Asymmetric Autocorrelation Function To Resolve Directional Ambiguity In PIV Images

Autocorrelation of a double-exposed image, unlike cross-correlation between two images, produces a correlation function that is symmetric about the origin. Thus, while it is possible to calculate the speed and direction of tracer particles in a particle image velocimetry (PIV) image using autocorrelation, it is impossible to tell whether the velocity is in the positive or negative direction. This ambiguity can be resolved by spatially shifting one exposure relative to the next or labeling exposures with color or polarization, but the complexity and limitations of these methods can be prohibitive. It is, however, possible to resolve the sign of the velocity from a triple-exposed image using unequal time intervals between exposures. Triple-exposed images, like double-exposed images, correlate symmetrically about zero. The directional ambiguity, however, can be resolved by calculating the probability that the three exposures occur in a specific temporal order; that is, by assuming that the correlation has a specific sign and testing to see if the assumption is correct. Traditional spectral and statistical correlation techniques are unable to accomplish this. Presented herein is a computationally efficient asymmetric correlation function that is able to differentiate the temporal order of triple exposed images. Included is a discussion of the limitations of this function and of difficulties in experimental implementation. NOMENCLATURE η Normalized asymmetric correlation function Φ Normalized correlation function o Φ Masked correlation function φ Non-normalized error correlation function [pixels] φ Non-normalized statistical correlation function [pixels] ∆ Correlation search length [pixels] τ Asymmetric exposure ratio Ι Pixel intensity i,j Image coordinates [pixels] m,n Data array indices M, N Interrogation image dimensions [pixels] x,y Pixel image coordinates [pixels] ∆i,∆j Pixel displacement [pixels] Asymmetric Autocorrelation to Resolve Directional Ambiguity inPIV Images Page 2 of 15 Y.M. Marzouk and D.P. Hart .