Processing local-stationary data

ABSTRACT

Stationarity of a data set guarantees spatially invariant statistical properties, such as mean, variance, dip or spectrum. Many image processing schemes assume stationary data and consequently do not apply to nonstationary data. However, these processes can be generalized to apply to local-stationary data sets, a type of nonstationary data. We split the local-stationary data into its stationary components, process them individually using the stationary process, and finally merge the components into a single output.

A process or image is called stationary, if the statistics - mean, variance, spectrum - of any subset accurately describe the statistics of the entire data. Stationarity is so helpful that scientists assume it, even when a given data is only approximately stationary. Unfortunately, many interesting things in life are nonstationary and predictions and prejudices based on small subsets are notoriously unreliable.

Stationarity and nonstationarity are well-known properties in statistics and image processing (). Gabor's windowed Fourier transform () and the later short-time Fourier transform () are early techniques that separate nonstationary images into stationary components. The components compactly localize events simultaneously in the time and frequency domain. In recent years, the time-frequency analysis and its applications of image compression, edge and feature detection, and texture analysis was considerably reformulated by multiresolution analysis, pyramid algorithms, and particularly wavelet transforms Castleman (1996). Local stationarity is a special case of nonstationarity: A local-stationary data set can be split into smaller stationary subsets Castleman (1996).

In this article, I first briefly define and illustrate stationarity and local-stationarity. Then I describe an algorithm that processes local-stationary data. by patching. The approach generalizes Claerbout's 1992a patching method and encapsulate it into a set of easy-to-use, object-oriented classes.