Folded catadioptric panoramic lens with an equidistance projection scheme

Transcription

1 Folded catadioptric panoramic lens with an equidistance projection scheme Gyeong-il Kweon, Kwang Taek Kim, Geon-hee Kim, and Hyo-sik Kim A new formula for a catadioptric panoramic lens with an equidistance projection scheme has been derived. The fabricated lens has a field of view that is wider than that of any previously reported panoramic lens, and the nonimaged region near the back of the camera has a constant volume with zero angular extension Optical Society of America OCIS codes: , , Introduction One obtains a panoramic image via the traditional method by stitching multiple images taken at a single viewpoint while aiming in different directions. 1,2 On the other hand, catadioptric panoramic lens permits monitoring in directions of a full 360 with a single camera. 3,4 The large field of view typically attainable with a catadioptric panoramic lens is not surpassed by that of the fish-eye lens. 5 7 Scenic (panorama) photography, security and surveillance, unmanned vehicles and robots, real estate and resort advertisement, and remote conferencing are among the promising application areas for the panoramic lens As schematically shown in Fig. 1, a typical catadioptric panoramic lens is composed of a standard camera lens and a rotationally symmetric panoramic mirror. 12 The symmetry axis the same as the z axis in the figure of the panoramic mirror coincides with the optical axis of the camera. We will designate the ray before the reflection at the panoramic mirror as the incident ray and the ray after the reflection at the mirror as the reflected ray. The angle of the incident ray and the angle of the reflected ray are measured as zenith angles. In other words, the incidence angle is the angle subtended by the positive z axis and the incident ray. Similarly, the reflection angle G.-i. Kweon (kweon@honam.ac.kr) and K. T. Kim are with the Department of Optoelectronics, Honam University, 59-1, Seobongdong, Gwangsan-gu, Gwangju, , South Korea. G.-h. Kim and H.-s. Kim are with the Operation and Research Division, Korea Basic Science Institute, 52, Yeoeun-dong, Yusung-gu, Daejeon, , South Korea. Received 20 August 2004; revised manuscript received 29 November 2004; accepted 11 December /05/ $15.00/ Optical Society of America is the angle subtended by the positive z axis and the reflected ray. The shape of the mirror and the distance between the panoramic mirror and the camera lens is such that an incident ray with a zenith angle is reflected at a point M on the panoramic mirror and passes through the nodal point N of the camera lens with a zenith angle. The nodal point is the location of the pinhole in the pinhole camera model. The nodal point N of a given lens can be found with the nodal slide method. It is said that the nodal point can be located with an accuracy better than 1 mm. Because of the geometric construction of the catadioptric panoramic lens, scenes from directions of a full 360 are captured within an annular region in the image sensor I, and this annular image can be subsequently unwarped with a dedicated software. At present, the most preferred mirror shape is the hyperbola in that it allows an effective single viewpoint. 13 One can obtain other popular mirror profiles, including the sphere, the cone, the parabola, and more complex shapes, by solving complex numerical equations. 14,15 However, the hyperbolic or other mirrors have varying image resolutions for different incidence angles. To remedy this problem, other researchers have investigated a mirror shape that has an equiangular projection scheme. 14 In this scheme the ratio between the differential increment in the angle of incidence and the differential increment in the angle of reflection is a constant: d g, (1) d where g is a constant defining the gain of the mirror. However, image sensors for artificial optical sys- 10 May 2005 Vol. 44, No. 14 APPLIED OPTICS 2759

2 Fig. 1. Schematic diagram illustrating the operational principle of the catadioptric panoramic lens. tems such as roll film, CCD, and complementary metal-oxide semiconductor sensors have only planar structures. Then the best projection scheme is the equidistance projection scheme rather than the equiangular projection scheme. 5 In the equidistance projection scheme, the ratio between the differential increment in the pixel distance on the image sensor and the differential increment in the angle of incidence is a constant: d m, (2) d where m is another constant. For this reason, the high-end fish-eye lens aims to achieve an ideal equidistance projection scheme. However, the cost is high, and the success is not without flaws. Another shortcoming of the current panoramic lenses is the fact that the nadir the region near the negative z axis is excluded from the imaged region because the camera occludes the view of the panoramic mirror. This can be a severe limitation for certain applications such as security and surveillance. The purpose of the present investigation is to design a panoramic lens whose field of view extends all the way to the nadir and still accomplishes the equidistance projection scheme. 2. Panoramic Mirror Profile with the Equidistance Projection Scheme Figure 2 shows the relevant variables needed for analyzing the panoramic mirror profile of the current design. Because the panoramic mirror is rotationally Fig. 2. Schematic diagram illustrating specular reflections in the panoramic mirror along with the variables needed for analyzing the mirror profile. symmetric in structure, only the right half of the mirror profile is shown. The shaded region is the cross section of the panoramic mirror, and the thicker curve in the cross section denotes the mirrored surface. As schematically shown in the figure, the panoramic mirror of the current design has a nonmirrored region around the symmetry axis. This nonmirrored region can exist as a circular hole around the center of the panoramic mirror or it can be a region of the mirror simply not used for imaging. In the latter case, the rays reflected from the center region are not used for panoramic image processing. The origin of the coordinate system coincides with the camera s nodal point, and the line of sight of the camera is along the positive z axis. The complete mirror profile can be written as a set of three curvilinear coordinates r,, in the spherical polar coordinate or as,, z in circular cylindrical coordinates. In this notation, r is the radial distance from the origin to point M on the mirror surface, is the zenith angle (i.e., polar angle), is the azimuth angle, is the perpendicular distance from the z axis, and z is the distance measured along the z axis. Owing to the rotationally symmetric structure, the mirror profile is degenerate about the azimuth angle, and therefore a set of r, or, z is sufficient for a complete description of the mirror profile. Because the polar angle is unique for every point on the mirror, the radial distance r can be parameterized in terms of [i.e., r r ]. Therefore becomes the independent variable, and r becomes the dependent variable. By a similar argument, the mirror profile can be alternatively given as z z. Furthermore, the two variables in the cylindrical coordinate can 2760 APPLIED OPTICS Vol. 44, No May 2005

3 also be parameterized in terms of the polar angle as r sin and z r cos. Owing to the presence of the central nonimaging region, the range of the polar variable has a lower limit and an upper limit denoted by 1 and 2 (i.e., 1 2 ). The corresponding inner and outer diameters of the mirror surface are given as 2 1 and , respectively. Tangent plane T at the point of reflection M subtends an angle of with the z axis: tan d dz. (3) Because the tangent function is diverging near 2, we invert Eq. (3) to obtain a cotangent function of the inclination angle : cot dz dz d d d d. (4) From the previous observations on the functional form of and z, Eq. (4) can be readily reduced to r cos r sin cot r sin r cos, (5) where r= is the derivative with respect to (i.e., r dr d ). Then we can easily separate the radial and the angular coordinates in Eq. (5): r sin cot ( )cos r cos cot ( )sin. (6) The left-hand side and the right-hand side of Eq. (6) can be independently integrated as dr r sin cot ( )cos d. (7) cos cot ( )sin We take i as the lower bound of the integration in the right-hand side. Although not limited to 1 and 2, i is usually chosen between 1 and 2. i and z i are the corresponding variables in the cylindrical coordinate with the polar angle i, and r i r i i 2 z i is the radial distance with the same polar angle. Then, for an arbitrary polar angle, the radial distance is given in the form of an indefinite integral: r( ) r( i )exp i sin cot ( )cos cos cot ( )sin d. (8) To evaluate the integration given in Eq. (8), one must know the functional relation between the inclination angle and the polar angle. The functional form of is not arbitrary, however, because a constraint is given from the law of specular reflections as ( ) 2. (9) Once we know, then we automatically know the zenith angle for the incident ray: ( ) 2 ( ). (10) In other words, by specifying the functional form of, we are specifying the projection scheme (i.e., versus ) for the panoramic lens. As schematically shown in Fig. 2, the distance from the camera s nodal point N to the image sensor I is close to the nominal camera focal length f. The reflected ray with a zenith angle is captured by a pixel in the image sensor with a distance from the z axis. Therefore the pixel distance is related to the zenith angle as f tan. (11) After the camera has been adjusted for a sharp focus, the distance f between the camera s nodal point N and the image sensor I is a constant. Therefore Eq. (11) combined with Eq. (2) dictates that tan be a linear function of as given by tan, (12) where and are constants. Making a comparison with Eq. (2), we note that m f. The range of the reflected ray is given as 1 2. We assume that the corresponding range of the incident ray is given as 1 2. Then the two constants in Eq. (12) are uniquely determined as tan 2 tan 1 2 1, (13) 2 tan 1 1 tan (14) Then, with Eq. (12), the inclination angle given in Eq. (9) can be written as a sole function of : ( ) tan. (15) 2 When the functional form of given in Eq. (15) is substituted into Eq. (8), the desired mirror profile can be easily obtained with canned numerical integration routines such as the MATLAB function cumtrapz. 3. Mirror Design Figure 3 shows a schematic diagram of the image formation. The image sensor is assumed as W mm wide and H mm high, and the distance from the camera s nodal point N to the image sensor I is denoted by 10 May 2005 Vol. 44, No. 14 APPLIED OPTICS 2761

4 Fig. 3. Camera field of view determined by the image sensor size and the focal length of the lens. f. Then the vertical field of view is given as V tan 1 (H 2f). (16) For most of the electronic image sensors, the width W is larger than the height H. Therefore rays having a polar angle larger than V will be cropped by the boundaries of the image sensor. For a complete circle to form in the image sensor, the largest angle of reflections 2 must be smaller than V. The camera we have chosen for the experiment is the Canon EOS 300D. It has a complementary metaloxide semiconductor image sensor that measures 22.7 mm wide and 15.1 mm high. The shortest focal length of the dedicated camera lens (EF-S, mm, f ) is 18 mm. When the camera is used with the widest field of view hence with the shortest focal length the vertical field of view is Allowing some safety margins, we have chosen 2 as 20, and 1 has been arbitrarily chosen as 10. If 1 is too large, the breadth of the annular image will be too thin, and the height of the unwarped image measured in pixels will be too low. If 1 is too small, on the other hand, then the width of the unwarped image measured in pixels will be narrow because the perimeter of the image circle with that polar angle is too short. Therefore a compromise between these two extremes is needed, and a value of 10 seems to be a reasonable choice. For the range of the incident rays, the minimum angle 2 is chosen to be 45, and the maximum angle 1 is chosen to be 180. Note that 1 is not necessarily smaller than 2. 1 is simply the incident angle corresponding to the reflected ray with the reflection angle 1, and, similarly, 2 is the incidence angle corresponding to the reflection angle 2. The nonimaging region in the center of the mirror is taken as to have a diameter of 10.0 cm (i.e., cm). This large diameter is chosen so that the camera lens barrel that measures approximately 8.0 cm can go through the center hole in the mirror (the reason for this particular choice will become clear in Section 4). Figure 4(a) shows the mirror profile obtained by use of Eq. (8) for the chosen design parameters. As can be seen from the figure, the minimum distance Fig. 4. (a) Profile of a panoramic mirror with the equidistance projection. The design parameters are 1 10, 2 20, 1 180, 2 45, and i cm. (b) The magnified view of the mirror in (a) near the center hole. Note that the figure is not to scale. between the mirror surface and the camera s nodal point the origin of the coordinate system is nearly 29.0 cm. In the numerical integration, 1 has been chosen as the lower bound of the integral (i.e., i 1 ). In that the inner radius of the mirror 1 is chosen to be 5.0 cm, then r i r 1 is given as r 1 1 sin 1. Figure 4(b) shows the details of the mirror profile near the inner rim. Note that the graph is not in scale. As is clear from the figure, the mirror profile is slightly wrapped up at the rim. This is necessary for an incident ray with zenith angle 1 180º to be reflected at the rim of the mirror and pass through the nodal point. The obtained mirror profile was fitted by use of the sixth-order polynomial given as z( ) n 0 6 Cn n, (17) where C n is the coefficient of the nth power term in. Table 1 shows all the fitting coefficients. The sixthorder polynomial was the lowest-order polynomial wherein the discrepancy between the designed and the fitted curves was less than 1 m throughout the whole interval between 1 and 2. Figure 5 shows the designed mirror profile (dotted curve) along with the polynomial fitting (solid curve). As can be inferred from the shape of the fitted curve, the mirror profile 2762 APPLIED OPTICS Vol. 44, No May 2005

5 Table 1. Fitting Coefficients of the Panoramic Mirror Profile by Use of Sixth-Order Polynomial in Variables Coefficients C C C C C C C is not a simple convex. Rather, the mirror surface looks like the lower part of a doughnut. The projection scheme of the designed mirror can be checked directly from the mirror profile. As schematically shown in Fig. 2, the input ray with a zenith angle is reflected at the mirror point (, r ) (, z ), and the reflected ray passes through the origin (i.e., the camera s nodal point) and is subsequently captured in the image sensor. This ray trajectory can be calculated with the mirror profile and the law of specular reflections. For this purpose, the range of the reflected ray angles 1 2 is equally divided into N 1 intervals. In our particular example, a 10 span between 10 and 20 is divided into six equal intervals, and therefore we have seven different reflection angles s between 10 and 20. Then the coordinates of the sampled mirror points in the cylindrical coordinate are obtained as s, z s ( s, z s ) (r s sin s, r s cos s ). By use of Eq. (17), the derivative of z with respect to at the sampled mirror point is obtained as dz d s n 1 6 ncn s n 1. (18) Fig. 5. Mirror profile in Fig. 4 (dotted curve) along with the best numerical fit (solid curve) by use of a sixth-order polynomial. Fig. 6. Calculated ray trajectories for the panoramic mirror in Fig. 4. The maximum allowable diameter of the camera is D. The inclination of the tangent plane at the sampled mirror point is given as ( s ) dz 2 tan 1 d s. (19) Finally, the incidence angle s can be obtained with Eq. (10). Figure 6 shows the ray trajectories along with the mirror profile. The reflected ray ends at the image plane not shown in the figure. The focal length of the camera is taken as 18 mm. From the graph we note that the incident rays are equally spaced in angles. In this orientation of the camera in which the optical axis is pointing toward the zenith, the vertical field of view is from the nadir (the foot of the camera) to 45 above the horizon. For the camera not to show up in the image, the cross section of the camera lens and the body should be contained within a circle having a diameter D. In this design the lens and the body should be smaller than 10 cm in diameter. Figure 7 shows the input ray directions and corresponding pixel distance in the image sensor. As can be seen, the mirror profile described by Eq. (8) has a perfect equidistance projection scheme. 4. Folded Mirror Design Oftentimes, it is helpful to fold the optical path. In our case, it is helpful to fold the optical path because it changes the imaged area from the back side of the camera to the front side, and therefore the camera is actually heading the imaging scene. 16 When 1 is equal to as in the case of Fig. 6, then the catadioptric lens can capture the rays coming from the opposite back of the camera. As has been discussed in Section 3, for the camera not to show up in the image, the largest diameter of the camera lens and body should be smaller than 2 1, where 1 is the 10 May 2005 Vol. 44, No. 14 APPLIED OPTICS 2763

6 Fig. 9. Schematic diagram illustrating the location and the size of the planar ring mirror in a folded design. Only the case with is considered. Fig. 7. Zenith angles of the incident rays versus the corresponding distances in the image sensor. The distance is measured from the optical axis to the pixel that the reflected ray hits. inner radius of the panoramic mirror. Therefore, for a given camera, the mirror should be designed big enough so that the entire cross section of the camera resides within a circle of 2 1 diameter. On the other hand, for a given mirror dimension, a small camera such as a bullet camera must be employed to ensure an unobstructed view. One can alleviate this problem by employing an inverting-type panoramic mirror. Figure 8 shows another example of a panoramic mirror with the equidistance projection along with the ray trajectories. The design parameters are given as 1 10, i 2 20, 1 70, 2 180, and cm. The outer radius of this mirror is the same as that of the mirror in Fig. 6. Note the vertical field of view is only 20 (i.e., 1 70 ) above the horizon. Incident rays with a smaller zenith angle will be blocked out by the outer rim of the mirror, and therefore the vertical field of view of an invertingtype panoramic mirror is narrower than that of the normal panoramic mirror. Besides this limitation in the field of view, this mirror has the same equidistance projection scheme. Because of the geometry of the inverting-type panoramic mirror, it is only necessary that the largest diameter of the mirror (i.e., 2 2 ) needs to be large enough to contain within its circle the entire cross section of the camera. In a different viewpoint, for the given mirror dimension, a larger camera can be tolerated in obtaining the view of the opposite back of the camera. In this case, the camera can be as large as 24.2 cm in diameter. Fig. 8. Inverting-type panoramic mirror profile along with the ray trajectories. The design parameters are 1 10, 2 20, 1 70, and i cm. Fig. 10. Schematic diagram of a folded panoramic mirror APPLIED OPTICS Vol. 44, No May 2005

7 Fig. 11. Profiles and locations of the curved and the planar mirrors in the folded panorama mirror. The curved mirror is identical to the one depicted in Fig. 4. Another effect of the inverting-type mirror is that the regions around the foot of the camera are imaged in the outer region of the annular image and hence by a larger number of pixels. Therefore this design can be useful when the region of interest is near the back of the camera. As has been shown, an inverting-type panoramic mirror can tolerate a larger camera size. Another way to accommodate a larger camera, especially a larger camera body, is to fold the optical path as schematically shown in Figs. 9 and 10. A folded panoramic mirror is composed of a curved mirror (previously referred to as a panoramic mirror) and a planar ring mirror. A normal-type panoramic mirror such as that shown in Fig. 4 is assumed in this configuration. Fig. 12. Photograph of the fabricated folded panoramic mirror along with the digital camera. When in operation, the camera lens should obtrude through the center hole of the curved mirror. Fig. 13. Schematic diagram of the experimental setup for verifying the projection scheme of the folded panoramic mirror. When 1 is different from 180, then the discussion becomes little more complicated. However, we believe the most interesting case is one in which the field of view starts from the opposite back of the camera (i.e., 1 ) and goes to the front side of the camera. Therefore we will restrict our discussion where 1 and 2 2. Figure 9 shows the location of the camera s nodal point N and the mirror profile. The ray trajectories having two extreme angles (i.e., 1 and 2 ) are also shown. The ensemble of rays having a zenith angle 2 defines a cone having a half-angle 2 with a vertex at the nodal point N. Similarly, the ensemble of rays having a zenith angle 1 defines another cone having a half-angle 1 with the vertex at the same nodal point N. To fold the optical path, one should position the planar mirror between the camera s nodal point N and the lowest point of the mirror (i.e., z z 1 ). At a particular axial position z, the smallest possible planar mirror should have a ring shape in which the inner radius I and outer radius O are defined by the cross section of the two cones with a plane perpendicular to the optical axis at z. In the presence of the planar mirror, the nodal point N will have its mirror image point N=, which is now closer to the curved mirror. By the law of specular reflections, the distance from the original nodal point N to the planar mirror is identical to the distance from the planar mirror to the new nodal point N=. At an axial position with z z o, the incident ray with a zenith angle 1 meets the reflected ray with a zenith angle 2. From the geometry shown in Fig. 9, one can note that z o 1 tan cot 2. (20) 10 May 2005 Vol. 44, No. 14 APPLIED OPTICS 2765

8 Fig. 15. Panoramic image of the university library. The camera is pointing toward the ceiling (i.e., zenith). by the planar mirror is farther from the curved mirror. Because the camera s nodal point lies somewhere within the lens barrel, the camera should obtrude a significant amount through the center hole of the curved mirror. Therefore it will be advantages to place the planar mirror at the shortest harmless distance from the curved mirror, and that distance is at z z o. The mirror profile of the curved mirror in a new coordinate system having the new nodal point N= as the new origin is given as Z( ) z o [z( ) z o ] 2z o z( ). (21) Fig. 14. (a) Exemplary panoramic image taken by use of the setup depicted in Fig. 13. (b) A magnified view of the right side of the planar ring mirror. When the planar mirror is placed between z o and z 1, then the outer radius O of the planar ring mirror should be larger than 1 to reflect all the reflected rays from the curved mirror. However, in this case, the incident ray having a zenith angle 1 will be blocked by the planar ring mirror and never get a chance to be reflected at the curved mirror. Therefore, in order not to diminish the vertical field of view of the curved mirror, the axial position of the planar ring mirror should be between z o and the camera s nodal point N. Once the planar mirror is placed between N and z o, then the planar mirror can have a circular shape with the outer radius smaller than 1 and larger than the radius of the outer cone at the position. The presence or absence of the center hole in the planar mirror does not alter the vertical field of view of the curved mirror. However, when the position of the planar mirror is closer to the nodal point N, then the distance between the curved mirror and the planar ring mirror becomes unnecessarily large. Also, the position of the new nodal point N= the image of the nodal point N The upper limit of the inner radius of the planar mirror is given as I z o tan 1, (22) and the outer radius O of the planar mirror should be identical to 1 : O z o tan 2 1. (23) When the outer radius of the planar mirror is given as in Eq. (23) and the inner radius is smaller than the one given in Eq. (22), then the field of view of the folded mirror will be identical to that of the original panoramic mirror. As schematically shown in Fig. 10, the camera in this folded design is hidden behind the panoramic mirror. Therefore in applications in which the interior of a store needs to be monitored, as an example, the imaging system can be placed into the ceiling with a minimum of obtrusive parts. Therefore it will have a better appearance as well as easier maintenance. 5. Experiment Figure 11 shows the mirror profile and the relative locations of the planar and the curved mirrors of the folded design corresponding to the panoramic mirror in Fig. 4. For a proper operation, the camera s nodal 2766 APPLIED OPTICS Vol. 44, No May 2005

9 Fig. 16. Panoramic image in Fig. 15 after the polar-to-rectangular transformation. point should reside at the origin of the coordinate system. The dimension of the mirror has been chosen such that the Canon EOS 300D camera can be used in conjunction with this mirror. The mirror is fabricated from A16061-T6 by use of an ultraprecision machining tool called Nanoform 600. Because of the large mirror size, the mirror surface has been cut with a root-mean-square surface roughness of 10 m. Figure 12 shows the fabricated folded mirror along with the digital camera. The camera is mounted in a sliding base that one can slide in through the center hole of the curved mirror. Figure 13 shows a schematic diagram of the experimental setup for testing the projection scheme of the folded panoramic mirror. A circle of 3 m radius is drawn with chalk on the floor of an auditorium. Then 24 chairs are placed along the perimeter of the circle at equal intervals. Therefore the angular separation between the neighboring chairs is 15. In the figure the chairs are denoted by circles with internal crosses. The nodal point of the camera has been found by use of the traditional nodal slide method well known in the fields of geometrical optics and panoramic photography. A best effort has been made to place the camera s nodal point as close to the center of the circle as possible. Because the field of view of the folded panoramic mirror is , eight chairs are in the view for each side of the camera. Figure 14(a) shows a sample image taken by use of the setup in Fig. 13, and Fig. 14(b) shows a magnified view of the right side of the planar ring mirror; indeed, eight chairs have been captured for each side of the ring. From the spacing of the chairs in the image, we note that this folded mirror has the equidistance projection scheme. Figure 15 shows a sample picture of the university library taken by use of the folded panoramic mirror. For this purpose, the optical axis of the imaging system is aligned perpendicular to the floor. Figure 16 shows the unwarped image by use of a commercial unwarping software. Not surprisingly, we can see from the ceiling to the floor in a single image. When video cameras are used instead of the still camera, a similar setup can be advantageously used in security and surveillance area. 6. Conclusion In conclusion, we have derived a new equation describing a panoramic mirror profile with the equidistance projection scheme. The test results from the fabricated prototype seem encouraging. We thank Dong-hyun Cha for his enthusiastic assistance during the experimental test of the fabricated panoramic lens. References 1. T. R. Halfhill, See you around, Byte May, (1995). 2. L. D. Paulson, Viewing the world through interactive panoramic images, Computer 37, 28 (2004). 3. H. Ishiguro, Development of low-cost compact omnidirectional vision sensors, in Panoramic Vision: Sensors, Theory, and Applications, R. Benosman and S. B. Kang, eds. (Springer, New York, 2001), pp I. Powell, Panoramic lens, Appl. Opt. 33, (1994). 5. C. Beck, Apparatus to photograph the whole sky, J. Sci. Instrum. 2, (1925). 6. J. M. Slater, Photography with the whole-sky lens, Am. Photographer October 1932, pp K. Miyamoto, Fish-eye lens, Appl. Opt. 54, (1964). 8. K. Yamazawa, Y. Yagi, and M. Yachida, Obstacle detection with omnidirectional image sensor hyperomni vision, in IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp J. S. Chahl and M. V. Srinivasan, Range estimation with a panoramic visual sensor, J. Opt. Soc. Am. A 14, (1997). 10. J. Zeil, M. I. Hofmann, and J. S. Chahl, Catchment areas of panoramic snapshots in outdoor scenes, J. Opt. Soc. Am. A 20, (2003). 11. S. Hrabar and G. S. Sukhatme, Omnidirectional vision for an autonomous helicopter, in Proceedings of the 2003 IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp J. Charles, R. Reeves, and C. Schur, How to build and use an all-sky camera, Astron. April 1987, pp S. Baker and S. K. Nayar, A theory of single-viewpoint catadioptric image formation, Int. J. Comput. Vision 35, (1999). 14. J. S. Chahl and M. V. Srinivasan, Reflective surfaces for panoramic imaging, Appl. Opt. 36, (1997). 15. R. A. Hicks and R. Bajcsy, Reflective surfaces as computational sensors, Image Vision Comput. 19, (2001). 16. S. K. Nayar and V. Peri, Folded catadioptric cameras, in Panoramic Vision: Sensors, Theory, and Applications, R. Benosman and S. B. Kang, eds. (Springer, New York, 2001), pp May 2005 Vol. 44, No. 14 APPLIED OPTICS 2767