In this paper we propose a new calibration method for non-central catadioptric cameras that use a conical mirror. This method consists of using parallel lines, extracted from a single omnidirectional image, instead of using the typical checkerboard to obtain the calibration parameters of the system

Bermúdez Cameo, Jesús

Bermúdez Vargas, James

Guerrero, José J.

Spanish

Universidad Zaragoza: Open Journal Systems

Revista “Jornada de Jóvenes Investigadores del I3A”, vol. 10 (Actas de la XI Jornada de Jóvenes Investigadores del I3A – 16 de junio de 2022). ISSN 2341-4790. Parallel Lines for Calibration of Non-Central Conical Catadioptric Cameras James Bermúdez Vargas, Jesús Bermúdez Cameo,  José J. Guerrero Afiliación: Robótica, Percepción y Tiempo Real (RoPeRT) Instituto de Investigación en Ingeniería de Aragón (I3A) Universidad de Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain. Tel. +34-976762707, e-mail: 792069@unizar.es  Abstract In this paper we propose a new calibration method for non-central catadioptric cameras that use a conical mirror. This method consists of using parallel lines, extracted from a single omnidirectional image, instead of using the typical checkerboard to obtain the calibration parameters of the system. Introduction Our goal is to propose a new calibration method for Non-Central Catadioptric Systems, specifically those that use a conical mirror.  And our main contribution is to be able to make this calibration from man made environments and not depend on the traditional calibration pattern (chessboard type) that is currently used. Environments like corridors and sidewalks, for example, where we can find long and parallel lines (see Figure 1). Image projection in Non-Central Catadioptric Systems [1], [2] differs a lot from Central Catadioptric Systems [3].  In our case, we work with those Non-Central who make use of a conical mirror, whose projective geometry has been previously studied in [4] (with the Unitary Torus Model), whose position is conditioned by the aperture angle of the mirror, explained in [5]. Our principal contribution is that from a single omnidirectional image (2D), which contains the projection of two parallel lines, we can obtain the calibration parameters of the entire non-central catadioptric system.  The main advantage of using lines is that they are present in many environments and we do not need a special pattern to perform the calibration. The Plücker coordinates of a 3D line is a homogeneous representation of a line L ∈ ℙ5. This representation can be organized in such a way that they form two vectors, L = (l, 𝑙 ̅ )T = (l1, l2, l3, 𝑙1̅, 𝑙2̅, 𝑙3̅)T.  One, the director vector (l), which represents the direction of the line; the other, the moment vector (𝑙)̅, which represents the normal to the plane formed by the line and the origin of the reference system. In conical catadioptric systems with the camera aligned to the axis of revolution of the mirror, the locus of the viewpoint is a circle of radius Rc, centered on the axis of revolution and at a distance Zc of the optical center of the camera [4] (see Figure 2, where the circle is on the mirror surface).  Calibration Method The first step is to identify two parallel lines within the omnidirectional 2D image, then we take at least five points from each one of them (see Figure 3) to start with their respective adjustment. We know from [6] that with five points we can linearly calculate the line-image (𝜌𝑗). 𝜌𝑗 = −?̂?𝑖,𝑗𝑝𝑖𝑛𝑣(?̂?𝑖,𝑗?̂?𝑛(𝑖,𝑗), ?̂?𝑖,𝑗?̂?𝑛(𝑖,𝑗), ?̂?𝑖,𝑗2 , ?̂?𝑛(𝑖,𝑗), ?̂?𝑛(𝑖,𝑗)), for  i = 1..5, and  j = 1..2 Introducing f in this last equation, our new line-image (𝜌j) is represented by (𝜔1𝑓2,𝜔2𝑓2,𝜔3𝑓2,𝜔4𝑓,𝜔5𝑓,𝜔6𝑓)𝑇 and expressing it in plücker coordinates we have the following: 𝜌𝑗 =(   𝜌𝑗,1𝜌𝑗,2𝜌𝑗,3𝜌𝑗,4𝜌𝑗,5𝜌𝑗,6)   =(       1𝑓2((1 − 𝑐𝑜𝑠2𝜏)𝑐𝑜𝑠2𝜏𝑍𝑣𝑙𝑗,2 − 𝑙?̅?,1)−1𝑓2((1 − 𝑐𝑜𝑠2𝜏)𝑐𝑜𝑠2𝜏𝑍𝑣𝑙𝑗,1 + 𝑙?̅?,2)𝑙3̅𝑡𝑎𝑛2𝜏𝑡𝑎𝑛2𝜏(𝑙1̅ + 𝑍𝑣𝑙2)𝑡𝑎𝑛2𝜏(𝑙2̅ − 𝑍𝑣𝑙1)𝑙3̅ )        Then solving our system of equations, we get Lj (in function of 𝜌𝑗) in Plücker coordinates. Revista “Jornada de Jóvenes Investigadores del I3A”, vol. 10 (Actas de la XI Jornada de Jóvenes Investigadores del I3A – 16 de junio de 2022). ISSN 2341-4790. Figure 1: Parallel lines in front of a Conical Catadioptric System. 𝐿𝑗 =(     𝑙𝑗,1𝑙𝑗,2𝑙𝑗,3𝑙?̅?,1𝑙?̅?,2𝑙?̅?,3)     =(          −𝑓2𝑠𝑖𝑛2𝜏𝜌𝑗,2 + 𝑓𝑐𝑜𝑠2𝜏𝜌𝑗,5𝑍𝑣𝑡𝑎𝑛2𝜏𝑓2𝑠𝑖𝑛2𝜏𝜌𝑗,1 + 𝑓𝑐𝑜𝑠2𝜏𝜌𝑗,4𝑍𝑣𝑡𝑎𝑛2𝜏𝑓2𝑐𝑜𝑠2𝜏(𝜌𝑗,2𝜌𝑗,4 − 𝜌𝑗,1𝜌𝑗,5)𝑍𝑣𝑡𝑎𝑛2𝜏𝜌𝑗,6𝑓(1 − 𝑐𝑜𝑠2𝜏)𝜌𝑗,4 − 𝑓2𝑠𝑖𝑛2𝜏𝜌𝑗,1𝑡𝑎𝑛2𝜏𝑓(1 − 𝑐𝑜𝑠2𝜏)𝜌𝑗,5 − 𝑓2𝑠𝑖𝑛2𝜏𝜌𝑗,2𝑡𝑎𝑛2𝜏𝑓𝜌𝑗,6 )           We know that two lines are parallel if and only if their direction vectors are linearly dependent: 𝑙1,1𝑙2,1=𝑙1,2𝑙2,2=𝑙1,3𝑙2,3 Taking the following equalities, (𝑙1,1𝑙2,1=𝑙1,3𝑙2,3) and (𝑙1,2𝑙2,2=𝑙1,3𝑙2,3), and replacing with their respective values, we get two equations respectively for the calibration parameters sought: 𝑓𝑡𝑎𝑛2𝜏 = −𝜌1,5𝜌1,6(𝑢) − 𝜌2,5𝜌2,6(𝑣)𝜌1,2𝜌1,6(𝑢) − 𝜌2,2𝜌2,6(𝑣) 𝑓𝑡𝑎𝑛2𝜏 = −𝜌1,4𝜌1,6(𝑢) − 𝜌2,4𝜌2,6(𝑣)𝜌1,1𝜌1,6(𝑢) − 𝜌2,1𝜌2,6(𝑣) where: u= 𝜌2,1𝜌2,5 – 𝜌2,2𝜌2,4;  v= 𝜌1,1𝜌1,5 – 𝜌1,2𝜌1,4 Also, from 𝜌1 and 𝜌2 we can deduce:  𝜌𝑗,3𝜌𝑗,6=𝑡𝑎𝑛2𝜏𝑓 As an example of our experimental results, we can show the error in the calculation of the aperture angle of the mirror (𝜏) measured in radians and compared with the ground (see the boxplot of the Figure 4). Conclusion In this paper we propose a new method for calibrating conical catadioptric images from the projection of two parallel lines.  We have shown that working in a selected projection it is possible to decouple the focal length of the camera f and the aperture angle of the mirror 𝜏 in an analytical procedure. Acknowledgments This work was supported by project RTI2018-096903-B-100 (MINECO/FEDER, UE), and the government of Panamá under the IFARHU–SENACYT scholarship program for PhD studies. REFERENCES [1]. BERMÚDEZ-CAMEO, J., LÓPEZ-NICOLÁS, G. and GUERRERO, J. J.  Fitting line projections in non-central catadioptric cameras with revolution symmetry.  Computer Vision and Image Understanding, vol. 167, pp. 134–152, 2018. [2]. AGRAWAL, A., TAGUCHI, Y. and RAMALINGAM, S.  Analytical forward projection for axial non-central dioptric and catadioptric cameras.  In European Conference on Computer Vision (ECCV 2010). Springer, 2010, pp. 129–143. [3]. BARRETO, J. P. and ARAUJO, H.  Geometric properties of central catadioptric line images and their application in calibration.  IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 8, pp. 1327–1333, 2005. [4]. LÓPEZ-NICOLÁS, G. and SAGÜÉS, C.  Unitary torus model for conical mirror based catadioptric system.  Computer Vision and Image Understanding, vol. 126, pp. 67–79, 2014. [5]. BAKER, S. and NAYAR, S. K.  A theory of single-viewpoint catadioptric image formation.  International Journal of Computer Vision, vol. 35, no. 2, pp. 175–196, 1999. [6]. BERMUDEZ-CAMEO, J., LÓPEZ-NICOLÁS, G., and GUERRERO, J. J. Lineimages in cone mirror catadioptric systems.  In 2014 22nd International Conference on Pattern Recognition. IEEE, 2014, pp. 2083–2088.         Figure 3: Synthetic Images with Parallel lines. Figure 2: Projection of a line in Conical Catadioptric System. Figure 4: Aperture angle (𝝉) error (in pixels) in function of the applied Gaussian Noise. 

Parallel Lines for Calibration of Non-Central Conical Catadioptric Cameras

https://papiro.unizar.es/ojs/index.php/jji3a/article/download/7007/5819

Parallel Lines for Calibration of Non-Central Conical Catadioptric Cameras

Abstract

Similar works

Full text

Available Versions

Universidad Zaragoza: Open Journal Systems