Why is the image plane often written as z = f in the pinhole camera model?
Asked 6/15/2020
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In the ideal pinhole camera model, why is the image plane commonly placed at z = f? I was trying to connect this to the thin-lens model and depth of field, thinking that as aperture diameter approaches zero a thin lens becomes a pinhole and everything becomes in focus. Mathematically, is z = f required by the pinhole model, or is it just a convenient choice of image-plane distance?
Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0
Photography Stack Exchange contributor
6y ago
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In the "ideal" pinhole camera (pinhole infinitely small), everything is always in focus, because point source of light in the universe outside the camera, the pinhole and the corresponding image point on the film/sensor/back are colinear. In other words there is a single path for light rays between the source and the image, so there is no need for different paths to converge on a single point like you have with a lens.
In you image above the distance between the image plane and the pinhole is arbitrary. The closer the image plane to the pinhole, the smaller the image.
Note that:
- putting the image plane farther from the pinhole makes the image bigger, this is equivalent to using a lens with a longer focal length. In a real-life pinhole camera it also means using a longer exposure since the same amount of light is spread over a bigger area (this is a square law so you can express this with something that looks a lot like a f-number)
- the theoretical pinhole camera has infinite DoF, but it also has an infinite f-number, so be ready for very, very long exposure times.
Originally by user75947. Source · Licensed CC BY-SA 4.0
user75947
6y ago
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Generated from our catalog & community — verify before relying on it.
In a pinhole camera model, the plane at z = f is usually a convention, not a physical requirement.
A pinhole camera is a projection model: light from a scene point passes through a single point (the pinhole) and hits the image plane. Because each scene point maps along one straight line through the pinhole, an ideal infinitesimal pinhole is always in focus. That is different from a lens, where many rays must be brought to a common focus, so thin-lens and depth-of-field formulas do not apply here.
Mathematically, you can place the image plane at any distance behind the pinhole. If the plane is farther away, the projected image is larger; if it is closer, the image is smaller. Writing the plane as z = f is just a convenient parameterization, since that distance plays the same scaling role that focal length plays in perspective projection. In computer vision and graphics, f often means the image-plane distance because it sets magnification / field of view.
So: z = f is not something that must be true for a pinhole camera. It is simply a chosen image-plane distance, often named f by analogy to focal length.
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