Determination of the most probable point from non-concurrent lines

Abstract

From a given camera position and orientation we can draw a ray from the camera focus through a point in the image plane (this gives two equations). To locate a three-dimensional point (three unknown coordinates) on the ray at least one more equation is required. When another camera or another perspective is used, the number of equations becomes four and the system is overdetermined, and is to be solved in terms of the best estimation. This work compares the application to the point triangulation problem of the least squares method as opposed to the frequently used linear regression method. Since all coefficients of the equations are subject to error, the linear regression method is not conceptually correct. However, it is commonly used because its straightforward computations provide reasonable solutions in most cases. In our approach the least squares method is formulated by equally weighting the coefficients, thereby obtaining a solution from the principal components of the scatter matrix of the coefficients. Since we have four equations for the three unknowns, we can have four exact solutions from combinations of three out of the four equations. Geometrically each equation represents a plane in 3-D space, and each exact solution represents a vertex of a tetrahedron formed by the four equations. After normali zing the solutions to be scale-invariant, we compute the volume of the normalized tetrahedron. When the volume is greater than one, our least squares approach is significantly different from the usual linear regression method. In other words, when the residual error sum is very large, the usual method fails to give a correct solution, and, therefore, should be considered valid only for relatively small or purely random errors. © 1986 SPIE.