Partition Mixture of 1D Wavelets for Multi-Dimensional Data with Application to Image Analysis

• Activity Number: 620 - Axles for Voxels: Recent Statistical Advances in Neuroimaging Data Analysis
• Type: Topic Contributed
• Date/Time: Thursday, August 2, 2018 : 8:30 AM to 10:20 AM
• Sponsor: ENAR
• Author(s): Li Ma* and Meng Li
• Companies: Duke University and Rice University
• Keywords: Bayesian inference; Multi-scale inference; Tree-based models; Signal processing; Wavelets; Recursive partitioning

Abstract

Traditional statistical wavelet analysis that carries out modeling and inference based on wavelet coefficients under a given, predetermined wavelet transform can quickly lose efficiency in multivariate problems, because such wavelet transforms, which are typically symmetric with respect to the dimensions, cannot adaptively exploit the energy distribution in a problem-specific manner. We introduce a principled probabilistic framework for incorporating such adaptivity---by (i) representing multivariate functions using one-dimensional (1D) wavelet transforms applied to a permuted version of the original function, and (ii) placing a prior on the corresponding permutation, thereby forming a mixture of permuted 1D wavelet transforms. Such a representation can achieve substantially better energy concentration in the wavelet coefficients. We demonstrate that with this framework even simple 1D Haar wavelets can achieve excellent performance in both 2D and 3D image processing via numerical experiments, outperforming state-of-the-art multidimensional wavelet-based methods especially in low signal-to-noise ratio settings, at a fraction of the computational cost.