Generalized Cross - Correlation Functions for Engineering Applications , Part I : Basic Theory

Traditional cross-correlation considers situations where two functions or data sets are linked by a constant shift either in time or space. Correlation provides estimates of such shifts even in the presence of considerable noise corruption. This makes the technique valuable in applications like sonar, displacement or velocity determination and pattern recognition. When regions are decomposed into patches in applications such as Particle Image Velocimerty it also allows estimates to be made of whole displacement/flow fields. The fundamental problem with traditional correlation is that patch size and hence statistical reliability must be compromised with resolution. This article develops a natural generalization of cross-correlation which removes the need for such compromises by replacing the constant shift with a function of time or space. This permits correlation to be applied globally to a whole domain retaining any long-range coherences present and dramatically improves statistical reliability by using all the data present in the domain for each estimate.