Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

A method is proposed for determining the motion of a body relative to a fixed environment using the changing image seen by a camera attached to the body. The optical flow in the image plane is the input, while the instantaneous rotation and translation of the body are the output. If optical flow could be determined precisely, it would only have to be known at a few places to compute the parameters of the motion. In practice, however, the measured optical flow will be somewhat inaccurate. It is therefore advantageous to consider methods which use as much of the available information as possible. We employ a least-squares approach which minimizes some measure of the discrepancy between the measured flow and that predicted from the computed motion parameters. Several different error norms are investigated. In general, our algorithm leads to a system of nonlinear equations from which the motion parameters may be computed numerically. However, in the special cases where the motion of the camera is purely translational or purely rotational, use of the appropriate norm leads to a system of equations from which these parameters can be determined in closed form. 1.

Solutions to the structure from motion problem have been shown to be very sensitive to measurement noise and the respective motion and geometry configuration. Statistical error analysis has become an invaluable tool in analyzing the sensitivity phenomenon. This paper presents a unifying approach to the problems of statistical bias, correlated noise, choice of error metrics, geometric instabilities and information fusion exploring several assumptions commonly used in motion estimation and reviews several promising techniques for motion estimation. The techniques are based on a small number of principles of statistics and perturbation theory. The analyticity of the approach enables the design of alternatives overcoming the observed instabilities.

Conics are widely accepted as one of the most fundamental image features together with points and line segments. The problem of space reconstruction and correspondence of two conics from two views is addressed in this paper. It is shown that there are two independent polynomial conditions on the corresponding pair of conics across two views, given the relative orientation of the two views. These two correspondence conditions are derived algebraically and one of them is shown to be fundamental in establishing the correspondences of conics. A unified closed-form solution is also developed for both projective reconstruction of conics in space from two views for uncalibrated cameras and metric reconstruction from calibrated cameras. Experiments are conducted to demonstrate the discriminality of the correspondence conditions and the accuracy and stability of the reconstruction both for simulated and real images. Keywords--- conic, stereo correspondence, reconstruction. I. Introduction In...

This paper studies the error sensitivity in the estimation of the 3D-motion and the normal of a planar surface from an instantaneous motion field. We use the statistical theory of the Cramer-Rao lower bound for the error covariance in the estimated motion and structure parameters which enables the derivation of results valid for any unbiased estimator under the assumption of Gaussian noise in the motion eld. The obtained lower-bound-matrix is studied analytically with respect to the measurement noise, size of the field of view and the motion-geometry configuration. The main result of this analysis is the coupling between translation and rotation which is exacerbated if the field of view and the slant of the plane become smaller and the deviation of the translation from the viewing direction becomes larger. By-products of this study are the relationships of the uncertainty bounds for every unknown motion parameter to the angle between translation and the plane-normal, the size of the field of view, the distance from the perceived plane and the translation magnitude.

An algorithm is described which rapidly verifies the potential rigidity of three dimensional point correspondences from a pair of two dimensional views under perspective projection. The output of the algorithm is a simple yes or no answer to the question "Could these corresponding points from two views be the projection of a rigid configuration?" Potential applications include 3D object recognition from a single previous view and correspondence matching for stereo or motion over widely separated views. Our analysis begins with the observation that it is often the case that two views cannot provide an accurate structure-frommotion estimate because of ambiguity and ill-conditioning. However, it is argued that an accurate yes/no answer to the rigidity question is possible and experimental results support this assertion with as few as six pairs of corresponding points over a wide range of scene structures and viewing geometries. Rigidity checking verifies point correspondences by using 3D ...

Abstract. We consider the self-calibration problem in the special context of a stereo head, where the two cameras are arranged on a lateral rig with coplanar optical axes, each camera being free to vary its angle of vergence. Under various constraints, we derive explicit forms for the epipolar equation, and show that a static stereo head constitutes a degenerate camera con guration for carrying out self-calibration in the sense of Hartley [4]. The situation is retrieved by consideration of a special kind of motion of the stereo head in which the baseline remains con ned to a plane. New closed-form solutions for self-calibration are thereby obtained, inspired by an earlier discrete motion analysis of Zhang et al. [11]. Key factors in our approach are the development of explicit, analytical forms of the fundamental matrix, and the use of the vergence angles in the parameterisation of the problem. Keywords: Self-calibration, stereo head, degeneracy, epipolar equation, fundamental matrix, ego-motion. 1

This paper shows how to upgrade the projective reconstruction of a scene to a metric one in the case where the only assumption made about the cameras observing that scene is that they have rectangular pixels (zero-skew cameras). The proposed approach is based on a simple characterization of zero-skew projection matrices in terms of line geometry, and it handles zero-skew cameras with arbitrary or known aspect ratios in a unified framework. The metric upgrade computation is decomposed into a sequence of linear operations, including linear leastsquares parameter estimation and eigenvalue-based symmetric matrix factorization, followed by an optional non-linear least-squares refinement step. A few classes of critical motions for which a unique solution cannot be found are spelled out. A MATLAB implementation has been constructed and preliminary experiments with real data are presented.

This paper examines the robustness of correspondence-based approaches to structure from motion. Unlike earJier studies it is proven in an algorithmindependent way, that small absolute errors in image displacements cause absolute errors in rotational motion parameters significant eno ugh to lead to large relative errors in the determination of environmental depth. Even if the exact motion parameters are known a priori small errors in image displacements still lead to large errors in depth for environmental points whose distance from the camera is greater than a few multiples of the total trans- ]ation in depth of the camera. In order to clarify the many issues on robustness that are raised in this paper, a new depth determination algorithm is developed and applied to dynamic image sequences of natural outdoor scen es

Abstract—There exists a large conceptual gap between symbolic models and emergent models for the mind. Many emergent models work on low-level sensory data, while many symbolic models deal with high-level abstract (i.e., action) symbols. There has been relatively little study on intermediate representations, mainly because of a lack of knowledge about how representations fully autonomously emerge inside the closed brain skull, using information from the exposed two ends (the sensory end and the motor end). As reviewed here, this situation is changing. A fundamental challenge for emergent modelsisabstraction,which symbolic models enjoy through human handcrafting. The term abstract refers to properties disassociated with any particular form. Emergent abstraction seems possible, although the brain appears to never receive a computer symbol (e.g., ASCII code) or produce such a symbol. This paper reviews major agent models with an emphasis on representation. It suggests two different ways to relate symbolic representations with emergent representations: One is based on their categorical definitions. The other considers that a symbolic representation corresponds to a brain’s outside behaviors observed and handcrafted by other outside human observers; but an emergent representation is inside the brain. Index Terms—Agents, attention, brain architecture, complexity, computer vision, emergent representation, graphic models, mental