Binary Operations in the Unit Ball: A Differential Geometry Approach

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space

Robust ... Analysis and Control of Linear Systems With Integral Quadratic Constraints

In this paper, we consider a class of uncertain linear systems which are subject to a general type of integral quadratic constraints (IQCs). Two problems are addressed: 1) robust H1 analysis and 2) robust H1 control. In the first problem, we determine if the system satisfies a desired H1 performance for all admissible uncertainties subject to the IQCs. In the second problem, we seek for a dynamic output feedback controller to achieve a desired robust H1 performance. We apply the wellknown S-procedure and show that these two problems can be effectively solved using linear matrix inequalities (LMIs). 1 Introduction This paper addresses two problems: robust H1 analysis and robust H1 control of a class of linear systems which are subject to an energy bounded (or L 2 bounded) exogenous input and several uncertainties involving the so-called integral quadratic constraints (IQCs). In the robust H1 analysis problem we determine the worst-case energy (or the induced L 2 norm) at an output, w..

Neural net based image matching

The paper describes a neural-based method for matching spatially distorted image sets. The matching of partially overlapping images is important in many applications - integrating information from images formed from different spectral ranges, detecting changes in a scene and identifying objects of differing orientations and sizes. Our approach consists of extracting contour features from both images, describing the contour curves as sets of line segments, comparing these sets, determining the corresponding curves and their common reference points, calculating the image-to-image co-ordinate transformation parameters on the basis of the most successful variant of the derived curve relationships. The main steps are performed by custom neural networks. The algorithms described in this paper have been successfully tested on a large set of images of the same terrain taken in different spectral ranges, at different seasons and rotated by various angles. In general, this experimental verification indicates that the proposed method for image fusion allows the robust detection of similar objects in noisy, distorted scenes where traditional approaches often fail

Neural net based image matching

The paper describes a neural-based method for matching spatially distorted image sets. The matching of partially overlapping images is important in many applications - integrating information from images formed from different spectral ranges, detecting changes in a scene and identifying objects of differing orientations and sizes. Our approach consists of extracting contour features from both images, describing the contour curves as sets of line segments, comparing these sets, determining the corresponding curves and their common reference points, calculating the image-to-image co-ordinate transformation parameters on the basis of the most successful variant of the derived curve relationships. The main steps are performed by custom neural networks. The algorithms described in this paper have been successfully tested on a large set of images of the same terrain taken in different spectral ranges, at different seasons and rotated by various angles. In general, this experimental verification indicates that the proposed method for image fusion allows the robust detection of similar objects in noisy, distorted scenes where traditional approaches often fail.</p