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Czech Technical University, Faculty of Electrical EngineeringDepartment of Control Engineering, Center for Machine Perception12135 Prague 2, Karlovo n�am�est�� 13, Czech Republicfax +4202 24357385, phone +4202 24357465, http://cmp.felk.cvut.czftp://cmp.felk.cvut.cz/pub/cmp/articles/svoboda/svobCVWW98.ps.gz.
Central Panoramic Cameras:Design and GeometryTom�a�s SvobodaTom�a�s PajdlaV�aclav Hlav�a�[email protected]�a�s Svoboda, Tom�a�s Pajdla and V�aclav Hlav�a�c. CentralPanoramic Cameras: Design and Geometry, third Computer Vi-sion Winter Workshop, Gozd Martuljek, Slovenia, February, 1998February 27, 1998
Central Panoramic Cameras: Design and Geometry �Tom�a�s Svoboda, Tom�a�s Pajdla and V�aclav Hlav�a�cCzech Technical UniversityCenter for Machine PerceptionKarlovo n�am. 13, 121 35 Prague, Czech Republictel. (+420 2) 2435 7458, fax: (+420 2) 2435 7385e-mail: fsvoboda,pajdla,[email protected] contribution gives the foundations of the useful panoramic cameras for stereovision. The approach to a perspective camera{hyperbolic mirror system design ispresented. The model of image formation by a central panoramic camera is de�ned.The analysis of epipolar geometry for panoramic cameras is the main growth of thispaper. We show that the panoramic cameras with convex hyperbolic or parabolicmirrors, central panoramic cameras, allow the epipolar geometry as perspectivecameras do. It is shown that the epipolar curves in central panoramic images areconics and their equation is derived. A simple adjustment procedure of a perspectivecamera and a hyperbolic mirror in order to form a proper central panoramic camerais proposed. This research was primarily motivated by looking for an improvementof the motion estimation from a pair of images but the results are also applicable forstructure reconstruction from panoramic stereo images. The theory is demonstratedby an experiment with real data which corroborates the conclusions drawn from thetheory.Keywords: computer vision, omnidirectional vision, epipolar geometry, panoramiccameras, hyperbolic mirror, stereo, catadioptric sensors.1 IntroductionIt is well known that egomotion estimation algorithms cannot, in some cases, well dis-tinguish a small pure translation of the camera from a small rotation. An example is atranslation parallel to the image plane and a rotation around an axis perpendicular to�This research was supported by the Czech Ministry of Education grant VS96049, the internalgrant of the Czech Technical University 3097472/335, the Grant Agency of the Czech Republic, grants102/95/1378, 102/97/0480, 102/97/0855 and European Union grant Copernicus CP941068.
the direction of the translation [6]. The confusion becomes dominant when the depthvariations in the scene are small or if the �eld of view is narrow.The confusion can be removed if a camera with a large �eld of view is used [6].Ideally, one would like to use a panoramic camera which has complete 360� �eld of viewand sees to all directions. Panoramic camera can, in principle, obtain correspondencesfrom everywhere independently of the direction of egomotion. The uncertainty of themotion estimation will therefore also become independent of the motion direction. Theabove intuitive reasoning has been formalized by the result of Brodsk�y et al. [3] who showthat the motion estimation is almost never ambiguous fro the spherical imaging surface.1.1 The problems to be solvedThe algorithm for egomotion estimation from a pair of perspective images has to (1) �ndthe corresponding points, (2) estimate the fundamental matrix describing the epipolargeometry, (3) calibrate the camera, and (4) decompose the fundamental matrix into atranslation and rotation components.Motion estimation from panoramic images requires: (1) to design a practical panoramiccamera with simple mathematical model and propose method for its calibration, (2) todevelop the epipolar geometry for panoramic images, essential mathematical tool dealingwith two and more images, and (3) to work out an algorithm for motion estimation. Thiscontribution mainly refers to the design of panoramic camera and the epipolar geometryof two panoramic cameras.1.2 Work related to panoramic stereoThough other researchers have already realized that the use of panoramic cameras im-proves egomotion estimation, little attention has been paid to developing the geometryof moving panoramic images as is the epipolar geometry [7] for perspective camera. Themost related works by others are by Benosman et al. [2], Yagi et al. [19, 20], Southwellet al. [16], Chahl et al. [4], an most recently by Nene et al. [14]. Benosman et al. usestwo 1024� 1 line cameras rotating around a vertical axis. He gets two panoramic imagesbut does not calculate epipolar geometry since the corresponding features lie trivially inthe same column. The disadvantage is in complicated construction of the sensor. Yagi atal. [19] developed a panoramic camera with a conic mirror. He uses it for detection of anazimuth of vertical lines. He integrates an acoustics sensor with the optical one and �ndsthe trajectory of a mobile robot. In [20] he uses a hyperbolic mirror, however he detectsswiveling motion analyzing an optical ow and develops no epipolar geometry. Southwellet al. [16] proposes an idea of the stereo with one camera and concentric double lobedmirror but he does not present solid mathematical background. Chahl and Srinivasan [4]draw a method for range estimation using panoramic sensor with conic mirror. Theyplay on range dependency of motion{induced deformation of the panoramic image. Mostrecently Nene and Nayar [14] o�er a stereo sensor built from one standard camera andtwo mirrors.
A number of authors use panoramic cameras for fast visualization of a complete sur-roundings of the observer or as a source of images in order to construct a scene represen-tation for virtual reality [5, 8, 9, 12, 13].2 Design of a panoramic cameraSeveral di�erent designs of panoramic camera emerged recently. We do not considerpanoramic vision systems with moving parts since they are not applicable for real timeimaging due to considerable time needed to capture a panoramic image. Fleck [8] useswide{angle lenses to capture a panoramic image covering a hemisphere. However, suchlenses are large and expensive [5]. A panoramic camera covering almost a whole imagingsphere can be obtained by combining a classical perspective (pinhole) camera with aconvex mirror. Yagi [19] uses a conic{shaped mirror and hyperbolic mirror [20]. Hamit [9]describes various approaches to getting panoramic images using di�erent types of mirrors.Nayar presents [13, 9] several prototypes of panoramic cameras using a parabolic mirror incombination with orthographic cameras. Chahl and Srinivasan [5] study re ective surfacespreserving a linear relationship between the angle of incidence of light on the surface andthe angle of re ection onto the imaging device. Most recently, independently of us, Bakerand Nayar [1] describe class of the mirrors preserving single e�ective viewpoint.The cameras based on convex mirrors seem to be very useful. They o�er large �eldof view (approx. 360� � 150�), instant image acquisition (at video rate), compact design,cheap (220$ for our mirror) production, and the freedom to choose the shape of the mirrorin order to yield a nice mathematical model of the camera.2.1 Shape of the mirrorA panoramic camera with a mirror, see Figure 1, consists of a perspective camera lookinginto a convex mirror. The ray p1 going from (or coming into) the camera is re ected bythe mirror onto a ray p01. Each ray p has to pass through the camera center of projectionC. Re ected rays p0 can but need not to intersect at the same point. Figure 1(a) shows apanoramic camera with a spherical mirror in which case the re ected rays do not intersectat the same point while Figure 1(b) shows the upper part of hyperboloid of two sheets,further called hyperbolic mirror, where all the re ected rays intersect at the focal point ofthe mirror F 0. The camera center of projection C coincides with the second mirror focusF . Figure 1(c) shows the parabolic mirror. Re ected rays p01; p02 intersect in the focalpoint F 0 of the mirror and the second mirror focus F is at in�nity thus the orthographicprojection has to be used.Here we focus on the case when all re ected rays intersect at a single point. This is animportant property which is necessary condition for the existence of epipolar geometrythat is inherent to the moving sensor and independent of the scene structure. Panoramiccameras which posses this property shall be called central panoramic cameras. In thiscase the model of the panoramic camera can be decomposed into a central projection
p1p2p10
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C(a) Spherical mirror anda perspective camera.The re ected opticalrays do not intersectin a unique point andthe sensor su�ers fromspherical aberration.C = F
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(b) Hyperbolic mirrorand a perspective cam-era. The re ected opti-cal rays intersect in thefocus of the hyperboloid.
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(c) Parabolic mirror anda perspective camera.The re ected opticalrays intersect in thefocus of the paraboloidwhen orthographicprojection is assumed.Figure 1: Three combinations lens{mirror.from space onto a curved surface of the mirror and a central projection from the surfaceof the mirror into the image plane.2.2 Model of a panoramic camera with a hyperbolic mirrorFigure 2 shows the composition of a perspective camera with a hyperbolic mirror sothat the camera projection center C coincide with the focal point of the mirror F . Theperspective camera can be modeled by an internal camera calibration matrix K whichrelates 3D coordinates X = [x; y; z]T into retinal coordinates q = [qu; qv; 1]T .q = 1zKX; (1)See [7] for more information about camera calibration.
Xh X�rtop
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C = Fqu
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Figure 2: The geometry of the mirror and the camera.In the \mirror coordinate system", which is centered at the focal point F 0, a hyperbolicmirror is de�ned by the equation(z +pa2 + b2)2a2 � x2 + y2b2 = 1; (2)where a, b are parameters of the mirror. The image formation can be expressed as acomposition of the coordinate transformations and projections. We want to �nd therelationship between a point X in the world coordinates and the camera point q in theretinal coordinate system. The derivation of the image formation is omitted here, detailscan be found in [17]. Complete model can be written concisely asq ' KRC�F�(RM(X � tM))RM(X � tM)� tC�; (3)where ' denotes equality up to similarity, RC ; tC characterize the transformation betweenthe mirror coordinate system and the camera coordinate system, RM ; tM denote thetransformation between the world and the mirror frame and where F�(RM(X � tM)) is
given by the following nonlinear function of a vector v = [v1; v2; v3]T :F�(v) = b2(�ev3 + ajjvjj)b2v23 � a2v21 � a2v22 : (4)There are 6 external calibration parameters (3 for tM and 3 for RM) and 9 internalparameters (two for the mirror, two for the rotation RC , and 5 for K). The matrix RChas only two free parameters as it is used to model the angle between the image plane andthe axis of the mirror. The translation vector tC = [0; 0;�2e]T is indispensable, since theprojection center of the camera has to coincide with the second focus of the hyperboloid.In order to establish the equations for the epipolar geometry, it is necessary to �nd thevector Xh for each image point q. The formula for computing Xh from pixel coordinatesq reads as: Xh = F+(RTCK�1q)RTCK�1q+ tC ; (5)where F+(RTCK�1q) is given by equation (4).2.3 Design of a useful hyperbolic mirrorThe shape of the real mirror has to be designed carefully since there are two main re-quirements: (1) The camera{mirror system has to be compact as much as possible sinceit is to be used on a mobile robot, (2) the spatial angle of view has to be close to wholesphere. The maximum value of the angle �, see Fig 2 is� = �2 + atan�h� 2ertop �; (6)where h is the distance between the camera center C and the top of the mirror and rtopis the radius of the mirror rim, Figure 2, which value has to be determined by a designer.Using mirror equation (2) h can be computed fromh = e+ as1 + r2topb2 : (7)It is obvious that � is a function of a; b. In order to achieve a good image resolution, theprojected mirror rim has to occupy the whole image, which claim determined projectedradius ru in normalized image coordinates u = K�1q. A designer draws up the valuesrtop and ru. Then height h can be computed from equations (6,7) ash = rtopru : (8)Knowing h, the ratio a=b can be determined from the equation (7). The maximum ofangle � increases with increasing a=b. However the size (height) of the mirror increasesas well. Looking at Fig 3, the mirror with a=b = 1:5 gives us good resolution in the areaclose to the horizontal direction of view. Although the maximum vertical angle of viewmight be not su�cient, a large angle of view, when a=b = 3, lacks good resolution close tothe horizontal direction. A compromise is necessary. In [17] we argue that suitable rangeis 2 < a=b < 3.
F
F 0 p1 p2p10 p20
(a) a = 30, b = 20,�max = 108�. F
F 0 p1 p2p10p20
(b) a = 30, b = 15,�max = 129�. C = F
F 0 p1 p2p10p20
(c) a = 30, b = 10,�max = 151�.Figure 3: The shape of hyperbolic mirror with di�erent ratio a=b. Camera parametersremain the same.3 Epipolar geometryIn perspective cameras, the corresponding points q1; q2 in the camera pair are related bywell-known relationship qT2Qq1 = 0; (9)where Q represents the fundamental matrix, see [7]. Epipolar geometry of perspectivecameras assigns to each point in one image an epipolar line in the second image. Thequestion arises what is the shape of the epipolar curves for central panoramic cameras.We de�ne the relationship between the coordinate frames of the mirrors in terms ofthe displacement between the sensors (between the local coordinate systems centered atmirror foci F 01 and F 02): the translation vector t and the rotation matrix R. Let the pointson the mirrors related to the one 3D point be denoted Xh1 resp. Xh2. These vectors arecoplanar with the translation vector t. This coplanarity can be written asXh2R(t ^Xh1) = 0; (10)where ^ denotes the vector product. Introducing an antisymmetry matrix SS = 264 0 �tz tytz 0 �tx�ty tx 0 375 ; (11)
II.
mirrorI.
mirror
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camera projection center
y
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x
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C2
F2’Xh2Xh1
π
Xn
e1’q1e1
e2q2
e2’
R,t
Figure 4: The epipolar geometry of two panoramic cameras with hyperbolic mirrors.we can rewrite the coplanarity constraint (10) in the matrix form asXhT2EXh1 = 0; (12)where matrix E = RS stands for the essential matrix. The essential matrix E can be usedhere instead of a fundamental matrix since vectors Xhi are metric entities. The vectorsXh1, Xh2 and t form the epipolar plane �. This plane intersects the mirrors shaping spaceconics. This conic is then mapped onto another conic in the image plane by a central(perspective) projection. To a point q1 in the �rst image a conic is uniquely assigned inthe second image qT2A2(E;q1) q2 = 0: (13)In the general case the matrix A2(E;q1) is a nonlinear function of the essential matrix E,the point q1, and the calibration parameters of the panoramic cameras and the mirrors.In this paper we suppose that the camera internal parameters, matrix K, are the samefor both sensors, but nothing precludes us from using K1 and K2 instead of just a singleK. All epipolar conics pass through two points which are the images of the intersectionof mirrors with the line F 01F 02. Therefore there are usually two epipoles, denoted e1 ande01 resp. e2 and e02 in Figure 4. The epipoles can degenerate into one double epipole if thecamera is translated along the axis of the mirror.After some intensive algebra, see [17, 18], the equation (13) can be rewritten asqT2K�TRCBRTCK�1q2 = 0; (14)
leaving us with A2 = K�TRCBRTCK�1, whereB = 264 �4s2a2e2 + p2b4 pqb4 psb2(�2e2 + b2)pqb4 �4s2a2e2 + q2b4 qsb2(�2e2 + b2)psb2(�2e2 + b2) qsb2(�2e2 + b2) s2b4 375 (15)is a nonlinear function of a, b, and[p; q; s]T = E(F+(RTCK�1q1)RTCK�1q1 + tC); (16)where F+(RTCK�1q1) is de�ned by equation (4). The equation (14) de�nes the curve onwhich the projected corresponding point has to lie and it is indeed an equation of a conicas alleged by equation (13).3.1 Using the epipolar geometryThe epipolar geometry presented above can be used similarly as in standard perspectivecameras. Finding correspondences is the one reason to use it. Once the epipolar geome-try between two panoramic images is established the search for correspondences is nicelyreduced to 1 DOF problem. At least 8 correspondences is needed to solve essential matrixlinearly (12), using method [11], for instance. Knowing the essential matrix E we cancompute epipolar conic for each point of interest in the �rst image, on which the corre-sponding point has to lie in the second image. We can then employ an iterative algorithmto establish epipolar geometry and to �nd correspondences more robustly [21].Let the essential matrix E be robustly estimated, from equation (12). Recall thatE = RS, where R is a rotation matrix and matrix S is formed by elements of thetranslation vector t, see equation (11). The motion parameters R and t can be recoveredusing the approach described in [10], for instance.4 ExperimentsBefore we had a real mirror we did test with synthetic images. Synthetic scene wasrendered by POV-ray software package1, in which we also de�ned the hyperbolic re ectivesurface and the perspective camera. We did tests with the \bar" scene taken from SCEDpackage2.A pair of panoramic images is shown in Figure 5. We can see three points of interestin the left image and the epipolar conics passing through the corresponding points in theright image. Epipolar conics intersect in two epipolar points, the normalized displacementis t = [1; 1; 1]T for this pair. Since we use an synthetic scene, the model perfectly matches.We can see that the epipolar conic goes exactly through the corresponding point and threeconics exactly intersect in the epipolar point, Figure 6.1http://www.povray.org2http://http.cs.berkeley.edu/~schenney/sced/sced.html
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300(c) The epipolar conics in-tersect in the epipole.Figure 6: Details from the synthetic images, see Figure 5.Real dataOur panoramic (omnidirectional) camera is dedicated to be used in our mobile robot. Thecamera is beneath the mirror. We designed the mirror to get high vertical angle of view.In [17] can be found the background for the values a = 28:1851 mm and b = 9:3950 mm.The mirror was manufactured by ASTRO|Telescope, P�rerov, Czech Republic.AssessmentWe calibrated the standard CCD camera o�{line, using the method and the equipmentfrom [15]. The focal length f is 8.5 mm and the camera resolution is 768 � 512. Toposition camera correctly, we projected the calibration circle and cross into image and
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550Figure 7: Assemblage of the sensor. The mirror �ts the projected calibration circle andthe calibration cross.tuned the position until the mirror �t, see Fig 7. The axis of the mirror has to �t thecalibration cross and the mirror rim has to �t the calibration circle.SequenceWe captured a motion sequence through our laboratory. The area, which sensor observes,is approximately 3�3 meters. The camera is below the mirror and the sensor was movingon the oor. Two consecutive images are shown in Fig 8. The points were selectedmanually. There is no rotation between the two positions, because it is di�cult to rotatethe real sensor in a controlled way. The translation between two positions is t = [60; 60; 0]cm. Although we used the very simple method to adjust the camera|mirror position,results are accurate, see Figure 9. The epipolar conics pass very close to the correspondingpoint. The inaccuracy is less than two pixels, which is su�cient to �nd correspondencesalong conics. Three conics intersect in the epipole exactly.5 SummaryThis paper presented the foundations of the central panoramic cameras. We presented theapproach how to design a useful panoramic camera and we proposed a simple adjustmentmethod. We de�ned the image formation function for such a camera. Our main contri-bution is the epipolar geometry for panoramic cameras. We have shown that panoramiccameras using convex hyperbolic or parabolic mirrors, so called central panoramic cam-eras, can be decomposed into two central projections and therefore allow for the sameepipolar geometry as perspective cameras. It has been shown that epipolar curves areconics and their equation was derived. The theory has been con�rmed by experimentswith synthetic and real data.
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550Figure 8: Two panoramic images. Translation in x and y axis. In the left image thepoints of interest are marked. On the right image the related epipolar conics are drawn.The conics pass through the corresponding points and intersect in two epipoles.Points of interest − image RealExp/Test07/seq2.bmp
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335(c) The epipolar conics in-tersect in the epipole.Figure 9: Details from the real images, see Figure 8.Even the very simple adjustment method of the sensor was used, results corroborateour theory. The epipolar conics lie close to the corresponding points. The errors are lessthan two pixels. Such accuracy is su�cient to �nd correspondences along conics usingsome correlation method.There are many questions which remain behind the scope of this paper. It is a questionhow lower resolution will a�ect the quality of motion estimation. Special care needs tobe paid to �nding an automatic calibration and adjustment method for real mirrors andperspective cameras. We will analyze how the bad adjustment of the camera and themirror in uences the precision of the epipolar geometry. Further, we will experiment withego-motion computation.
References[1] Simon Baker and Shree K. Nayar. A theory of catadioptric image formation. Sub-mitted to IJCV, 1997. Also appears in ICCV98, and exists as TR CUCS-015-97, andit was presented in Darpa IUW 1997.[2] R. Benosman, T. Maiere, and J. Devars. Multidirectional stereovision sensor, calibra-tion and scenes reconstruction. In International Conference on Pattern Recognition1996, Vienna, Austria, pages 161{165. IEEE Computer Society Press, September1996.[3] T. Brodsk�y, C. Fernm�uller, and Y. Aloimonos. Directions of motion �elds are hardlyever ambiguous. In Buxton. B. and R. Cipola, editors, 4th European Conference onComputer Vision, Cambridge UK, volume 2, pages 119{128, 1996.[4] J.S. Chahl and M.V. Srinivasan. Range estimation with a panoramic visual sensor.Journal of the Optical Society of America, 14(9):2144{2151, September 1997.[5] J.S. Chahl and M.V. Srinivasan. Re ective surfaces for panoramic imaging. AppliedOptics, 36(31):8275{8285, November 1997.[6] K. Daniilidis and H.-H. Nagel. The coupling of rotation and translation in mo-tion estimation of planar surfaces. In IEEE Conf. on Computer Vision and PatternRecognition, pages 188{193, New York, NY, June 1993.[7] Olivier Faugeras. 3-D Computer Vision, A Geometric Viewpoint. MIT Press, 1993.[8] Margaret M. Fleck. Perspective projection: The wrong imaging model. Researchreport 95-01, Department of Computer Science, University of Iowa, Iowa City, USA,1995.[9] Francis Hamit. New video and still cameras provide a global roaming viewpoint.Advance Imaging, pages 50{52, March 1997.[10] Richard I. Hartley. Euclidean reconstruction from uncalibrated views. In Workshopon Invariants in Computer Vision, pages 187 { 202, October 1993.[11] Richard I. Hartley. In defence of the 8-point algorithm. In Fifth International Con-ference on Computer Vision, MIT Cambridge Massachusetts, pages 1064{1070. IEEECopmuter Society Press, 1995.[12] Hiroshi Ishiguro, Masashi Yamamoto, and Saburo Tsuji. Omni{directional stereo.IEEE Transactions on Pattern Analysis and Machine Inteligence, 14(2):257{262,February 1992.[13] Shree K. Nayar. Catadioptric omnidirectional camera. In International Conferenceon Computer Vision and Pattern Recognition 1997, Puerto Rico USA, pages 482{488.IEEE Computer Society Press, June 1997.
[14] Sameer A. Nene and Shree K. Nayar. Stereo with mirrors. In Sharat Chandranand Uday Desai, editors, The Sixth International Conference on Computer Vision inBombay, India, pages 1087{1094, 6 Community Centre, Panscheel Park, New Delhi110 017, January 1998. N. K. Mehra for Narosa Publishing House.[15] Tom�a�s Pajdla. BCRF - Binary Illumination Coded Range Finder: Reimplementa-tion. ESAT MI2 Technical Report Nr. KUL/ESAT/MI2/9502, Katholieke Univer-siteit Leuven, Belgium, April 1995.[16] David Southwell, Anup Basu, Mark Fiala, and Jereome Reyda. Panoramic stereo. InInternational Conference on Pattern Recognition 1996, Vienna, Austria, volume A,pages 378{382. IEEE Computer Society Press, August 1996.[17] Tom�a�s Svoboda, Tom�a�s Pajdla, and V�aclav Hlav�a�c. Central panoramic cameras: Ge-ometry and design. Research report K335/97/147, Czech Technical University, Fac-ulty of Electrical Engineering, Center for Machine Perception, December 1997. Avail-able at ftp://cmp.felk.cvut.cz/pub/cmp/articles/svoboda/TR-K335-97-147.ps.gz.[18] Tom�a�s Svoboda, Tom�a�s Pajdla, and V�aclav Hlav�a�c. Epipolar geometry for panoramiccameras. In the �fth European Conference on Computer Vision, Freiburg, Germany,June 1998. appears in 6/1998.[19] Yasushi Yagi, Yoshimitsu Nishizawa, and Masahiko Yachida. Map-based navigationfor a mobile robot with omnidirectional images sensor COPIS. IEEE Transaction onRobotics and Automation, 11(5):634{648, October 1995.[20] Yasushi Yagi, Kazuma Yamazawa, and Masahiko Yachida. Rolling motion estimationfor mobile robot by using omnidirectional image sensor hyperomnivision. In Inter-national Conference on Pattern Recognition 1996, Vienna, Austria, pages 946{950.IEEE Computer Society Press, September 1996.[21] Zhengyou Zhang, Rachid Deriche, Olivier Faugeras, and Quang-Tuang Luog. Arobust technique for matching two uncalibrated images through the recovery of theunknown epipolar geometry. Arti�cial Inteligence, 78:87{119, October 1995. AlsoResearch Report No.2273, INRIA Sophia-Antipolis.
