Theoretical analysis of the measurement precision in particle image velocimetry

An analytical expression for the error in the sub-pixel displacement is derived. The expression relates the error to the second- and fourth-order moments of the intensity probability density function, which can both be expressed in terms of the source density. Scaling relations for the error as a function of the particle-image diameter are determined by means of the Cramèr–Rao lower bound, for both constant image density and constant source density. The analytical expression is evaluated for the error as a function of the sub-pixel displacement, the particle-image diameter, and the image density, and the results are validated against results from Monte-Carlo simulations. It is demonstrated that results obtained from synthetic PIV images that do not represent the correct image intensity probability density function systematically underestimate the displacement measurement error.