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Bayesian fMRI Analysis Using Wavelet-Based Spatial Priors for Voxel-Specific GLM

This paper presents a variational Bayesian approach for voxel-specific generalized linear models (GLM) in fMRI analysis, employing wavelet-based spatial basis function priors. Key elements include a data-driven estimation of smoothing levels tailored to each regressor, ensuring adaptability to local or nonstationary features. The orthonormal discrete wavelet transform (DWT) is utilized for efficient signal denoising through thresholding techniques. This method enhances the modeling of spatial variations in smoothness, facilitating improved analysis of regression coefficients within fMRI datasets.

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Bayesian fMRI Analysis Using Wavelet-Based Spatial Priors for Voxel-Specific GLM

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

  2. 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

  3. 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 levelsFewer wavelet coefficients

  4. Orthonormal Discrete Wavelet Transform (DWT) • Wavelet transform: Wavelet coefficients [Nx1] Data [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

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

  6. 3D denoising of a regression coefficient map Histogram of the wavelet coefficients

  7. 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

  8. Generative model

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

  10. 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…

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

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