Blurred Correlation Versus Correlation Blur

Abstract

We discuss the topic of correlation in a scale space setting. Correlation involves two distinct scales. The “outer scale” is the scale of the region over which the correlation will be calculated. Classically this is the whole space of interest, but in many cases one desires the correlation over some region of interest. The “inner scale” is the scale at which the signals to be correlated are represented. Classically this means infinite precision. For our purposes we define “correlation” as the point–wise product of two signals, “blurred correlation” as the integration of this correlation over the region of interest, and “correlation blur” as this point–wise correlation applied to the signals represented at the inner scale. For generic purposes we are interested in “blurred correlation blur”. We discuss a well known (and practically important) example of blurred correlation for essentially zero inner scale. Such a situation leads to apparently paradoxical results. We then discuss correlation blur, which can be understood as a form of “regularized” correlation, leading to intuitively acceptable results even for the case of point sets (e.g., temporal events or point sets in space). We develop the formal structure and present a number of examples.