the stretch of a foliation and geometric superrigidity we consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type such objects can be considered as compact models for the geometry of the symmetric space based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions to achieve this we consider the heat flow equation along the leaves of a foliation a bochner formula on foliations and a geometric invariant for foliations with leafwise riemannian metrics called the stretch we obtain as applications a metric rigidity theorem for foliations and a rigidity type result for riemannian manifolds whose geometry is only partially symmetric

regularized bundle adjustment to model heads from image sequences without calibration data we address the structure from motion problem in the context of head modeling from video sequences for which calibration data is not available this task is made challenging by the fact that correspondences are difficult to establish due to lack of texture and that a quasi euclidean representation is required for realism we have developed an approach based on regularized bundle adjustment it takes advantage of our rough knowledge of the head s shape in the form of a generic face model it allows us to recover relative head motion and epipolar geometry accurately and consistently enough to exploit a previously developed stereo based approach to head modeling in this way complete and realistic head models can be acquired with a cheap and entirely passive sensor such as an ordinary video camera with minimal manual intervention we chose to demonstrate and evaluate our technique mainly in the context of head modeling we do so because it is the application for which all the tools required to perform the complete reconstruction are available to us we will however argue that the approach is generic and could be applied to other tasks such as body modeling for which generic facetized models exist

gauges and gauge transformations for uncertainty description of geometric structure with indeterminacy this paper presents a consistent theory for describing indeterminacy and uncertainty of three dimensional 3 d reconstruction from a sequence of images first we give a group theoretical analysis of gauges and gauge transformations we then discuss how to evaluate the reliability of the solution that has indeterminacy and extend the cramer rao lower bound to incorporate internal indeterminacy we also introduce the free gauge approach and define the normal form of a covariance matrix that is independent of particular gauges finally we show simulated and real image examples to illustrate the effect of gauge freedom on uncertainty description

uniqueness of nash equilibria in a quantum cournot duopoly game a quantum cournot game whose classical form game has multiple nash equilibria is examined although the classical equilibria fail to be pareto optimal the quantum equilibrium exhibits the following two properties i if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large the multiplicity of equilibria vanishes and ii the more strongly the strategic variables are entangled the more closely the unique equilibrium approaches to the optimal one

stabilizing the focal length computation for 3 d reconstruction from two uncalibrated views in order to reconstruct 3 d shape from two uncalibrated views one needs to resolve two problems i the computed focal lengths can be imaginary ii the computation fails for fixated images we present a practical remedy for these by subsampling feature points and fixing the focal length we first summarize theoretical backgrounds and then do simulations which reveal a rather surprising fact that when the focal length is actually fixed not using that knowledge yields better results for non fixated images we give an explanation to this seeming paradox and derive a hybrid method switching the computation by judging whether or not the images are fixated doing simulations and real image experiments we demonstrate the effectiveness of our method

generalised correlation for multi feature correspondence computing correspondences between pairs of images is fundamental to all structures from motion algorithms correlation is a popular method to estimate similarity between patches of images in the standard formulation the correlation function uses only one feature such as the gray level values of a small neighbourhood research has shown that different features such as colour edge strength corners texture measures work better under di0erent conditions we propose a framework of generalized correlation that can compute a real valued similarity measure using a feature vector whose components can be dissimilar the framework can combine the e0ects of di0erent image features such as multi spectral features edges corners texture measures etc into a single similarity measure in a 3exible manner additionally it can combine results of di0erent window sizes used for correlation with proper weighting for each relative importances of the features can be estimated from the image itself for accurate correspondence in this paper we present the framework of generalised correlation provide a few examples demonstrating its power as well as discuss the implementation issues 2002 published by elsevier science ltd on behalf of pattern recognition society

globally convergent autocalibration using interval analysis we address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed in general to address this problem we have developed a method where an interval branch and bound method is employed for numerical minimization thanks to the properties of interval analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy and the only input information it requires from the user are a set of point correspondences and a search interval the cost function is based on the huang faugeras constraint of the essential matrix a recently proposed interval extension based on bernstein polynomial forms has been investigated to speed up the search for the solution finally experimental results are presented

quasi perspective projection model theory and application to structure and motion factorization from uncalibrated image sequences this paper addresses the problem of factorization based 3d reconstruction from uncalibrated image sequences previous studies on structure and motion factorization are either based on simplified affine assumption or general perspective projection the affine approximation is widely adopted due to its simplicity whereas the extension to perspective model suffers from recovering projective depths to fill the gap between simplicity of affine and accuracy of perspective model we propose a quasi perspective projection model for structure and motion recovery of rigid and nonrigid objects based on factorization framework the novelty and contribution of this paper are as follows firstly under the assumption that the camera is far away from the object with small lateral rotations we prove that the imaging process can be modeled by quasi perspective projection which is more accurate than affine model from both geometrical error analysis and experimental studies secondly we apply the model to establish a framework of rigid and nonrigid factorization under quasi perspective assumption finally we propose an extended cholesky decomposition to recover the rotation part of the euclidean upgrading matrix we also prove that the last column of the upgrading matrix corresponds to a global scale and translation of the camera thus may be set freely the proposed method is validated and evaluated extensively on synthetic and real image sequences and improved results over existing schemes are observed

sequential updating of projective and affine structure from motion a structure from motion algorithm is described which recovers structure and camera position modulo a projective ambiguity camera calibration is not required and camera parameters such as focal length can be altered freely during motion the structure is updated sequentially over an image sequence in contrast to schemes which employ a batch process a specialisation of the algorithm to recover structure and camera position modulo an affine transformation is described together with a method to periodically update the affine coordinate frame to prevent drift over time we describe the constraint used to obtain this specialisation structure is recovered from image corners detected and matched automatically and reliably in real image sequences results are shown for reference objects and indoor environments and accuracy of recovered structure is fully evaluated and compared for a number of reconstruction schemes a specific application of the work is demonstrated affine structure is used to compute free space maps enabling navigation through unstructured environments and avoidance of obstacles the path planning involves only affine constructions

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

implicit meshes for effective silhouette handling using silhouettes in uncontrolled environments typically requires handling occlusions as well as changing or cluttered backgrounds which limits the applicability of most silhouette based methods for the purpose of 3 d shape modeling we show that representing generic 3 d surfaces as implicit surfaces lets us effectively address these issues this desirable behavior is completely independent from the way the surface deformations are parame trized to show this we demonstrate our technique in three very different cases modeling the deformations of a piece of paper represented by an ordinary triangulated mesh reconstruction and tracking a person s shoulders whose deformations are expressed in terms of dirichlet free form deformations reconstructing the shape of a human face parametrized in terms of a principal component analysis model

compact kahler manifolds with nonpositive bisectional curvature in this article we prove that for any compact kahler manifold mn with real analytic metric and nonpositive bisectional curvature there exists a finite cover m of m such that m is a holomorphic and metric fiber bundle over a compact kahler manifold n with nonpositive bisectional curvature and c1 n 0 and the fiber is a flat complex torus this partially confirms a conjecture of yau

camera calibration using symmetric objects this paper proposes a novel method for camera calibration using images of a mirror symmetric object assuming unit aspect ratio and zero skew we show that interimage homographies can be expressed as a function of only the principal point by minimizing symmetric transfer errors we thus obtain an accurate solution for the camera parameters we also extend our approach to a calibration technique using images of a 1 d object with a fixed pivoting point unlike existing methods that rely on orthogonality or pole polar relationship our approach utilizes new interimage constraints and does not require knowledge of the 3 d coordinates of feature points to demonstrate the effectiveness of the approach we present results for both synthetic and real images

an improved algorithm for two image camera self calibration and euclidean structure recovery using absolute quadric we present an improved algorithm for two image camera self calibration and euclidean structure recovery where the effective focal lengths of both cameras are assumed to be the only unknown intrinsic parameters by using the absolute quadric it is shown that the effective focal lengths can be computed linearly from two perspective images without imposing scene or motion constraints moreover a quadratic equation derived from the absolute quadric is proposed for solving the parameters of the plane at infinity from two images which upgrades a projective reconstruction to a euclidean reconstruction

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2

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

an efficient solution to the five point relative pose problem an efficient algorithmic solution to the classical five point relative pose problem is presented the problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points the algorithm consists of computing the coefficients of a tenth degree polynomial in closed form and subsequently finding its roots it is the first algorithm well suited for numerical implementation that also corresponds to the inherent complexity of the problem we investigate the numerical precision of the algorithm we also study its performance under noise in minimal as well as overdetermined cases the performance is compared to that of the well known 8 and 7 point methods and a 6 point scheme the algorithm is used in a robust hypothesize and test framework to estimate structure and motion in real time with low delay the real time system uses solely visual input and has been demonstrated at major conferences

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

on nonpositively curved compact riemannian manifolds with degenerate ricci tensor in this article we prove that if the ricci tensor of a compact nonpositively curved manifold m n is nowhere negative definite then it admits local flat factors to be more precise let u c m be the open subset where the ricci tensor has maximum rank r then for any connected component u a of u its universal cover u a is isometric to r n r n r a where n r a is a nonpositively curved manifold with negative ricci curvature in particular if m n is compact nonpositively curved without euclidean de rham factor and both the manifold and the metric are real analytic then its ricci tensor must be negative definite somewhere

social optimality in quantum bayesian games a b s t r a c t a significant aspect of the study of quantum strategies is the exploration of the gametheoretic solution concept of the nash equilibrium in relation to the quantization of a game pareto optimality is a refinement on the set of nash equilibria a refinement on the set of pareto optimal outcomes is known as social optimality in which the sum of players payoffs is maximized this paper analyzes social optimality in a bayesian game that uses the setting of generalized einstein podolsky rosen experiments for its physical implementation we show that for the quantum bayesian game a direct connection appears between the violation of bell s inequality and the social optimal outcome of the game and that it attains a superior socially optimal outcome

camera calibration from surfaces of revolution this paper addresses the problem of calibrating a pinhole camera from images of a surface of revolution camera calibration is the process of determining the intrinsic or internal parameters i e aspect ratio focal length and principal point of a camera and it is important for both motion estimation and metric reconstruction of 3d models in this paper a novel and simple calibration technique is introduced which is based on exploiting the symmetry of images of surfaces of revolution traditional techniques for camera calibration involve taking images of some precisely machined calibration pattern such as a calibration grid the use of surfaces of revolution which are commonly found in daily life e g bowls and vases makes the process easier as a result of the reduced cost and increased accessibility of the calibration objects in this paper it is shown that two images of a surface of revolution will provide enough information for determining the aspect ratio focal length and principal point of a camera with fixed intrinsic parameters the algorithms presented in this paper have been implemented and tested with both synthetic and real data experimental results show that the camera calibration method presented is both practical and accurate

geometric measures of entanglement and the schmidt decomposition in the standard geometric approach the entanglement of a pure state is sin 2 where is the angle between the entangled state and the closest separable state of products of normalized qubit states we consider here a generalization of this notion by considering separable states that consist of products of unnormalized states of different dimensions the distance between the target entangled state and the closest unnormalized product state can be interpreted as a measure of the entanglement of the target state the components of the closest product state and its norm have an interpretation in terms of respectively the eigenvectors and eigenvalues of the reduced density matrices arising in the schmidt decomposition of the state vector for several cases where the target state has a large degree of symmetry we solve the system of equations analytically and look specifically at the limit where the number of qubits is large

stochastic approximation and rate distortion analysis for robust structure and motion estimation recent research on structure and motion recovery has focused on issues related to sensitivity and robustness of existing techniques one possible reason is that in practical applications the underlying assumptions made by existing algorithms are often violated in this paper we propose a framework for 3d reconstruction from short monocular video sequences taking into account the statistical errors in reconstruction algorithms detailed error analysis is especially important for this problem because the motion between pairs of frames is small and slight perturbations in its estimates can lead to large errors in 3d reconstruction we focus on the following issues physical sources of errors their experimental and theoretical analysis robust estimation techniques and measures for characterizing the quality of the final reconstruction we derive a precise relationship between the error in the reconstruction and the error in the image correspondences the error analysis is used to design a robust recursive multi frame fusion algorithm using stochastic approximation as the framework since it is capable of dealing with incomplete information about errors in observations rate distortion analysis is proposed for evaluating the quality of the final reconstruction as a function of the number of frames and the error in the image correspondences finally to demonstrate the effectiveness of the algorithm examples of depth reconstruction are shown for different video sequences

on the geometric rank of homogeneous spaces of nonpositive curvature in this paper we investigate differential geometric properties of homogeneous hadamard manifolds and prove a rigidity result in this class of spaces a hadamard manifold h is a simply connected complete riemannian manifold of nonpositive curvature if in addition h is homogeneous then by the work of all aw1 aw2 and hei we know that h is isometric to a solvable lie group s a n with lie algebra s of type nc see aw1 and a left invariant metric here a exp s s denotes an abelian complement to the nilpotent commutator subgroup n exp s s assuming that h has a trivial flat de rham factor the number k dim a only depends on h whereas s need not be uniquely determined and is called the aloebraic rank of h k is an upper bound for the oeometric rank r rk h dr3 i e the maximal integer such that every geodesic is contained in a r fiat a totally geodesic copy of in h the geometric meaning of rk m is well understood for a huge class of hadamard manifolds m thanks to the following remarkable result by ballmann and burns spatzier we quote a generalisation from e3 and eh

generalized quantization scheme for two person non zero sum games we proposed a generalized quantization scheme for non zero sum games which can be reduced to the two existing quantization schemes under an appropriate set of parameters some other important situations are identified which are not apparent in the two existing quantization schemes

quantum stackelberg duopoly in the presence of correlated noise we study the influence of entanglement and correlated noise using correlated amplitude damping depolarizing and phase damping channels on the quantum stackelberg duopoly our investigations show that under the influence of an amplitude damping channel a critical point exists for an unentangled initial state at which firms get equal payoffs the game becomes a follower advantage game when the channel is highly decohered two critical points corresponding to two values of the entanglement angle are found in the presence of correlated noise within the range of these limits of the entanglement angle the game is a follower advantage game in the case of a depolarizing channel the payoffs of the two firms are strongly influenced by the memory parameter the presence of quantum memory ensures the existence of the nash equilibrium for the entire range of decoherence and entanglement parameters for both the channels a local maximum in the payoffs is observed which vanishes as the channel correlation increases moreover under the influence of the depolarizing channel the game is always a leader advantage game furthermore it is seen that the phase damping channel does not affect the outcome of the game

self calibration and metric reconstruction inspite of varying and unknown intrinsic camera parameters in this paper the theoretical and practical feasibility of self calibration in the presence of varying intrinsic camera parameters is under investigation the paper s main contribution is to propose a self calibration method which efficiently deals with all kinds of constraints on the intrinsic camera parameters within this framework a practical method is proposed which can retrieve metric reconstruction from image sequences obtained with uncalibrated zooming focusing cameras the feasibility of the approach is illustrated on real and synthetic examples besides this a theoretical proof is given which shows that the absence of skew in the image plane is sufficient to allow for self calibration a counting argument is developed which depending on the set of constraints gives the minimum sequence length for self calibration and a method to detect critical motion sequences is proposed

quantum game theory based on the schmidt decomposition we present a novel formulation of quantum game theory based on the schmidt decomposition which has the merit that the entanglement of quantum strategies is manifestly quantified we apply this formulation to 2 player 2 strategy symmetric games and obtain a complete set of quantum nash equilibria apart from those available with the maximal entanglement these quantum nash equilibria are extensions of the nash equilibria in classical game theory the phase structure of the equilibria is determined for all values of entanglement and thereby the possibility of resolving the dilemmas by entanglement in the game of chicken the battle of the sexes the prisoners dilemma and the stag hunt is examined we find that entanglement transforms these dilemmas with each other but cannot resolve them except in the stag hunt game where the dilemma can be alleviated to a certain degree

visual modeling with a hand held camera in this paper a complete system to build visual models from camera images is presented the system can deal with uncalibrated image sequences acquired with a hand held camera based on tracked or matched features the relations between multiple views are computed from this both the structure of the scene and the motion of the camera are retrieved the ambiguity on the reconstruction is restricted from projective to metric through self calibration a flexible multi view stereo matching scheme is used to obtain a dense estimation of the surface geometry from the computed data different types of visual models are constructed besides the traditional geometry and image based approaches a combined approach with view dependent geometry and texture is presented as an application fusion of real and virtual scenes is also shown

appearance based structure from motion using linear classes of 3 d models in this paper we address the problem of recovering 3 d models from sequences of partly calibrated images with unknown correspondence to that end we integrate tracking structure from motion with geometric constraints specifically in the form of linear class models in a single framework the key to making the proposed approach work is the use of appearance based model matching and refinement which updates the estimated correspondences on each iteration of the algorithm another key feature is the matching of a 3 d model directly with the input images without the conventional 2 step approach of stereo data recovery and 3 d model fitting initialization of the linear class model to one of the input images the reference image is currently partly manual this synthesis and refine approach or appearance based constrained structure from motion abcsfm is especially useful in recovering shapes of objects whose general structure is known but which may have little discernable texture in significant parts of their surfaces we applied the proposed approach to 3 d face modeling from multiple images to create new 3 d faces for decface a synthetic talking head developed at cambridge research laboratory digital equipment corporation the decface model comprises a collection of 3 d triangular and rectangular facets with nodes as vertices in recovering the decface model we assume that the sequence of images is taken with a camera with unknown focal length and pose the geometric constraints used are of the form of linear combination of prototypes of 3 d faces of real people results of this approach show its good convergence properties and its robustness against cluttered backgrounds

adjusting shape parameters using model based optical flow residuals we present a method for estimating the shape of a deformable model using the least squares residuals from a model based optical flow computation this method is built on top of an estimation framework using optical flow and image features where optical flow affects only the motion parameters of the model using the results of this computation our new method adjusts all of the parameters so that the residuals from the flow computation are minimized we present face tracking experiments that demonstrate that this method obtains a better estimate of shape compared to related frameworks

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

constructing quantum games from nonfactorizable joint probabilities a probabilistic framework is developed that gives a unifying perspective on both the classical and quantum versions of two player games we suggest exploiting peculiar joint probabilities involved in einstein podolsky rosen epr experiments to construct a quantum game when the corresponding classical game is obtained from factorizable joint probabilities we analyze how nonfactorizability changes nash equilibria in three well known games of prisoner s dilemma stag hunt and chicken in this framework we find that for the game of prisoner s dilemma even nonfactorizable epr joint probabilities cannot be helpful to escape from the classical outcome of the game for a particular version of the chicken game however we find that the two nonfactorizable sets of joint probabilities which maximally violate the clauser holt shimony horne sum of correlations indeed result in new nash equilibria