Multiscale Methods for Data on Graphs and Irregular Multidimensional Situations

Abstract

For regularly spaced one-dimensional data, wavelet shrinkage has proven to be a compelling method for non-parametric function estimation. We create three new multiscale methods that provide wavelet-like transforms both for data arising on graphs and for irregularly spaced spatial data in more than one dimension. The concept of scale still exists within these transforms, but as a continuous quantity rather than dyadic levels. Further, we adapt recent empirical Bayesian shrinkage techniques to enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular spatial data. We demonstrate that our methods perform very well when compared with several other methods for spatial regression for both real and simulated data. Although we concentrate on multiscale shrinkage (regression) we present our new 'wavelet transforms' as generic tools intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data. Copyright (c) 2009 Royal Statistical Society.

A wavelet lifting approach to long-memory estimation

“…Areas of densely sampled time locations are thus associated with sets of shorter intervals. The LOCAAT algorithm, as designed in Jansen et al (2009), has both the initial and dual scaling basis functions given by suitably scaled characteristic functions over these intervals, but, in general, this is not a requirement.…”

“…where the weights (b n i ) i∈I n are obtained from the requirement that the algorithm preserves the signal mean value (Jansen et al 2001(Jansen et al , 2009). The interval lengths associated with the neighbouring points are also updated to account for the decreasing number of unlifted coefficients that remain.…”

“…Our Hurst exponent estimation method makes use of a recently developed lifting transform called the lifting one coefficient at a time (LOCAAT) transform proposed by Jansen et al (2001Jansen et al ( , 2009) which works as follows.…”

Multi-scale freeform surface texture filtering using a mesh relaxation scheme

“…Many researchers have proposed the use of the lifting schemes to analyse irregular data represented by meshes or graphs [16,[29][30][31]. Recently, the authors have also proposed a freeform filtering algorithm based on the lifting wavelets transform to filter freeform surfaces represented by 3D irregular meshes.…”

“…The introduction of the second generation wavelets and lifting scheme [22][23][24] made the extension of wavelets and MRA possible for irregular data sets and for different types of 3D meshes and graphs and a few algorithms have been proposed [21,[25][26][27][28][29][30][31][32][33].…”