An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

We study an energy functional for computing optical flow that combines three assumptions: a brightness constancy assumption, a gradient constancy assumption, and a discontinuity-preserving spatio-temporal smoothness constraint. In order to allow for large displacements, linearisations in the two data terms are strictly avoided. We present a consistent numerical scheme based on two nested fixed point iterations. By proving that this scheme implements a coarse-to-fine warping strategy, we give a theoretical foundation for warping which has been used on a mainly experimental basis so far. Our evaluation demonstrates that the novel method gives significantly smaller angular errors than previous techniques for optical flow estimation. We show that it is fairly insensitive to parameter variations, and we demonstrate its excellent robustness under noise.

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High Accuracy Optical Flow Estimation Based on a Theory for Warping

We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

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A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

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Anisotropic diffusion in image processing

http://pancamr.lecture.ub.ac.id/files/2012/02/A-survey-on-vision-based-human-action-recognition_Image-and-Vision-Computing_2010.pdf

A survey on vision-based human action recognition

The updated 2nd edition of this book presents a variety of image analysis applications, reviews their precise mathematics and shows how to discretize them. For the mathematical community, the book shows the contribution of mathematics to this domain, and highlights unsolved theoretical questions. For the computer vision community, it presents a clear, self-contained and global overview of the mathematics involved in image procesing problems. The second edition offers a review of progress in image processing applications covered by the PDE framework, and updates the existing material. The book also provides programming tools for creating simulations with minimal effort.

Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations

1: Introduction 2: From Gaussian Convolution to Bilateral Filter 3: Applications 4: Efficient Implementation 5: Relationship between BF and Other Methods or Framework 6: Extensions of Bilateral Filtering 7: Conclusions. Acknowledgements. References.

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Bilateral filtering : theory and applications

Defined as the apparent motion in a sequence of images, the optical flow is very important in the computer vision community where its accurate estimation is necessary for many applications. It is o...

Computing optical flow via variational techniques

Image-based modeling and photo editing Oh et al. ACM SIGGRAPH conference (c) 2001, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/383259.383310 Fast bilateral filtering for the display of high-dynamic-range images Durand and Dorsey ACM SIGGRAPH conference (c) 2002, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/566570.566574 Bilateral mesh denoising Fleishman et al. ACM SIGGRAPH conference (c) 2003, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1201775.882368 Non-iterative, feature-preserving mesh smoothing Jones et al. ACM SIGGRAPH conference (c) 2003, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1201775.882367 Flash photography enhancement via intrinsic relighting Eisemann et al. ACM SIGGRAPH conference (c) 2004, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1186562.1015778 Digital photography with flash and no-flash image pairs Petschnigg et al. ACM SIGGRAPH conference (c) 2004, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1186562.1015777 Video enhancement using per-pixel virtual exposures Bennett and McMillan ACM SIGGRAPH conference (c) 2005, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1186822.1073272 Fast median and bilateral filtering Weiss ACM SIGGRAPH conference (c) 2006, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1179352.1141918 Real-time video abstraction Winnemoller et al. ACM SIGGRAPH conference (c) 2006, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1179352.1142018 Two-scale tone management for photographic look Bae et al. ACM SIGGRAPH conference (c) 2006, Association for Computing Machinery, Inc. Reprinted by permission. http://doi.acm.org/10.1145/1141911.1141935

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A gentle introduction to bilateral filtering and its applications

This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background Usually, motion segmentation and image restoration are considered separately in image sequence restoration Moreover, motion segmentation is often noise sensitive In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and vice-versa This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements The model is theoretically justified Existence and unicity are proved in the space of bounded variations A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated Experimental results obtained on noisy synthetic data and real images will illustrate the capabilities of this original and promising approach

Pierre Kornprobst

Papers

Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations

Bilateral filtering : theory and applications

Computing optical flow via variational techniques

A gentle introduction to bilateral filtering and its applications

Image Sequence Analysis via Partial Differential Equations