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### Planar Mirrors

In Solution (16), if we set k = 2 and c > 0, we get the cross-section of a planar mirror:
 (17)
As shown in Figure 2, this plane is the one which bisects the line segment joining the viewpoint and the pinhole.

The converse of this result is that for a fixed viewpoint and pinhole , there is only one planar solution of the fixed viewpoint constraint equation. The unique solution is the perpendicular bisector of the line joining the pinhole to the viewpoint:
 (18)
To prove this, it is sufficient to consider a fixed pinhole , a planar mirror with unit normal ,and a point on the mirror. Then, the fact that the plane is a solution of the fixed viewpoint constraint implies that there is a single effective viewpoint . To be more precise, the effective viewpoint is the reflection of the pinhole in the mirror; i.e. the single effective viewpoint is:
 (19)
Since the reflection of a single point in two different planes is always two different points, the perpendicular bisector is the unique planar solution.

An immediate corollary of this result is that for a single fixed pinhole, no two different planar mirrors can share the same viewpoint. Unfortunately, a single planar mirror does not enhance the field of view, since, discounting occlusions, the same camera moved from to and reflected in the mirror would have exactly the same field of view. It follows that it is impossible to increase the field of view by packing an arbitrary number of planar mirrors (pointing in different directions) in front of a conventional imaging system, while still respecting the fixed viewpoint constraint. On the other hand, in applications such as stereo where multiple viewpoints are a necessary requirement, the multiple views of a scene can be captured by a single camera using multiple planar mirrors. See, for example, [Goshtasby and Gruver, 1993] and [Nene and Nayar, 1998].

This brings us to the panoramic camera proposed by Nalwa [Nalwa, 1996]. To ensure a single viewpoint while using multiple planar mirrors, Nalwa [Nalwa, 1996] arrived at a design that uses four separate imaging systems. Four planar mirrors are arranged in a square-based pyramid, and each of the four cameras is placed above one of the faces of the pyramid. The effective pinholes of the cameras are moved until the four effective viewpoints (i.e. the reflections of the pinholes in the mirrors) coincide. The result is a sensor that has a single effective viewpoint and a panoramic field of view of approximately . The panoramic image is of relatively high resolution since it is generated from the four images captured by the four cameras. This sensor is straightforward to implement, but requires four of each component: i.e. four cameras, four lenses, and four digitizers. (It is possible to use only one digitizer but at a reduced frame rate.)

Next: Conical Mirrors Up: Specific Solutions of the Previous: Specific Solutions of the
Simon Baker
1/22/1998