Particle image pattern mutual information and uncertainty estimation for particle image velocimetry

In this work we introduce a new measure for particle image velocimetry (PIV) cross-correlation quality and establish analytically its connection to the basic PIV theory. This metric, which we term 'mutual information' (MI), can be used to estimate the number of correlated particles and connect to the PIV measurement uncertainty quantification. In PIV the number of particles in common between two consecutive frames forms the basis of the cross-correlation operation that yields the velocity measurement. Since the particle image pattern intensity distribution within each image represents the available signal, the inherent number of common particle pairs between the cross-correlated images, which can be thought of as the amount of mutual information, governs the potential accuracy of the PIV measurement. The number of common particle pairs between the images can be expressed by the product of the image density N_I, and the fraction of particles that leave the frame due to in-plane and out-of-plane motion F_I and F_O, respectively. It has previously been shown that this parameter, N_IF_IF_O, directly relates to the validity of a PIV measurement. However, in real experiments, N_IF_IF_O is unknown and difficult to calculate. Here we propose to overcome this limitation by introducing a new metric (MI), which directly computes the apparent amount of common information contained in the particle patterns of two consecutive images without prior knowledge of the particle field. Both theoretical derivation and experimental results are provided to show that MI and N_IF_IF_O represent the same characteristics of a PIV measurement. Subsequently, MI is used to develop a model for PIV uncertainty estimation. This metric and the corresponding uncertainty model presented herein are applied to both standard and a filtered phase-only (robust phase correlation) correlation methods. These advancements lead to robust uncertainty estimation models, which are tested against both synthetic benchmark data as well as real experimental measurements. For all cases considered here, $U_{68.5}$ and $U_{95}$ uncertainties demonstrated coverage factors approximately equal to the theoretically expected values of 68.5% and 95%, which reflect 1σ and 2σ levels in a normal distribution model respectively. (paper)