Estimating the Jacobian of the Singular Value Decomposition: Theory and applications

The Singular Value Decomposition (SVD) of a matrix is a linear algebra tool that has been successfully applied to a wide variety of domains. The present paper is concerned with the problem of estimating the Jacobian of the SVD components of a matrix with respect to the matrix itself. An exact analytic technique is developed that facilitates the estimation of the Jacobian using calculations based on simple linear algebra. Knowledge of the Jacobian of the SVD is very useful in certain applications involving multivariate regression or the computation of the uncertainty related to estimates obtained through the SVD. The usefulness and generality of the proposed technique is demonstrated by applying it to the estimation of the uncertainty for three different vision problems, namely self-calibration, epipole computation and rigid motion estimation.