Triple correlations

The (auto)triple correlation l⁽³⁾(t1, t2) is defined as the triple function integral, applied to the signal l(t) l⁽³⁾(t1, t2) = ∫ l(t)l(t + t1)l(t + t2) dt. The triple correlation l⁽³⁾(t1, t2) is less popular than the standard (double) correlation l⁽²⁾(t1) for several reasons: l⁽²⁾is sometimes easier to observe and to process, l⁽³⁾is small for many bipolar or complex signals, the mathematics associated with l⁽²⁾is better known. On the other hand, the triple correlation l⁽³⁾knows more about the signal l than does the ordinary autocorrelation l⁽²⁾. Also l⁽³⁾is in some ways more sensitive, in other ways less sensitive to noise, to bias drifts, etc. Hence, there are situations, where it is quite favorable to evaluate one-dimensional signals or two-dimensional pictures by means of their triple correlations. We will review the underlying mathematical tools and report on our projects where triple correlations were employed for studying laser pulse shapes, sound quality, halftone print statistics, mobility of bacteria, and astronomical speckle interferometry. We will mention also how others have used the triple correlation for ocean waves, engine noises, intensity interferometry, and other optical signal processing tasks.