Processing local-stationary data

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 (1992) patching method and encapsulate it into a set of easy-to-use, object-oriented classes. Definition of stationarity A random process y(x) is strict-sense stationary if the joint distribution of any set of samples does not depend on the sample’s placement. Consequently, first order cumulative distribution functions, e.g., mean and variance, of y(x) are constant. Furthermore, second order cumulative distribution functions (such as autocorrelation and autocovariance) depend only on the distance in placement, x2 − x1. For example, a Gaussian process is strict-sense stationary since it is completely specified by its mean and covariance function. If the mean is constant and the autocovariance is a function that depends only on the distance in placement, then we call the data wide-sense stationary or simply stationary.