Equi-areal Catadioptric Sensors

Transcription

1 Equi-areal Catadioptric Sensors R. Anew Hicks Ronald K. Perline Department of Mathematics and Computer Science Drexel University Philadelphia, PA 904 ahicks, Abstract A prominent characteristic of most catadioptric sensors is their lack of uniformity of resolution. While several sensors previously appearing in the literature have been described as equi-resolution, this is not the case, because they do not incorporate the concept of solid angle. We describe catadioptric sensors whose associated projections from the viewing sphere to the image plane are area preserving (equi-areal), and so truly are equi-resolution. We show that in the orthographic case the catoptric component must be a surface of revolution of constant Gaussian curvature. We compare these equi-areal sensors in both the perspective and orthographic cases with other sensors that have been proposed earlier for treating the uniformity-ofresolution problem.. Introduction A catadioptric sensor may be thought of as a physical realization of a projection from 3-space or a given surface to a plane. There are many such sensors, and the corresponding projections have been well studied in some examples. Among the best known catadioptric sensors are those whose catoptrics (mirrors) are conic sections. Such conic sensors have particularly nice properties that can be best described in terms of their projections. For example, for a sensor consisting of a parabolic catoptric coupled with an orthographic dioptric (camera), the projection may be factored as a projection onto a sphere, followed by a stereographic mapping (see Geyer and Daniilidis [7]). From a design perspective, one might start with a given property to be required of the catadioptric sensor, and then attempt to construct a sensor satisfying the design constraint. This is equivalent to demanding that the projection satisfy some specific mathematical condition. One approach to this problem is to consider systems with a fixed dioptric component, then use differential methods to derive a mirror shape with the required properties. A common feature of traditional catadioptric sensors is lack of uniformity of resolution. This can be measured experimentally as follows: A picture is taken of an object, then the object is rotated about the sensor, where the rotation is skew with respect to the optical axis. A second picture of the object is then taken. A count of the number of pixels representing the object in the two images will in general differ significantly from one image to the next. This is a familiar phenomenon: an object that is viewed with a curved security mirror will appear smaller at the perimeter of the mirror then if viewed in the middle. In this paper we describe a family of mirrors that deal with this difficulty - they fairly allocate pixels to the regions of space that they image. In a precise sense, that we describe below, these mirrors are optimal with respect to uniformity of resolution. Mathematically, this is equivalent to their projections being area preserving. We will refer to these sensors as equi-areal. Remarkably, the resulting surfaces have close contact with certain model surfaces in classical differential geometry. Consider figure (), which displays a collection of uniformly spaced spheres about a curved mirror. If an orthographic projection is used to observe the sensor from below, several of the spheres will be imaged, with distortion dependent upon the shape of the mirror. In the right of figure () is depicted the resulting image if a parabolic mirror is used. Notice how the size of each sphere depends on it s elevation. Our goal is to find a mirror shape such that the area of all of these spheres are the same. A basic result that we derive is that in the orthographic case, the sphere coupled with an orthographic projection induces an area preserving projection (see figure ()). Finally, a comment regarding terminology should be made. Several catadioptric sensors previously appearing in the literature have been called equi-angular or equiresolution. None of these sensors are equi-resolution in the sense described above. The reason for this is that none of these models incorporate the concept of solid angle. Due

2 Figure. On the left we see a collection of spheres uniformly placed about a curved mirror. On the right we see the image formed when the mirror is parabolic and the projection used is orthographic. The number of pixels used to represent each sphere varies greatly. Figure. If a spherical mirror is observed with an orthographic projection, the projections of the spheres in the scene will now all have the same area. to this confusion regarding terminology, we choose to refer to our sensors as equi-areal or area preserving, since this accurately describes one of their geometric properties. θ. Related Work Here we consider only past work on catadioptric sensors that is directly related to the sensors described in this paper. For more comprehensive overviews on the subject, see Bogner [3], Baker and Nayar [], Yagi [3] and Bakstein and Pajdla []. The problem of designing a catadioptric sensor that results in improved uniformity of resolution (as compared to conventional sensors) has been approached by several authors (in the following discussion, all dioptric components will considered to be either perspective or orthographic, and all catoptric components rotationally symmetric). In [4], Chahl and Srinivasan employ differential methods to derive a mirror shape such that the radial angle θ and the angle of elevation (see figure 3) are linearly related. This relationship results in a second-order ordinary differential equation whose solutions are given in implicit form dθ sin A θ K Br K Here, d K, and A and B are constants arising from the fact that the differential equation derived in [4] is secondorder. In [], Ollis et al. argue that improved resolution uniformity is obtained if one demands that the angle of elevation be proportional to the tangent of the radial angle θ; geometrically, tan θ represents the distance of the image point to the optical axis. Thus, as the angle θ sweeps out some Figure 3. Chahl and Srinivasan consider a mirror in which d dθ is a constant K. range of angle, a proportional length is swept out in the image plane (note that the line of reasoning here is essentially one dimensional). The profiles for the resulting mirrors satisfy the differential equation π dθ r cot K tan θ The authors then apply these mirrors to mobile robot navigation and panoramic stereo. A variant of the Ollis mirror is described by Conroy and Moore in [5]. Here the authors use the same parameters as Chahl and Srinivasan and generalize their equations by showing that if the mirror cross-section satisfies the equation d dθ α θ (i.e., α is an arbitrary function of θ) then the cross-section will satisfy the equation dθ r cot α θ dθ () Essentially, equation () is a convenient way to specify in polar coordinates the profile of any mirror with a given α θ. The authors then calculate what the appropriate α should be in order that the angle be proportional to the corresponding area (disk) in the image, rather than its radius, as was the case in [] (see figure (4)). Substitution of this α θ into equation () gives an ordinary differential equation that may be solved numerically.

3 θ ftan( θ) f Figure 4. Conroy and Moore consider a mirror for which is proportional to the area of the corresponding disk in the image plane. Here f is the focal length of the camera. We now give a geometric interpretation of the problem that will help to clarify the construction described in this paper. The inclination angle is simply a parameter for the reflection vector at a point on the mirror. The reflection vector can be thought of as a point on the unit sphere S, and so the mirror induces a correspondence between points in the image plane and the sphere S of all reflection vectors (the reflection sphere). Each of the previously described sensors implicitly prescribe a particular correspondence between the image plane and the reflection sphere. Viewed in this manner, it is natural to construct a sensor so that the correspondence between points in the image plane and the reflection sphere is uniformly distributed; in other words, the induced sphere-to-plane map should be area preserving. We refer to this sensor as the equi-areal sensor. There is another line of reasoning that leads to the derivation of the equi-areal sensor. Consider any object sphere with radius sufficiently large to enclose the sensor apparatus. One can derive the equations describing a family of mirrors such that the corresponding projections from the object sphere to the image plane is equi-areal. It turns out that the resulting sensor does not have a single effective viewpoint; thus the only sphere that is imaged equi-areally is our original object sphere. One way to alleviate this difficulty is to let the radius of the object sphere go to infinity; the resulting equations simplify considerably, and in fact coincide with the equations associated with the equi-areal sensor defined in the previous paragraph. In practice the equiareal sensor, that by definition images the infinity object sphere in a uniform manner, does an excellent job imaging finite spheres whose radius is large with respect to the dimensions of the sensor. It should be pointed out that one may choose a surface other than a sphere when defining the term equi-areal. For example, one could design a sensor that uniformly images planes rather than spheres. In this vein, Hicks and Bajcsy describe a sensor in [9] which will uniformly image planes. Sensors that uniformly image cylinders were investigated by Hicks et al. in [0] and Gaechter and Pajdla in [6]. The central quantity of interest in this paper is the magnification factor. This quantity is considered by Nayar and Baker in [], where it is referred to as the resolution of the sensor. For this reason, the sensors described below may also be called equi-resolution. Nayer and Baker compute the resolution of an arbitrary catadioptric sensor and compare it to a conventional camera. In [], Swaminathan et al. examine the resolution of non-single viewpoint catadioptrics that employ conic catoptrics. 3. Rotationally Symmetric Equi-Areal Mirrors 3. The Orthographic Case Suppose that the cross-section of a rotationally symmetric mirror is given by the graph of some function y f x and that the dioptric component of our sensor realizes an orthographic projection (see figure (5)). The idea of the derivation is to compute the infinitesimal magnification factor of the inverse of the projection from the reflection sphere to the image plane in terms of the cross-section f, and then to require that this quantity be constant. We consider the inverse of the projection rather than the projection in order to simplify computations. Consider two concentric circles of radii x and x x in the image plane, centered about the optical axis (which is also the axis of symmetry of the mirror) of the sensor. The annulus between the two circles is mapped to a topological annulus on the reflection sphere, bounded by and. The area of the planar annulus is just π x x πx ; and the area of the spherical annulus is cos cos (here we are using the fact that the area of a spherical cap is cos - see figure (6)). We define the magnification factor at x as the ratio of these two quantities as x 0, i.e. m f x! cos cos lim x" 0 π x x πx () (Here is the change in corresponding to x.) Since as x 0, x d dx lim x" 0 we have that m f x! cos cos x x sin π x x x πx arctan f #$ x (since the angle of d (3) dx Using the fact that incidence equals the angle of reflection - see figure (7)) we have This quantity is, of course, simply the determinant of the Jacobian of the mapping. 3

4 x x ψ ψ Figure 5. The basic form of a catadioptric sensor whose catadioptric is well modeled by an orthographic projection. Figure 7. Why arctan f #$ x : the dotted line represents the normal to the mirror crosssection at x & f x. Thus ψ arctan f #( x. arctan f #$ x, so where C is a constant of integration. To interpret C geometrically we give an alternative approach to the problem in which we can directly derive a first order differential equation. Figure 6. The area of a cap on a sphere of radius, corresponding to the angle is cos. m f x! so our basic equation is f #% x f # # x πx f# x & (4) f # x f # # x πx f # x K (5) where K is the prescribed magnification constant. Note that there is no restriction on the sign of K; K ' 0 simply means that the projection is orientation reversing. Remarkably, (5) is the differential equation for the cross section of a surface of revolution of constant Gaussian curvature Kπ (see for example [8], page 464). Since a sphere has constant curvature, it follows that a spherical mirror coupled with an orthographic projection will be an equi-areal sensor. Notice that if we write equation (5) as f #% x f # #( x Kπx (6) f # x that the left-hand side is a perfect derivative. Consequently we may integrate both sides to obtain C f # x Kπx (7) Figure 8. It may be that the mirror is not smooth at its lowest point. We let 0 denote the angle between the normal at the lowest point and the vertical axis. The derivative of the mirror cross-section may not be zero on its axis of symmetry, so assume that the normal to the mirror at the point where it crosses its axis of symmetry may make an angle of 0 with the axis. With the notation of figure (8), we see that the region on the sphere corresponding to the angle should be proportional to the area of a disk of radius x in the image plane. In this case we have then that our fundamental equation is x cos ) cos 0! Kπx (8) Here, arctan f #% x. Substituting this expression into equation (8) and simplifying gives cos 0 f # x Kπx (9) * cos 0. Comparing equation (7) and (9) gives that C To obtain solutions, equation (9) can be integrated in terms of elliptic functions. Sample solution surfaces are shown in figure (9). On the left we see a surface of constant positive curvature, and on the left a surface of constant negative curvature, known as the pseudosphere. The pseudosphere is well known as a model for Lobachevsksy s hyperbolic geometry. 4

5 3.3 The Perspective Case Figure 9. Surfaces of revolution of constant positive (on the left) and negative (on the right) Gaussian curvature. In the orthographic case, these are the equi-resolution mirrors. 3. Comparison in the Orthographic Case It is natural to compare the equi-areal sensors with other catadioptric sensors with respect to uniformity of resolution. To do so, we reparameterize m f by and normalize by dividing by its maximum. For the sphere, and the surfaces of revolution corresponding to the graphs of x and x 3, this quantity is plotted in figure (0). Notice that the magnification factor for the paraboloid decreases by 75% as one passes from below the sensor to the horizon line ( 90 degrees). One observes that, for 0, the magnification factor for the cubic is zero; this is due to the zero curvature of the cubic at its intersection with the optical axis. Normalized magnification Sphere Cubic Paraboloid Figure 0. A comparison of the magnification factors of three mirrors, where each magnification factor has been normalized, i.e. divided by its maximum. The cubic has zero magnification at 0 since both of its derivatives are zero there. Note that his does not mean that a portion of the scene is omitted. For the perspective case, we could make use of equation (), but instead we present another form of the equation using a different derivation. Using the notation of figure () we seek to find a differential equation in polar coordinates for a mirror with a prescribed θ. Assume that the center of projection of the camera is placed at the origin, and that a ray of light entering the camera is incident with the mirror at r & θ. Then if the cross section of the mirror is given parametrically as θ + r θ sin θ & r θ cos θ, then the slope of the tangent line at r & θ is dy m dx dy dθ dx dθ dθ cos θ, r sin θ sin θ dθ r cos θ On the other hand, ψ arctan m and θ ψ - θ. Thus arctan m! dθ θ. Solving this equation for dθ gives r sin θ cos θ tan θ cos θ sin θ tan θ (0) θ θ ψ Figure. For the projection to be area preserving we require that cos. cos 0! Kπ f tan θ () (see figure (4)). Solving for in equation () and substituting into equation (0) gives a differential equation that may be solved numerically. The solution in the case where K π & f & r 0 / and 0 0 (i.e. the mirror is smooth at its lowest point) can be seen in figure (). 3.4 Comparison in the Perspective Case We now study the uniformity-of-resolution properties of the three mirror models discussed in the introduction, as compared to our equi-areal mirror. Each of these models is characterized by its dependence of on θ. Calculating in much the same way as in the orthographic case, one can see that the magnification factor in the perspective case is m θ 0 sin cos 3 θ π f sin θ d () dθ 5

6 Normalized magnification Hicks&Perline Ollis Figure. The cross-section of an equi-areal mirror based on dioptrics that are well modeled by perspective projection Conroy Chahl The three models of interest here are: ) Chahl and Srinivasan: Kθ 0 ) Conroy and Moore: Kπ f tan θ 0 3) Ollis et al.: K tan θ 0 Note that the perspective equi-areal model discussed in the last subsection is defined by cos cos 0 3 Kπ f tan θ. Without loss of generality, we take f. The magnification factors for the above models then, expressed as functions of, are ) Chahl and Srinivasan: K sin cos 3 K πsin K ) Conroy and Moore: K sin 3) Ollis et al.: K sin π A plot of these magnification factors (normalized) appears in figure (3). The parameter values chosen for our comparison allow one to see the qualitative behavior and differences between the various models. In all cases, we took the mirror to be smooth at its lowest point (that is, 0 0), and f. The value of K used was 3 for the Conroy and Ollis mirrors, but 5 for Chahl mirror. The reason for this discrepancy is that a choice of K 3 requires a prohibitively wide-angle catoptric in the simulations that are presented in figure (4). In this figure we see the effect of the different magnification factors using the scene from figure (). Note that because we have considered the inverse of the projection map, if a mirror has a decreasing magnification factor, then the spheres closer to the edge of the mirror are smaller then those near the center. 4. Experimental Verification We tested the area preserving property of a spherical sensor by imaging 5 objects (red, rectangular squares) of the same size about the sensor, which was oriented on a table so that its optical axis was horizontal with respect to an observer. A single point was chosen, near the bottom of the mirror, and using a meter stick a circle marked out on the Figure 3. A comparison of normalized magnification factors of the three equi-angular mirrors, and the magnification factor of an equiresolution mirror. table. Then the objects were all placed at a distance of 75 cm from the lowest point on mirror. Each rectangle was placed so as to be tangent to the traced out circle. A sample image appears in figure (5). To measure the number of pixels representing each object, each object was circumscribed by a polygon to approximate its boundary. The angles of elevation for the objects () were 30 & 45 & 60 & 75 and 90 degrees and the corresponding numbers of pixels were 3960, 3950, 4040, 3990, and 3830, which should be considered reasonably constant since the objects are approximately quailaterals with sides varying from about 50 to 00 pixels. Hence an error of one pixel in estimating the boundaries can lead to an error of several huned pixels when estimating the total number of pixels representing one of the objects. Note that despite the fact that the objects are less than one meter from the sensor, this distance was in fact sufficiently large for the sensor to exhibit its equi-areal nature. 5. Conclusions We have described a family of catadioptric sensors that image space in a uniform fashion. Further experimental work needs to be performed regarding the implementation of the sensors, especially those with Also needing to be adessed are optical issues, such as defocus blur. These sensors may be an appropriate choice when performing such tasks as tracking or stereo. 6

7 Figure 5. Five objects, all representing the same solid angle with respect to a sensor employing a spherical mirror, placed at 30 & 45 & 60 & 75 and 90 degrees. Figure 4. Comparison of four different mirror designs in the perspective case. References [] S. Baker and S. Nayar. A theory of catadioptric image formation. In Proc. International Conference on Computer Vision, pages 35 4, 998. [] H. Bakstein and T. Padjla. Non-central cameras: A review. In Proceedings of Computer Vision Winter Workshop, Ljubljana, Slovenian Pattern Recorgnition Society, pages 3 33, 00. [3] S. Bogner. Introduction to panoramic imaging. In Proceedings of the IEEE SMC Conference, pages , 995. [4] J.S. Chahl and M.V. Srinivasan. Reflective surfaces for panoramic imaging. Applied Optics, 36: , 997. [5] T. Conroy and J. Moore. Resolution invariant surfaces for panoramic vision systems. In Proc. International Conference on Computer Vision, pages , 999. [6] S. Gaechter and T. Pajdla. Mirror design for an omnidirectional camera with space variant imager. In Proc. of the Workshop on Omnidirectional Vision Applied to Robotic Orientation and Nondestructive Testing (NDT), Budapest, 00. [7] C. Geyer and K. Daniilidis. A unifying theory for central panoramic systems and practical applications. In Proc. of the European Conference on Computer Vision, pages , 000. [8] A. Gray. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, 998. [9] R. A. Hicks and R. Bajcsy. Catadioptic sensors that approximate wide-angle perspective projections. In Proc. Computer Vision Pattern Recognition, pages , 000. [0] R. A. Hicks, R. Perline, and M. Coletta. Catadioptric sensors for panoramic viewing. In Proc. Int. Conf. on Computing and Information Technology, 00. [] M. Ollis, H. Herman, and Sanjiv Singh. Analysis and design of panoramic stereo vision using equi-angular pixel cameras. Technical Report, The Robotics Institute, Carnegie Mellon University, 5000 Forbes Avenue Pittsburgh, PA 53, 999. [] R. Swaminathan, M. Grossberg, and S. Nayar. Caustics of catadioptrics cameras. In Proc. International Conference on Computer Vision, pages II: 9, 00. [3] Y. Yagi. Omnidirectional sensing and its applications. IEICE Trans. on Information and Systems, E8-D(3), pages ,