Equiresolution catadioptric sensors

Abstract

A prominent characteristic of most catadioptric sensors is their lack of uniformity of resolution. We describe catadioptric sensors whose associated projections from the viewing sphere to the image plane have constant Jacobian determinants and so are equiresolution in the sense that any two equal solid angles are allocated the same number of pixels in the image plane. We show that in the orthographic case the catoptric component must be a surface of revolution of constant Gaussian curvature. We compare these equiresolution sensors in both the perspective and orthographic cases with other sensors that were proposed earlier for treating the uniformity-of-resolution problem.

Equitable mirrors

“…This quantity compares the area of a region on the object sphere to the area of its projection on the image plane. Hicks and Perline compute the magnification factor of a sensor under orthographic projection by considering two concentric circles of radii r and r + ∆r around the optical axis [7]. The infinitesimal annulus with area π((r + ∆r) 2 − r 2 ) is then mapped to the topological annulus on the sphere, bounded by φ and φ + ∆φ, whose area is π[(1 − cos (φ + ∆φ))) − (1 − cos (φ)).…”

“…Much like cartographers' maps, then, cataidoptric sensors are designed based on the needs of their users. When these needs involve minimizing a single kind of distortion much can be said [3,4,6,7]. However minimizing multiple types of distortion has not been as popular.…”

“…Hicks and Perline present the family of equi-areal mirrors, so named because they induce projections that uniformly rescale area. [7].…”

Computational Cameras: Convergence of Optics and Processing

“…As a result, the computationally recovered image often has spatially uneven resolution, which is undesirable in many applications. Various configurations of mirrors have been proposed to produce images of uniform resolution (e.g., in [64]- [66]). …”

Krill-eye: Superposition Compound Eye for Wide-Angle Imaging via GRIN Lenses

“…Catadioptric systems (mirror-based optics) [27] could be included in this category. Though they have different characteristics for the geometry of projection, they do not have the uniformity for all of resolution, distortion and luminance [13,21].…”