self calibration of a 1d projective camera and its application to the self calibration of a 2d projective camera we introduce the concept of self calibration of a 1d projective camera from point correspondences and describe a method for uniquely determining the two internal parameters of a 1d camera based on the trifocal tensor of three 1d images the method requires the estimation of the trifocal tensor which can be achieved linearly with no approximation unlike the trifocal tensor of 2d images and solving for the roots of a cubic polynomial in one variable interestingly enough we prove that a 2d camera undergoing planar motion reduces to a 1d camera from this observation we deduce a new method for self calibrating a 2d camera using planar motions both the self calibration method for a 1d camera and its applications for 2d camera calibration are demonstrated on real image sequences

understanding positioning from multiple images it is possible to recover the three dimensional structure of a scene using only correspondences between images taken with uncalibrated cameras faugeras 1992 the reconstruction obtained this way is only defined up to a projective transformation of the 3d space however this kind of structure allows some spatial reasoning such as finding a path in order to perform more specific reasoning or to perform work with a robot moving in euclidean space euclidean or affine constraints have to be added to the camera observations such constraints arise from the knowledge of the scene location of points geometrical constraints on lines etc first this paper presents a reconstruction method for the scene then it discusses how the framework of projective geometry allows symbolic or numerical information about positions to be derived and how knowledge about the scene can be used for computing symbolic or numerical relationships implementation issues and experimental results are discussed

two way ambiguity in 2d projective reconstruction from three uncalibrated 1d images we show that there is in general a two way ambiguity for 2d projective reconstruction from three uncalibrated 1d views independent of the number of point correspondences the two distinct projective reconstructions are exactly related by a quadratic transformation with the three camera centers as fundamental points unique 2d reconstruction is possible only when the three camera centers are aligned by carlsson duality 1995 there is a dual two way ambiguity for 2d projective reconstruction from six point correspondences independent of the number of 1d views the theoretical results are demonstrated on numerical examples

oscillation of certain second order nonlinear differential equations abstract in this paper we consider the second order nonlinear differential equation formula where 0 is any quotient of odd integers a c 1 r 0 q c r r f c r r xf x 0 f x 0 for x 0 some new sufficient conditions for the oscillation of all solutions of are obtained several examples that dwell upon the importance of our results are also included

places of algebraic function fields in arbitrary characteristic we consider the zariski space of all places of an algebraic function field f k of arbitrary characteristic and investigate its structure by means of its patch topology we show that certain sets of places with nice properties e g prime divisors places of maximal rank zero dimensional discrete places lie dense in this topology further we give several equivalent characterizations of fields that are large in the sense of f pop s annals paper embedding problems over large fields we also study the question whether a field k is existentially closed in an extension field l if l admits a k rational place in the appendix we prove the fact that the zariski space with the zariski topology is quasi compact and that it is a spectral space

kronecker function rings and abstract riemann surfaces abstract let r be an integral domain x r the abstract riemann surface of r and r b the kronecker function ring of the integral closure r of r it is proved that there exists a homeomorphism natural in r between x r and spec r b ideal theoretic and topological results are given for the extension j r r b notably that r is a prufer domain if and only if r r and j is universally going down it is also proved that each spectral space x is a closed spectral image of a treed spectral space y if x is irreducible y can be taken as an abstract riemann surface

1d camera geometry and its application to the self calibration of circular motion sequences this paper proposes a novel method for robustly recovering the camera geometry of an uncalibrated image sequence taken under circular motion under circular motion all the camera centers lie on a circle and the mapping from the plane containing this circle to the horizon line observed in the image can be modelled as a 1d projection a 2times2 homography is introduced in this paper to relate the projections of the camera centers in two 1d views it is shown that the two imaged circular points of the motion plane and the rotation angle between the two views can be derived directly from such a homography this way of recovering the imaged circular points and rotation angles is intrinsically a multiple view approach as all the sequence geometry embedded in the epipoles is exploited in the estimation of the homography for each view pair this results in a more robust method compared to those computing the rotation angles using adjacent views only the proposed method has been applied to self calibrate turntable sequences using either point features or silhouettes and highly accurate results have been achieved

geometry of single axis motions using conic fitting previous algorithms for recovering 3d geometry from an uncalibrated image sequence of a single axis motion of unknown rotation angles are mainly based on the computation of two view fundamental matrices and three view trifocal tensors we propose three new methods that are based on fitting a conic locus to corresponding image points over multiple views the main advantage is that determining only five parameters of a conic from one corresponding point over at least five views is simpler and more robust than determining a fundamental matrix from two views or a trifocal tensor from three views it is shown that the geometry of single axis motion can be recovered either by computing one conic locus and one fundamental matrix or by computing at least two conic loci a maximum likelihood solution based on this parametrization of the single axis motion is also described for optimal estimation using three or more loci the experiments on real image sequences demonstrate the simplicity accuracy and robustness of the new methods

affine structure from line correspondences with uncalibrated affine cameras this paper presents a linear algorithm for recovering 3d affine shape and motion from line correspondences with uncalibrated affine cameras the algorithm requires a minimum of seven line correspondences over three views the key idea is the introduction of a one dimensional projective camera this converts 3d affine reconstruction of line directions into 2d projective reconstruction of points in addition a line based factorization method is also proposed to handle redundant views experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

weighted means and oscillation conditions for second order matrix differential equations on considere le systeme d equations differentielles d ordre 2 p t y t q t y t 0 pour t 0 ou p et q sont des matrices symetriques n n reelles sur 0 et y un vecteur colonne on suppose que p t est definie positive pour tout t tel que p t 1 on donne des conditions d oscillations

integer valued polynomials and prufer v multiplication domains abstract let d be a domain with quotient field k we consider the ring int d f k x f d d of integer valued polynomial rings over d we completely characterize the domains d for which int d is a prufer v multiplication domain

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

oscillation and nonoscillation for second order linear differential equations abstract new oscillation and nonoscillation theorems are obtained for the second order linear differential equation u p t u 0 where p t c 0 and p t 0 conditions only about the integrals of p t on every interval 2 n t 0 2 n 1 t 0 n 1 2 for some fixed t 0 0 are used in the results

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

radial multi focal tensors the 1d radial camera maps all points on a plane containing the principal axis onto the radial line which is the intersection of that plane and the image plane it is a sufficiently general model to express both central and non central cameras since the only assumption it makes is of known center of distortion in this paper we study the multi focal tensors arising out of 1d radial cameras there exist no two view constraints like the fundamental matrix for 1d radial cameras however the 3 view and 4 view cases are interesting for the 4 view case we have the radial quadrifocal tensor which has 15 d o f and 2 internal constraints for the 3 view case we have the radial trifocal tensor which has 7 d o f and no internal constraints under the assumption of a purely rotating central camera this can be used to do a non parametric estimation of the radial distortion of a 1d camera even in the case of a non rotating camera it can be used to do parametric estimation assuming a planar scene finally we examine the mixed trifocal tensor which models the case of two 1d radial cameras and one standard pin hole camera of the above radial multifocal tensors only the radial trifocal tensor is useful practically since it doesn t require any knowledge of the scene and is extremely robust we demonstrate results based on real images for this for the quadrifocal tensor too we present a way to do a metric reconstruction of the scene and to undistort the image given a sufficiently dense set of point correspondences we also show results on synthetic images however it must be noted that currently the quadrifocal and mixed trifocal tensors are useful only from a theoretical stand point

self calibration of a moving camera from pointcorrespondences and fundamental matrices we address the problem of estimating three dimensional motion and structure from motion with an uncalibrated moving camera we show that point correspondences between three images and the fundamental matrices computed from these point correspondences are sufficient to recover the internal orientation of the camera its calibration the motion parameters and to compute coherent perspective projection matrices which enable us to reconstruct 3 d structure up to a similarity in contrast with other methods no calibration object with a known 3 d shape is needed and no limitations are put upon the unknown motions to be performed or the parameters to be recovered as long as they define a projective camera the theory of the method which is based on the constraint that the observed points are part of a static scene thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic is first detailed several algorithms are then presented and their performances compared by means of extensive simulations and illustrated by several experiments with real images

the tree of quadratic transforms of a regular local ring of dimension two abstract let d be a 2 dimensional regular local ring and let q d denote the quadratic tree of 2 dimensional regular local overrings of d we explore the topology of the tree q d and the family r d of rings obtained as intersections of rings in q d if a is a finite intersection of rings in q d then a is noetherian and the structure of a is well understood however other rings in r d need not be noetherian the two main goals of this paper are to examine topological properties of the quadratic tree q d and to examine the structure of rings in the set r d

irredundant intersections of valuation overrings of two dimensional noetherian domains abstract let d be a two dimensional noetherian domain let r be an overring of d and let and be collections of valuation overrings of d we consider circumstances under which v v r w w r implies that we show that if r is integrally closed these representations are strongly irredundant and every member of has krull dimension 2 then if in addition and are noetherian subspaces of the zariski riemann space of the quotient field of d e g if and have finite character then the restriction that the members of have krull dimension 2 can be omitted an example shows that these results do not extend to overrings of three dimensional noetherian domains

circular motion geometry using minimal data circular motion or single axis motion is widely used in computer vision and graphics for 3d model acquisition this paper describes a new and simple method for recovering the geometry of uncalibrated circular motion from a minimal set of only two points in four images this problem has been previously solved using nonminimal data either by computing the fundamental matrix and trifocal tensor in three images or by fitting conics to tracked points in five or more images it is first established that two sets of tracked points in different images under circular motion for two distinct space points are related by a homography then we compute a plane homography from a minimal two points in four images after that we show that the unique pair of complex conjugate eigenvectors of this homography are the image of the circular points of the parallel planes of the circular motion subsequently all other motion and structure parameters are computed from this homography in a straightforward manner the experiments on real image sequences demonstrate the simplicity accuracy and robustness of the new method

oscillation criteria for second order nonlinear perturbed differential equations abstract we present new oscillation criteria for the second order nonlinear perturbed differential equations these criteria are of a high degree of generality and they extend and unify a number of existing results

interval oscillation criteria for second order nonlinear perturbed differential equations abstract we present new interval oscillation criteria for certain classes of second order nonlinear perturbed differential equations which are based on the information only on a sequence of subintervals of t o rather than on the whole half line our results are sharper than those known in the literature and deal with the cases which are not covered by the known criteria in particular our results can be applied to extreme cases such as q s ds finally two examples which dwell upon the sharpness of the conditions of our results are also illustrated

the constructible topology on spaces of valuation domains we consider properties and applications of a compact hausdorff topology called the ultrafilter topology defined on an arbitrary spectral space and we observe that this topology coincides with the constructible topology if k is a field and a a subring of k we show that the space zar k a of all valuation domains having k as quotient field and containing a endowed with the zariski topology is a spectral space by giving in this general setting the explicit construction of a ring whose zariski spectrum is homeomorphic to zar k a we extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings as a very particular case we prove that two collections of valuation domains of k with the same ultrafilter closure represent as an intersection the same integrally closed domain

elementary properties of minimal and maximal points in zariski spectra we investigate connections between arithmetic properties of rings and topological properties of their prime spectrum any property that the prime spectrum of a ring may or may not have defines the class of rings whose prime spectrum has the given property we ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense answers are provided for a variety of different properties of prime spectra e g normality or complete normality hausdorffness of the space of maximal points compactness of the space of minimal points

reconstruction from projections using grassmann tensors in this paper a general procedure is given for reconstruction of a set of feature points in an arbitrary dimensional projective space from their projections into lower dimensional spaces this extends the methods applied in the well studied problem of reconstruction of scene points in 3 given their projections in a set of images in this case the bifocal trifocal and quadrifocal tensors are used to carry out this computation it is shown that similar methods will apply in a much more general context and hence may be applied to projections from n to m which have been used in the analysis of dynamic scenes and in radial distortion correction for sufficiently many generic projections reconstruction of the scene is shown to be unique up to projectivity except in the case of projections onto one dimensional image spaces lines in which case there are two solutions projections from n to 2 have been considered by wolf and shashua in international journal of computer vision 48 1 53 67 2002 where they were applied to several different problems in dynamic scene analysis they analyzed these projections using tensors but no general way of defining such tensors and computing the projections was given this paper settles the general problem showing that tensor definition and retrieval of the projections is always possible

a principal ideal theorem for compact sets of rank one valuation rings abstract let f be a field and let zar f be the space of valuation rings of f with respect to the zariski topology we prove that if x is a quasicompact set of rank one valuation rings in zar f whose maximal ideals do not intersect to 0 then the intersection of the rings in x is an integral domain with quotient field f such that every finitely generated ideal is a principal ideal to prove this result we develop a duality between a quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to 0 and b one dimensional prufer domains with nonzero jacobson radical and quotient field f the necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to 0 is motivated by settings in which the valuation rings considered all dominate a given local ring

interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations abstract we present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are sharper than some previous results and handle the cases which are not covered by known criteria in particular several examples that dwell upon the sharp conditions of our results are also included

motion from point matches multiple of solutions in this paper we study the multiplicity of solutions of the motion problem given n point matches between two frames how many solutions are there to the motion problem we show that the maximum number of solutions is 10 when 5 point matches are available this settles a question that has been around in the computer vision community for a while we follow two tracks we show that the two approaches are compatible and yield the same result we then describe a computer implementation of the second approach that uses maple a language for symbolic computation the program allows us to compute exactly the solutions for any configuration of 5 points some experiments are described

oscillation criteria for second order nonlinear perturbed differential equations abstract some criteria are obtained which guarantee that all solutions of the second order nonlinear perturbed differential equations r t y t p t y t q t y t h t y t y t are oscillatory

algebraic functions for recognition in the general case a trilinear relationship between three perspective views is shown to exist the trilinearity result is shown to be of much practical use in visual recognition by alignment yielding a direct reprojection method that cuts through the computations of camera transformation scene structure and epipolar geometry moreover the direct method is linear and sets a new lower theoretical bound on the minimal number of points that are required for a linear solution for the task of reprojection the proof of the central result may be of further interest as it demonstrates certain regularities across homographics of the plane and introduces new view invariants experiments on simulated and real image data were conducted including a comparative analysis with epipolar intersection and the linear combination methods with results indicating a greater degree of robustness in practice and a higher level of performance in reprojection tasks

a visual feedback based time optimal motion policy for capturing an unpredictable evader in this paper we address the pursuit evasion problem of capturing an unpredictable omnidirectional evader using a differential drive robot ddr in an obstacle free environment we present three main contributions 1 we provide a state feedback based time optimal motion policy for the ddr the motion policy is based on a partition of the state space one main contribution of this paper is to provide algebraic equations of the regions boundaries of this partition in terms of the state space coordinates 2 we estimate the state of the evader based on images using the one dimensional trifocal tensor we propose a new formulation of the estimation of the evader s state relative to the pursuer 3 we present a bound for conventional cameras over the pursuer s field of view that guarantees that if the evader is initially visible it will remain visible inside the camera s view regardless of its motion strategy until the capture condition is achieved we also present an implementation of the pursuer

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

the strong ultrafilter topology on spaces of ideals abstract the patch constructible refinement of the zariski topology on the prime spectrum of a commutative ring is well known and well studied recently fontana and loper gave an equivalent definition of this topology using ultrafilters in this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring we study various properties of these topologies as applications we use the ultrafilters to classify all the compact subsets of a spectral space in the zariski topology and we classify grothendieck s retrocompact spaces again using ultrafilters

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

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

outward looking circular motion analysis of large image sequences this work presents a novel and simple method of analyzing the motion of a large image sequence captured by a calibrated outward looking video camera moving on a circular trajectory for large scale environment applications previous circular motion algorithms mainly focus on inward looking turntable like setups they are not suitable for outward looking motion where the conic trajectory of corresponding points degenerates to straight lines the circular motion of a calibrated camera essentially has only one unknown rotation angle for each frame the motion recovery for the entire sequence computes only one fundamental matrix of a pair of frames to extract the angular motion of the pair using laguerre s formula and then propagates the computation of the unknown rotation angles to the other frames by tracking one point over at least three frames finally a maximum likelihood estimation is developed for the optimization of the whole sequence extensive experiments demonstrate the validity of the method and the feasibility of the application in image based rendering

some topological considerations on semistar operations abstract we consider properties and applications of a new topology called the zariski topology on the space sstar a of all the semistar operations on an integral domain a we prove that the set of all overrings of a endowed with the classical zariski topology is homeomorphic to a subspace of sstar a the topology on sstar a provides a general theory through which we see several algebraic properties of semistar operation as very particular cases of our construction moreover we show that the subspace sstar f a of all the semistar operations of finite type on a is a spectral space

geometrically invariant watermarking using feature points this paper presents a new approach for watermarking of digital images providing robustness to geometrical distortions the weaknesses of classical watermarking methods to geometrical distortions are outlined first geometrical distortions can be decomposed into two classes global transformations such as rotations and translations and local transformations such as the stirmark attack an overview of existing self synchronizing schemes is then presented theses schemes can use periodical properties of the mark invariant properties of transforms template insertion or information provided by the original image to counter geometrical distortions thereafter a new class of watermarking schemes using the image content is presented we propose an embedding and detection scheme where the mark is bound with a content descriptor defined by salient points three different types of feature points are studied and their robustness to geometrical transformations is evaluated to develop an enhanced detector the embedding of the signature is done by extracting feature points of the image and performing a delaunay tessellation on the set of points the mark is embedded using a classical additive scheme inside each triangle of the tessellation the detection is done using correlation properties on the different triangles the performance of the presented scheme is evaluated after jpeg compression geometrical attack and transformations results show that the fact that the scheme is robust to these different manipulations finally in our concluding remarks we analyze the different perspectives of such content based watermarking scheme