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Computer Vision is a rapidly growing field of research investigating computational and algorithmic issues associated with image acquisition, processing, and understanding. It serves tasks like manipulation, recognition, mobility, and communication in diverse application areas such as manufacturing, robotics, medicine, security and virtual reality. This volume contains a selection of papers devoted to theoretical foundations of computer vision covering a broad range of fields, e.g. motion analysis, discrete geometry, computational aspects of vision processes, models, morphology, invariance, image compression, 3D reconstruction of shape. Several issues have been identified to be of essential interest to the community: non-linear operators; the transition between continuous to discrete representations; a new calculus of non-orthogonal partially dependent systems.

Inhaltsverzeichnis

Frontmatter

Attentive Visual Motion Processing: Computations in the Log-Polar Plane

Abstract
Attentive Visual Motion Processing: Computations in the Log-Polar Plane. Attentive vision is characterized by selective sensing in space and time as well as selective processing with respect to a specific task. Selection in space involves the splitting of the visual field in a high resolution area—the fovea—and a space-variant resolution area—the periphery. Both in neurobiology and in robot vision, models of the resolution decrease towards the image boundaries have been established. The most convincing model is the theory of log-polar mapping where very high data compression rates are achieved. In combination with the complexity reduction we believe that the log-polar mapping has further computational advantages which we elaborate in this study. Based on the optical flow we study the computation of 3D motion and structure globally and locally. We present a global method to compute the Focus of Expansion in the case of pure translation. By fixating on an object we show how to estimate ego motion in the presence of translation and rotation of the observer from the flow in the log-polar periphery. Then, we turn to local differential computations and we establish both approximate and exact expressions for the time to collision.
K. Daniilidis

Invariant Thinning and Distance Transform

Abstract
Invariant Thinning and Distance Transform. Thinning is a preprocessing method which is applied to binary (i.e. black-and-white) digital (i.e. discretized) images. The goal of thinning is to reduce the sets of black points in the image to “thin” sets while retaining the “topology” of them as well as “form” properties. Usually thinning methods are organized in an iterative way by “peeling off” outer layers of the sets under consideration. This implies that thinning is an extremely time-consuming task. Recently, Neusius and Olszewski [12] proposed a thinning method which is based on a distance transform. This idea is indeed not new (see e.g. [6]), but Neusius and Olszewski were the first to treat it in a systematic way. Since the distance transform can be calculated efficiently by a two-sweep method, this approach looks attractive. The aim of this paper is to show that under certain assumptions a ‘classical’ thinning method [4], which has invariance with respect to motions as a distinctive feature, also can be interpreted as a distance-transform-based method.
U. Eckhardt, L. Latecki

Recognition of Images Degraded by Linear Motion Blur without Restoration

Abstract
Recognition of Images Degraded by Linear Motion Blur without Restoration. The paper is devoted to the feature-based description of images degraded by linear motion blur. The proposed features are invariant with respect to motion velocity, are based on image moments and are calculated directly from the blurred image. In that way, we are able to describe the original image without the PSF identification and image restoration. In many applications (such as in image recognition against a database) our approach is much more effective than the traditional “blind-restoration” one. The derivation of the motion blur invariants is a major theoretical result of the paper. Numerical experiments are presented to illustrate the utilization of the invariants for blurred image description. Stability of the invariants with respect to additive random noise is also discussed and is shown to be sufficiently high. Finally, another set of features which are invariant not only to motion velocity but also to motion direction is introduced.
J. Flusser, T. Suk, S. Saic

Symmetric Bi- and Trinocular Stereo: Tradeoffs between Theoretical Foundations and Heuristics

Abstract
Symmetric Bi- and Trinocular Stereo: Tradeoffs between Theoretical Foundations and Heuristics. Tradeoffs between theoretical and heuristic sides of ill-posed problems of the intensity-based computational stereo are discussed, as applied to the previously proposed symmetric approach for solving this problem. The heuristics are needed to deal with discontinuities in stereo images due to partial occlusions of observed surface. Basically, it is these discontinuities that cause the ill-posedness of the stereo problems. Theoretical base of the symmetric stereo is refined here by introducing a novel probabilistic model of the surface geometry and by deducing compound Bayesian decision rules to be implemented by dynamic programming techniques, as in the case of simple MAP-decision with the maximum a posteriori probability. Also, several heuristics are proposed to regularize the binocular stereo.
G. L. Gimel’farb

Surface from Motion—without and with Calibration

Abstract
Surface from Motion—without and with Calibration. For uncalibrated and calibrated imaging situations, results on “surface from motion” are given in a systematic order, also described by an abbreviating rule notation. A few new theoretical results for surface from motion are included. Experimental evaluations of reconstruction steps are sketched for some case studies (as optical flow computation, integrative approach to shape from motion, or surface from motion using calibration).
R. Klette

Properties of Pyramidal Representations

Abstract
Properties of Pyramidal Representations. The categorization of different components generalizes the classical concept of image pyramids and provides a powerful tool for efficient image analysis. Three aspects of image pyramids are distinguished: their structure, the contents of their cells and the processes that operate on them. The properties of these three aspects of a pyramidal system are discussed and illustrated by examples. The general theory covers the most recent results, e.g. structure preserving irregular pyramids, sigmoid pyramids, the concept of equivalent interpretation, and the fuzzy curve pyramid.
W. G. Kropatsch

A Robust Approach to Estimation of Parametric Models

Abstract
A Robust Approach to Estimation of Parametric Models. This article presents a robust method for estimation of parametric models. The method consists of two procedures: model-recovery and model-selection. The model-recovery procedure systematically recovers a redundant set of parametric models in a local-to-global fashion, iteratively combining data classification and parameter estimation. The model-selection procedure, defined as a quadratic Boolean problem, then searches for the subset of the recovered models which produce the simplest global description. To achieve a computationally efficient method the model-recovery and the model-selection are combined in an iterative way. The main features of the method are a high degree of resistance to outliers and the insensitivity to incorrect initial estimates. The method has been successfully applied to linear as well as nonlinear parameter estimation problems, e.g. for recovering variable-order bivariate polynomials and superquadric models in range images, and parametric curve models in edge images.
A. Leonardis

Computer Vision and Mathematical Morphology

Abstract
Computer Vision and Mathematical Morphology. Mathematical morphology as originally developed by Matheron and Serra is a theory of set mappings, modeling binary image transformations, which are invariant under the group of Euclidean translations. This framework turns out to be too restricted for many applications, in particular for computer vision where group theoretical considerations such as behavior under perspective transformations and invariant object recognition play an essential role. So far, symmetry properties have been incorporated by assuming that the allowed image transformations are invariant under a certain commutative group. This can be generalized by dropping the assumption that the invariance group is commutative. To this end we consider an arbitrary homogeneous space (the plane with the Euclidean translation group is one example, the sphere with the rotation group another), i.e. a set X on which a transitive but not necessarily commutative transformation group Г is defined. As our object space we then take the Boolean algebra P(X) of all subsets of this homogeneous space. Generalizations of dilations, erosions, openings and closings are defined and several representation theorems can be proved. We outline some of the limitations of mathematical morphology in its present form for computer vision and discuss the relevance of the generalizations discussed here.
J. B. T. M. Roerdink

A Variational Approach to the Design of Early Vision Algorithms

Abstract
A Variational Approach to the Design of Early Vision Algorithms. A mathematical model for the design of early vision processing stages is presented. The model comprises a “generic” class of abstract minimization problems from which specific nonlinear diffusion processes can be derived for various kinds of visual data. Each smoothing process results in a one-parameter family of segmentations of the underlying domain and thus provides a basis for the segmentation of “general” scenes. A wide range of numerical realizations can be implemented using standard Galerkin discretization. Theoretically, each algorithm can also be implemented as a globally convergent network using analog VLSI-hardware. The approach is illustrated by deriving nonlinear diffusion schemes for the processing of greyvalue data and for the processing of locally computed image motion data.
C. Schnörr, R. Sprengel, B. Neumann

Banach Constructor and Image Compression

Abstract
Banach Constructor and Image Compression. The Banach constructor is defined as a concept unifying special cases of deterministic fractal modeling. The fractal compression of digital images is presented as a Banach constructor defined by a patchwork. The patchwork concept is a formal mathematical model which allowed for: a compact definition of the fractal operator, specification of a condition for its contractivity (for all v norms, 1 ≤ v ≤ ∞), and formulating conditions ensuring the required fidelity of the reconstructed image. Fast fractal compression algorithm (FFC) is based on patchworks which are affine (with contrast and scaling fixed), sparse, and local. Formulas for the best fit, affine, contrast fixed transforms which perform the best fit of two digital patches, are given for v norms with v = 1, v = 2, and v = ∞. Experiments confirm superiority of quadratic norm at quality-time tradeoff.
W. Skarbek

Piecewise Linear Approximation of Planar Jordan Curves and Arcs: Theory and Applications

Abstract
Piecewise Linear Approximation of Planar Jordan Curves and Arcs: Theory and Applications. Piecewise linear approximation of planar Jordan curves and arcs based on the shortest polygonal path in a polygonally bounded compact set and on the geodesic diameter in a polygon is described and applications involving gridding techniques are highlighted.
F. Sloboda, B. Zat’ko

Segmentation with Volumetric Part Models

Abstract
Segmentation with Volumetric Part Models. Volumetric models are top-level shape representation in computer vision applications. Volumetric models are especially suited for part-level representation on which manipulation, recognition and other reasoning can be based. The two most popular types of volumetric models in computer vision are generalized cylinders and superquadrics. This paper gives an overview of recovery and segmentation methods applying these two types of volumetric models. Methods of segmentation into parts are analyzed and advantageous properties of part-models discussed.
F. Solina

Theoretical Foundations of Anisotropic Diffusion in Image Processing

Abstract
Theoretical Foundations of Anisotropic Diffusion in Image Processing. A frequent problem in low-level vision consists of eliminating noise and small-scale details from an image while still preserving or even enhancing the edge structure. Nonlinear anisotropic diffusion filtering may be one possibility to achieve these goals. The objective of the present paper is to review the author’s results on a scale-space interpretation of a class of diffusion filters which comprises also several nonlinear anisotropic models. It is demonstrated that these models—which use an adapted diffusion tensor instead of a scalar diffusivity—offer advantages over isotropic filters. Most of the restoration and scale-space properties carry over from the continuous to the discrete case. Applications are presented ranging from preprocessing of medical images and postprocessing of fluctuating numerical data to visualizing quality relevant features for the grading of wood surfaces and fabrics.
J. Weickert

Stability and Likelihood of Views of Three Dimensional Objects

Abstract
Stability and Likelihood of Views of Three Dimensional Objects. Evaluating the representative power of two dimensional images of three dimensional objects has come up in a number of applications, mostly concerning 3D object recognition. We address the question of image characterization independently of these applications. We first introduce the basic concepts of view stability and likelihood for general objects. The stability and likelihood functions are then used to quantitatively characterize the repre-sentability of images. For objects composed of localized features, we develop explicit expressions through which the stability and likelihood functions of any view of a general object can be evaluated from its three principal second moments. This permits a quantitative characterization of the viewing sphere of objects. By way of qualitative analysis we compute the most stable and most likely views, which can identify the characteristic views of objects. We show that the most stable and most likely views of an object are the same and are often unique. This view is the “flattest” view of the object, obtained when the three dimensional object has its minimal spread along the viewing direction. We demonstrate these results using images of familiar objects.
D. Weinshall, M. Werman, N. Tishby

Backmatter

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