A Fisher-Rao metric for paracatadioptric images of lines

In a central paracatadioptric imaging system a perspective camera takes an image of a scene reflected in a paraboloidal mirror. A 360° field of view is obtained, but the image is severely distorted. In particular, straight lines in the scene project to circles in the image. These distortions make it diffcult to detect projected lines using standard image processing algorithms. The distortions are removed using a Fisher-Rao metric which is defined on the space of projected lines in the paracatadioptric image. The space of projected lines is divided into subsets such that on each subset the Fisher-Rao metric is closely approximated by the Euclidean metric. Each subset is sampled at the vertices of a square grid and values are assigned to the sampled points using an adaptation of the trace transform. The result is a set of digital images to which standard image processing algorithms can be applied. The effectiveness of this approach to line detection is illustrated using two algorithms, both of which are based on the Sobel edge operator. The task of line detection is reduced to the task of finding isolated peaks in a Sobel image. An experimental comparison is made between these two algorithms and third algorithm taken from the literature and based on the Hough transform

Geometric Properties of Central Catadioptric Line Images and Their Application in Calibration

In central catadioptric systems, lines in a scene are projected to conic curves in the image. This work studies the geometry of the central catadioptric projection of lines and its use in calibration. It is shown that the conic curves where the lines are mapped possess several projective invariant properties. From these properties, it follows that any central catadioptric system can be fully calibrated from an image of three or more lines. The image of the absolute conic, the relative pose between the camera and the mirror, and the shape of the reflective surface can be recovered using a geometric construction based on the conic loci where the lines are projected. This result is valid for any central catadioptric system and generalizes previous results for paracatadioptric sensors. Moreover, it is proven that systems with a hyperbolic/elliptical mirror can be calibrated from the image of two lines. If both the shape and the pose of the mirror are known, then two line images are enough to determine the image of the absolute conic encoding the camera’s intrinsic parameters. The sensitivity to errors is evaluated and the approach is used to calibrate a real camer

Fast Central Catadioptric Line Extraction

International audienceLines are particularly important features for different tasks such as calibration, structure from motion, 3D reconstruction in computer vision. However, line detection in catadioptric images is not trivial because the projection of a 3D line is a conic eventually degenerated. If the sensor is calibrated, it has been already demonstrated that each conic can be described by two parameters. In this way, some methods based on the adaptation of conventional line detection methods have been proposed. However, most of these methods suffer from the same disadvantages than in the perspective case (computing time, accuracy, robustness, ...). In this paper, we then propose a new method for line detection in central catadioptric image comparable to the polygonal approximation approach. With this method, only two points of a chain allows to extract with a very high accuracy a catadioptric line. Moreover , this algorithm is particularly fast and is applicable in realtime. We also present experimental results with some quantitative and qualitative evaluations in order to show the quality of the results and the perspectives of this method

Uncalibrated Visual Compass from Omnidirectional Line Images with Application to Attitude MAV Estimation

International audienceThis paper presents a new algorithm based on previous results of the authors, for the estimation of the yaw angle of an omnidirectional camera robot undergoing a 6-DoF rigid motion. Our real-time algorithm is uncalibrated, robust to noisy data, and it only relies on the projection of 3-D parallel lines as image features. Numerical and real-world experiments conducted with an eye-in-hand robot manipulator, which we used to simulate the 3-D motion of a Micro unmanned Aerial Vehicle (MAV), show the accuracy and reliability of our estimation algorithm

Plane-Based Calibration of Central Catadioptric Cameras

International audienceWe present a novel calibration technique for all central catadioptric cameras using images of planar grids. We adopted the well-known sphere camera model to describe the catadioptric projection. We show that, using the so-called lifted coordinates, a linear relation mapping the grid points to the corresponding points on the image plane can be written as a 6 × 6 matrix Hcata , which acts like the classical 3 × 3 ho- mography for perspective cameras. We show how to compute the image of the absolute conic (IAC) from at least 3 homo- graphies and how to recover from it the intrinsic parameters of the catadioptric camera. In the case of paracatadioptric cameras one such homography is enough to estimate the IAC, thus allowing the calibration from a single image

Relating vanishing points to catadioptric camera calibration

This paper presents the analysis and derivation of the geometric relation between vanishing points and camera parameters of central catadioptric camera systems. These vanishing points correspond to the three mutually orthogonal directions of 3D real world coordinate system (i.e. X, Y and Z axes). Compared to vanishing points (VPs) in the perspective projection, the advantages of VPs under central catadioptric projection are that there are normally two vanishing points for each set of parallel lines, since lines are projected to conics in the catadioptric image plane. Also, their vanishing points are usually located inside the image frame. We show that knowledge of the VPs corresponding to XYZ axes from a single image can lead to simple derivation of both intrinsic and extrinsic parameters of the central catadioptric system. This derived novel theory is demonstrated and tested on both synthetic and real data with respect to noise sensitivity

Nonlinear Control and Estimation Techniques with Applications to Vision-based and Biomedical Systems

This dissertation is divided into four self-contained chapters. In Chapter 1, a new estimator using a single calibrated camera mounted on a moving platform is developed to asymptotically recover the range and the three-dimensional (3D) Euclidean position of a static object feature. The estimator also recovers the constant 3D Euclidean coordinates of the feature relative to the world frame as a byproduct. The position and orientation of the camera is assumed to be measurable unlike existing observers where velocity measurements are assumed to be known. To estimate the unknown range variable, an adaptive least squares estimation strategy is employed based on a novel prediction error formulation. A Lyapunov stability analysis is used to prove the convergence properties of the estimator. The developed estimator has a simple mathematical structure and can be used to identify range and 3D Euclidean coordinates of multiple features. These properties of the estimator make it suitable for use with robot navigation algorithms where position measurements are readily available. Numerical simulation results along with experimental results are presented to illustrate the effectiveness of the proposed algorithm. In Chapter 2, a novel Euclidean position estimation technique using a single uncalibrated camera mounted on a moving platform is developed to asymptotically recover the three-dimensional (3D) Euclidean position of static object features. The position of the moving platform is assumed to be measurable, and a second object with known 3D Euclidean coordinates relative to the world frame is considered to be available a priori. To account for the unknown camera calibration parameters and to estimate the unknown 3D Euclidean coordinates, an adaptive least squares estimation strategy is employed based on prediction error formulations and a Lyapunov-type stability analysis. The developed estimator is shown to recover the 3D Euclidean position of the unknown object features despite the lack of knowledge of the camera calibration parameters. Numerical simulation results along with experimental results are presented to illustrate the effectiveness of the proposed algorithm. In Chapter 3, a new range identification technique for a calibrated paracatadioptric system mounted on a moving platform is developed to recover the range information and the three-dimensional (3D) Euclidean coordinates of a static object feature. The position of the moving platform is assumed to be measurable. To identify the unknown range, first, a function of the projected pixel coordinates is related to the unknown 3D Euclidean coordinates of an object feature. This function is nonlinearly parameterized (i.e., the unknown parameters appear nonlinearly in the parameterized model). An adaptive estimator based on a min-max algorithm is then designed to estimate the unknown 3D Euclidean coordinates of an object feature relative to a fixed reference frame which facilitates the identification of range. A Lyapunov-type stability analysis is used to show that the developed estimator provides an estimation of the unknown parameters within a desired precision. Numerical simulation results are presented to illustrate the effectiveness of the proposed range estimation technique. In Chapter 4, optimization of antiangiogenic therapy for tumor management is considered as a nonlinear control problem. A new technique is developed to optimize antiangiogenic therapy which minimizes the volume of a tumor and prevents it from growing using an optimum drug dose. To this end, an optimum desired trajectory is designed to minimize a performance index. Two controllers are then presented that drive the tumor volume to its optimum value. The first controller is proven to yield exponential results given exact model knowledge. The second controller is developed under the assumption of parameteric uncertainties in the system model. A least-squares estimation strategy based on a prediction error formulation and a Lyapunov-type stability analysis is developed to estimate the unknown parameters of the performance index. An adaptive controller is then designed to track the desired optimum trajectory. The proposed tumor minimization scheme is shown to minimize the tumor volume with an optimum drug dose despite the lack of knowledge of system parameters. Numerical simulation results are presented to illustrate the effectiveness of the proposed technique. An extension of the developed technique for a mathematical model which accounts for pharmacodynamics and pharmacokinetics is also presented. Futhermore, a technique for the estimation of the carrying capacity of endothelial cells is also presented

Uncalibrated Visual Compass from Omnidirectional Line Images with Application to Attitude MAV Estimation

International audienceThis paper presents a new algorithm based on previous results of the authors, for the estimation of the yaw angle of an omnidirectional camera robot undergoing a 6-DoF rigid motion. Our real-time algorithm is uncalibrated, robust to noisy data, and it only relies on the projection of 3-D parallel lines as image features. Numerical and real-world experiments conducted with an eye-in-hand robot manipulator, which we used to simulate the 3-D motion of a Micro unmanned Aerial Vehicle (MAV), show the accuracy and reliability of our estimation algorithm

Adaptative Markov Random Fields for Omnidirectional Vision

International audienceImages obtained with catadioptric sensors contain significant deformations which prevent the direct use of classical image treatments. Thus, Markov Random Fields (MRF) whose usefulness is now obvious for projective image processing , can not be used directly on catadioptric images because of the inadequacy of the neighborhood. In this paper, we propose to define a new neighborhood for MRF by using the equivalence theorem developed for central catadioptric sensors. We show the importance of this adaptation for a motion detection application

A Camera Self-Calibration Method Based on Plane Lattice and Orthogonality

 The calibration using orthogonal line is one of the basic approaches of camera calibration, but it requires the orthogonal line be accurately detected, which makes results of error increases. This paper propose a novel camera self-calibration technique using plane lattices and virtual orthogonal line. The rigorous analytical relations among the feature point coordinates of the plane lattice, the corresponding image point coordinate, intrinsic parameters, relative pose are induced according to homography matrix of the central projection. Let a slope of non-parallel and non-orthogonal virtual line in the lattice plane, and the slope of its orthonormal line can be calculated. In at least three photographs taken, vanishing points can be solved in two groups of orthogonal directions by using the homography matrix, so the camera intrinsic parameters are linearly figured out. This method has both simple principle and convenient pattern manufacture, and does not involve image matching, besides having no requirement concerning camera motion. Simulation experiments and real data show that this algorithm is feasible, and provides a higher accuracy and robustness