pinholemodel

The pinhole model is an idealized geometric model of a camera used in computer vision and photography. It treats the camera as a device with a single small aperture through which light rays pass and project onto an image plane. In this model, a 3D point in world coordinates is transformed into camera coordinates, then projected onto the image plane to produce 2D image coordinates.

Mathematically, a world point Xw = [X, Y, Z, 1]^T is mapped to camera coordinates by Xc = R Xw + t, where R is the rotation and t is the translation that describe the camera pose relative to the world. The projection onto the image plane involves normalizing by depth: x = Xc/Zc, y = Yc/Zc. These normalized coordinates are then mapped to pixel coordinates using the intrinsic matrix K, which encodes the focal lengths, skew, and principal point: [u; v; 1] ∝ K [x; y; 1], where K = [ fx s cx; 0 fy cy; 0 0 1 ]. In common notation, the relation can be written as s [u v 1]^T = K [R|t] [Xw; Yw; Zw; 1]^T, with the final pixel coordinates given by u = (K [R|t] Xw)_1 / (K [R|t] Xw)_3 and v = (K [R|t] Xw)_2 / (K [R|t] Xw)_3.

The pinhole model assumes no lens distortion. It is foundational for camera calibration, 3D reconstruction, and many computer vision algorithms. Distortions from real lenses are typically treated separately with additional distortion models. Limitations include inaccuracies for wide-angle lenses and scenes where the ideal projection deviates from perspective geometry.