A theory of self-calibration of a moving camera

There is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement. The epipolar transformation imposes two algebraic constraints on the camera calibration. If two epipolar transformations, arising from different camera displacements, are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four. The curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces, or by a curve of degree six in the dual of the image plane. The curve in the dual plane has one singular point of order three and three singular points of order two.If three epipolar transformations are available, then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration. The two curves have a common singular point of order three.Experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation. The real intersections of the two dual curves are found by locating the zeros of a function defined on the interval [0, 2]. The intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations.     

Motion of points and lines in the uncalibrated case

In the present paper we address the problem of computing structure and motion, given a set point and/or line correspondences, in a monocular image sequence, when the camera is not calibrated.Considering point correspondences first, we analyse how to parameterize the retinal correspondences, in function of the chosen geometry: Euclidean, affine or projective geometry. The simplest of these parameterizations is called the FQs-representation and is a composite projective representation. The main result is that considering N+1 views in such a monocular image sequence, the retinal correspondences are parameterized by 11 N–4 parameters in the general projective case. Moreover, 3 other parameters are required to work in the affine case and 5 additional parameters in the Euclidean case. These 8 parameters are calibration parameters and must be calculated considering at least 8 external informations or constraints. The method being constructive, all these representations are made explicit.Then, considering line correspondences, we show how the the same parameterizations can be used when we analyse the motion of lines, in the uncalibrated case. The case of three views is extensively studied and a geometrical interpretation is proposed, introducing the notion of trifocal geometry which generalizes the well known epipolar geometry. It is also discussed how to introduce line correspondences, in a framework based on point correspondences, using the same equations.Finally, considering the F Qs-representation, one implementation is proposed as a motion module, taking retinal correspondences as input, and providing and estimation of the 11 N–4 retinal motion parameters. As discussed in this paper, this module can also estimate the 3D depth of the points up to an affine and projective transformation, defined by the 8 parameters identified in the first section. Experimental results are provided.     

Neural mass activity, bifurcations, and epilepsy

In this letter, we propose a general framework for studying neural mass models defined by ordinary differential equations. By studying the bifurcations of the solutions to these equations and their sensitivity to noise, we establish an important relation, similar to a dictionary, between their behaviors and normal and pathological, especially epileptic, cortical patterns of activity. We then apply this framework to the analysis of two models that feature most phenomena of interest, the Jansen and Rit model, and the slightly more complex model recently proposed by Wendling and Chauvel. This model-based approach allows us to test various neurophysiological hypotheses on the origin of pathological cortical behaviors and investigate the effect of medication. We also study the effects of the stochastic nature of the inputs, which gives us clues about the origins of such important phenomena as interictal spikes, interictal bursts, and fast onset activity that are of particular relevance in epilepsy.     

Editorial

International Journal of Computer Vision -     

A Unifying and Rigorous Shape from Shading Method Adapted to Realistic Data and Applications

We propose a new method for the Lambertian Shape From Shading (SFS) problem based on the notion of Crandall-Lions viscosity solution. This method has the advantage of requiring the knowledge of the solution (the surface to be reconstructed) only on some part of the boundary and/or of the singular set (the set of the points at maximal intensity). Moreover it unifies in an unique mathematical formulation the works of Rouy et al. [34, 50], Falcone et al. [21], Prados et al. [46, 48, 49], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [17] dealing with classical solutions and value functions. Also, it allows to generalize their results to the “perspective SFS” problem recently simultaneously introduced in [13,46,55]. While the theoretical part has been developed in [44], in this paper we give some stability results and we describe numerical schemes for the SFS based on this method. We construct provably convergent and robust algorithms. Finally, we apply our SFS method to real images and we suggest some real-life applications.     

New signal processing methods and information technologies for the real time control of JET reactor relevant plasmas

A general trend in the experimental programmes of present day Tokamaks, and of JET in particular, is the constant increase in the number of parameters to be controlled in real time, to satisfy the machine protection requirements on the one hand and to improve performance on the other. Since the amount of data collected is also increasing at least at a rate compatible with the Moore law, significant developments are required in the field of real time algorithms particularly for magnetic reconstructions, disruption prediction and image processing. A new real time equilibrium code called EQUINOX, using internal and external measurements of the magnetic fields, has been qualified on JET. It can provide reconstructed accurate equilibria about every 50 ms on a 2 GHz PC. An advanced disruption predictor, based on machine learning tools, has been deployed using inputs selected with a genetic algorithm. Its success rate remains of the order of 94% for up to 170 ms before the occurrence of the disruption. Nonextensive entropies, which are more sensitive to long range correlations, seem to be useful in detecting vibrations in the videos of JET cameras, both visible and infrared.     

The Critical Sets of Lines for Camera Displacement Estimation: A Mixed Euclidean-Projective and Constructive Approach

The problem of the recovery of the motion, and the structure from motion is relevant to many computer vision applications. Many algorithms have been proposed to solve this problem. Some of these use line correspondences. For obvious practical reasons, it is important to study the limitation of such algorithms. In this paper, we are concerned with the problem of recovering the relative displacements of a camera by using line matches in three views. In particular, we want to know whether there exist sets of 3D lines such that no matter how many lines we observe there will always be several solutions to the relative displacement estimation problem. Such sets of lines may be called critical in the sense that they defeat the corresponding algorithm. This question has been studied in detail in the case of point matches by early-century Austrian photogrammeters and, independently, in the mid-seventies and early-eighties by computer vision scientists. The answer lies in the idea of a critical surface.The case of lines has been much less studied. Recently, Buchanan (1992a, 1992b) provided a first analysis of the problem in which he gave a positive answer: there exist critical sets of lines and they are pretty big (² lines). In general these sets are algorithm dependent, for example the critical set of lines for the Liu-Huang algorithm introduced in (Buchanan, 1992a), but Buchanan has shown that there is a critical set that defeats any algorithm. This paper is an attempt to build on his work and extend it in several directions. First, we cast his purely projective analysis in a more euclidean framework better suited to applications and, currently, more familiar to most of the computer vision community. Second, we clearly relate his critical set to those of previously published algorithms, in particular (Liu and Huang, 1988a, 1988b). Third, we provide an effective, i.e., computational, approach for describing these critical sets in terms of simple geometric properties. This has allowed us to scrutinize the structure of the critical sets which we found to be both intricate and beautiful.