Statistical Analysis of Geometric Computation

This paper studies the statistical behavior of errors involved in fundamental geometric computations. We first present a statistical model of noise in terms of the covariance matrix of the N-vector. Using this model, we compute the covariance matrices of N-vectors of lines and their intersections. Then, we determine the optimal weights for the least-squares optimization and compute the covariance matrix of the resulting optimal estimate. The result is then applied to line fitting to edges and computation of vanishing points and focuses of expansion. We also point out that statistical biases exist in such computations and present a scheme called renormalization, which iteratively removes the bias by automatically adjusting to noise without knowing noise characteristics. Random number simulations are conducted to confirm our analysis.