General Linear Cameras with Finite Aperture

Published in EGSR 2007

Abstract
A pinhole camera selects a two-dimensional set of rays from the four-dimensional light field. Pinhole cameras are a type of general linear camera, defined as planar 2D slices of the 4D light field. Cameras with finite apertures can be considered as the summation of a collection of pinhole cameras. In the limit they evaluate a two-dimensional integral of the four-dimensional light field. Hence a general linear camera with finite aperture factors the 4D light field into two integrated dimensions and two imaged dimensions. We present a simple framework for representing these slices and integral projections, based on certain eigenspaces in a two-plane parameterization of the light field. Our framework allows for easy analysis of focus and perspective, and it demonstrates their dual nature. Using our framework, we present analogous taxonomies of perspective and focus, placing within them the familiar perspective, orthographic, cross-slit, and bilinear cameras; astigmatic and anastigmatic focus; and several other varieties of perspective and focus.

Point Perspective. The width and height of shrink with depth, while maintaining a fixed ratio.

Orthographic. The size of the checkerboard squares does not vary with depth.

Cross Slit. The width of checkerboard squares recedes at a different rate to their height.

Pushbroom. The width of checkerboard squares decreases with depth, but their height does not.

Pencil. Each scanline comes from a unique point perspective image. The horizontal location of the eye for a particular scanline is determined by that scanline's y coordinate.

Twisted Orthographic. Each scanline comes from a unique orthographic image. The viewing direction for a particular scanline is determined by that scanline's y coordinate.

Bilinear. Anything goes.

Focused. There is a depth at which objects are sharp.

Focused at Infinity. Objects become increasingly sharp with depth.

Astigmatic. Objects are vertically and horizontally sharp at two different depths.

Astigmatic at Infinity. Objects are horizontally sharp at a near depth, and vertically sharp at infinity.

Partially Afocal. Objects become sharp in one direction only at some real depth.

Partially Afocal at Infinity. Objects become sharp in one direction only at infinity.

Afocal. Objects are never perfectly sharp in any direction at any depth.

Figure 1. This figure illustrates the set relationships between the various types of linear cameras on the left, and the various types of focus on the right. The number of degrees of freedom is indicated with color. Click on any thumbnail at the top or region in the venn diagram to see an example scene rendered with that type of camera. When viewing an image, use the left or right keys to scroll through the other images. If you wish to inspect multiple images side by side, you can drag an open one out of the way and open a second. Click an image a second time to close it.

Paper
Adobe Acrobat PDF (5 MB)

Slides from talk at EGSR 2007
Powerpoint Slides (3.5 MB)