papers in the area of image processing. Professor Choraś is a member of the editorial boards of “Machine Vision and Graphics.” He is the editor-in-chief of “Image Processing and Communications Journal.” He has served on numerous conference committees, e.g., Visualization, Imaging, and Image Processing (VIIP), IASTED International Conference on Signal Processing, Pattern Recognition and Applications, International Conference on Computer Vision and Graphics, ICINCO International Conference on Informatics in Control, Automation and Robotics, ICETE International Conference on E-business and Telecommunication Networks, and CORES International Conference on Computer Recognition Systems, and many others. Prof Choraś is a member of the IASTED, WSEAS, various Committees of the Polish Academy of Sciences, TPO. When not working on academic ventures, Professor Choraś likes to relax with activities such as walking, tennis, and swimming.     

Object tracking in video-surveillance

This paper faces the automatic object tracking problem in a video-surveillance task. A previously selected and then identified target has to be retrieved in the scene under investigation because it is lost due to masking, occlusions, or quick and unexpected movements. A two-step procedure is used, firstly motion detection is used to determine a candidate target in the scene, secondly using a semantic categorization and Content Based Image Retrieval techniques, the candidate target is identified whether it is the one that was lost or not. The use of Content Based Image Retrieval serves as support to the search problem and is performed using a reference data base which was populated a priori. The article is published in the original. Davide Moroni (Magenta, 1977), M.Sc. in Mathematics honours degree from the University of Pisa in 2001, dipl. at the Scuola Normale Superiore of Pisa in 2002, Ph.D. in Mathematics at the University of Rome ‘La Sapienza’ in 2006, is a research fellow at the Institute of Information Science and Technologies of the Italian National Research Council, in Pisa. His main interests include geometric modelling, computational topology, image processing and medical imaging. At present he is involved in a number of European research projects working in discrete geometry and scene analysis. Gabriele Pieri (Pescia, 1974), M.Sc. (2000) in Computer Science from the University of Pisa, since 2001 joined the “Signals and Images” Laboratory at ISTI-CNR, Pisa, working in the field of image analysis. His main interests include neural networks, machine learning, industrial diagnostics and medical imaging. He is author of more than twenty papers.     

An efficient method for geo-referenced video mosaicing for environmental monitoring

Environmental monitoring applications require seamless registration of optical data into large area mosaics that are geographically referenced to the world frame. Using frame-by-frame image registration alone, we can obtain seamless mosaics, but it will not exhibit geographical accuracy due to frame-to-frame error accumulation. On the other hand, the 3D geo-data from GPS, a laser profiler, an INS system provides a globally correct track of the motion without error propagation. However, the inherent (absolute) errors in the instrumentation are large for seamless mosaicing. The paper describes an effective two-track method for combining two different sources of data to achieve a seamless and geo-referenced mosaic, without 3D reconstruction or complex global registration. Experiments with real airborne video images show that the proposed algorithms are practical in important environmental applications. Zhigang Zhu received his B.E., M.E. and Ph.D. degrees, all in computer science from Tsinghua University, Beijing, in 1988, 1991 and 1997, respectively. He is currently an associate professor in the Department of Computer Science, the City College of the City University of New York. Previously, he was an associate professor at Tsinghua University, and a senior research fellow at the University of Massachusetts, Amherst. His research interests include 3D computer vision, HCI, virtual/augmented reality, video representation, and various applications in education, environment, robotics, surveillance and transportation. He has published over 90 technical papers in the related fields. He is a member of IEEE and ACM. Edward M. Riseman received his B.S. degree from Clarkson College of Technology in 1964 and his M.S. and Ph.D. degrees in electrical engineering from Cornell University in 1966 and 1969, respectively. He joined the Computer Science Department at UMass-Amherst as assistant professor in 1969, has been a professor since 1978, and served as chairman of the department from 1981 to 1985. Professor Riseman has conducted research in computer vision, artificial intelligence, learning, and pattern recognition, and has more than 200 publications. He has co-directed the Computer Vision Laboratory since its inception in 1975. Professor Riseman has been on the editorial boards of Computer Vision and Image Understanding (CVIU) from 1992 to 1997 and of the International Journal of Computer Vision (IJCV) from 1987 to the present. He is a senior member of IEEE, and a fellow of AAAI. Allen R. Hanson received his B.S. degree from Clarkson College of Technology in 1964 and his M.S. and Ph.D. degrees in electrical engineering from Cornell University in 1966 and 1969, respectively. He joined the Computer Science Department at UMass-Amherst as an associate professor in 1981, and has been a professor there since 1989. Professor Hanson has conducted research in computer vision, artificial intelligence, learning, and pattern recognition, and has more than 150 publications. He is co-director of the Computer Vision Laboratory at UMass-Amherst, and has been on the editorial boards of the following journals: Computer Vision, Graphics and Image Processing 1983–1990, Computer Vision, Graphics, and Image Processing—Image Understanding 1991–1994, and Computer Vision and Image Understanding 1995–present. Howard Schultz received a M.S. degree in physics from UCLA in 1974 and a Ph.D. in physical oceanography from the University of Michigan in 1982. Currently, he is a senior research fellow with the Computer Science Department at the University of Massachusetts, Amherst. His research interests include quantitative methods for image understanding and remote sensing. The current focus of his research activities are on developing automatic techniques for generating complex, 3D models from sequences of images. This research has found application in a variety of programs including real-time terrain modeling and video aided navigation. He is a member of the IEEE, the American Geophysical Union, and the American Society of Photogrammetry and Remote Sensing.     

Computational model for recognition of scene objects with an active sensor

In the paper, a computational model for recognition of objects in a scene image is presented. The model is based on the use of an active sensor. The structure of the object model (OM) is described. This structure is a component that stores different representations of the object and puts at user’s disposal an interface whose operations are used in the scene recognition process. Semen Yu. Sergunin. Born 1980. Graduated from the Faculty of Mathematics and Mechanics of Moscow State University in 2002. Finished postgraduate course of the Department of Computational Mathematics of the same faculty. Scientific interests include image recognition. Author of about 20 papers. Mikhail I. Kumskov. Graduated from the Faculty of Computational Mathematics and Cybernetics of Moscow State University in 1978. Received his candidate’s degree (in Physics and Mathematics) in 1981 and doctoral degree in 1997. In 1981–1997 taught at the Faculty of Computational Mathematics and Cybernetics in the special seminar on Computer Graphics and Image Processing of the Department of Automation of Systems of Computational Complexes. Since 1992 works at the laboratory of Mathematical Chemistry of the Zelinsky Institute of Organic Chemistry of the Russian Academy of Sciences. Since 1997 teaches at the Department of Computational Mathematics of the Faculty of Mathematics and Mechanics of Moscow State University. Author of more than 50 papers. Scientific interests include prediction of properties of chemical compounds, optimization of structural object representation for classification problems, and image understanding.     

Edge-Forming Methods for Image Zooming

The article is concerned with edge-forming methods to be applied as a post-process for image zooming. Image zooming via standard interpolation methods often produces the so-called checkerboard effect, in particular, when the magnification factor is large. In order to remove the artifact and to form reliable edges, a nonlinear semi-discrete model and its numerical algorithm are suggested along with anisotropic edge-forming numerical schemes. The algorithm is analyzed for stability and choices of parameters. For image zooming by integer factors, a few iterations of the algorithm can form clear and sharp edges for gray-scale images. Various examples are presented to show effectiveness and efficiency of the newly-suggested edge-forming strategy. The work of this author is supported in part by NSF grant DMS–0312223. Youngjoon Cha received his B.Sc. (1988) and M.Sc. (1990) from Mathematics, Seoul National University, Seoul, South Korea; and Ph.D. (1996) from Mathematics, Purdue University, working on mathematical epidemiology, under a guidance of Prof. Fabio Milner. He was a post-doctoral researcher at Purdue University, and Seoul National University, South Korea, from 1996 to 1997 and from 1997 to 1998, respectively. He is currently an associate professor in the Department of Applied Mathematics, Sejong University, South Korea. His research interests include image processing, mathematical and numerical modeling for waves, and mathematical epidemiology. Seongjai Kim received his B.Sc. (1988) and M.Sc. (1990) from Mathematics, Seoul National University, Seoul, South Korea; and Ph.D. (1995) from Mathematics, Purdue University, working on computational fluid dynamics, under a guidance of Prof. Jim Douglas, Jr. After two years of post-doctoral research on seismic inversion at Rice University, he worked for Shell E&P Tech. Co., Houston, for a year and the Department of Mathematics, University of Kentucky, for seven years. He is currently an associate professor in the Department of Mathematics and Statistics, Mississippi State University. His research interests are in mathematical and numerical modeling for wave propagation in highly heterogeneous media, seismology, and image processing for challenging images.     

On Semismooth Newton’s Methods for Total Variation Minimization

In [2], Chambolle proposed an algorithm for minimizing the total variation of an image. In this short note, based on the theory on semismooth operators, we study semismooth Newton’s methods for total variation minimization. The convergence and numerical results are also presented to show the effectiveness of the proposed algorithms. The research of this author is supported in part by Hong Kong Research Grants Council Grant Nos. 7035/04P and 7035/05P, and HKBU FRGs. The research of this author is supported in part by the Research Grant Council of Hong Kong. This work was started while the author was visiting Department of Applied Mathematics, The Hong Kong Polytechnic University. The research of this author is supported in part by The Hong Kong Polytechnic University Postdoctoral Fellowship Scheme and the National Science Foundation of China (No. 60572114). Michael Ng is a Professor in the Department of Mathematics at the Hong Kong Baptist University. As an applied mathematician, Michael’s main research areas include Bioinformatics, Data Mining, Operations Research and Scientific Computing. Michael has published and edited 5 books, published more than 140 journal papers. He is the principal editor of the Journal of Computational and Applied Mathematics, and the associate editor of SIAM Journal on Scientific Computing. Liqun Qi received his B.S. in Computational Mathematics at Tsinghua University in 1968, his M.S, and Ph.D. degree in Computer Sciences at University of Wisconsin-Madison in 1981 and 1984, respectively. Professor Qi has taught in Tsinghua University, China, University of Wisconsin-Madison, USA, University of New South Wales, Australia, and The Hong Kong Polytechnic University. He is now Chair Professor of Applied Mathematics at The Hong Kong Polytechnic University. Professor Qi has published more than 140 research papers in international journals. He established the superlinear and quadratic convergence theory of the generalized Newton method, and played a principal role in the development of reformulation methods in optimization. Professor Qi’s research work has been cited by the researchers around the world. According to the authoritative citation database ISIHighlyCited.com, he is one of the world’s most highly cited 300 mathematicians during the period from 1981 to 1999. Yu-Fei Yang received the B.Sc., M.S. and Ph.D. degrees in mathematics from Hunan University, P. R. China, in 1987, 1994 and 1999, respectively. From 1999 to 2001, he stayed at the University of New South Wales, Australia as visiting fellow. From 2002 to 2005, he held research associate and postdoctoral fellowship positions at the Hong Kong Polytechnic University. He is currently professor in the College of Mathematics and Econometrics, at Hunan University, P. R. China. His research interests includes optimization theory and methods, and partial differential equations with applications to image analysis. Yu-Mei Huang received her M.Sc. in Computer science from Lanzhou University in 2000. She is now pursuing her doctoral studies in computational mathematics in Hong Kong Baptist University. Her research interests are in image processing and numerical linear algebra.     

Variational Image Binarization and its Multi-Scale Realizations

A variational approach for image binarization is discussed in this paper. The approach is based on the interpolation of surface. This interpolation is computed using edge points as interpolating points and minimizing an energy functional which interpolates a smooth threshold surface. A globally convergent Sequential Relaxation Algorithm (SRA) is proposed for solving the optimization problem. Moreover, our algorithm is also formulated in a multi-scale framework. The performance of our method is demonstrated on a variety of real and synthetic images and compared with traditional techniques. Examples show that our method gives promising results.This research is partially supported by HKBU Faculty Research Grant FRG/02-03/II-04 and NSF of China Grant. C.S. Tong received a BA degree in Mathematics and a Ph.D. degree (on Mathematical Modelling of Intermolecular Forces) both from Cambridge University. After graduation, he joined the Signal and Image Processing division of GEC-Marconis Hirst Research Centre as a Research Scientist, working on image restoration and fractal image compression. He then moved to the Department of Mathematics at Hong Kong Baptist University in 1992, becoming Associate Professor since 2002.He is a member of the IEEE, a Fellow of the Institute of Mathematics and Its Application, and a Chartered Mathematician. His current research interests include image processing, fractal image compression, and neural networks. Yongping Zhang received the M. S. degree from Department of Mathematics at Shaanxi Normal University, Xian, China, in 1988 and received the Ph.D. degree from The Institute of Artificial Intelligence and Robotics at Xian Jiaotong University, Xian, China, in 1998.In 1988 he joined Department of Mathematics at Shaanxi Normal University, where he became Associate Professor in July 1987. He held postdoctoral position at Northwestern Polytechnic University during the 1999–2000 academic years. Currently he is a research associate in the Bioengineering Institute at the University of Auckland, New Zealand. His research interests are in Computer Vision and Pattern Recognition, and include Wavelets, Neural Networks, PDE methods and variational methods for image processing. Nanning Zheng received the M.S. degree from Xian Jiaotong University, Xian, China, in 1981 and the Ph.D. degree from Keio University, Japan, in 1985. He is an academician of Chinese Engineer Academy, and currently a Professor at Xian Jiaotong University. His research interest includes Signal Processing, Machine Vision and Image Processing, Pattern Recognition and Virtual Reality.This revised version was published online in June 2005 with correction to CoverDate     

A Numerical Scheme for the Curvature Equation Near the Singularities

In this paper we propose a modification in the usual numerical method for computing the solutions of the curvature equation in the plane . This modification takes place near the singularities of the image. We propose to use zero as the vertical speed at a saddle point and, at an extremum, the geometric mean of the eigenvalues of the Hessian matrix. This modification is theoretically justified and the preliminary experimental results show that it makes the algorithm more reliable.Marcos Craizer has a degree in mathematics from UFRJ (Rio de Janeiro), a M.Sc. from IMPA (Rio de Janeiro) and received his Ph.D. in mathematics also from IMPA, in 1989. His research interests in image processing includes image representation, curve evolution and PDE applications. Since 1988, he has been working at the math department of PUC-Rio, Brazil.Sinésio Pesco is an Assistant Professor of the Department of Mathematics at Pontifical Catholic University of Rio de Janeiro (PUCRio). He received his Ph.D. and MS degree in Applied Mathematics at PUC-Rio and a B.S. degree in mathematics from State University of Maringa Brasil. He has visiting positions at Lawrence Livermore National Laboratory, CSE/OGI School of Science and Engineering (Oregon Health & Science University) and Scientific Computation and Imaging Insititute (University of UTAH). His main research interests are in Computational Topology, Image Processing and Scientific Visualization. Since 1991, he has been working in the development of a CAD system for petroleum reservoir modeling.Ralph Teixeira has a degree in Computer Engineering from IME (in Rio), a M.Sc. from IMPA (also in Rio) and received his Ph.D. in Mathematics from Harvard University in 1998. His research interests in Computer Vision include shape representations by skeletons (medial axis and similar objects), curve evolutions and PDE applications. Since 2001, he has been working at Fundação Getulio Vargas in Rio de Janeiro, Brazil.     

Detecting textured objects using convex hull

In this paper, we present a methodology of locating 3D objects of known shapes from a single gray-scale image, in particular objects with rich textures on the surface. While traditional approaches identify objects by grouping and matching local features, we locate the object in the image using its convex hull, a high level feature not given much attention in the image using literature. A “direct line detection” algorithm is developed to detect line segments directly from the gray-scale image divided in small blocks. Lines are clustered and convex hull of a single or group of clusters is computed and edited to extract the 2D contour of the object. Successful experiments on rectangular boxes and cylinders show the effectiveness of the convex hull approach and its potential usage in industrial applications. Part of the work discussed in this paper was performed when both authors were affiliated with Symbol Technologies.     

The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Uncertainty principles can be derived by using a group theoretic approach. This approach yields also a formalism for finding functions which are the minimizers of the uncertainty principles. A general theorem which associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups. This study is concerned with the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the Affine and the Affine-Weyl-Heisenberg groups. We explore the relationship between the two-dimensional affine group and the SIM (2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the Affine-Weyl-Heisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored. Chen Sagiv, finished her B.Sc studies in Physics and Mathematics in 1990 and her M.Sc. studies in Physics in 1995 both in the Tel-Aviv University. After spending a few years in the high-tech industry, she went back to pursue her PhD in Applied Mathematics at the Tel Aviv university. Her main research interests are Gabor analysis and the applications of differential geometry in image processing, especially for texture segmentation. Nir A. Sochen completed his B.Sc. studies in Physics, 1986, and his M.Sc. in theoretical physics, 1988, at the University of Tel-Aviv. He received his Ph.D. in Theoretical physics, 1992, from the Université de Paris-Sud while conducting his research in the Service de Physique Théorique at the Centre d’Etude Nucleaire at Saclay, France. He was the recipient of the Haute Etude Scientifique Fellowship, and pursued his research for one year at the Ecole Normal Superieure, Paris. He was subsequently an NSF Fellow in Physics at the University of California, Berkeley, where focus of research and interest shifted from quantum field theories and integrable models, related to high-energy physics and string theory, to computer vision and image processing. Upon returning to Israel he spent one year with the Physics Dept., Tel-Aviv University and two years with the Department of Electrical Engineering of the Technion. He is currently a Senior Lecturer in the Department of Applied mathematics, Tel-Aviv University, and a member of the Ollendorff Minerva Center, Technion. His main research interests are the applications of differential geometry and statistical physics in image processing and computational vision. Yehoshua Y. (Josh) Zeevi is the Barbara and Norman Seiden Professor of Computer Sciences, Department of Electrical Engineering, Technion, where he served as the Dean 1994–1999. He is the Head of the Ollendorff Minerva Center for Vision and Image Sciences, and of the Zisapel Center for Nano-Electronics. He received his Ph.D. from U.C. Berkeley, was a Visiting Scientist at Lawrence Berkeley Lab; a Vinton Hayes Fellow at Harvard University, a Fellow-at-large at the MIT-NRP, a Visiting Senior Scientist at NTT, an SCEEE Fellow USAF on a joint appointment with MIT, a Visiting Professor at MIT, Harvard, Rutgers and Columbia Universities. His work on automatic gain control in vision led to the development of the Adaptive Sensitivity algorithms and Camera that mimics the eye, and he was a co-founder of i Sight Inc. He was also involved in development of Gabor representations and texture generators for helmet-mounted flight simulators. He is an Editor-in-Chief of J. Visual Communication and Image Representation, Elsevier, and the editor of three books. He has served on boards and international committees, including the Technion Board of Governors and Council, and the IEEE technical committee of Image and Multidimensional Signal Processing.     

Construction and Application of Hybrid Wavelet and Other Parametric Orthogonal Transforms

We introduce new methods for construction and implementation of various parametric and hybrid orthogonal transforms, including generalized Haar-like, Daubechies, and Coiflet wavelet transforms. The corresponding fast algorithms of computations are briefly discussed and the variance properties of these transforms in analyzing 1-st order Markov processes are investigated. The designed hybrid transforms can be useful in various specific signal processing applications where combining properties of Hadamard and wavelet transforms may be of particular benefit. We also present some numerical results pertaining to image zonal and threshold coding using these hybrid transforms and compare their efficacy with those of traditional orthogonal transforms.Hakob Sarukhanyan received his M.S. degree in Applied Mathematics from Yerevan State University in 1973, and his Ph.D. and D.Sc. degrees in Technical Sciences from the National Armenian Academy of Sciences (NAAS) in 1982 and 1999 accordingly. He has worked as a faculty in the Department of Applied Mathematics at Yerevan State University in 1968–73, and as a junior and senior researcher in the Laboratory of Image Processing Systems at the Institute for Informatics and Automation Problems (IIAP) of the NAAS in 1973–93. He has been the Head of the above Laboratory since 1993 and has been elected a member of the Doctoral Council at IIAP in 2000. He has been a visiting professor at the Tampere Institute of Technology, Finland, in 1999–2001. He is a recipient of research grants from various European funding agencies as well as from the US Civilian and Research Foundation (sponsored by the NSF and the US Department of State). His main research interests are in construction of Hadamard matrices and their applications in wireless communications, combinatorics theory, and fast orthogonal transforms for image processing. He is the author of more than 70 scientific publications in major scientific media.Arthur Petrosian received his M.S. Summa Cum Laude degree in Mathematics from Moscow State University in 1983, and a Ph.D. in Applied Mathematics from the Institute for Problems of Informatics & Automation of the National Armenian Academy of Sciences in 1989. He was a visiting scientist at the Institute of General and Physical Chemistry at Belgrade University, Yugoslavia (1991), an NIH supported postdoctoral fellow at the University of Michigan, Ann Arbor (1992–93), and a research instructor at the Medical College of Ohio, Toledo (1993–94). He joined Texas Tech University Health Sciences Center as an Assistant Professor in 1994, and was appointed as an Adjunct Professor of Mathematics and Electrical and Computer Engineering at Texas Tech University in 1995. He was promoted to the Associate Professor level at Texas Tech University Health Sciences Center in 2000. While at Texas Tech, he received a number of research grant awards to conduct research in EEG signal processing and in biomedical signal/image compression, including from the Federal Administration on Aging, Alzheimers Association, and the US Civilian and Research Foundation (sponsored by the NSF and the US Department of State, to promote cooperative research between the wavelet theory groups in United States and ex-USSR). He is a Senior Member of IEEE and a past member of the New York Academy of Sciences. He is currently a Scientific Review Administrator in the Surgery, Biomedical Imaging, and Bioengineering integrated review group at the National Institutes of Health, US Department of Health and Human Services.     

Shape Estimation from Support and Diameter Functions

We address the problem of reconstructing a planar shape from a finite number of noisy measurements of its support function or its diameter function. New linear and non-linear algorithms are proposed, based on the parametrization of the shape by its Extended Gaussian Image. This parametrization facilitates a systematic statistical analysis of the problem via the Cramér-Rao lower bound (CRLB), which provides a fundamental lower bound on the performance of estimation algorithms. Using CRLB, we also generate confidence regions which conveniently display the effect of parameters like eccentricity, scale, noise, and measurement direction set, on the quality of the estimated shapes, as well as allow a performance analysis of the algorithms. Supported in part by U.S. National Science Foundation grants CCR-9984246 and DMS-0203527. Amyn Poonawala received the B.E. degree from the University of Mumbai, India, in 2001, and the M.S. degree from the University of California, Santa Cruz (UCSC), in 2004, both in computer engineering. He is currently pursuing the Ph.D. degree in computer engineering at UCSC. His technical interests include statistical signal and image processing and inverse problems in microlithography. Peyman Milanfar received the B.S. degree in electrical engineering/mathematics from the University of California, Berkeley, in 1988, and the S.M., E.E., and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology, in 1990, 1992, and 1993, respectively. Until 1999, he was a Senior Research Engineer at SRI International, Menlo Park, CA. He is currently Associate Professor of Electrical Engineering at the University of California, Santa Cruz. He was a Consulting Assistant Professor of computer science at Stanford University from 1998-2000, and a visiting Associate Professor there in 2002. His technical interests are in statistical signal and image processing, and inverse problems. He won a National Science Foundation CAREER award in 2000, was associate editor for the IEEE Signal Processing Letters from 1998 to 2001, and is a Senior member of the IEEE. Richard Gardner holds B.Sc. and Ph.D. degrees in mathematics from University College London and was awarded a D.Sc. degree from the University of London in 1988 for contributions to measure theory and convex geometry. He has held positions at universities and research institutions in several countries and has been Professor of Mathematics at Western Washington University since 1991. He founded geometric tomography, an area of geometric inverse problems involving data concerning sections by and projections on lines or planes, and published a book on the subject in 1995.     

On the Foundations of Vision Modeling III. Noncommutative Monoids of Occlusive Preimages

A significant cue for visual perception is the occlusion pattern in 2-D retinal images, which helps humans or robots navigate successfully in the 3-D environments. There have been many works in the literature on the modeling and analysis of the occlusion phenomenon, most of which are from the analytical or statistical points of view. The current paper presents a new theory of occlusion based on the simple topological definitions of preimages and a binary operation on them called “occlu.” We study numerous topological as well as algebraic structures of the resultant noncommutative preimage monoids (a monoid is a semigroup with identity). Some implications of the new theory in terms of real vision research are also addressed. Research is partially supported by NSF (USA) under the grant number DMS-0202565. Jianhong (Jackie) Shen received the Ph.D degree in Applied Mathematics from the Massachusetts Institute of Technology in 1998, and the B.S. degree from the University of Science and Technology of China (USTC) in 1994. He was a CAM (Computational and Applied Mathematics) Assistant Professor at UCLA from 1998 to 2000. He is currently an Assistant Professor of Applied Mathematics in the University of Minnesota, MN, USA. His current research interests include image, signal, and information processing, vision modeling and computation, as well as multiscale and stochastic modeling in medical and biological sciences. His new book: Image Processing and Analysis - variational, PDE, wavelets, and stochastic methods, coauthored with Prof. Tony F. Chan (Dean of Physical Sciences, UCLA), will be published by the SIAM (Soc. Ind. Appl. Math.) Publisher in September 2005. Most of his research and teaching activities could be found at .     

Automatic diatom identification using contour analysis by morphological curvature scale spaces

A method for automatic identification of diatoms (single-celled algae with silica shells) based on extraction of features on the contour of the cells by multi-scale mathematical morphology is presented. After extracting the contour of the cell, it is smoothed adaptively, encoded using Freeman chain code, and converted into a curvature representation which is invariant under translation and scale change. A curvature scale space is built from these data, and the most important features are extracted from it by unsupervised cluster analysis. The resulting pattern vectors, which are also rotation-invariant, provide the input for automatic identification of diatoms by decision trees and k-nearest neighbor classifiers. The method is tested on two large sets of diatom images. The techniques used are applicable to other shapes besides diatoms. Andrei C. Jalba received his B.Sc. (1998) and M.Sc. (1999) in Applied Electronics and Information Engineering from “Politehnica” University of Bucharest, Romania. He recently obtained a Ph.D. degree at the Institute for Mathematics and Computing Science of the University of Groningen, where he now is a postdoctoral researcher. His research interests include computer vision, pattern recognition, image processing, and parallel computing. Michael Wilkinson obtained an M.Sc. in astronomy from the Kapteyn Laboratory, University of Groningen (RuG) in 1993, after which he worked on image analysis of intestinal bacteria at the Department of Medical Microbiology, RuG. This work formed the basis of his Ph.D. at the Institute of Mathematics and Computing Science (IWI), RuG, in 1995. He was appointed as researcher at the Centre for High Performance Computing (also RuG) working on simulating the intestinal microbial ecosystem on parallel computers. During that time he edited the book “Digital Image Analysis of Microbes” (John Wiley, UK, 1998) together with Frits Schut. After this he worked as a researcher at the IWI on image analysis of diatoms. He is currently assistant professor at the IWI. Jos B.T.M. Roerdink received his M.Sc. (1979) in theoretical physics from the University of Nijmegen, the Netherlands. Following his Ph.D. (1983) from the University of Utrecht and a 2-year position (1983--1985) as a Postdoctoral Fellow at the University of California, San Diego, both in the area of stochastic processes, he joined the Centre for Mathematics and Computer Science in Amsterdam. There he worked from 1986-1992 on image processing and tomographic reconstruction. He was appointed associate professor (1992) and full professor (2003), respectively, at the Institute for Mathematics and Computing Science of the University of Groningen, where he currently holds a chair in Scientific Visualization and Computer Graphics. His current research interests include biomedical visualization, neuroimaging and bioinformatics. Micha Bayer graduated from St. Andrews University, Scotland, with an M.Sc. in Marine Biology in 1994. He obtained his Ph.D. in Marine Biology from there in 1998, and then followed this up with two postdoctoral positions at the Royal Botanic Garden Edinburgh, Scotland, first on the ADIAC and then on the DIADIST project. In both of these projects he was responsible for establishing the collections of diatom training data to be used for the pattern recognition systems. From 2002–2003 he was enrolled for an M.Sc. in information technology at the University of Glasgow, Scotland, and is now working as a grid developer at the National e-Science Centre at Glasgow University. Stephen Juggins is a senior lecturer at the School of Geography, Politics and Sociology, University of Newcastle. His research focuses on the use of diatoms for monitoring environmental change and on the analysis of ecological and palaeoecological data. He has worked in Europe, North America and Central Asia on problems of river water quality, historical lake acidification, coastal eutrophication and Quaternary climate change.     

Segmentation of a Vector Field: Dominant Parameter and Shape Optimization

Vector field segmentation methods usually belong to either of three classes: methods which segment regions homogeneous in direction and/or norm, methods which detect discontinuities in the vector field, and region growing or classification methods. The first two classes of method do not allow segmentation of complex vector fields and control of the type of fields to be segmented, respectively. The third class does not directly allow a smooth representation of the segmentation boundaries. In the particular case where the vector field actually represents an optical flow, a fourth class of methods acts as a detector of main motion. The proposed method combines a vector field model and a theoretically founded minimization approach. Compared to existing methods following the same philosophy, it relies on an intuitive, geometric way to define the model while preserving a general point of view adapted to the segmentation of potentially complex vector fields with the condition that they can be described by a finite number of parameters. The energy to be minimized is deduced from the choice of a specific class of field lines, e.g. straight lines or circles, described by the general form of their parametric equations. In that sense, the proposed method is a principled approach for segmenting parametric vector fields. The minimization problem was rewritten into a shape optimization and implemented by spline-based active contours. The algorithm was applied to the segmentation of precomputed optical flow fields given by an external, independent algorithm. Tristan Roy graduated from Ecole Centrale de Lille, France, in 2001. He is currently a Ph.D. student in mathematics at UCLA. His current research interests are variational analysis, optimization problems and PDEs. Fields of application are image segmentation and restoration. Eric Debreuve received his Ph.D. in Image Processing from the University of Nice-Sophia Antipolis, France, in 2000. He was a postdoctoral fellow at the Medical Imaging Research Laboratory (now UCAIR), University of Utah, Salt Lake City, for two years. He is currently a research scientist of the CNRS (a national research institute of France) at Laboratory I3S, University of Nice-Sophia Antipolis, France. His current research interests are image and video segmentation using active contours. Michel Barlaud received his These d'Etat from the University of Paris XII and Agregation de Physique (ENS Cachan). He is currently a Professor of Image Processing at the University of Nice-Sophia Antipolis, and the leader of the Image Processing group of I3S. His research topics are: Image and Video coding using Scan Based Wavelet Transform, Inverse problem using Half Quadratic Regularization and, Image and Video Segmentation using Region Based Active Contours and PDE's. He is a regular reviewer for several journals, a member of the technical committees of several scientific conferences. He leads several national research and development projects with French industries, and participates in several international academic collaborations: European Network of Excellence SCHEMA and SIMILAR (Louvain La Neuve (Belgium), ITI Greece, Imperial College …) and NSF-CNRS Funding (Universities of Stanford and Boston). He is the author of a large number of publications in the area of image and video processing, and the Editor of the book “Wavelets and Image Communication” Elsevier, 1994. Gilles Aubert received the These d'Etat es-sciences Mathematiques from the University of Paris 6, France, in 1986. He is currently professor of mathematics at the University of Nice-Sophia Antipolis and member of the J.A.Dieudonne Laboratory at Nice, France. His research interests are calculus of variations, nonlinear partial differential equations and numerical analysis; fields of application including image processing and, in particular, restoration, segmentation, optical flow and reconstruction in medical imaging.     

Preoperative planning of pelvic and lower limbs surgery by CT image processing

This paper outlines main functions of a system for preoperative planning of pelvic and lower limbs surgery. A generic scheme of operation planning using the system as well as the main system features and methods applied at the time of development are discussed. Eight steps of planning procedure using the presented system are described. The article is published in the original. Vasil Hancharenka. Born in 1979, graduated from Minsk State Higher Radioengineering College in 2002. 2002–2005: PhD student of the United Institute of Informatics Problems of the National Academy of Sciences of Belarus. At present: junior researcher at the United Institute of Informatics Problems of the National Academy of Sciences of Belarus. Professional interest: CT image processing, image segmentation, watershed transformation, development of systems for computer support in radiology and surgery. Alexander Tuzikov. Graduated from the Belarus State University (Minsk, 1980), received the candidate of physics-mathematical sciences degree of Institute of Mathematics (1985) and doctor of physics-mathematical sciences degree of the Institute of Engineering Cybernetics of the National Academy of Sciences of Belarus (2000). He is a Deputy General Director of the United Institute of Informatics Problems of the National Academy of Sciences of Belarus. His research subjects include image processing and analysis, medical imaging, stereo image reconstraction, mathematical morphology, discrete applied mathematics. Viachaslau Arkhipau. Born in 1982, graduated from Belarusian State University in 2006. At present he is a PhD student and junior researcher (part of time) in the United Institute of Informatics Problems of the National Academy of Sciences of Belarus. His interest is; medical image processing, image registration, tomography image visualization. In 1994 Aleh Kryvanos receved a dyploma at the Belarusian State University, Faculty of Mathematics and Informatics. In 2002 he defended a PhD thesis at the University of Mannheim, Germany. His field of specialization is medical Image Processing, Image Analysis, Operation Planning, Surgical Navigation, Image Restoration. His research activities are currently directed to medical equipment for diagnosing, surgery supporting, and archiving.     

Parallel algorithms for a hypercomplex discrete fourier transform

Methods for the parallel computation of a multidimensional hypercomplex discrete Fourier transform (HDFT) are considered. The basic idea consists in the application of the properties of the hypercomplex algebra in which this transform is performed. Additional possibilities for increasing the efficiency of the algorithm are provided by the natural parallelism of the multidimensional Cooley-Tukey scheme. Marat Vyacheslavovich Aliev. Born 1978. Graduated from the Adygeya State University in 2000. Received candidate’s degree in physics and mathematics in 2004. Presently he is a senior lecturer at the Department of Applied Mathematics and Information Technologies, Adygeya State University. Scientific interests: image processing, fractals, fast algorithms of discrete transforms, and finite-dimensional algebras. Author of 14 publications, including 7 papers. Member of the Russian Association of Pattern Recognition and Image Analysis. Aleksandr Mikhailovich Belov. Born 1980. Graduated from the Samara State Aerospace University in 2002. In the same year, he entered postgraduate courses with the specialty 05.13.18: mathematical modeling, numerical methods, and program complexes. Presently he is a postgraduate student at the Department of Geoinformatics, Samara State Aerospace University, and a trainee at the Laboratory of Mathematical Methods of Image Processing, Image Processing Systems Institute, Russian Academy of Sciences. Scientific interests: discrete orthogonal transforms, fast algorithms of discrete orthogonal transforms, and theory of canonical systems of calculus. Author of 13 publications, including 5 papers. Member of the Russian Association of Pattern Recognition and Image Analysis. Aleksei Vladimirovich Ershov. Born 1983. In 2000, he graduated from the Samara Lyceum of Economics and entered the Faculty of Mechanics and Mathematics, Samara State University, to specialize in the field of Organization and Technology of Information Security. In 2001, he started his training within an additional educational program and was qualified as a translator in the field of professional communication. Presently he is a fifth-year student at Samara State University. The title of his diploma work is “Control of the Flows of Confidential Information.” He is an active participant in the translation of the monograph Principia Mathematica, Cambridge University Press, 1927, by A. Whitehead and B. Russell. Author of four publications, including two papers. Marina Aleksandrovna Chicheva. Born 1964. Graduated from the Kuibyshev Aviation Institute (now Samara State Aerospace University) in 1987. Received candidate’s degree in Engineering in 1998. Presently she is a senior researcher at the Image Processing Systems Institute, Russian Academy of Sciences. Scientific interests: image processing, compression, and fast algorithms of discrete transforms. Author of more than 50 publications, including 18 papers and 1 monograph. Member of the Russian Association of Pattern Recognition and Image Analysis.