Correcting Radial Distortion by Circle Fitting

The process and definition of camera calibration have change greatly over recent years. Aerial metric cameras calibration, for which laboratory and field calibration procedures were a separate process, was performed before and independent of any actual mapping data collection using precise calibration fixtures with an assumption that the camera parameters determined would remain valid for a significant period. In contrast, non-metric cameras are characterized by unstable intrinsic parameters over the times and they are vulnerable to the engine and other vibrations during flight data acquisitions. Moreover, there is no standard calibration procedures exist for these cameras. But, since non-metric camera self-calibration augments the concept of calibration as a part of the measurement process, it can determine the camera intrinsic parameters at the time of the data acquisition as long as highly-convergent geometry and the use of multiple exposures are employed. Therefore, this paper investigates variations of the lens distortion components with object distance within the photographic field by using the self-calibration method. The use of redundant flight paths and tilted camera is also incorporated to ascertain the stability of the principal distance and the principal points during the flight mission. During the experiments, a series of flight mission is conducted to photograph test field areas from over a relatively flat area to highly mountainous one. It is revealed that the radial, decentering, and affinity distortion parameters are more stable than that of the principal distance and principal points against vibrations.

In this paper, we present a robust technique of stereo calibration using homography constraints. Our method is novel as stereo calibration is performed by solving a polynomial equation system including two radial distortion parameters, using a minimal number of five image point correspondences. This enables us to calibrate from only a pair of stereo images of a planar scene, and to provide the exact algebraic solution to the stereo calibration problem. The minimal case solution is useful to reduce the computation time and increase the calibration robustness when using random sample consensus (RANSAC) from the correspondences of the stereo image pair. Further, a non-linear parameter optimization for the intrinsic and extrinsic parameters of stereo cameras is performed using the inliers, which are determined after RANSAC. In addition, our method can achieve more robust calibration results with multiple stereo image pairs by performing joint optimization. In contrast to the previous stereo calibration methods, our method works without requiring any special hardware and has no problems with one stereo image pair, even corrupted by severe radial distortions. Finally, by evaluating our method on both synthetic and real scene data, we demonstrate that our method is both efficient and accurate for stereo calibration.

In cameras with radial distortion, straight lines in space are in general mapped to curves in the image. Although epipolar geometry also gets distorted, there is a set of special epipolar lines that remain straight, namely those that go through the distortion center. By finding these straight epipolar lines in camera pairs we can obtain constraints on the distortion center(s) without any calibration object or plumb line assumptions in the scene. Although this holds for all radial distortion models we conceptually prove this idea using the division distortion model and the radial fundamental matrix which allow for a very simple closed form solution of the distortion center from two views (same distortion) or three views (different distortions). The non-iterative nature of our approach makes it immune to local minima and allows finding the distortion center also for cropped images or those where no good prior exists. Besides this, we give comprehensive relations between different undistortion models and discuss advantages and drawbacks.

Increased use of digital imagery has facilitated the opportunity to use features, in addition to points, in photogrammetric applications. Straight lines are often present in object space, and prior research has focused on incorporating straight–line constraints into bundle adjustment for frame imagery. In the research reported in this paper, object–space straight lines are used in a bundle adjustment with self–calibration. The perspective projection of straight lines in the object space produces straight lines in the image space in the absence of distortions. Any deviations from straightness in the image space are attributed to various distortion sources, such as radial and decentric lens distortions. Before incorporating straight lines into a bundle adjustment with self–calibration, the representation and perspective transformation of straight lines between image space and object space should be addressed. In this investigation, images of straight lines are represented as a sequence of points along the image line. Also, two points along the object–space straight line are used to represent that line. The perspective relationship between image– and object–space lines is incorporated in a mathematical constraint. The underlying principle in this constraint is that the vector from the perspective centre to an image point on a straight–line feature lies on the plane defined by the perspective centre and the two object points defining the straight line. This constraint has been embedded in a software application for bundle adjustment with self–calibration that can incorporate point as well as straight–line features. Experiments with simulated and real data have proved the feasibility and the efficiency of the algorithm proposed.

Increased use of digital imagery has facilitated the opportunity to use features, in addition to points, in photogrammetric applications. Straight lines are often present in object space, and prior research has focused on incorporating straight‐line constraints into bundle adjustment for frame imagery. In the research reported in this paper, object‐space straight lines aw used in a bundle adjustment with self‐calibration. The perspective projection of straight lines in the object space produces straight lines in the image space in the absence of distortions. Any deviations from straightness in the image space are attributed to various distortion sources, such as radial and decentric lens distortions. Before incorporating straight lines into a bundle adjustment with self‐calibration, the representation and perspective transformation of straight lines between image space and object space should be addressed. In this investigation. images of straight lines are represented as a sequence of points along the image line. Also, two points along the object‐space straight line are used to represent that line. The perspective relationship between image‐ and object‐space lines is incorporated in a mathematical constraint. The underlying principle in this constraint is that the vector from the perspective centre to an image point on a straight‐line feature lies on the plane defined by the perspective centre and the two object points defining the straight line. This constraint has been embedded in a software application for bundle adjustment with self‐calibration that can incorporate point as well as straight‐line features. Experiments with simulated and real data have proved the feasibility and the eficiency of the algorithm proposed.RésuméLe développement de l'imagerie numérique a fourni l'occasion de recourir davantage aux détails des objets, et non pas seulement aux points qui les constituent, duns toute application photogrammétrique. C'est ainsi que les objets présentent souvent des lignes droites qu'il était tentant, dans une recherche antérieure, d'introduire pour contraindre la compensation par faisceaux d'imageries photographiques. On présente dans cet article cette recherche où les lignes droites de l'espace objet sont utilisées dans une compensation par faisceaux avec auto‐étalonnage. En l'absence de distorsions, la projection de lignes droites de l'espace objet dans l'espace image s'opère également sous forme de lignes droites. Tout écart à une droite sur l'image peut done être attribuéà toutes sortes de distorsions, comme la distorsion radiale ou cello due au décentrement de l'objectif. Avant d'utiliser ces lignes droites dans une compensation par faisceaux avec auto‐étalonnage, il faut efectuer la représentation et la transformation perspective des lignes droites entre les espaces objet et image. Dans cette démarche, on considère les images des lignes droites comme constituées d'une suite de points jalonnant cette image tandis que duns l'espace objet cette ligne droite n'est définie que par deux points seulement. La relation de perspective qui relie les droites des espaces objet et image est alors introduite comme contrainte mathématique. Le principe de base de cette contrainte est que le vecteur issu du centre perspectif vers un point image d'une ligne droite de l'objet appartient au plan défini par re centre perspectif et les deux points retenus dans la définition de cette droite. On a incorporé cette contrainte dans un logiciel appliquéà la compensation par faisceaux avec auto‐étalonnage prenant en compte les points ainsi que les éléments en ligne droite de l'objet. Des essais avec des données simulées puis réelles ont montré la faisabilité et l'efficacité de l'algorithme proposé.ZusummenfussungDurch die zunehmende Nutzung digitaler Bilder wurde die Möglichkeit geschaffen, neben Punkten auch Objektmerkmale in photogrammetrischen Anwendungen zu nutzen. Oftmals finden sich Geraden in Objektraum, und frühere Forschung hat sich darauf konzentriert, Linienbedingungen in die Bündelausgleichung für Flächenkameras zu entwickeln. In den hier vorgestellten Forschungen werden Geraden im Objektraum in einer Bündelausgleichung mit Selbstkalibrierung eingesetzt. Wenn keine Verzeichnungen vorliegen, werden bei einer perspektiven Abbildung Geraden im Objektraum in Geraden im Bildraum abgebildet. Jegliche Abweichung von einer Gerden m Bildraum kann mit verschiedenen Ursachen für Verzeichnung in Verbindung gebracht werden, wie zum Beispiel radiale order asymmetrische Objektivverzeichung. Bevor Gerden in die Bündelausgleichung mit Selbstakalibrierung eingehen können, sollte die Repräsentation und die perspektive Transformation der Gerden zwischen Bild‐ und Objektraum geklärt werden. In dieser Untersuchung werden die Abbildungen der Geraden als eine Sequenz von Punkten entlang einer Bildlinie dargestellt. Zwei Punkte entlang der Objektgeraden werden genutzt, um diese darzustellen. Die perspektive Beziehung zwischen Bild‐ und Objektgeraden wird mit Hilfe einer mathematischen Bedingung formuliert. Das Prinzip, das dieser Beziehung zugrunde liegt, geht davon aus, dass ein Vektor vom Projektionszentrum zu einem Bildpunkt auf einer Geraden auf einer Ebene liegt, die durch das Projektionszentrum und die zwei Objektpunkte, die die Gerade definieren, bestimmt wird. Diese Bedingung wurde in ein Anwendungsprogramm zur Bündelausgleichung mit Selbstakalibrierung eingbaut, das sowohl Punkt‐ als auch Geradenmerkmale verarbeiten kann. Experimente mit simulierten und echten Datensätzen belegen die Anwendbarkeit und die Effizienz des vorgeschlagenen Algorithmus.

In the vertical parking process, the issue of turning in place due to discontinuities in path curvature is addressed by proposing an optimal reference path planning method based on circular arcs, straight lines, and a multi-objective evaluation function. This method first analyzes the obstacle avoidance constraints between the vehicle’s outer contour and the parking space, as well as the vehicle’s kinematic constraints. The feasible driving region’s upper and lower boundaries are determined by tangent circular arcs and straight lines. Subsequently, a multi-objective evaluation function is designed, which integrates path curvature, adjustable margins at any given moment, and path length, to obtain the optimal circular arc and straight line combination within the feasible region. Finally, the path is fitted using a polynomial curve to form the optimal reference path. Simulation results demonstrate that the planned path satisfies both the continuity of path curvature and the vehicle’s kinematic constraints.

In this paper, we propose an image-based technique for panoramic novelview generation using three uncalibrated wide-angle images as its input. State of the art in novel view generation presumes the calibration and removal of radial distortion or any other deformation resulting from t he geometry of a non-central camera. We propose a method which replaces this calibration with the assumption that the epipole corresponding to the novel viewpoint is at the center of radial distortion and that it is known. 1 Introduction In novel view synthesis, given an image pair, the intensity of each ray in the novel view is determined by finding corresponding point matches betwee n those images, using the epipolar geometries between the two views and with respect to the novel view. This process requires that the given images obey the standard perspective model, with a single viewpoint and no radial distortion. Such assumptions make the creation of novel panoramas a process of acquiring tens of images, each with relatively narrow field-of-view, with special apparatus, and applying calibration and estimatio n of circular motion [9]. In this paper, we propose an approach where a novel view can be synthesized from multiple views which might be highly radially distorted, and even non-central, without compensating explicitly for the resulting image deformations. We shed new light on the problem (Fig. 1) of “how a scene looks from a scene point?”[8] by eliminating the need for a reference plane and facilitating synthesis of omnidre ctional novel views. We assume that, for n ≥ 3 views, the location of the epipole of the novel view in each image is known, and, in the case of radially distorted views, that it is coincident with the centre of radial distortion in the image. This assumption is either enforced (for example by choosing a visible scene point as the centre of projection of the view to be synthesized, and actually fixating on this point in the case of radially dis torted views), or else it may naturally be satisfied for certain wide-angle imaging syste ms (for example multiple views of a spherical mirror). Given such a configuration, the 2D sta r of virtual rays at the novel viewpoint are mapped to a 1D star of lines in each image, which we model as a projection from P 2 (the domain of the novel view) to P 1 (the angles of the line pencil in image plane). We make use of the well known fact that three projections of a line in space (a virtual line in our case) yield a trifocal constraint, from w hich we extract the projection matrices required for the view synthesis. The main contributions of this paper are:

In many AR applications, 2D-3D projective mapping is usually modeled with the linear pin-hole camera model. However, the images of real cameras suffer from lens distortion. We present a method to correct the radial distortion caused by camera lens. As the radial distortion correction is achieved via a non-iterative process, the proposed method is computationally efficient and very fast. The accuracy and robustness of the proposed system are verified on the experimental results of several indoor and outdoor images.

Improving radial lens distortion correction with multi-task learning

Recently developed cameras in the low-cost sector exhibit lens distortion patterns that cannot be handled well with established models of radial lens distortion. This study presents an approach that divides the image sensor and distortion modeling into two concentric zones for the application of an extended radial lens distortion model. The mathematical model is explained in detail and it was validated on image data from a DJI Mavic Pro UAV camera. First, the special distortion pattern of the camera was examined by decomposing and analyzing the residuals. Then, a novel bi-radial model was introduced to describe the pattern. Eventually, the new model was integrated in a bundle adjustment software package. Practical tests revealed that the residuals of the bundle adjustment could be reduced by 63% with respect to the standard Brown model. On the basis of external reference measurements, an overall reduction in the residual errors of 40% was shown.

The Hough transform is a well known technique for detecting straight lines within images, especially in the presence of noise, or where there is incomplete data (gaps or occlusions). When subjected to lens distortion, straight lines become curved, and indeed this can be used to identify and correct lens distortion. However, curved lines distort and blur the peaks within the Hough transform, making the lines more difficult to detect. However, by analysing the distortion within the Hough transform, it is possible to directly estimate the lens distortion parameters enabling the distortion to be corrected in real time. The proposed technique uses a Terasic DE1-SoC FPGA board (Cyclone V FPGA) to fit a parabola to the distorted peak using Hough's original transform, and from the parabola coefficients directly estimates the lens distortion parameter. This enables the following frame to be corrected in parallel with curve detection.

Lens distortion is the result of radial variations in the lens magnification. It occurs with most low cost wide angle cameras and affects image quality. As a result, straight lines, particularly those in the periphery, appear curved. One dominant lens distortion correction approach is based on linear features within the image. This kind of correction method usually requires detecting edges or lines, and the Hough transform is a powerful tool to do so. In the Hough transform space, straight and curved lines yield patterns with different shapes. In this paper, a proof of concept is explored whereby lens distortion parameters are derived directly from analysing the shape of peaks within Hough transform space. Subsequently, by using this information, the lens distortion can be corrected. This is demonstrated both through simulation and correction of real world images.

The images captured by real cameras often suffer from lens distortion which is a nonlinear and radial distortion. In this work, we present a method to correct the radial distortion caused by camera lens. In particular, we discuss a framework to determine distortion vectors and distortion function directly without employing approximation approach which often causes incorrect warping problem. The proposed framework is entirely non-iterative, and therefore is computationally efficient. We also compute the center of distortion, which is important to obtain optimal results. The accuracy and robustness of the proposed system are verified on the experimental results of several indoor and outdoor images.

This paper presents an effective method to automatically segment and fit the edge contour curve data of a picture into straight line segments, circular arc segments or elliptical arc segments. Firstly, the curvature of each point is calculated, and the curvature change breakpoints are found to distinguish between straight lines, circular arcs and elliptical arcs; then, circular arc segments and elliptical arcs are distinguished by the change degree of curvature of adjacent points, and the automatic segmentation of contour edges is realized. Finally, the circular arc segment and elliptic arc segment are fitted by least squares method, based on which the length of the straight line, the direction angle of the circular arc, the rotation angle of the elliptic arc and other related information are calculated. The results show that our model can better identify the feature texture information of image edges.

This article presents research results of the signal-to-interference ratio dependence (S/I) on the spatial structure of the radiation field of a three-element linear sparse antenna array in communication systems with spatial signal separation. As a result of the research, it was stated that when the receiving linear antenna is located in the region of the zero order maximum of the field of the useful signal, the formation of the S/I ratio does not depend on whether the field distribution is considered on a straight line or on a circular arc. The location of the receiving antenna in the region of the first-order maximum of the field of the useful signal gives the best results in the formation of the S/I ratio if the field distribution is used on circular arc and not on a straight line.