A 2-D Extension of the Trend Concept Applicable to Images, Surfaces and Space-Time Signals

The analysis of highly irregular processes is often based on the search for a trend, an average curve representing the general pattern of the phenomenon observed. This concept seems intuitive, but determining it objectively raises a number of problems. As the moving average method is one of the most widely used for finding one-dimensional trends, we propose here to extend it to two-dimensional data sets, and to study some of the properties of these 2-D moving averages. We then show that they can be used to detect characteristic observation windows (uniform or self-adaptive), leading to structural trends in the analyzed signal. The method is first applied to functions f(x,y) whose variables have the same physical dimension (length,…). It is then generalized to cases of different dimensions (e.g. space-time signals). Applications cover a wide range of fields (turbulent structures in fluid mechanics, image analysis, characterization of surfaces, optimization of an observation process, multi-scale modelling, ...).