AU2424900A - Resolution invariant panoramic imaging - Google Patents

Description

WO 00/42470 PCT/AU00/00022 -1 RESOLUTION INVARIANT PANORAMIC IMAGING FIELD OF THE INVENTION This invention relates to generating wide angle images of spaces, generally referred to as 5 panoramic imaging. BACKGROUND ART Panoramic imaging is becoming an important tool in the area of mobile robotics and machine 10 vision. There are many documented methods for recording a panoramic view of a scene. One simple method involves having a series of cameras mounted on a ring to give views around the entire 3600 of horizon. This involves, say, four cameras if they each have a field of view of 900 and some integration of images. There are also a number of single camera methods for panoramic imaging, including rotating a camera about its vertical axis and taking pictures 15 continuously to obtain a full panoramic view. Another approach uses wide angle lenses to achieve a large field of view, but these lenses are heavy, expensive and distort the image. An attractive approach to panoramic imaging is to mount a single fixed camera under a curved reflective surface covering a hemisphere such as with a conical, spherical, hyperboloidal, or 20 other profile. The optical axis of the camera is aligned with the central axis of the mirror. A known family of constant gain reflective surfaces have the advantage that they can produce large fields of view such as for a hemispherical or hyperboloidal mirror yet preserve a linear relationship between changes in angles of incidence and reflection of light rays viewed by the camera. This linear relationship simplifies image processing and ensures constant elevational 25 resolution of the image. The shape of the surface is determined by the gain of the linear relationship. For a unity gain, the surface is a cone; for higher gains, the surface is specified by a family of polynomial functions. For ease of explanation in this specification the panoramic plane will be considered as being horizontal and the field of view as vertical as would be the case for a robot moving in a horizontal plane. It will be apparent that in the 30 general case orientation of the planes is arbitrary.

wU UU/4247U PCT/AU00/00022 -2 All the mirror shapes mentioned above share a common draw back. That is that the CCD cameras used for imaging invariably have uniform Cartesian arrays of pixels to capture the polar image of the scene, and so the pixel density per solid angle increases with the radius of the polar image. The unwarping process transforms the image from polar to Cartesian 5 coordinates so that the angular coordinate in the original polar image maps to the x-coordinate in the unwarped image while the radial coordinate maps to the y-coordinate. Thus the pixel density in the unwarped image varies from low for small x values which correspond to the centre of the original image to high for large x values which correspond to the outer rim of the polar image. This is illustrated in Figure 1 which shows the unwarping of an image 10 captured with a hyperboloidal mirror. The variation in image quality is clearly evident in the unwarped version. One way to circumvent this problem is to use a specially designed CCD camera with a polar array of pixels with a pixel density which decreases with radius. There are alignment 15 problems with such an approach. DISCLOSURE OF THE INVENTION In a first aspect this invention provides a panoramic imaging system including an imaging device having an image plane and a first field of view, a first reflective surface having at least 20 one circularly symmetric portion convex in a radial direction disposed in said first field of view to provide an expanded panoramic second field of view, the profile of the or each convex portion providing a varying gain between the fields of view in the radial direction to limit variation in the solid angle of view across the image plane of the imaging device. 25 Preferably, the profile of the convex portion provides a substantially uniform solid angle of view across the image plane. That is, the shape ensures that the resolution in the image is invariant to changes in elevation. Thus, where the imaging system involves a device with an array of uniformly spaced pixels in the image plane, the shape of the reflective surfaces results in solid angle pixel density invariance. 30 vvw v iuu o / u PCT/AU00/00022 -3 The profile of the reflective surface in polar coordinates is preferably determined by solving the equation dr -r cot -V1/2 (1 + a (0)) dO dO where r is the radial distance from the reflective surface to the imaging device 5 0 is the angle from the optical axis of the imaging device a (0) is the mirror gain given by a (0) = B, [tan (0) + tan (0)] Ba 2( Bt = tan 2 (O) - tan 2 () 10 < and # are the maximum and minimum elevations viewed U and 0 are the maximum and minimum radial angles imaged. In one approach r can be plotted against 0 at selected intervals to describe the profile by 15 solving the above equation for selected values of 0. For example determining values of r for incremental values of 0 of about 1/50 has been found to produce a sufficiently accurate profile for practical application. There are a number of methods for panoramic range finding. One method uses a cone mirror 20 above a camera. The camera mirror assembly is either displaced during image collection, or two camera mirror assemblies are used to obtain the two views necessary for range finding. Although this method provides range information in the horizontal plane at video rates, its drawbacks are that no range information is available in the vertical (elevation) direction, objects must be more than a minimum distance from the camera and there may be a blind spot ... ..... 'II/AUUU/UUUZZ -4 due to the second camera system. A discontinuous, axially symmetric mirror, which is in essence a coaxial mirror pair, mounted above a camera to obtain two views of a panoramic scene for stereo disparity range finding 5 is known. There are however no proposals concerning specific mirror shapes to achieve specific desirable properties. Additionally, known constant gain mirror profiles have been generalised to derive a family of such coaxial mirror pair profiles for panoramic stereo imaging and processing based on disparities in the vertical plane. 10 In another aspect this invention provides for range finding using a panoramic imaging system containing two resolution invariant mirrors. Preferably the mirror or reflector surface has at least two of said convex portions arranged to respectively provide at least partially overlapping panoramic second fields of view for range determination. The second fields of view are preferably substantially co-incident. In the preferred form of the invention the two 15 convex portions form a continuous mirror or reflective surface. In a further aspect this invention provides a design for a back to back stereo mirror system with the desirable property of equal pixel sharing between two cameras and thus the two stereo images. The stereo cone in this case is preferably symmetric in the directions 20 orthogonal to the camera axis which is a desirable property for some applications. In this aspect of the invention the imaging system preferably includes two first reflective surfaces each having an associated image plane with corresponding first fields of view, and at least one convex portion of each first reflective surface providing respective panoramic second fields of view, said first reflective surface being arranged back to back such that said reflective 25 second fields of view at least partially overlap. A second reflective surface can, in some applications be interposed between the image plane and the second reflective surface. This allows positioning of the imaging device for example behind the first reflective surface. In some variations an apperture can be provided in the first 30 reflective surface to provide the first field of view from the imaging device.

wu U/4L4-/U PCT/AU00/00022 -5 In another aspect this invention provides a reflective surface for use in a panoramic imaging system including an imaging device having an imaging plane and a first field of view, said reflective surface having at least one circularly symmetric portion convex in a radial direction with a profile providing varying gain in the radial direction between an expanded panoramic 5 second field of view provided by the reflective surface and the first field of view to limit variation in the solid angle of view across the image plane of the imaging device. In yet a further aspect this invention provides mirrors having minimal intrusive designs, which intrude to a minimal extent into the viewing "hemisphere". These are also termed forward 10 facing designs. They involve an additional planar mirror and camera relocation within the primary reflective surface. The attraction of this arrangement is that the first reflective mirror surface profile is the same design as in a more conventional arrangement. The invention will be further described, by way of example only with reference to the 15 accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 illustrates an unwarping process for a prior art panoramic imaging system; Figure 2 schematically shows the relationship between camera image and horizontal 20 view direction in a panoramic imaging system; ,uu V .a't/u PCT/AU00/00022 -6 Figure 3 illustrates geometric relationships between a reflecting surface and a camera used to derive mirror profiles according to this invention; Figures 4A and 4B, are graphs showing a comparison of a constant gain mirror with 5 a variable gain mirror used in the imaging system according to this invention; Figures 5A and 5B, shows ray traced scenes respectively reflected in constant and variable gain mirrors; Figures 6A and 6B, graphically illustrates a comparison of panoramic imaging systems respectively utilising double constant and variable gain mirror configurations; 10 Figures 7A and 7B, shows raced traced images of scenes respectively corresponding to panoramic imaging systems utilising double constant and variable gain mirror configurations; Figure 8 schematically illustrates relationships between camera and reflective surfaces used in range calculation utilising a resolution invariant double mirror according to this 15 invention; Figure 9 schematically illustrates a back to back mirror configuration according to this invention; Figure 10 schematically illustrates a double back to back mirror configuration according to this invention; 20 Figure 11 schematically illustrates a forward looking panoramic imaging system according to this invention; and Figure 12 shows a system utilising a combination of the arrangements in Figures 10 and 11. 25 BEST MODE FOR CARRYING OUT THE INVENTION The various aspects of this invention will, for clarity, be described under separate subheadings. 1 Resolution Invariant Mirror Families 30 Vv Tu/sa w u PC'T/AU00/00022 -7 This section describes a family of mirror designs that achieve the objective of resolution invariance, or equivalently solid angle pixel density invariance. 5 1.1 Constant Image Pixel Density - The Variable Gain (a) Mirror In accordance with one aspect of this invention resolution invariance is achieved by adjusting the mirror profile to image relatively less of the scene in the centre of the image and relatively more at the perimeter. That is, a mirror profile is selected to maintain a constant relationship 10 between the pixel density and the angle of elevation in the scene or more precisely, the solid angle. The mirror gain a, is the relationship between the change in elevation of rays incident on the mirror and the change in the angle of rays reflected into the camera as follows a - = 4(1) 15 where 60 is the change in vertical elevation and c8 is the change in angle of reflected rays received by the camera. With resolution invariance a becomes a function of image angle 0 which is related to the radial coordinate in the image, p, as shown in Figure 2. Figure 3 schematically shows an imaging system including an imaging device in the form of 20 a camera having an image plane and a first field of view. A reflective surface or mirror is ... in the first field of view to provide an expanded panoramic second field of view. The surface is circularly symmetric and convex in a radial direction. Consider a mirror profile (r, 0) in polar coordinates where r is the radial distance to the 25 camera and 0 is the angle from the optical axis of the camera to the point on the mirror surface as shown in Fig. 3. The angle of incidence of a light ray relative to the mirror is y and the angle of an incoming light ray with respect to the vertical is d. Then WO 00/42470 PCT/AU00/00022 -8 y = tan- ( rd (2) dr subject to the geometric constraint (from the law of reflection) 2y + 0 + = 7t (3) 5 Differentiating (2) and (3) with respect to 0 dy- d tan- (rd0) From (2) d6 d6 dr dy -I1 From (3) dO 2 d6 so, substituting a from (1) gives dtan' rd)- - (l +a) (4) d6 dr 2 10 Now, for a variable gain mirror, a is a function of image angle 0 (related to the radial coordinate in the image, p) so (4) becomes dtan-' - - 2(1 +a (0)) (5) dA drI= or, rearranging 15 WU UU/42470 PCT/AU00/00022 -9 dr-= r cot[-/2f(1 +a(0))d6] (6) The equation for the mirror gain, a (0) to achieve pixel density invariance can be found using the following theory. 5 1.1.1 Pixel Density Invariance Profiles There are p(p) pixels in an area of radius p in the image. More formally, there are p(p) =7Kp 2 10 pixels in an area of radius p , where K is the number of pixels per unit area, a constant. Differentiating by p gives ap(p) = 27Tp (7) ap Now, the radius in the image, p is related to the radial angle of a ray reflected from the 15 mirror, 0 by the focal length of the camera, f (a constant) p =ftan(O) (8) so differentiating p(p) by 0 and substituting (7) and (8) gives 20 -10 8p(p) _ p(p) ap 00 ap 86 2 8Kpf tan(O) (9) 00 S2tKf 2 tan(0)(1 +tan 2 (0)) Now, it is required that the image pixel density be invariant to angle of elevation in the scene which leads to more of the scene being imaged towards the perimeter, so 5 p(p) = p# + C() (10) where P3 and C(4) are constants. Differentiating both sides of (10) by # and substituting (1) and (9) gives 0 8 p(p) SOp(p) o 86 8 (11) 10 Rearranging (11) gives a(0) = 27rf2K tan(0)[1 + tan 2 (0)] P ) (12) -= B. [tan(0) + tan 3 (0)] where B. is a constant. Integrating this expression for a(0) by 0 gives an expression for 15 (see (1)), the elevation of an object imaged at angle 0. That is WO 00/42470 PCT/AU00/00022 -11 B = - tan 2 (0) + 4)(0 = 0) (13) 2 where # (0 = 0) is a constant of integration. The constants B, and 4) (0 = 0) can be determined from the maximum and minimum values 5 of 0 and $ which are known for a desired mirror configuration, using (13). BU = tan 2 (O) - tan 2 (0) (14) B (0 = 0) = - tan 2 () 2 It appears not possible to find an analytical solution to (6) if a is a function of 0, so there is no explicit equation for the mirror shape. Instead, a differential equation solver is needed to 10 find solutions to (6) over the range of 0 (the mirror surface). Figures 4A and 4B show for comparison a constant gain mirror and a variable gain mirror with the same camera field of view and range of elevations imaged. The rays shown are constantly spaced in 0, with about 20 between each ray. It is clear from Fig. 4A that in the 15 constant gain case these rays are constantly spaced in 4, with about 8.5' between each ray, and from Fig. 4, that the spacing between the rays in the variable gain case increases with increasing 4. So, in the variable gain case, a greater proportion of the scene is imaged towards the outer edge of the polar image. This is also shown in Figures 5A and 5B, ray traced images reflected in a constant gain and variable gain mirror with the same range of 20 elevations visible.

VVwu uu/q~. /u PCT/AU00/00022 - 12 1.2 Panoramic Stereo Using a Variable Gain Mirror A mirror with two convex portions or a double mirror is required. The radial profile of a 5 double mirror is shown in Figure 8. The mirror arrangement for panoramic stereo with variable gain mirrors will necessarily be different than for constant gain mirrors due to the variation of the mirror gain, a. The gain must vary in a constant fashion over the entire double mirror so that the constant pixel density theorem will hold over the entire image. If the minimum and maximum elevations viewed (4) 10 and b ) are to be equal for both mirrors in the double mirror system, the range of reflected angles (6 - 0 ) cannot be equal for the two mirrors. The minimum and maximum angles of reflected rays captured by the camera over the entire mirror surface are known from camera geometry. Therefore the minimum ray reflected from the lower mirror (0 ) and the maximum ray reflected from the upper mirror (2 ) are known. So, since (12) holds over the 15 entire mirror, B. is constant, and from (14) 2( - ) 2( -) (tan 2 (1) - tan (6)) tan 2 (-) - tan 2 ( 0) tan 2 (0) - tan 2 (0) = tan 2

(U

2 ) - tan 2 (02) (15) tan 2 ( ) + tan 2 ( 0) = tan 2 (,) + tan 2 () It is desirable to minimise the gap in the radial direction between the images from the two mirrors so as to maximise usage of the camera field of view. For minimum gap , = 0 2 , so 2 tan 2 () = tan 2 (02) + tan 2 (0) 1 22 =tan tan 2 (0 2 ) + tan2(O--)J (16) 20 = tan 202 WO 00/42470 PCT/AU00/00022 - 13 Figures 6A, 6B and 7A, 7B show graphical and ray traced comparisons of constant and variable gain double mirror systems viewing the same scene. 2.4 Calculation of Range for a Variable Gain Panoramic Stereo System 5 The information available for range calculation are the image angles for a single object reflected in both mirrors, 01 and 02 as shown in Figure 8. The two mirrors 0, and 02 form a reflective surface. The differential equations (6) for the surfaces are known. In the calculations that follow only the lower mirror is examined as the results are identical for the 10 upper mirror. In order to find the position of object P, the equations of the incident beams from P to each mirror reflection point (r,, 0) and (r 2 , 02) must be found. These equations can then be solved simultaneously to give the position of object P, (x,, yp). 15 Yp 1 -m 11

C

1 1 m12 n 11 (17)

I

2 II I 1 2

M

11

C

11 1 1 C 2 m 12 - ml 1 m1 2

-

H

WO 00/42470 PCT/AU00/00022 -14 where mn is the gradient of the incident beam to the lower mirror and C,, is the equation constant. The equation constant is given by

C

11 =Yl - m 1 x (18) where 5 x, = r 1 sin 0 Yl = r, cos 01 (19) are the Cartesian coordinates of the reflection point (r 1 , 01). The gradient of the incident 10 beam is found using the law of reflection m = tataan d +tan- ' -tan-' dYl (20) n I MR1 dy I where mR, is the gradient of the reflected beam from the lower mirror to the camera and 15 dy 1 /dx, is the gradient of the lower mirror profile at the reflection point. The gradient of the reflected beam is mR = tan 6 1 (21) The gradient of the mirror profile for the lower variable gain mirror is found as in the constant gain case, from 20 vv" uuuo PCT/AU00/00022 - 15 dy 1 dy, dx 1 dx 1 dO 1 dO 1 dr 1 ±- cos6 1 -r, sin6 1 (22) dO I dr - sinO, +r cosO dO 1 AI where dr/dO for either mirror of the variable gain mirror configuration is found by integrating (5) and substituting (12). fd tan-' r d

-

) -If (1 + a (0)) d ( dr 2 tan-l rd = _ 10 B- f (tan(O)+ tan 3 (O))dO (23) ,dr 2 2 IB = -- 0 - metan 2 (0) + D 2 4 where D is a constant of integration. Rearranging (23) gives 5 dr t Ba 2 -r co - - tan 2 (0) + D (24) dO 2 4 Now from (23) and (2), 1 Ba D= y - + - tan 2 (0) (25) 2 4 10 so, for the lower variable gain mirror profile WO 00/42470 PCT/AU00/00022 -16 IB DI + - 1 + tan 2 () similarly for D 2 , for the upper variable gain profile. So, by substituting (24) into (22) gives the gradient of the variable gain mirror profiles at any 5 point. Note that as in the constant gain case, the gradient depends only on 0. The equation constants for the incident beam equations from (18) require the polar coordinates of the reflection points from each mirror, (r 1 , 0 1 ) and (r 2 , 02) . Since the variable gain mirror equations are not known exactly, r, and r 2 must be found using a differential equation solver 10 to find solutions to (26) at 0 and 02. 2 Back-to-back Stereo Mirror Families A key disadvantage of single camera stereo panoramic systems is that since there are two 15 images of the "same" scene, the pixels assigned to each image is half that for non stereo panoramic imaging and the two images do not share an equal number of pixels in constant gain schemes. Actually, the panoramic stereo double mirror method typically causes the view of a scene in one radial direction to be compressed into around 1/4 the field of view of the camera. 20 A method to achieve panoramic stereo with less image compression is to use two cameras and two single curved mirror surfaces back to back, as shown in Fig. 9. This method compresses the imaged scene into 1/2 the field of view of the camera, and indeed each image has an equal share of the total number of pixels available. There are, however, possible alignment 25 problems with this system as with any stereo system using two cameras to capture two views of a scene. An advantage of the scheme proposed in Fig. 9 is that the stereo cone can be symmetric about WU 00/42470 PCT/AU00/00022 -17 the horizon using two cameras with equal fields of view and the maximum and minimum angles of elevation reflected by the two mirrors being equal. The angle covered by the stereo cone in this case is 2i - r. Fig.9 shows the general case where the maximum and minimum angles of elevations viewed by each camera need not be equal. The range of elevations must 5 still be equal for the fields of view to be aligned. The number of free parameters to be specified are reduced here as the minimum angle of elevation (4 ) and from one mirror must be parallel to the maximum angle of elevation (;) from the other mirror. This is to ensure that the fields of view are parallel. So, with reference 10 to Fig. 9 (26) In the scheme of Fig. 9, the mirror families can be either constant gain or resolution invariant. 15 2.1 The Use of Double Mirrors in a Back to Back Design Fig. 10 shows a back to back design incorporating double mirrors. Although the figure shows 20 constant gain mirrors, the double mirror can also have a variable gain. The advantage to this system is that the stereo cone from the back to back configuration combines with the stereo cones from the double mirror configuration to increase the total area imaged in stereo. In this configuration, the fields of view of each double mirror pair need not be aligned as in previous examples. For symmetry about the horizon A = , A = 2 , 25 4 = and Cr = . The constraints WO 00/42470 PCT/AU00/00022 - 18 4 1 align the three stereo cones. It is also possible to increase the total stereo cone further by allowing the mirror pairs to have 5 different gains. 3 Forward Looking Mirror Design An example of a forward looking mirror design is shown in Fig. 11. For many applications, 10 it is desirable to have a panoramic camera looking out from, say, a hemisphere, somewhat as an eye of a bird, or perhaps two such on either side of a "nose cone". There are aerodynamic considerations or other protrusion considerations which motivate such a "forward looking" system. This configuration is termed forward looking because the camera faces towards the scene. Either a constant or variable gain mirror (double or single) could be used for the 15 curved mirror in the system. The planar mirror is an annulus or circle interposed such that all rays reflected from the curved mirror are reflected into camera o positioned behind the curved mirror. The dotted lines in Fig. 11 show where the reflected rays would converge if the planar mirror was removed and the dotted camera shows the camera o' for an equivalent system without the planar mirror. 20 In order for the rays reflected by the planar mirror to converge at the new camera position, the planar mirror must be the perpendicular bisector of the line joining the old and new camera locations. Hence the distance between the camera locations is 2D where D is defined in Fig. 11 as the distance from either camera to the planar mirror. The introduction of the 25 planar mirror into the system does increase the possibility of alignment difficulties as the planar mirror must be perpendicular to the camera axis and also be positioned so as to reflect all rays from the curved mirror into the camera without occluding the view of the curved mirror.

WU UU/42470 PCT/AU00/00022 - 19 The maximum value for D,D , is when the maximum beam reflected from the mirror system (the U beam reflected at point b on the planar mirror) into camera o grazes the curved profile at c. In this case 5 r cos (t) [tan (6) + tan (0)] (27) D = -_(27) 2 tan (0) D , defines the minimum height for the mirror system, H . In practice, the value for D needs to be slightly smaller to avoid occlusion, leading to a larger mirror system height. The general equation for the height of the mirror system is 10 H = r cos (0) - D (28) It should also be noted that 0 must be greater than zero for camera o to be located behind the curved mirror. Also, ) if the minimum elevation ray 4) is not to be occluded by the planar mirror. 15 Fig. 12 shows a design that incorporates the ideas of Sections 2 and 3. It consists of two forward looking systems back to back, giving a design reminiscent of a eye mounted on a stalk, such as a crab's eye. The "stalk" for this system would be hidden from view by the lower planar mirror. In this arrangement portions are provided in the curved mirror to 20 provide for reflection of rays from the curved surface to the camera by the plane mirrors.

F - i ItIUUUUUUAL -20 The foregoing describes only some aspects of the present invention and modifications can be made without departing from the scope of the invention. 5

Claims (21)

1. A panoramic imaging system including an imaging device having an image plane and a first field of view, a first reflective surface having at least one circularly symmetric portion 5 convex in a radial direction disposed in said first field of view to provide an expanded panoramic second field of view, the profile of the or each convex portion providing a varying gain between the fields of view in the radial direction to limit variation in the solid angle of view across the image plane of the imaging device. 10

2. A panoramic imaging system as claimed in claim 1 wherein the profile of the or each convex portion provides a substantially uniform solid angle of view across the image plane.

3. A panoramic imaging system as claimed in claim 1 or claim 2 wherein the profile of the or each convex portion at least approximates a profile defined in polar co-ordinates by the 15 equation: dr - r cot -2 (1 + a (0))dOl where r is the radial distance from the reflective surface to the imaging device 0 is the angle from the optical axis of the imaging device a (0) is the mirror gain given by 20 a (0) = Be [tan (0) + tan 3 (0)] B = 2( tan2() - tan 2 (0) Sand 4 are the maximum and minimum elevations viewed 0 and 0 are the maximum and minimum radial angles imaged. T , uuVw41&t u PCT/AU00/00022 - 22

4. A panoramic imaging system as claimed in claim 3 wherein the profile of the or each convex portion includes by a series spaced apart points defined by determining distance r for selected values of angle 0.

5 5. A panoramic imaging system as claimed in claim 4 wherein the selected values of 0 are separated by about 1/50.

6. A panoramic imaging system as claimed in any one of claims 1 to 5 including a first reflector surface having at least two of said convex portions arranged to respectively provide 10 at least partially overlapping panoramic second fields of view for range determination.

7. A panoramic imaging system as claimed in claim 6 wherein said panoramic second fields of view are substantially co-incident. 15

8. A panoramic imaging system as claimed in claim 7 wherein said at least two convex portions form a continuous reflective surface.

9. A panoramic imaging system as claimed in any one of claims 1 to 8 including two of said first reflective surfaces each having an associated image plane with corresponding first 20 fields of view, and at least one convex portion of each first reflective surface providing respective panoramic second fields of view, said first reflective surface being arranged back to back such that said reflective second fields of view at least partially overlap.

10. A panoramic imaging system as claimed in any one of claims 1 to 9 further including 25 a second reflective surface interposed between the image plane and said second reflective surface.

11. A panoramic imaging system as claimed in claim 10 wherein the imaging device is positioned behind the second reflective surface. 30 WO 00/42470 PCT/AU00/00022 - 23

12. A panoramic imaging system as claimed in claim 11 wherein an aperture is provided in said first reflective surface to provide said first field of view from the imaging device. 5

13. A panoramic imaging system as claimed in any of claims 10 to 12 wherein said second reflective surface is substantially planar.

14. A reflective surface for use in a panoramic imaging system including an imaging device having an imaging plane and a first field of view, said reflective surface having at least 10 one circularly symmetric portion convex in a radial direction with a profile providing varying gain in the radial direction between an expanded panoramic second field of view provided by the reflective surface and the first field of view to limit variation in the solid angle of view across the image plane of the imaging device.

15 15. A reflective surface as claimed in claim 14 wherein the profile of the or each convex portion provides a substantially uniform solid angle of view across the image plane.

16. A reflective surface as claimed in claim 14 or claim 18 wherein the profile of the or each convex portion at least approximates a profile defined in polar co-ordinates by the 20 equation: dr -r cot - f ( + c (0)) dO d6 where r is the radial distance from the reflective surface to the imaging device 0 is the angle from the optical axis of the imaging device a (0) is the mirror gain given by 25 a (0) = Ba [tan (0) + tan 3 (0)] B 2( a tan 2 () - tan 2 (_) WO 00/42470 PCT/AU00/00022 - 24 Sand 4) are the maximum and minimum elevations viewed 0 and 0 are the maximum and minimum radial angles imaged.

17. A reflective surface as claimed in claim 16 wherein the profile of the or each convex 5 portion includes by a series spaced apart points defined by determining distance r for selected values of angle 0.

18. A reflective surface as claimed in claim 17 wherein the selected values of 0 are separated by about 1/50. 10

19. A reflective surface as claimed in any one of claims 14 to 18 including a first reflector surface. having at least two of said convex portions arranged to respectively provide at least partially overlapping panoramic second fields of view for range determination. 15

20. A reflective surface as claimed in claim 19 wherein said panoramic second fields of view are substantially co-incident.

21. A reflective surface as claimed in claim 20 wherein said at least two convex portions form a continuous reflective surface.