Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi

Line geometry and camera autocalibration

We provide a completely new rigorous matrix formulation of the absolute quadratic complex (AQC), given by the set of lines intersecting the absolute conic. The new results include closed-form expressions for the camera intrinsic parameters in terms of the AQC, an algorithm to obtain the dual absolute quadric from the AQC using straightforward matrix operations, and an equally direct computation of a Euclidean-upgrading homography from the AQC. We also completely characterize the 6×6 matrices acting on lines which are induced by a spatial homography. Several algorithmic possibilities arising from the AQC are systematically explored and analyzed in terms of efficiency and computational cost. Experiments include 3D reconstruction from real images

Method for 3D modelling based on structure from motion processing of sparse 2D images

A method based on Structure from Motion for processing a plurality of sparse images acquired by one or more acquisition devices to generate a sparse 3D points cloud and of a plurality of internal and external parameters of the acquisition devices includes the steps of collecting the images; extracting keypoints therefrom and generating keypoint descriptors; organizing the images in a proximity graph; pairwise image matching and generating keypoints connecting tracks according maximum proximity between keypoints; performing an autocalibration between image clusters to extract internal and external parameters of the acquisition devices, wherein calibration groups are defined that contain a plurality of image clusters and wherein a clustering algorithm iteratively merges the clusters in a model expressed in a common local reference system starting from clusters belonging to the same calibration group; and performing a Euclidean reconstruction of the object as a sparse 3D point cloud based on the extracted parameters

Linear Camera Autocalibration with Varying Parameters

We provide a new technique for the Euclidean upgrading of a projective calibration for a set of ten or more cameras with known skew angle and aspect ratio and arbitrary varying focal length and principal point. The proposed algorithm, which is purely linear and thus of very low computational cost and not suffering from initialization problems, is based on the geometric object given by the set of lines incident with the absolute conic. We include experiments which show the good performance of the technique

The Absolute Line Quadric and Camera Autocalibration

We introduce a geometrical object providing the same information as the absolute conic: the absolute line quadric (ALQ). After the introduction of the necessary exterior algebra and Grassmannian geometry tools, we analyze the Grassmannian of lines of P^3 from both the projective and Euclidean points of view. The exterior algebra setting allows then to introduce the ALQ as a quadric arising very naturally from the dual absolute quadric. We fully characterize the ALQ and provide clean relationships to solve the inverse problem, i.e., recovering the Euclidean structure of space from the ALQ. Finally we show how the ALQ turns out to be particularly suitable to address the Euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters, providing new linear and non-linear algorithms for this problem. We also provide experimental results showing the good performance of the techniques

Calibration routine for a telecentric stereo vision system considering affine mirror ambiguity

A robust calibration approach for a telecentric stereo camera system for three-dimensional (3-D) surface measurements is presented, considering the effect of affine mirror ambiguity. By optimizing the parameters of a rigid body transformation between two marker planes and transforming the two-dimensional (2-D) data into one coordinate frame, a 3-D calibration object is obtained, avoiding high manufacturing costs. Based on the recent contributions in the literature, the calibration routine consists of an initial parameter estimation by affine reconstruction to provide good start values for a subsequent nonlinear stereo refinement based on a Levenberg–Marquardt optimization. To this end, the coordinates of the calibration target are reconstructed in 3-D using the Tomasi–Kanade factorization algorithm for affine cameras with Euclidean upgrade. The reconstructed result is not properly scaled and not unique due to affine ambiguity. In order to correct the erroneous scaling, the similarity transformation between one of the 2-D calibration plane points and the corresponding 3-D points is estimated. The resulting scaling factor is used to rescale the 3-D point data, which then allows in combination with the 2-D calibration plane data for a determination of the start values for the subsequent nonlinear stereo refinement. As the rigid body transformation between the 2-D calibration planes is also obtained, a possible affine mirror ambiguity in the affine reconstruction result can be robustly corrected. The calibration routine is validated by an experimental calibration and various plausibility tests. Due to the usage of a calibration object with metric information, the determined camera projection matrices allow for a triangulation of correctly scaled metric 3-D points without the need for an individual camera magnification determination

Affine Approximation for Direct Batch Recovery of Euclidean Motion From Sparse Data

We present a batch method for recovering Euclidian camera motion from sparse image data. The main purpose of the algorithm is to recover the motion parameters using as much of the available information and as few computational steps as possible. The algorithmthus places itself in the gap between factorisation schemes, which make use of all available information in the initial recovery step, and sequential approaches which are able to handle sparseness in the image data. Euclidian camera matrices are approximated via the affine camera model, thus making the recovery direct in the sense that no intermediate projective reconstruction is made. Using a little known closure constraint, the FA-closure, we are able to formulate the camera coefficients linearly in the entries of the affine fundamental matrices. The novelty of the presented work is twofold: Firstly the presented formulation allows for a particularly good conditioning of the estimation of the initial motion parameters but also for an unprecedented diversity in the choice of possible regularisation terms. Secondly, the new autocalibration scheme presented here is in practice guaranteed to yield a Least Squares Estimate of the calibration parameters. As a bi-product, the affine camera model is rehabilitated as a useful model for most cameras and scene configurations, e.g. wide angle lenses observing a scene at close range. Experiments on real and synthetic data demonstrate the ability to reconstruct scenes which are very problematic for previous structure from motion techniques due to local ambiguities and error accumulation

Projector Self-Calibration using the Dual Absolute Quadric

The applications for projectors have increased dramatically since their origins in cinema. These include augmented reality, information displays, 3D scanning, and even archiving and surgical intervention. One common thread between all of these applications is the nec- essary step of projector calibration. Projector calibration can be a challenging task, and requires significant effort and preparation to ensure accuracy and fidelity. This is especially true in large scale, multi-projector installations used for projection mapping. Generally, the cameras for projector-camera systems are calibrated off-site, and then used in-field un- der the assumption that the intrinsics have remained constant. However, the assumption of off-site calibration imposes several hard restrictions. Among these, is that the intrinsics remain invariant between the off-site calibration process and the projector calibration site. This assumption is easily invalidated upon physical impact, or changing of lenses. To ad- dress this, camera self-calibration has been proposed for the projector calibration problem. However, current proposed methods suffer from degenerate conditions that are easily en- countered in practical projector calibration setups, resulting in undesirable variability and a distinct lack of robustness. In particular, the condition of near-intersecting optical axes of the camera positions used to capture the scene resulted in high variability and significant error in the recovered camera focal lengths. As such, a more robust method was required. To address this issue, an alternative camera self-calibration method is proposed. In this thesis we demonstrate our method of projector calibration with unknown and uncalibrated cameras via autocalibration using the Dual Absolute Quadric (DAQ). This method results in a significantly more robust projector calibration process, especially in the presence of correspondence noise when compared with previous methods. We use the DAQ method to calibrate the cameras using projector-generated correspondences, by upgrading an ini- tial projective calibration to metric, and subsequently calibrating the projector using the recovered metric structure of the scene. Our experiments provide strong evidence of the brittle behaviour of existing methods of projector self-calibration by evaluating them in near-degenerate conditions using both synthetic and real data. Further, they also show that the DAQ can be used successfully to calibrate a projector-camera system and reconstruct the surface used for projection mapping robustly, where previous methods fail

Camera calibration of long image sequences with the presence of occlusions

Camera calibration is a critical problem in applications such as augmented reality and image based model reconstruction. When constructing a 3D model of an object from an uncalibrated video sequence, large amounts of frames and self occlusions of parts of the object are common and difficult problems. In this paper we present a fast and robust algorithm that uses a divide and conquer strategy to split the video sequence into sub-sequences containing only the most relevant frames. Then a robust stratified linear based algorithm is able to calibrate each of the subsequences to a metric structure and finally the subsequences are merged together and a final non-linearoptimization refines the solution. Examples of real datareconstructions are presented.Postprint (author’s final draft