D. Mondal and D. B. Percival (2012),
`Wavelet Variance Analysis for Random Fields on a Regular Lattice,'
IEEE Transactions on Image Processing,
21, no. 2, pp. 537-549.
Summary
There has been considerable recent interest
in using wavelets to analyze time series and images
that can be regarded as realizations
of certain one- and two-dimensional stochastic processes.
Wavelets give rise to the concept of the
wavelet variance (or wavelet power spectrum),
which decomposes the variance of a stochastic process
on a scale-by-scale basis.
The wavelet variance has been applied to a variety of time series,
and a statistical theory for estimators of this variance
has been developed.
While there have been applications of the wavelet variance
in the two-dimensional context
(in particular in works by Unser, 1995,
on wavelet-based texture analysis for images
and by Lark and Webster, 2004, on analysis of soil properties),
a formal statistical theory for such analysis has been lacking.
In this paper, we develop the statistical theory
by generalizing and extending
some of the approaches developed for time series,
thus leading to a large sample theory for estimators
of two-dimensional wavelet variances.
We apply our theory to simulated data
from Gaussian random fields with exponential covariances
and from fractional Brownian surfaces.
We also use our methodology to analyze images
of four types of clouds observed over the south-east Pacific ocean.
Key Words
Analysis of variance;
Cloud data;
Daubechies filters;
Fractional Brownian surface;
Semi-variogram
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