2.8. Appendix: Estimating Homography Between the Model Plane and Its Image

There are many ways to estimate the homography between the model plane and its image. Here, we present a technique based on maximum likelihood criterion. Let M_i and m_i be the model and image points respectively. Ideally, they should satisfy (2.18). In practice, they do not because of noise in the extracted image points. Let's assume that m_i is corrupted by Gaussian noise with mean 0 and covariance matrix . Then, the maximum likelihood estimation of H is obtained by minimizing the following functional

where , with being the i-th row of H.

In practice, we simply assume = σ²I for all i. This is reasonable if points are extracted independently with the same procedure. In this case, the above problem becomes a nonlinear least-squares one: . The nonlinear minimization is conducted with the Levenberg-Marquardt algorithm, as implemented in Minpack [23]. This requires an initial guess, which can be obtained as follows.

Let . Then equation (2.18) can be rewritten as

When we are given n points, we have n such equations, which can be written in matrix equation as Lx = 0, where L is a 2n x 9 matrix. As x is defined up to a scale factor, the solution is well known to be the right singular vector of L associated with the smallest singular value (or equivalently, the eigenvector of L^TL associated with the smallest eigenvalue). In L, some elements are constant 1, some are in pixels, some are in world coordinates, and some are multiplications of both. This makes L poorly conditioned numerically. Much better results can be obtained by performing a simple data normalization prior to running the above procedure.