Visual Servoing Path Planning for Cameras Obeying the Unified Model

Abstract

This paper proposes a path planning visual servoing strategy for a class of cameras that includes conventional perspective cameras, fisheye cameras and catadioptric cameras as special cases. Specifically, these cameras are modeled by adopting a unified model recently proposed in the literature and the strategy consists of designing image trajectories for eye-in-hand robotic systems that allow the robot to reach a desired location while satisfying typical visual servoing constraints. To this end, the proposed strategy introduces the projection of the available image features onto a virtual plane and the computation of a feasible image trajectory through polynomial programming. Then, the computed image trajectory is tracked by using an image-based visual servoing controller. Experimental results with a fisheye camera mounted on a 6-d.o.f. robot arm are presented in order to illustrate the proposed strategy.     

Epipolar Geometry for Central Catadioptric Cameras

Central catadioptric cameras are cameras which combine lenses and mirrors to capture a very wide field of view with a central projection. In this paper we extend the classical epipolar geometry of perspective cameras to all central catadioptric cameras. Epipolar geometry is formulated as the geometry of corresponding rays in a three-dimensional space. Using the model of image formation of central catadioptric cameras, the constraint on corresponding image points is then derived. It is shown that the corresponding points lie on epipolar conics. In addition, the shape of the conics for all types of central catadioptric cameras is classified. Finally, the theory is verified by experiments with real central catadioptric cameras.     

Structure from motion with wide circular field of view cameras

This paper presents a method for fully automatic and robust estimation of two-view geometry, autocalibration, and 3D metric reconstruction from point correspondences in images taken by cameras with wide circular field of view. We focus on cameras which have more than 180/spl deg/ field of view and for which the standard perspective camera model is not sufficient, e.g., the cameras equipped with circular fish-eye lenses Nikon FC-E8 (183/spl deg/), Sigma 8 mm-f4-EX (180/spl deg/), or with curved conical mirrors. We assume a circular field of view and axially symmetric image projection to autocalibrate the cameras. Many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping. Examples are the above-mentioned fish-eye lenses and properly assembled catadioptric cameras with conical mirrors. We show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem. This allows the use of efficient RANSAC robust estimation to find the image projection model, the epipolar geometry, and the selection of true point correspondences from tentative correspondences contaminated by mismatches. Real catadioptric cameras are often slightly noncentral. We show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3D scene reconstruction. Noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors.     

Catadioptric Projective Geometry

Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications. We show that a central catadioptric projection is equivalent to a two-step mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lens-based perspective cameras are also central catadioptric devices with a virtual planar mirror and are, thus, covered by the unifying model. We prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images (conics). It turns out that planar and parabolic mirrors build a dual catadioptric projection pair. As a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system.     

Catadioptric Visual Servoing From 3-D Straight Lines

In this paper, we consider the problem of controlling a 6 DOF holonomic robot and a nonholonomic mobile robot from the projection of 3-D straight lines in the image plane of central catadioptric systems. A generic central catadioptric interaction matrix for the projection of 3-D straight lines is derived using an unifying imaging model valid for an entire class of cameras. This result is exploited to design an image-based control law that allows us to control the 6 DOF of a robotic arm. Then, the projected lines are exploited to control a nonholonomic robot. We show that as when considering a robotic arm, the control objectives are mainly based on catadioptric image feature and that local asymptotic convergence is guaranteed. Simulation results and real experiments with a 6 DOF eye-to-hand system and a mobile robot illustrate the control strategy.     

Analysis and evaluation of a general camera model

A versatile General Camera Model, GCM, has been developed, and is described in detail. The model is general in the sense that it can capture both fisheye and conventional as well as catadioptric cameras in a unified framework. The camera model includes efficient handling of non-central cameras as well as compensations for decentring distortion. A novel way of analysing radial distortion functions of camera models leads to a straightforward improvement of conventional models with respect to generality, accuracy and simplicity. Different camera models are experimentally compared for two cameras with conventional and fisheye lenses, and the results show that the overall performance is favourable for the GCM.     

Self-calibration of hybrid central catadioptric and perspective cameras

Hybrid central catadioptric and perspective cameras are desired in practice, because the hybrid camera system can capture large field of view as well as high-resolution images. However, the calibration of the system is challenging due to heavy distortions in catadioptric cameras. In addition, previous calibration methods are only suitable for the camera system consisting of perspective cameras and catadioptric cameras with only parabolic mirrors, in which priors about the intrinsic parameters of perspective cameras are required. In this work, we provide a new approach to handle the problems. We show that if the hybrid camera system consists of at least two central catadioptric and one perspective cameras, both the intrinsic and extrinsic parameters of the system can be calibrated linearly without priors about intrinsic parameters of the perspective cameras, and the supported central catadioptric cameras of our method can be more generic. In this work, an approximated polynomial model is derived and used for rectification of catadioptric image. Firstly, with the epipolar geometry between the perspective and rectified catadioptric images, the distortion parameters of the polynomial model can be estimated linearly. Then a new method is proposed to estimate the intrinsic parameters of a central catadioptric camera with the parameters in the polynomial model, and hence the catadioptric cameras can be calibrated. Finally, a linear self-calibration method for the hybrid system is given with the calibrated catadioptric cameras. The main advantage of our method is that it cannot only calibrate both the intrinsic and extrinsic parameters of the hybrid camera system, but also simplify a traditional nonlinear self-calibration of perspective cameras to a linear process. Experiments show that our proposed method is robust and reliable.     

Calibration of Central Catadioptric Cameras Using a DLT-Like Approach

In this study, we present a calibration technique that is valid for all single-viewpoint catadioptric cameras. We are able to represent the projection of 3D points on a catadioptric image linearly with a 6×10 projection matrix, which uses lifted coordinates for image and 3D points. This projection matrix can be computed from 3D–2D correspondences (minimum 20 points distributed in three different planes). We show how to decompose it to obtain intrinsic and extrinsic parameters. Moreover, we use this parameter estimation followed by a non-linear optimization to calibrate various types of cameras. Our results are based on the sphere camera model which considers that every central catadioptric system can be modeled using two projections, one from 3D points to a unitary sphere and then a perspective projection from the sphere to the image plane. We test our method both with simulations and real images, and we analyze the results performing a 3D reconstruction from two omnidirectional images.     

Photometric visual servoing for omnidirectional cameras

2D visual servoing consists in using data provided by a vision sensor for controlling the motions of a dynamic system. Most of visual servoing approaches has relied on the geometric features that have to be tracked and matched in the image acquired by the camera. Recent works have highlighted the interest of taking into account the photometric information of the entire image. This approach was tackled with images of perspective cameras. We propose, in this paper, to extend this technique to central cameras. This generalization allows to apply this kind of method to catadioptric cameras and wide field of view cameras. Several experiments have been successfully done with a fisheye camera in order to control a 6 degrees of freedom robot and with a catadioptric camera for a mobile robot navigation task.     

Efficient generic calibration method for general cameras with single centre of projection

Generic camera calibration is a non-parametric calibration technique that is applicable to any type of vision sensor. However, the standard generic calibration method was developed such that both central and non-central cameras can be calibrated within the same framework. Consequently, existing parametric calibration techniques cannot be applied for the common case of cameras with a single centre of projection (e.g. pinhole, fisheye, hyperboloidal catadioptric). This paper proposes improvements to the standard generic calibration method for central cameras that reduce its complexity, and improve its accuracy and robustness. Improvements are achieved by taking advantage of the geometric constraints resulting from a single centre of projection in order to enable the application of established pinhole calibration techniques. Input data for the algorithm is acquired using active grids, the performance of which is characterised. A novel linear estimation stage is proposed that enables a well established pinhole calibration technique to be used to estimate the camera centre and initial grid poses. The proposed solution is shown to be more accurate than the linear estimation stage of the standard method. A linear alternative to the existing polynomial method for estimating the pose of additional grids used in the calibration is demonstrated and evaluated. Distortion correction experiments are conducted with real data for both an omnidirectional camera and a fisheye camera using the standard and proposed methods. Motion reconstruction experiments are also undertaken for the omnidirectional camera. Results show the accuracy and robustness of the proposed method to be improved over those of the standard method.     

A new linear algorithm for calibrating central catadioptric cameras

In this paper, a novel linear calibration algorithm based on lines is presented for central catadioptric cameras. We firstly derive the relationship between the projection on the viewing sphere of a space point and its catadioptric image. And then by the relationship we establish a group of linear constraints on the catadioptric parameters from the catadioptric projections of spatial lines. By using these linear constraints, any central catadioptric camera can be fully calibrated from a single view of three or more lines without prior knowledge on the camera. Extensive experiments show this algorithm can improve the calibration's robustness.     

A General Framework for the Selection of World Coordinate Systems in Perspective and Catadioptric Imaging Applications

An imaging system with a single effective viewpoint is called a central projection system. The conventional perspective camera is an example of central projection system. A catadioptric realization of omnidirectional vision combines reflective surfaces with lenses. Catadioptric systems with an unique projection center are also examples of central projection systems. Whenever an image is acquired, points in 3D space are mapped into points in the 2D image plane. The image formation process represents a transformation from ³ to ², and mathematical models can be used to describe it. This paper discusses the definition of world coordinate systems that simplify the modeling of general central projection imaging. We show that an adequate choice of the world coordinate reference system can be highly advantageous. Such a choice does not imply that new information will be available in the images. Instead the geometric transformations will be represented in a common and more compact framework, while simultaneously enabling newer insights. The first part of the paper focuses on static imaging systems that include both perspective cameras and catadioptric systems. A systematic approach to select the world reference frame is presented. In particular we derive coordinate systems that satisfy two differential constraints (the compactness and the decoupling constraints). These coordinate systems have several advantages for the representation of the transformations between the 3D world and the image plane. The second part of the paper applies the derived mathematical framework to active tracking of moving targets. In applications of visual control of motion the relationship between motion in the scene and image motion must be established. In the case of active tracking of moving targets these relationships become more complex due to camera motion. Suitable world coordinate reference systems are defined for three distinct situations: perspective camera with planar translation motion, perspective camera with pan and tilt rotation motion, and catadioptric imaging system rotating around an axis going through the effective viewpoint and the camera center. Position and velocity equations relating image motion, camera motion and target 3D motion are derived and discussed. Control laws to perform active tracking of moving targets using visual information are established.     

Visual navigation of wheeled mobile robots using direct feedback of a geometric constraint

Many applications of wheeled mobile robots demand a good solution for the autonomous mobility problem, i.e., the navigation with large displacement. A promising approach to solve this problem is the following of a visual path extracted from a visual memory. In this paper, we propose an image-based control scheme for driving wheeled mobile robots along visual paths. Our approach is based on the feedback of information given by geometric constraints: the epipolar geometry or the trifocal tensor. The proposed control law only requires one measurement easily computed from the image data through the geometric constraint. The proposed approach has two main advantages: explicit pose parameters decomposition is not required and the rotational velocity is smooth or eventually piece-wise constant avoiding discontinuities that generally appear in previous works when the target image changes. The translational velocity is adapted as demanded for the path and the resultant motion is independent of this velocity. Furthermore, our approach is valid for all cameras with approximated central projection, including conventional, catadioptric and some fisheye cameras. Simulations and real-world experiments illustrate the validity of the proposal.     

Identical Projective Geometric Properties of Central Catadioptric Line Images and Sphere Images with Applications to Calibration

Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. Lines and spheres in space are all projected into conics in the central catadioptric image planes, and such conics are called line images and sphere images, respectively. We discovered that there exists an imaginary conic in the central catadioptric image planes, defined as the modified image of the absolute conic (MIAC), and by utilizing the MIAC, the novel identical projective geometric properties of line images and sphere images may be exploited: Each line image or each sphere image is double-contact with the MIAC, which is an analogy of the discovery in pinhole camera that the image of the absolute conic (IAC) is double-contact with sphere images. Note that the IAC also exists in the central catadioptric image plane, but it does not have the double-contact properties with line images or sphere images. This is the main reason to propose the MIAC. From these geometric properties with the MIAC, two linear calibration methods for central catadioptric cameras using sphere images as well as using line images are proposed in the same framework. Note that there are many linear approaches to central catadioptric camera calibration using line images. It seems that to use the properties that line images are tangent to the MIAC only leads to an alternative geometric construction for calibration. However, for sphere images, there are only some nonlinear calibration methods in literature. Therefore, to propose linear methods for sphere images may be the main contribution of this paper. Our new algorithms have been tested in extensive experiments with respect to noise sensitivity.     

Catadioptric camera calibration using geometric invariants

Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. In this paper, we propose a novel method for the calibration of central catadioptric cameras using geometric invariants. Lines and spheres in space are all projected into conics in the catadioptric image plane. We prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two. From these invariants, constraint equations for the intrinsic parameters of catadioptric camera are derived. Therefore, there are two kinds of variants of this novel method. The first one uses projections of lines and the second one uses projections of spheres. In general, the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration. One important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines. The performances of our method are demonstrated by both the results of simulations and experiments with real images.     

一种反射折射摄像机的简易标定方法

Central catadioptric cameras are widely used in virtual reality and robot navigation,and the camera calibration is a prerequisite for these applications.In this paper,we propose an easy calibration method for central catadioptric cameras with a 2D calibration pattern.Firstly,the bounding ellipse of the catadioptric image and field of view (FOV) are used to obtain the initial estimation of the intrinsic parameters.Then,the explicit relationship between the central catadioptric and the pinhole model is used to initialize the extrinsic parameters.Finally,the intrinsic and extrinsic parameters are refined by nonlinear optimization.The proposed method does not need any fitting of partial visible conic,and the projected images of 2D calibration pattern can easily cover the whole image,so our method is easy and robust.Experiments with simulated data as well as real images show the satisfactory performance of our proposed calibration method.     

一种反射折射摄像机的简易标定方法

Central catadioptric cameras are widely used in virtual reality and robot navigation, and the camera calibration is a prerequisite for these applications. In this paper, we propose an easy calibration method for central catadioptric cameras with a 2D calibration pattern. Firstly, the bounding ellipse of the catadioptric image and field of view (FOV) are used to obtain the initial estimation of the intrinsic parameters. Then, the explicit relationship between the central catadioptric and the pinhole model is used to initialize the extrinsic parameters. Finally, the intrinsic and extrinsic parameters are refined by nonlinear optimization. The proposed method does not need any fitting of partial visible conic, and the projected images of 2D calibration pattern can easily cover the whole image, so our method is easy and robust. Experiments with simulated data as well as real images show the satisfactory performance of our proposed calibration method.     

环视鱼眼图像处理深度学习研究进展

视觉环境感知在自动驾驶汽车发展中起着关键作用,在智能后视镜、倒车雷达、360°全景、行车记录仪、碰撞预警、红绿灯识别、车道偏移、并线辅助和自动泊车等领域也有着广泛运用。传统的环境信息获取方式是窄角针孔摄像头,视野有限有盲区,解决这个问题的方法是环境信息感知使用鱼眼镜头,广角视图能够提供整个180°的半球视图,理论上仅需两个摄像头即可覆盖360°,为视觉感知提供更多信息。处理环视图像目前主要有两种途径：一是对图像先纠正,去失真,缺点是图像去失真会损害图像质量,并导致信息丢失;二是直接对形变的鱼眼图像进行建模,但目前还没有效果比较好的建模方法。此外,环视鱼眼图像数据集的缺乏也是制约相关研究的一大难题。针对上述挑战,本文总结了环视鱼眼图像的相关研究,包括环视鱼眼图像的校正处理、环视鱼眼图像中的目标检测、环视鱼眼图像中的语义分割、伪环视鱼眼图像数据集生成方法和其他鱼眼图像建模方法等,结合自动驾驶汽车的环境感知应用背景,分析了这些模型的效率和这些处理方法的优劣,并对目前公开的环视鱼眼图像通用数据集进行了详细介绍,对环视鱼眼图像中待解决的问题与未来研究方向做出预测和展望。