Bayesian fMRI analysis with Spatial Basis Function Priors Variational Bayes scheme for voxel-specific GLM using wavelet-based spatial priors for the regression coefficients Guillaume Flandin & Will Penny SPM Homecoming, Nov. 11 2004

Spatial prior using a kernel Spatial prior over regression and AR coefficients Data-driven estimation of the amount of smoothing (different for each regressor) Does not handle spatial variations in smoothness  spatial basis set prior Penny et al, NeuroImage, 2004

Orthonormal Discrete Wavelet Basis Set Decomposition of time series/spatial processes on an orthonormal basis set with: Multiresolution: time-frequency/scale-space properties Natural adaptivity to local or nonstationary features Good properties: Decorrelation / Whitening, Sparseness / Compaction, Fast implementation with a pyramidal algorithm in O(N) complexity Increased levels Fewer wavelet coefficients

Orthonormal Discrete Wavelet Transform (DWT) Data [Nx1] Wavelet coefficients [Nx1] Set of wavelet basis functions [NxN] Inverse transform: Multidimensional transform No need to build V in practice, thanks to Mallat’s pyramidal algorithm. Daubechies Wavelet Filter Coefficients

Wavelet shrinkage or nonparametric regression Signal denoising technique based on the idea of thresholding wavelet coefficients. DWT Thresh. IDWT Nonlinear operator  DWT => Threshold 

3D denoising of a regression coefficient map Histogram of the wavelet coefficients

Bayesian Wavelet Shrinkage Wavelet coefficients are a priori independent, The prior density of each coefficient is given by a mixture of two zero-mean Gaussian. Consider each level separately Applied only to detail levels Negligible coeffs. Significant coeffs. Estimation of the parameters via an Empirical Bayes algorithm

Generative model

Approximate posteriors Variational Bayes Iteratively updating Summary Statistics to maximise a lower bound on evidence

Summary / Future Variational Bayes scheme for voxel-specific GLM using wavelet-based spatial priors for the regression coefficients Replace the mono scale Gaussian filtering (=> anisotropic smoothing + amount of smoothness estimated from data) Lower the quantity of data to deal with in the iterative algorithm Implementation => spm_vb_* (2D vs. 3D, level-dependent parameters, Gibbs-like oscillations, …) General framework which allows lots of adaptations and improvements…

Wavelet denoising Signal denoising technique based on the idea of thresholding wavelet coefficients: