Statistical Parametric Maps for Functional MRI Experiments in R: The Package fmri
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1 Weierstrass Institute for Applied Analysis and Stochastics Statistical Parametric Maps for Functional MRI Experiments in R: The Package fmri Karsten Tabelow UseR!2011 Mohrenstrasse Berlin Germany Tel August 18, 2011
2 Images and Noise The Package fmri August 18, 2011 Page 2 (19)
3 Images and Noise The Package fmri August 18, 2011 Page 2 (19)
4 Image Denoising Image Denoising Methods Kernel estimators Wavelets, Curvelets,... MCMC, SANN Diffusion methods Scale-space methods Scanner upgrade The Package fmri August 18, 2011 Page 3 (19)
5 Image Denoising Image Denoising Methods Kernel estimators Wavelets, Curvelets,... MCMC, SANN Diffusion methods Scale-space methods Scanner upgrade Structural Adaptive Smoothing Local parametric model for data Iterative estimation of parameters and homogeneity regions Dimension-free, structure preserving/enhancing The Package fmri August 18, 2011 Page 3 (19)
6 Structural adaptive smoothing: Setup General setup / Local parametric model Design: x 1,..., x n X IR p Observations: Y 1,..., Y n Y IR q Y i P θ(xi ) θ : IR p Θ (i.i.d.) The Package fmri August 18, 2011 Page 4 (19)
7 Structural adaptive smoothing: Setup General setup / Local parametric model Design: x 1,..., x n X IR p Observations: Y 1,..., Y n Y IR q Y i P θ(xi ) θ : IR p Θ (i.i.d.) Structural assumption Partitioning X = M m=1 Xm such that θ(x) = θ(x i) m : x X m x i X m i.e. θ constant on each X m Some components of θ may be global parameters. The Package fmri August 18, 2011 Page 4 (19)
8 Structural adaptive smoothing: General idea General idea Determine structure and estimates in an iterative procedure Weighting scheme W (x i) = (w i1,..., w in) W (x i) describes a local model w (k) ij = K loc ( x i x j 2 /h 2 k) K s(s (k) ij /λ) s ij measures the difference between estimates ˆθ(x i) and ˆθ(x j) Four steps: A) Initialize estimates ˆθ(x i) and describe initial partitioning W (x i) ideally by observed values and w j(x i) = w ij = δ ij B) Learn about θ(x) from information on partitioning C) Learn about partitioning from estimates ˆθ(x) D) Iterate while inspecting scale space from local to global The Package fmri August 18, 2011 Page 5 (19)
9 Structural adaptive smoothing: Theoretical results Results Propagation under homogeneity: For a homogeneous situation and µ > 4 the last step k estimate ˆθ i fulfills P ( (k N ) i K(ˆθ i, θ) > µ log(n) ) 2k /n. Propagation under local homogeneity: Similar results for interior points of local homogeneous regions. Stability of estimates: The quality of the best intermediate result holds (up to a constant) for the final estimate. Optimal adaptive rate of estimation under smoothness conditions on f θ. Separation property The Package fmri August 18, 2011 Page 6 (19)
10 Structural adaptive smoothing: Parameters Parameters λ can be selected by a propagation condition, independ of data k determines the maximum bandwidth c h = /p provides exponential growth of sum of location weights Kernels K loc (z) = (1 z) + and K s(z) = min(1, 2(1 z)) + Propagation condition ˆθ (k) (x) PS estimate of θ(x) from iteration k, θ (k) (x) Kernel estimate of θ(x) using bandwidth h (k) Let θ(x) θ. Select parameters such that k and some α > 0 E ˆθ (k) (x) θ (1 + α) E θ (k) (x) θ. The Package fmri August 18, 2011 Page 7 (19)
11 Images and Noise The Package fmri August 18, 2011 Page 8 (19)
12 Images and Noise The Package fmri August 18, 2011 Page 8 (19)
13 What application? Human Eye vs. Structural adaptive smoothing Human eye is a very good denoiser Experience: Structural adaptive smoothing compares well with eyes. Limited use for visual inspection and for volumetric (integral) quantities Applications Suitable for higher dimensional data Structure enhancement, structure identification. functional MRI - package fmri The Package fmri August 18, 2011 Page 9 (19)
14 Smoothing fmri data fmri data = 3D + T How to smooth fmri data? Blurring effect in Gaussian filter T2-volumes have limited structure. Local tests inefficient in 3D+T space Interesting structure is activation structure We have a model about HRF The Package fmri August 18, 2011 Page 10 (19)
15 fmri data analysis Data Preparation Registration Motion correction Normalization Smoothing The Package fmri August 18, 2011 Page 11 (19)
16 fmri data analysis Data Preparation Registration Motion correction Normalization Smoothing Linear Modeling Y = Xβ + ɛ Prewhitening Ỹ = Xβ + ɛ The Package fmri August 18, 2011 Page 11 (19)
17 fmri data analysis Data Preparation Linear Modeling Smoothing Thresholding Registration Y = Xβ + ɛ t-statistic... Motion correction Prewhitening multiple test problem Normalization Ỹ = Xβ + ɛ no filter The Package fmri August 18, 2011 Page 11 (19)
18 fmri data analysis Data Preparation Linear Modeling Smoothing Thresholding Registration Y = Xβ + ɛ t-statistic Motion correction Prewhitening FDR Normalization Ỹ = Xβ + ɛ no filter The Package fmri August 18, 2011 Page 11 (19)
19 fmri data analysis Data Preparation Linear Modeling Smoothing Thresholding Registration Y = Xβ + ɛ t-statistic Motion correction Prewhitening RFT Normalization Ỹ = Xβ + ɛ no filter The Package fmri August 18, 2011 Page 11 (19)
20 fmri data analysis Data Preparation Linear Modeling Smoothing Thresholding Registration Y = Xβ + ɛ t-statistic Motion correction Prewhitening RFT Normalization Ỹ = Xβ + ɛ Gaussian filter The Package fmri August 18, 2011 Page 11 (19)
21 fmri data analysis Data Preparation Linear Modeling Smoothing Thresholding Registration Y = Xβ + ɛ t-statistic Motion correction Prewhitening RFT Normalization Ỹ = Xβ + ɛ adaptive! The Package fmri August 18, 2011 Page 11 (19)
22 Artificial fmri data (a) Signal location in Phantom data (3D) with different signal size 26 slices with 15 containing activations The Package fmri August 18, 2011 Page 12 (19)
23 Artificial fmri data (a) Signal location in Phantom data (3D) with different signal size 26 slices with 15 containing activations (b) Relative proportion of detection using non adaptive smoothing The Package fmri August 18, 2011 Page 12 (19)
24 Artificial fmri data (a) Signal location in Phantom data (3D) with different signal size 26 slices with 15 containing activations (b) Relative proportion of detection using non adaptive smoothing (c) Relative proportion of detection using Structural Adaptive Smoothing The Package fmri August 18, 2011 Page 12 (19)
25 Finger tapping fmri data at different resolutions 64 x 64, non-adaptive 128 x 128, non-adaptive 64 x 64, adaptive 128 x 128, adaptive anatomy Slice 15 Slice 14 Slice 12 The Package fmri August 18, 2011 Page 13 (19)
26 Structural adaptive segmentation How to integrate smoothing and testing in on step? H : max i V T (ˆΓ H )=max max h H i V γi δ (or max γi δ). i V ( ) ˆγ (h) i δ b n(h)(ν) a a n(h) (ν) n(h) (ν) ˆDˆγ (h) i Structural adaptation Distribution of T (ˆΓ H ) under hypotheses by simulation, determine critical values Inspect scale space sequentially and segment as soon as critical value is exceeded Use adaptive smoothing as long as Hypothesis not rejected The Package fmri August 18, 2011 Page 14 (19)
27 Structural adaptive segmentation: Results Structural Structural adaptive segmentation adaptive smoothing δ=0, α=0.05 δ=2, α=0.05 δ=0, α=0.01 Anatomic Slice 15 Slice 14 Slice 13 The Package fmri August 18, 2011 Page 15 (19)
28 Performing an FMRI Analysis FMRI analysis library(fmri);library(adimpro); ds <- read.nifti("imagination.nii") anatomic <- extract.data(ds)[,,,1] scans <- 105 onsets <- c(16, 46, 76) duration <- 15 tr <- 2 hrf <- fmri.stimulus(scans, onsets, duration, tr) x <- fmri.design(hrf) spm <- fmri.lm(ds, x) spm.segment <- fmri.smooth(spm, hmax = 4, adaptation = "segment") ds.ana <- read.nifti("mprageco.nii") for (slice in 2:30) { img <- plot(spm.segment, ds.ana, slice = slice) write.image(make.image(img, gammatype="itu"), file=paste("result", slice, ".png", sep="")) } The Package fmri August 18, 2011 Page 16 (19)
29 Conclusions Package fmri Structural adaptive smoothing for fmri (using RFT) Structural adaptive segmentation for fmri Package fmri: I/O (cf. oro.nifti) Package fmri: Linear Modeling Package fmri: Smoothing Package fmri: Signal detection Package fmri: Publication ready images Package fmri: No group analysis Cf. packages arf3ds4, AnalyzeFMRI, neurosim, cudabayesreg The Package fmri August 18, 2011 Page 17 (19)
30 Collaborations Joint Work with: Jörg Polzehl and Vladimir Spokoiny, WIAS and MATHEON Henning U. Voss, Weill Medical College, Cornell University Cooperation: Citigroup Biomedical Imaging Center, Weill Medical College, Cornell University BNIC, Charité, Berlin R-Community: CRAN Task View: Medical Image Analysis Acknowledgments: We thank the Weill Medical College, Cornell University (HUV), for providing functional and diffusion-weighted MR datasets. The Package fmri August 18, 2011 Page 18 (19)
31 Bibliography K. Tabelow, J. Polzehl, H.U. Voss, V. Spokoiny (2006). Analyzing fmri experiments with structural adaptive smoothing procedures. Neuroimage, 33(1): K. Tabelow, V. Piëch, J. Polzehl, H.U. Voss (2009). High-resolution fmri: Overcoming the signal-to-noise problem. Journal of Neuroscience Methods, 178(2): J. Polzehl, H.U. Voss, K. Tabelow (2010). Structural adaptive segmentation for statistical parametric mapping. Neuroimage, 52(2): K. Tabelow, J. Polzehl (2011). Statistical parametric maps for functional MRI experiments in R: The package fmri. Journal of Statistical Software, 44(11): 1 21 Special Volume of Journal of Statistical Software MRI in R. The Package fmri August 18, 2011 Page 19 (19)
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