Propagating Covariance in Computer Vision

This paper describes how to propagate approximately additive random perturbations through any kind of vision algorithm step in which the appropriate random perturbation model for the estimated quantity produced by the vision step is also an additive random perturbation. We assume that the vision algorithm step can be modeled as a calculation (linear or non-linear) that produces an estimate that minimizes an implicit scaler function of the input quantity and the calculated estimate. The only assumption is that the scaler function be non-negative, have finite first and second partial derivatives, that its value is zero for ideal data, and that the random perturbations are small enough so that the relationship between the scaler function evaluated at the ideal but unknown input and output quantities and evaluated at the observed input quantity and perturbed output quantity can be approximated sufficiently well by a first order Taylor series expansion. The paper finally discusses the issues of verifying that the derived statistical behavior agrees with the experimentally observed statistical behavior. 1 Intelligent Systems Laboratory, Department of Electrical Engineering, University of Washington, Seattle, WA 98195, USA Propagating Covariance in Computer Vision Robert M. Haralick Intelligent Systems Laboratory Department of Electrical Engineering University of Washington Seattle, WA 98195, USA Abstract: This paper describes how to propagate approximately additive random perturbations through any kind of vision algorithm step in which the appropriate random perturbation model for the estimated quantity produced by the vision step is also an additive random perturbation. We assume that the vision algorithm step can be modeled as a calculation (linear or non-linear) that produces an estimate that minimizes an implicit scaler function of the input quantity and the calculated estimate. The only assumption is that the scaler function be non-negative, have nite rst and second partial derivatives, that its value is zero for ideal data, and that the random perturbations are small enough so that the relationship between the scaler function evaluated at the ideal but unknown input and output quantities and evaluated at the observed input quantity and perturbed output quantity can be approximated su ciently well by a rst order Taylor series expansion. The paper nally discusses the issues of verifying that the derived statistical behavior agrees with the experimentally observed statistical behavior. This paper describes how to propagate approximately additive random perturbations through any kind of vision algorithm step in which the appropriate random perturbation model for the estimated quantity produced by the vision step is also an additive random perturbation. We assume that the vision algorithm step can be modeled as a calculation (linear or non-linear) that produces an estimate that minimizes an implicit scaler function of the input quantity and the calculated estimate. The only assumption is that the scaler function be non-negative, have nite rst and second partial derivatives, that its value is zero for ideal data, and that the random perturbations are small enough so that the relationship between the scaler function evaluated at the ideal but unknown input and output quantities and evaluated at the observed input quantity and perturbed output quantity can be approximated su ciently well by a rst order Taylor series expansion. The paper nally discusses the issues of verifying that the derived statistical behavior agrees with the experimentally observed statistical behavior.