Fundamental Matrix: Digital camera calibration and Essential Matrix parameters (original) (raw)

Digital Camera Calibration, Relative Orientation and Essential Matrix Parameters

WSEAS Transactions on Signal Processing archive, 2017

The fundamental matrix, based on the co-planarity condition, even though it is very interesting for theoretical issues, it does not allow finding the camera calibration parameters, and the base and rotation parameters altogether. In this work we present an easy calibration method for calculating the internal parameters: pixel dimensions and image center pixel coordinates. We show that the method is slightly easier if the camera rotation angles, in relation with the general referential system, are small. The accuracy of the four calibration parameters are evaluated by simulations. In addition, a method to improve the accuracy is explained. When the calibration parameters are known, the fundamental matrix can be reduced to the essential matrix. In order to find the relative orientation parameters in stereo vision, there is also presented a new method to extract the base and the camera rotation by means of the essential matrix. The proposed method is simple to implement. We also includ...

Self-Calibration of a Moving Camera from Point Correspondences and Fundamental Matrices

International Journal of Computer Vision, 1997

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.

Camera self-calibration using the singular value decomposition of the fundamental matrix: From point correspondences to 3D measurements

1999

This paper deals with a fundamental problem in motion and stereo analysis, namely that of determining the camera intrinsic calibration parameters. A novel method is proposed that follows the autocalibration paradigm, according to which calibration is achieved not with the aid of a calibration pattern but by observing a number of image features in a set of successive images. The proposed method relies upon the Singular Value Decomposition of the fundamental matrix, which leads to a particularly simple form of the Kruppa equations. In contrast to the classical formulation that yields an over-determined system of constraints, the derivation proposed here provides a straightforward answer to the problem of determining which constraints to employ among the set of available ones. Moreover, the derivation is a purely algebraic one, without a need for resorting to the somewhat non-intuitive geometric concept of the absolute conic. Apart from the fundamental matrix itself, no other quantities that can be extracted from it (e.g. the epipoles) are needed for the derivation. Experimental results from extensive simulations and several image sequences demonstrate the effectiveness of the proposed method in accurately estimating the intrinsic calibration matrices. It is also shown that the computed intrinsic calibration matrices are sufficient for recovering 3D motion and performing metric measurements from uncalibrated images.

Linear and Non-Linear Camera Calibration Techniques

Journal of Global Research …, 2011

This Paper deals with calibrate a camera to find out the intrinsic and extrinsic camera parameters which are necessary to recover the depth estimation of an object in stereovision system.

Closed-Form Solutions for the Euclidean Calibration of a Stereo Rig

1998

In this paper we describe a method for estimating the internal parameters of the left and right cameras associated with a stereo image pair. The stereo pair has known epipolar geometry and therefore 3-D projective reconstruction of pairs of matched image points is available. The stereo pair is allowed to move and hence there is a collineation relating the two projective reconstructions computed before and after the motion. We show that this collineation has similar but different parameterizations for general and ground-plane rigid motions and we make explicit the relationship between the internal camera parameters and such a collineation. We devise a practical method for recovering four camera parameters from a single general motion or three camera parameters from a single ground-plane motion. Numerous experiments with simulated, calibrated and natural data validate the calibration method.

Retrieval of the Calibration Matrix from the 3-D Projective Camera Model

By relating the projective camera model to the perspective one, the intrinsic camera parameters give rise to what is called the calibration matrix. This paper presents two new methods to retrieve the calibration matrix from the projective camera model. In both methods, a collective approach was adopted, using matrix representation. The calibration matrix was retrieved from a quadratic matrix term. The two methods were framed around a correct utilization of Cholesky factorization to decompose the quadratic matrix term. The first method used an iterative Cholesky factorization to retrieve the calibration matrix from the quadratic matrix term. The second method used Cholesky factorization to factor the quadratic matrix term but after its inversion. The basic argument behind the two methods is that: the direct use of Cholesky factorization does not reveal the correct decomposition due to the missing matrix structure in terms of lower- upper ordering. This study presents two new algorith...

Calibration and Epipolar Geometry of Generic Heterogenous Camera Systems

The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 2012

The application of perspective camera systems in photogrammetry and computer vision is state of the art. In recent years nonperspective and especially omnidirectional camera systems were increasingly used in close-range photogrammetry tasks. In general perspective camera model, i. e. pinhole model, cannot be applied when using non-perspective camera systems. However, several camera models for different omnidirectional camera systems are proposed in literature. Using different types of cameras in a heterogeneous camera system may lead to an advantageous combination. The advantages of different camera systems, e. g. field of view and resolution, result in a new enhanced camera system. If these different kinds of cameras can be modeled, using a unified camera model, the total calibration process can be simplified. Sometimes it is not possible to give the specific camera model in advance. In these cases a generic approach is helpful. Furthermore, a simple stereo reconstruction becomes possible using a fisheye and a perspective camera for example. In this paper camera models for perspective, wide-angle and omnidirectional camera systems are evaluated. The crucial initialization of the model's parameters is conducted using a generic method that is independent of the particular camera system. The accuracy of this generic camera calibration approach is evaluated by calibration of a dozen of real camera systems. It will be shown, that a unified method of modeling, parameter approximation and calibration of interior and exterior orientation can be applied to derive 3D object data.

Theory and Calibration for Axial Cameras

Proceedings of the 7th Asian Conference on Computer Vision Volume Part I, 2006

Although most works in computer vision use perspective or other central cameras, the interest in non-central camera models has increased lately, especially with respect to omnidirectional vision. Calibration and structure-from-motion algorithms exist for both, central and non-central cameras. An intermediate class of cameras, although encountered rather frequently, has received less attention. So-called axial cameras are non-central but their projection rays are constrained by the existence of a line that cuts all of them. This is the case for stereo systems, many non-central catadioptric cameras and pushbroom cameras for example. In this paper, we study the geometry of axial cameras and propose a calibration approach for them. We also describe the various axial catadioptric configurations which are more common and less restrictive than central catadioptric ones. Finally we used simulations and real experiments to prove the validity of our theory.

On projection matrix identification for camera calibration

Computer Vision Theory and Applications, 2007

The projection matrix identification problem is considered with application to calibration of intrinsic camera parameters. Physical and orthogonal intrinsic camera models in context of 2D and 3D data are discussed. A novel nonlinear goal function is proposed for homographic calibration method having the fast convergence of Levenberg-Marquardt optimization procedure. Three models (linear, quadratic, and rational) and four optimization procedures for their identification were compared wrt their time complexity, the projection accuracy, and the intrinsic parameters accuracy. The analysis has been performed for both, the raw and the calibrated pixel data, too. The recommended technique with the best performance in all used quality measures is the Housholder QR decomposition for the linear least square method of the linear form of projection equations.

High precision camera calibration in vision measurement

Optics & Laser Technology, 2007

In order to ensure the measurement precision for shape recovery from a stereo vision system, a novel high precision calibration method is presented. Some standard points acquired from the object of interest are used as standard world points. There are four rotation matrices and translation matrices that need to be calibrated, the first is between the left camera and the world coordinate, the second is between the right camera and the world coordinate, the third is between the left camera and the right camera, and the last is between the right camera and the left camera. The lens distortion parameters include radial and tangential distortion. The merits of the calibration method are its high precision, easy of operation, and high reliability. r