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

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

Fitting line projections in non-central catadioptric cameras with revolution symmetry

Line-images in non-central cameras contain much richer information of the original 3D line than line projections in central cameras. The projection surface of a 3D line in most catadioptric non-central cameras is a ruled surface, encapsulating the complete information of the 3D line. The resulting line-image is a curve which contains the 4 degrees of freedom of the 3D line. That means a qualitative advantage with respect to the central case, although extracting this curve is quite difficult. In this paper, we focus on the analytical description of the line-images in non-central catadioptric systems with symmetry of revolution. As a direct application we present a method for automatic line-image extraction for conical and spherical calibrated catadioptric cameras. For designing this method we have analytically solved the metric distance from point to line-image for non-central catadioptric systems. We also propose a distance we call effective baseline measuring the quality of the reconstruction of a 3D line from the minimum number of rays. This measure is used to evaluate the different random attempts of a robust scheme allowing to reduce the number of trials in the process. The proposal is tested and evaluated in simulations and with both synthetic and real images

Proyecciones cónicas de rectas en sistemas catadióptricos para percepción visual en entornos construidos por el hombre

Los sistemas de visión omnidireccional son dispositivos que permiten la adquisición de imágenes con un campo de vista de 360º en un eje y superior 180º en el otro. La necesidad de integrar estas cámaras en sistemas de visión por computador ha impulsado la investigación en este campo profundizando en los modelos matemáticos y la base teórica necesaria que permite la implementación de aplicaciones. Existen diversas tecnologías para obtener imágenes omnidireccionales. Los sistemas catadióptricos son aquellos que consiguen aumentar el campo de vista utilizando espejos. Entre estos, encontramos los sistemas hiper-catadióptricos que son aquellos que utilizan una cámara perspectiva y un espejo hiperbólico. La geometría hiperbólica del espejo garantiza que el sistema sea central. En estos sistemas adquieren una especial relevancia las rectas del espacio, en la medida en que, rectas largas son completamente visibles en única imagen. La recta es una forma geométrica abundante en entornos construidos por el hombre que además acostumbra a ordenarse según direcciones dominantes. Salvo construcciones singulares, la fuerza de la gravedad fija una dirección vertical que puede utilizarse como referencia en el cálculo de la orientación del sistema. Sin embargo el uso de rectas en sistemas catadióptricos implica la dificultad añadida de trabajar con un modelo proyectivo no lineal en el que las rectas 3d son proyectadas en cónicas. Este TFM recoge el trabajo que se presenta en el artículo "Significant Conics on Catadioptric Images for 3D Orientation and Image Rectification" que pretendemos enviar a "Robotics and Autonomous Systems". En él se presenta un método para calcular la orientación de un sistema hiper-catadióptrico utilizando las cónicas que son proyecciones de rectas 3D. El método calcula la orientación respecto del sistema de referencia absoluto definido por el conjunto de puntos de fuga en un entorno en que existan direcciones dominantes

A Caustic Approach of Panoramic Image Analysis

In this Conference, the problem of blur in a panoramic image from a catadioptric camera is analyzed through the determination of the virtual image. This determination is done first with an approximative method, and second through the caustic approach. This leads us to a general caustic approach of panoramic image analysis, where equations of virtual images are given. Finally, we give some direct applications of our analysis, such as depth of field (blur) or image resolution