Abstract
In this article we propose a novel framework for the modelling of non-stationary multivariate lattice processes. Our approach extends the locally stationary wavelet paradigm into the multivariate two-dimensional setting. As such the framework we develop permits the estimation of a spatially localised spectrum within a channel of interest and, more importantly, a localised cross-covariance which describes the localised coherence between channels. Associated estimation theory is also established which demonstrates that this multivariate spatial framework is properly defined and has suitable convergence properties. We also demonstrate how this model-based approach can be successfully used to classify a range of colour textures provided by an industrial collaborator, yielding superior results when compared against current state-of-the-art statistical image processing methods.
Highlights
Wavelet methods have enjoyed popularity for many years for statistical data analysis due to their ability to provide efficient representations of signals and images
For the multivariate setting on which we focus in this article, this leads to the consideration of (i) spatially localised wavelet spectra which represent the structure within a single channel, and (ii) cross-spectra which capture the structure across channels
To illustrate the potential of the LS2Wmv feature vector described in Algorithm 1 in texture classification tasks, we simulate a number of colour textures of dimension 256×256 with different colour texture structure, and use the classification procedure described above (Sect. 4) to classify 50 sampled sub-images of each texture, using another 50 subimages as a training set
Summary
Wavelet methods have enjoyed popularity for many years for statistical data analysis due to their ability to provide efficient representations of signals and images. Recently a significant body of work has emerged in the area locally stationary wavelet time series. This stems from the seminal work of Nason et al (2000). Notable contributions include work on forecasting (Fryzlewicz et al 2003), changepoint analysis (Killick et al 2013), signal classification (Fryzlewicz and Ombao 2009) and alias detection (Eckley and Nason 2014)
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