Decomposition of transformation matrices for robot vision

TLDR

The mapping from 3‑D world points to 2‑D image points is expressed by a 3×4 transformation matrix, which can be determined experimentally from known points or analytically from camera parameters, yet inverting it to recover camera pose is a challenging nonlinear problem. The paper seeks to provide a simple analytical method for this inversion. The method analytically solves the nonlinear equations to recover camera location and orientation from the transformation matrix. The technique works well in practice and offers valuable applications for motion tracking.

Abstract

The relationship between the three-dimensional coordinates of a point and the corresponding two-dimensional coordinates of its image, as seen by a camera, can be expressed in terms of a 3 by 4 matrix using the homogeneous coordinate system. This matrix is known more generally as the transformation matrix and can be determined experimentaily by measuring the image coordinates of six or more known paoints in space. Such a transformation can also be derived analytically from knowledge of the camera position, orientation, focal length and scaling and translation parameters in the image plane. However, the inverse problem of computing the camera location and orientation from the transformation matrix involves solution of simultaneous nonlinear equations in several variables and is considered difficult. In this paper we present a new and simple analytical technique that accomplishes this inversion rather easily. This technique works very well in practice and has considerable applications for motion tracking.