Geometric Derivation of Camera Equations

Photogrammetry, camera matching, and model-based image matching are commonly used techniques to analyze photographs and video for accident reconstruction and other forensic applications. Investigators are often tasked with taking measurements from photographs or determining speed from a video. All such calculations are based on fundamental geometric principles governing image projection inside a camera. Most treatments in the literature express the image projection equations in matrix notation rather than closed-form solutions. The purpose of this paper is to present a geometric derivation of the image projection equations in closed form that can be readily applied by a qualified investigator without the need for specialized software. In addition, a simple brute force optimization procedure is described to perform camera matching and model-based image matching. Examples are provided to demonstrate the method.