Estimation of geometric parameters in 3D reconstruction problems using linear matrix inequalities

This paper proposes methods for estimating geometrical parameters, such as coordinate vectors of feature points, camera positions, camera orientations, and so on, in 3D reconstruction problems. In particular, we consider the following two problems: for given camera projection matrices and coordinate vectors of projected feature points on the image planes, find coordinate vectors of feature points; and, for given coordinate vectors of feature points and coordinate vectors of projected feature points on image planes, find camera projection matrices. These problems are formulated as the matrix optimization problem of finding the matrix that minimizes the matrix norm of a certain matrix. The matrix optimization problem is reduced to a convex optimization problem with linear matrix inequalities (LMI), which is widely used in control engineering. The proposed method is evaluated through numerical examples.