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 homographies 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 re-projection tasks. Keywords--- Visual Recognition, Al...

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call relative affine structure. Via a number of corollaries of our main results we show that our framework unifies previous work -- including Euclidean, projective and affine -- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

Regular structures, flat and non-flat, are perceived as regular in a wide range of viewing angles and under varying illumination. In this papers, we exploit this simple observation and develop an invariant measure of pattern regularity.

Part I of this paper investigates the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. The projective extension is then derived, showing that it is necessary only for tasks for which a reference view is not available --- such as happens when updating scene structure from a moving stereo rig. The geometric difference between the two proposed invariants are that the affine invariant measures the relative deviation from a single reference plane, whereas the projective invariant measures the relative deviation from two reference planes. The affine invariant can be computed from three correspondin...

Foundation for Research and Technology—Hellas

We propose two methods for visual control of a robotic system which do not require the formulation of an explicit calibration between image space and the world coordinate system. Calibration is known to be a difficult and error prone process. By extracting control information directly from the image, we free our techniques from the errors normally associated with a fixed calibration. The two algorithms we propose both utilize feedback from a simple geometric effect, rotational invariance, to control the positioning servo loop. We attach a camera system to a robot such that the camera system and the robot's gripper rotate simultaneously. We also constrain the camera to lie in a position where it can observe the gripper's rotational axis. As the camera system rotates about the gripper's rotational axis, the circular path traced out by a point-like feature projects to an elliptical path in image space. We gather the projected feature points over part of a rotation (=2 radians) ...

This thesis presents a methodology for scene reconstruction that is based on the principles of projective geometry, while dealing with uncertainty at a fundamental level. Uncertainty in geometric features is represented and manipulated using probability density functions on projective space, allowing geometric constructions to be carried out via statistical inference. The main contribution of this thesis is the development of stochastic projective geometry, a formalism for performing uncertain geometric reasoning during the scene reconstruction process. The homogeneous coordinates of points and lines in the projective plane are represented by antipodal pairs of points on the unit sphere, and geometric uncertainty in their location is represented...

This paper demonstrates an approach which exploits an active camera as a projective pointing mechanism. The optical centre of a static camera is notionally substituted by the centre of rotation of the active camera, as is the image plane by a frontal plane, a plane perpendicular to the optical axis of the active camera in its resting direction. Algorithms devised for 3D motion and 3D structure recovery using a single passive camera become immediately applicable to the active camera without need for reformulation. Furthermore, because the active camera can access a panoramic field of view, instabilities which may arise when the field of view is small, or because the shared field of view between successive views after movement is small, are lessened. Two quite different applications of the idea are presented. In the first, the homography between a planar surface in the scene and the frontal plane is recovered and used to recover scene trajectories. In the second, the essential matrix bet...

ABSTRACT: Identifying PSL(2, C) as a projective group for patterns in the conformal camera model, the projective harmonic analysis on its double covering group SL(2, C) is presented in the noncompact and compact pictures–the pictures used to study different aspects of irreducible unitary representations of semisimple Lie groups. Bypassing technicalities of representation theory, but stressing the motivation and similarities with Euclidean Fourier analysis, each constructed picture of the projective Fourier analysis includes the Fourier transform, Plancherel’s theorem and convolution property. Projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern (after removing conformal distortions) by using only one projective Fourier transform of the original pattern, what is demonstrated in a computer simulation. The convolution properties in both pictures must by used to develop algorithms for projectively-invariant matching of patterns. Work in progress on fast algorithms for computing with projective Fourier transforms and for rendering image projective transformations is discussed. Efficient computations of the convolutions would follow from the both fast projective Fourier transforms and their inverses.

In this paper we show that Ullman and Basri’s linear combination (LC) representation, which was originally proposed for alignment-based object recognition, can be used for outlier detection in motion tracking with an affine camera. For this task LC can be realized either on image frames or feature trajectories, and therefore two methods are developed which we call linear combination of frames and linear combination of trajectories. For robust estimation of the linear combination coefficients, the support vector regression (SVR) algorithm is used and compared with the RANSAC method. SVR based on quadratic programming optimization can efficiently deal with more than 50 percent outliers and delivers more consistent results than RANSAC in our experiments. The linear combination representation can use SVR in a straightforward manner while previous factorization-based or subspace separation methods cannot. Experimental results are presented using real video sequences to demonstrate the effectiveness of our LC + SVR approaches, including a quantitative comparison of SVR and RANSAC. 1.