Advances in Computer Vision PDF
Preview Advances in Computer Vision
Advances in Computing Science Advisory Board R. F. Albrecht, Innsbruck Z. Bubnicki, Wroclaw R. E. Burkard, Graz A. G. Butkovskiy, Moscow C. H. Cap, Zurich M. Dal Cin, Erlangen W. Hackbusch, Kiel G. R. Johnson, Fort Collins W. Kreutzer, Christchurch W. G. Kropatsch, Wien 1. Lovrek, Zagreb G. Nemeth, Budapest H. J. Stetter, Wien Y. Takahara, Chiba W. Tornig, Darmstadt 1. Troch, Wien H. P. Zima, Wien F. Solina W. G. Kropatsch R. Klette R. Bajcsy (eds.) Advances in Computer Vision SpringerWienNe wYork Prof. Dr. Franc Solina Faculty of Computer and Infonnation Science, University of Ljubljana, Ljubljana, Slovenia Prof. Dr. Walter G. Kropatsch Abt. Mustererkennung und Bildverarbeitung, TV Wien, Vienna, Austria Prof. Dr. Reinhard Klette Computer Science Department, Auckland University, Auckland, New Zealand Prof. Dr. Ruzena Bajcsy Computer and Infonnation Science, GRASP Laboratory, University of Pennsylvania, Philadelphia, USA This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. © 1997 Springer-VerlaglWien Typesetting: Camera-ready by authors Graphic design: Ecke Bonk Printed on acid-free and chlorine-free bleached paper SPIN: 10631594 With 96 Figures ISSN 1433-0113 ISBN-13:978-3-211-83022-2 e-ISBN-13:978-3-7091-6867-7 DOl: 10.1007/978-3-7091-6867-7 Preface Computer vision used to be a rather small and exclusive research area fo cused mainly on theoretical issues and on solving problems for large and wealthy costumers such as manufacturing companies and the military who could afford the high costs. But expensive special image capture and com puting hardware is now no longer required. Low cost video cameras, power ful personal computers and highspeed computer networks are making images ubiquitous in every possible application domain. Besides more traditional application domains such as manufacturing, robotics, medicine and security, newer ones such as virtual reality, tele-presence and image databases I'l-re in vogue. Computer vision solutions used to be very specific and difficult to adapt to other or even unforeseen situations. The current development is calling for simple to use yet robust applications that could be employed in various situations. This trend requires the reassessment of some theo retical issues in computer vision. A better general understanding of vision processes, new insights and better theories are needed. This volume contains a selection of papers presented at the eight overall "Theoretical Foundations of Computer Vision" meeting and the second in the castle of Dagstuhl in March 1996. The aim of this meeting was to bring together scientists in computer vision from the West and from the former eastern block countries. The organizers believed that there was still a certain ignorance of each other's work and that such face to face meetings are beneficial to all participants and to the whole computer vision field. The organizers feel that this goal was achieved and that the road to more direct contacts and exchanges between researchers and students is now open. Due to this goal the meeting covered a broad variety of computer vi sion topics. As the title of the meeting suggests most of the papers have a strong theoretical flavor but with some very real world implications. It was not easy to organize them in a linear fashion. The volume starts with pa pers dealing with 2D images (scale space, morphology, segmentation, neural networks, Hough transform, texture, pyramids) followed by papers on recov ering the 3D structure (shape from shading, optical flow, 3D object recogni tion). Finally, the last few papers are on how vision is integrated into a larger task-driven framework (hand-eye calibration, navigation, perception-action cycle). March 1997 Franc Solina, Walter G. Kropatsch, Reinhard Klette, Ruzena Bajcsy Contents Joachim Weickert and Brahim Benhamouda A semi discrete nonlinear scale-space theory and its relation to the Perona-Malik paradox .................................................. 1 Ulrich Eckhardt and Eckart Hundt Topological approach to mathematical morphology .................... 11 Jos B.T.M. Roerdink and Arnold Meijster Segmentation by watersheds: definition and parallel implementation ... 21 Herbert Jahn A graph network for image segmentation .............................. 31 Wladyslaw Skarbek Associative memory for images by recurrent neural subnetworks ....... 41 Matevz Kovacic, Bojan Kverh and Franc Solina Optimal models for visual recognition ................................. 51 Atsushi Imiya Order of points on a line segment ..................................... 61 Souheil Ben-Yacoub Subjective contours detection ......................................... 71 Dmitry Chetverikov Texture feature based interaction maps: potential and limits ........... 79 Georgy L. Gimel'Farb Non-Markov Gibbs image model with almost local pairwise pixel interactions ........................................................... 89 Walter G. Kropatsch Equivalent contraction kernels to build dual irregular pyramids ........ 99 Christophe Duperthuy and Jean-Michel Jolion Towards a generalized primal sketch .................................. 109 Jean-Michel Jolion Categorization through temporal analysis of patterns .................. 119 VITI Vito Di Gesu and Cesare Valenti Detection of regions of interest via the Pyramid Discrete Symmetry 'Transform ........................................................... 129 Andreas Koschan and Volker Rodehorst Dense depth maps by active color illumination and image pyramids ... 137 Karsten Schluns and Reinhard Klette Local and global integration of discrete vector fields .................. 149 Vladimir A. Kovalevsky A new approach to shape from shading ...... : ........................ 159 Ryszard Kozera Recent uniqueness results in shape from shading ...................... 169 John L. Barron and Roy Eagleson Computation of time-varying motion and structure parameters from real image sequences ................................................. 181 Steven S. Beauchemin and John L. Barron A theory of occlusion in the context of optical flow ................... 191 Michail Schlesinger Algebraic method for solution of some best matching problems ....... 201 Michael Schubert, Klaus Voss Determining the attitude of planar objects with general curved contours from a single perspective view ............................... 211 Bjorn Krebs and Friedrich M. Wahl CAD based 3D object recognition on range images ................... 221 Konstantinos Daniilidis Dual quaternions for absolute orientation and hand-eye calibration .... 231 Ruzena Bajcsy, Henrik I. Christensen and Jana Kosecka Segmentation of behavioral spaces for navigation tasks ............... 241 Gerald Sommer, Eduardo Bayro-Corrochano and Thomas Bulow Geometric algebra as a framework for the perception-action cycle ..... 251 List of contributors ................................................... 261 A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox Joachim Weickert and Brahim Benhamouda 1 Introduction Although much effort has been spent in the recent decade to establish a theoret ical foundation of certain partial differential equations (PDEs) as scale-spaces, it is almost never taken into account that, in practice, images are sampled on a fixed pixel gridl . For nonlinear PDE-based filters, usually straightforward finite difference discretizations are applied in the hope that they reflect the nice prop erties of the continuous equations. Since scale-spaces cannot perform better than their numerical discretizations, however, it would be desirable to have a genuinely discrete nonlinear framework which reflects the discrete nature of digital images. In this paper we discuss a semi discrete scale-space framework for nonlinear diffu sion filtering. It keeps the scale-space idea of having a continuous time parameter, while taking into account the spatial discretization on a fixed pixel grid. It leads to nonlinear systems of coupled ordinary differential equations. Conditions are established under which one can prove existence of a stable unique solution which preserves the average grey level. An interpretation as a smoothing scale-space transformation is introduced which is based on an extremum principle and the ex istence of a large class of Lyapunov functionals comprising for instance p-norms, even central moments and the entropy. They guarantee that the process is not only simplifying and information-reducing, but also converges to a constant image as the scale parameter t tends to infinity. This semi discrete framework gives an answer to one of the central problems related to nonlinear diffusion scale spaces: the surprising practical success of the Perona-Malik (PM) filter in spite of its theoretical doubtfulness (Perona-Malik paradox [13]). Recently Kichenassamy [12, 13] has made significant contributions to the understanding of this phenomenon for the continuous PM equation. In our paper we contrast these results by applying our (semi-)discrete theory for explaining this effect. In particular, we shall see that the PM equation - whose continuous formulation is generally regarded to be ill-posed - leads to a well posed semidiscrete scale-space. Within its stability range, an explicit (Euler forward) time discretization inherits these semi discrete well-posedness and scale space properties to the fully discrete setting. Moreover, we prove that its 1-D variant is monotonicity preserving. Thus, a sigmoid-like edge cannot develop oscillations, and the practically observed instabilities are restricted to staircasing effects. The paper is organized as follows: Section 2 explains the continuous PM filter and presents an m-dimensional semi discrete formulation. In Section 3 we discuss a lOne exception is Lindeberg's semi discrete linear diffusion scale-space [14]. F. Solina et al. (eds.), Advances in Computer Vision © Springer-Verlag/Wien 1997 2 semidiscrete well-posedness and scale-space theory for nonlinear diffusion filters, which we apply in Section 4 for establishing well-posedness of the semi discrete PM scale-space. Section 5 is devoted to fully discrete results, especially the proof of monotonicity preservation. The paper concludes with a summary in Section 6. 2 The Perona-Malik filter 2.1 Continuous formulation We consider an m-dimensional rectangular image domain n = (0, ad x ... x (0, am) with boundary an, and a (grey-value) image which is given by a bounded mapping f : n -t JR. In order to avoid the blurring and localization problems of linear diffusion filtering, Perona and Malik proposed a nonlinear diffusion method [16]. Their nonuniform process (which they name anisotropic2) reduces the diffusivity at those locations which have a larger likelihood to be edges, since they reveal larger gradients. Perona and Malik obtain a filtered image u(x, t) as solution of a nonlinear diffusion equation with the original image as initial condition and reflecting boundary conditions (an denotes the derivative normal to the image boundary an): n OtU = div (g(lV'uI2) V'u) on x (0,00), (1) u(x, 0) = f(x) on n, (2) ° an OnU = on x (0,00). (3) Among the diffusivities they propose is3 2 1 g(lV'ul ) = 1 + lV'ul2 / A2 (A> 0). (4) The experiments of Perona and Malik were visually impressive [16]: edges re mained stable over a very long time. It was demonstrated that edge detection based on this process clearly outperforms the linear Canny edge detector. However, the PM approach reveals some serious problems: It is not hard to see [16] that the PM equation is of forward parabolic type only for lV'ul ~ A. Regions with lV'ul > A are identified as edges, where it may act like a backward diffusion equation across the edge. The forward-backward diffusion behaviour is explicitly intended in the PM method, since it gives the desirable result of blurring small fluctuations and sharpening edges. On the other hand, backward diffusion is well-known to be an ill-posed process where the solution (if it exists) is highly sensitive even to the slightest perturba tions of the initial data. 2In our terminology the PM filter is regarded as an isotropic model, since it uses a scalar valued diffusivity and not a diffusion tensor. For models with a diffusion tensor, see e.g. [19,20]. 3For smoothness reasons we write g(lV'uI2) instead of g(lV'ul). 3 The current understanding of the PM process is not complete, but there is very much theoretical and practical evidence that such forward-backward processes are ill-posed as well [18, 6, 10, 15, 4, 2, 7, 3, 17]. As one possibility to under stand the behaviour of this process it has been suggested to study regularizing approximations where the regularization parameter tends to zero [17, 8, 9]. In this field, however, conjectures are still dominating over established convergence results. Recently Kichenassamy [12, 13] proved that the PM filter does not even have weak solutions. He introduced a notion of generalized solutions to the PM process, which are piecewise linear and contain jumps, and he analyzed their moving and merging. The current opinion is that, for these solutions, one should neither expect uniqueness nor stability with respect to the initial image [4, 17, 13]. Interestingly, all practically observed instabilities are less severe than one would expect from theory: The main observed instability in simple implementations is the so-called staircasing effect, where a smoothed step edge evolves into piecewise linear segments which are separated by jumps. Contributions to the explanation and avoidance of staircasing can be found in [23, 3, 1, 5, 12, 13]. In particular, it should be noted that Kichenassamy's generalized solutions are in accordance with these numerical results. The staircasing effect, however, is mainly visible for fine spatial discretizations and for slowly varying ramp-like edges. Under practical situations, this is hardly observed, and it is an experimental fact that discretizations of the PM are not very unstable. To find an explanatic;)fi for this phenomenon, let us now investigate a spatially discretized version of this process. 2.2 Semidiscrete formulation A discrete m-dimensional image can be regarded as a vector f E lRN , whose components Ii, i E J := {I, ... , N} display the grey values at the N pixels. Pixel i represents the location Xi. Let hi denote the grid-size in 1 direction. By Ui and gi we denote approximations to U(Xi' t) and g(IV'U(Xi' t)12), respectively. Then, a consistent4 spatial discretization of the PM equation with reflecting boundary conditions can be written as (5) where Ni,(i) consist of the two neighbours of pixel i along the direction 1 (bound ary pixels may have only one neighbour) and L L gi := 9 ( "12 m (UP_2hUl q) 2) (6) 1=1 p,qEN,(i) 4The originally in [16] proposed scheme is not consistent, which may cause severQ deviations from rotational invariance, as can be seen in [15].
The list of books you might like
