Elsevier

Pattern Recognition

Volume 45, Issue 10, October 2012, Pages 3636-3647
Pattern Recognition

A new robust algorithmic for multi-camera calibration with a 1D object under general motions without prior knowledge of any camera intrinsic parameter

https://doi.org/10.1016/j.patcog.2012.04.006Get rights and content

Abstract

In computer vision, camera calibration is a necessary process when the retrieval of information such as angles and distances is required. This paper addresses the multi-camera calibration problem with a single dimension calibration pattern under general motions. Currently, the known algorithms for solving this problem are based on the estimation of vanishing points. However, this estimate is very susceptible to noise, making the methods unsuitable for practical applications. Instead, this paper presents a new calibration algorithm, where the cameras are divided into binocular sets. The fundamental matrix of each binocular set is then estimated, allowing to perform a projective calibration of each camera. Then, the calibration is updated for the Euclidean space, ending the process. The calibration is possible without imposing any restrictions on the movement of the pattern and without any prior information about the cameras or motion. Experiments on synthetic and real images validate the new method and show that its accuracy makes it suitable also for practical applications.

Highlights

► We address the multi-camera calibration problem with a 1D object that moves freely. ► Camera calibration is necessary when the retrieval of 3D information is required. ► The proposed algorithm does not require prior knowledge of the camera parameters. ► Experiments show that the method is suitable even in the presence of noise.

Introduction

Mathematically, in the image creation process, the camera accomplishes a mapping between a 3D space and the image plane. During this process, some information, e.g., angles, distances and volume are lost. If these information are needed, it becomes necessary to estimate the intrinsic and extrinsic camera parameters, through a procedure known as calibration. Usually, during this procedure, the camera captures images from an object with well known dimensions and shape (known as the calibration pattern). Afterwards, the relation between some points of the calibration pattern and their respective projections in the image plane are used to determine the camera parameters.

The first calibration algorithms to become widely used were based on 3D patterns [21], [14]. Over the years, new calibration methods have been proposed using 2D patterns [19], [23]. More recently, Zhang [24] proposed the calibration using a single dimension pattern (consisting of a set of points in a straight line). In this, the 1D pattern should perform unknown displacements, while the camera captures images of it. The only restriction is that one of the pattern points must remain fixed during the image acquisition. However, Zhang's work cannot be applied in practice because the method is very susceptible to noise. Therefore, only with the work of de França et al. [3], the calibration with 1D patterns could be used in practice, with an accuracy comparable to other traditional methods. This is because it was shown that the accuracy of the original method of Zhang [24] is significantly improved simply by analysing the mathematical formulation of the problem. Thus, it was possible to improve the numerical conditioning by performing a simple data normalization.

The main advantage of using 1D patterns to make the calibration is the possibility to calibrate several cameras at the same time. This is because the points of the 1D pattern can be “captured” simultaneously by cameras even in very different points of view. However, in spite of Zhang has highlighted the advantages of using 1D patterns for calibration of multiple cameras, only after the work of Kojima et al. [13] this became possible in fact. This is because, in the case of one set of cameras, the algorithm of Kojima also estimates the extrinsic parameters of all cameras. However, unfortunately, it is necessary that at least one of the cameras, called the “reference camera”, is already calibrated.

Most calibration methods that use 1D calibration patterns assume special pattern moves or prior knowledge of some of the camera parameters. In particular, most require that a point of the pattern is always fixed. This, besides being a limitation, is one additional source of errors, because in practice it is not always easy to keep a point of the pattern fixed as the pattern moves. The work of Wang et al. [22] was the first to propose a multi-camera calibration algorithm based on a 1D pattern that moves freely and without prior knowledge of the parameters of any of the cameras. As in the algorithm of Kojima et al. [13], pattern projections are used to estimate a set of vanishing points correspondences. Thus, the infinite homography from all cameras in the set is estimated and an affine calibration can be performed. Then, the affine camera matrices can be updated to the Euclidean space, completing the calibration. However, as demonstrated empirically in Section 5.1, the estimation of vanishing points is very susceptible to noise, making these algorithms unsuitable for practical use.

In this work, a different approach is proposed, where vanishing points are not necessary. Although the calibration pattern can move freely, the proposed method also does not assume prior knowledge of any intrinsic parameter. Moreover, as in any calibration algorithm using 1D calibration patterns, the proposed method has an advantage over those based on 2D plane calibration patterns, which is the possibility to calibrate two or more cameras simultaneously even if they are in very different points of view. To make this possible, in the proposed technique, the fundamental matrix of the system is initially estimated and a projective calibration of the binocular set is accomplished. Then, the plane at infinity and the intrinsic parameters of the reference camera are estimated. Thus, calibration can be updated to an Euclidean calibration. Experimental results performed on synthetic and real images show that the proposed method produces good results even in the presence of noise and thus provides a convenient and flexible approach to calibrate one or more binocular sets.

Section snippets

Notation

In a set with a K+1 cameras, the reference camera is called “camera 0”. For simplicity but without loss of generality, it is considered that the reference camera is at the origin of the world coordinate system. The other cameras are indexed by k, with k=1,…,K, and referred to as “camera k”. In the case of matrices, an index is used to indicate which camera it refers to. For example, A0 is related to “camera 0”, while A1 is always associated with the “camera 1”. Moreover, the homogeneous

Background

Fig. 2 outlines a set of K+1 cameras that capture images from a 1D calibration pattern with n points, while it performs N unrestricted displacements (indexed by i). Considering pairs of cameras always composed by the reference camera and another camera k in Fig. 2, there are a total of K binocular sets (indexed by k). In this case, for each binocular set, there is a point correspondence (m0jimkji) for each point Mji belonging to the pattern, where j=1,2,…,n.

The cross ratio between four points (

Solving multi-camera calibration with 1D objects

Again, considering the set of K+1 cameras in Fig. 2 and a 1D pattern with n points, after N displacements, for each binocular set k, we obtain nN correspondences (m0jimkji). Since the coordinates of such correspondences can be obtained accurately, the fundamental matrix, Fk, of each binocular set can be estimated with good accuracy. To do this, it suffices to use one of the several methods already well known in the literature [1].

From the work of Hartley et al. [7] and Faugeras [4], it is

Experimental results

The accuracy of the algorithm discussed in this paper was analysed using both synthetic and real data. The experiments performed with each data type are described below. It was always considered that the noise present in the data follows a normal distribution with zero mean and standard deviation σ. Also, when necessary, the fundamental matrix and the infinite homography have always been estimated using maximum likelihood algorithms [9]. The projective reconstruction was performed using the

Conclusions

The calibration technique with 1D patterns is indicated mainly for calibration of multiple cameras. Although some algorithms have already been proposed to solve this problem, they have limitations, e.g., restrictions on the pattern movements or requirement of prior knowledge about the parameters of the cameras. Moreover, they are based on vanishing points, whose estimation, in practice, is very susceptible to noise. Instead, this paper proposes a new method, which is based on the estimation of

Acknowledgments

The authors would like to thank CAPES, UFSC and PROPPG/UEL for the financial support for the research.

José Alexandre de França received the BS degree in electronic engineering from the Universidade Federal de Campina Grande (UFCG), Brazil, in 1995, the MS degree in electronic engineering from the UFCG, in 1997, the PhD degree in electronic engineering from the Universidade Federal de Santa Catarina, Brazil, in 2005. Now, he is professor and leader of the Automation and Intelligent Instrumentation Laboratory of the Londrina State University. His current research interests include camera

References (24)

  • J. Heikkila

    Geometric camera calibration using circular control points

    IEEE Transactions on Pattern Analysis and Machine Intelligence

    (2000)
  • R. Horaud et al.

    Stereo calibration from rigid motions

    IEEE Transactions on Pattern Analysis and Machine Intelligence

    (2000)
  • Cited by (0)

    José Alexandre de França received the BS degree in electronic engineering from the Universidade Federal de Campina Grande (UFCG), Brazil, in 1995, the MS degree in electronic engineering from the UFCG, in 1997, the PhD degree in electronic engineering from the Universidade Federal de Santa Catarina, Brazil, in 2005. Now, he is professor and leader of the Automation and Intelligent Instrumentation Laboratory of the Londrina State University. His current research interests include camera calibration and 3D reconstruction, geometric primitive extraction, vision guided robot navigation, measurement and instrumentation, precision agriculture.

    Marcelo Ricardo Stemmer received the BS degree in electric engineering from the Federal University of Santa Catarina (UFSC), Brazil, in 1982, the MS degree in control, automation and industrial informatics from UFSC in 1985, the PhD degree in industrial automation from WZL/RWTH-Aachen, Germany, in 1991. Now, he is professor and leader of the Intelligent Industrial Systems group at UFSC.

    Maria Bernadete de Morais França received the BS degree in electronic engineering from the Universidade Federal de Campina Grande (UFCG), Brazil, in 1996, the MS degree in electronic engineering from the UFCG, Brazil, in 1998. Now, she is professor and member of the Automation and Intelligent Instrumentation Laboratory of the Londrina State University. Her current research interests include camera calibration and 3D reconstruction, measurement and instrumentation, precision agriculture.

    Juliani Chico Piai received the BS degree in electronic engineering from the Universidade Estadual de Londrina (UEL), Brazil, in 2006, the MS degree in electronic engineering from the UEL, Brazil, in 2009. Now, she is professor and member of the Automation and Intelligent Instrumentation Laboratory of the Londrina State University. Her current research interests include camera calibration and 3D reconstruction, measurement and instrumentation, precision agriculture, traffic control.

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