Numerical Results for the Hyperboloid

  For our numerical experiments we set c=1 meter, used the hyperboloid mirror with k=4, and assumed the radius of the lens to be 5 centimeters. Initially, we just considered the point $\mathbf{m} = (0.125,0.0,0.125)$ on the mirror and set the distance from the viewpoint to the world point $\mathbf{w}$ to be l=5 meters. In Figure 10 we plot the area of the blur region (on the ordinate) against the distance to the focused plane v (on the abscissa). The smaller the area of the blur region, the better focused the image will be. We see from Figure 10 that the area never reaches exactly 0, and so an image formed using this catadioptric sensor can never be perfectly focused. However, note that the minimum area is very small, and in practice there is no problem focusing the image for a single world point. Moreover, is is possible to use additional corrective lenses to compensate for most of this effect [Hecht and Zajac, 1974].

  

Figure 10: The area of the blur region plotted against the distance to the focused plane $v = \frac{f\cdot u}{u-f}$ for a point $\mathbf{m} = (0.125,0.0,0.125)$ on the hyperboloid mirror with k=4. In this example, we have c=1 meter, the radius of the lens 5 centimeters, and the distance from the viewpoint to the world point l=5 meters. So, $\mathbf{m} =(7.07,0.0,7.07)$.The area never becomes exactly 0 and so the image can never be perfectly focused. However, the area does become very small and so focusing on a single point is not a problem in practice. There are two minima in the area which correspond to the two different foldings of the blur region illustrated in Figures 11. Also note that the distance at which the image will be best focused (1.05-1.2 meters) is much less than the distance from the pinhole to the world point (approximately 1 meter from the pinhole to the mirror plus 5 meters from the mirror to the world point.) The reason is that the mirror is convex and so tends to increase the divergence of rays of light coming from the world point $\mathbf{w}$.

Also note that the distance at which the image of the world point will be best focused (i.e. somewhere in the range 1.05-1.2 meters) is much less than the distance from the pinhole to the world point (approximately 1 meter from the pinhole to the mirror plus 5 meters from the mirror to the world point). The reason for this effect is that the mirror is convex. Hence, it tends to increase the divergence of rays coming from the world point. For the rays to converge and focus the image, a larger distance to the image plane u is needed. A larger value of u corresponds to a smaller value of v, the distance to the focused plane.

Next, we provide an explanation of the fact that there are two minima of the blur area in Figure 10. As mentioned before, for an isolated conventional lens the blur region is a circle. In this case, as the focus setting is

  

Figure 11: The variation in the shape of the blur region as the distance to the focused plane is varied. All of the blur regions in this figure are relatively well focused. Note that the scale of the 6 figures are all different and that the scale on the two axes is, in general, different.

adjusted to focus the lens, all points on the blur circle move towards the center of the blur circle at a rate which is proportional to their distance from the center of the blur circle. Hence, the blur circle steadily shrinks until the blur region has area and the lens is perfectly focused. If the focus setting is moved further in the same direction, the blur circle grows again as all the points on it move away from the center. For a catadioptric sensor using a curved mirror, the speed with which points move is dependent on their position in a more complex way. Moreover, the direction in which the points are moving is not constant. This behavior is illustrated in Figures 11(a)-(f).

In Figure 11(a) the focused plane is 1.07 meters from the pinhole and the image is quite defocused. As the focused plane moves to 1.08 meters away in Figure 11(b), the points in the left half of Figure 11(a) are moving upwards more rapidly than those in the right half (the points in the right half are also moving upwards.) Further, the points in the left half are moving upwards more rapidly than they are moving towards the right. This effect continues in Figures 11(c) and (d). In Figure 11(d) all the points are still moving horizontally towards the center of the blur region, but vertically they are now moving away from the center. The points continue to move horizontally towards the center of the blur region, but with those in the left half again moving faster. This causes the blur region to overlap itself as seen in Figure 11(e). Finally, for the focused plane at 1.185 meters in Figure 11(f), all points are moving away from the center of the blur region in both directions. The blur region is expanding and the image becoming more defocused.

  

Figure 12: The focus setting which minimizes the area of the blur region in Figure 10 plotted against the angle $\theta$ which the world point $\mathbf{w}$ makes with the plane z = 0. We see that the best focus setting for $\mathbf{w}$ varies considerably across the mirror from around 1.45 meters at the periphery of the mirror $\theta = 0^{\circ}$, to only about 1.05 meters at its center $\theta = 90^{\circ}$. In practice, these results mean that it is very difficult to focus the entire scene at the same time, unless additional compensating lenses are used to compensate for the field curvature [Hecht and Zajac, 1974].

Finally, we investigated how the focus setting which minimizes the area of the blur region (see Figure 10) changes with the angle $\theta$ which the world point $\mathbf{w}$ makes with the plane z = 0. The results are displayed in Figure 12. As before, we set c=1 meter, used the hyperboloid mirror with k=4, assumed the radius of the lens to be 5 centimeters, and fixed the world point $\mathbf{w}$ to be l=5 meters from the effective viewpoint $\mathbf{v} = (0,0,0)$. We see that the best focus setting for $\mathbf{w}$ varies considerably across the mirror from around 1.45 meters at the periphery of the mirror $\theta = 0^{\circ}$ where the curvature is the least, to only about 1.05 meters at its center $\theta = 90^{\circ}$ where the curvature is the greatest. In practice, these results, often referred to as ``field curvature'' [Hecht and Zajac, 1974], mean that it is very difficult to focus the entire scene at the same time. Either the center of the mirror is well focused or the points around the periphery are well focused. Fortunately, it is possible to introduce additional lenses which compensate for the field curvature caused by the curved mirror [Hecht and Zajac, 1974].