Regularised estimation of 2D-locally stationary wavelet processes

Abstract

Locally Stationary Wavelet processes provide a flexible way of describing the time/space evolution of autocovariance structure over an ordered field such as an image/time-series. Classically, estimation of such models assume continuous smoothness of the underlying spectra and are estimated via local kernel smoothers. We propose a new model which permits spectral jumps, and suggest a regularised estimator and algorithm which can recover such structure from images. We demonstrate the effectiveness of our method in a synthetic experiment where it shows desirable estimation properties. We conclude with an application to real images which illustrate the qualitative difference between the proposed and previous methods.

Case study: shipping trend estimation and prediction via multiscale variance stabilisation

“…Alternatively, a model using wavelets instead of Fourier basis functions was introduced by Nason et al [50] as the locally stationary wavelet processes. These, and their associated estimation methods were further improved and extended by Fryzlewicz [19], Van Bellegem and von Sachs [62], Fryzlewicz [20], Fryzlewicz and Nason [24], Triantafyllopoulos and Nason [61], Van Bellegem and von Sachs [63], Fryzlewicz and Ombao [25], for forecasting in Fryzlewicz et al [26], Xie et al [66] and for extensions to 2D processes and texture modelling, see [17] or [28] or [59] and references therein.…”

The locally stationary dual-tree complex wavelet model

“…This work was subsequently followed by methodology applied to non-stationarity detection in image textures by Taylor et al (2014) and segmentation of imagery into stationary regions by Nunes et al (2014). A regularised smoothing strategy for the two-dimensional LSW process is developed in Gibberd and Nelson (2016). Very recently, Taylor et al (2017) combined the lattice and multivariate extensions to formulate an LSW model for multivalued image data.…”