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Diffusion-based spatial priors for imaging.

Harrison LM, Penny W, Ashburner J, Trujillo-Barreto N, Friston KJ - Neuroimage (2007)

Bottom Line: This can furnish a non-stationary smoothing process that preserves features, which would otherwise be lost with a fixed Gaussian kernel.We describe a Bayesian framework that incorporates non-stationary, adaptive smoothing into a generative model to extract spatial features in parameter estimates.Critically, this means adaptive smoothing becomes an integral part of estimation and inference.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London, WC1N 3BG, UK. l.harrison@fil.ion.ucl.ac.uk

ABSTRACT
We describe a Bayesian scheme to analyze images, which uses spatial priors encoded by a diffusion kernel, based on a weighted graph Laplacian. This provides a general framework to formulate a spatial model, whose parameters can be optimized. The application we have in mind is a spatiotemporal model for imaging data. We illustrate the method on a random effects analysis of fMRI contrast images from multiple subjects; this simplifies exposition of the model and enables a clear description of its salient features. Typically, imaging data are smoothed using a fixed Gaussian kernel as a pre-processing step before applying a mass-univariate statistical model (e.g., a general linear model) to provide images of parameter estimates. An alternative is to include smoothness in a multivariate statistical model (Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. Neuroimage 24, 350-362). The advantage of the latter is that each parameter field is smoothed automatically, according to a measure of uncertainty, given the data. In this work, we investigate the use of diffusion kernels to encode spatial correlations among parameter estimates. Nonlinear diffusion has a long history in image processing; in particular, flows that depend on local image geometry (Romeny, B.M.T., 1994. Geometry-driven Diffusion in Computer Vision. Kluwer Academic Publishers) can be used as adaptive filters. This can furnish a non-stationary smoothing process that preserves features, which would otherwise be lost with a fixed Gaussian kernel. We describe a Bayesian framework that incorporates non-stationary, adaptive smoothing into a generative model to extract spatial features in parameter estimates. Critically, this means adaptive smoothing becomes an integral part of estimation and inference. We illustrate the method using synthetic and real fMRI data.

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Real data: twelve contrast images of a slice showing bilateral response in posterior cingulated gryi (pCG) during a study of coherent motion (Harrison et al., 2007). (a) Twelve samples (b) estimated conditional means using EGL and GGL (left and right). (c) Inset of panel b with contour plot of a local diffusion kernel overlaid. Distinguishing borders between regions of high/low parameter estimates is difficult due to smoothing by the EGL. However, borders are easily seen on the right. (d) Posterior probability maps; where white regions indicate p(w > 0.5) > 0.95. (e) Inset of panel d for GSP, EGL and GGL. Active regions using EGL are characterized by rounded edges, i.e., blobs, while for GGL the shape of bilateral response are elongated in fitting with the anatomy of pCG, (f–h) surface plots of conditional means from inset. Note vertical scale, especially for GSP, which shows large shrinkage compared to EGL and GGL, (i and h) graph plot (gplot.m) of second and third eigenvectors of EGL and GGL. Heterogeneous graph weights of GGL are easily observed compared to EGL.
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fig5: Real data: twelve contrast images of a slice showing bilateral response in posterior cingulated gryi (pCG) during a study of coherent motion (Harrison et al., 2007). (a) Twelve samples (b) estimated conditional means using EGL and GGL (left and right). (c) Inset of panel b with contour plot of a local diffusion kernel overlaid. Distinguishing borders between regions of high/low parameter estimates is difficult due to smoothing by the EGL. However, borders are easily seen on the right. (d) Posterior probability maps; where white regions indicate p(w > 0.5) > 0.95. (e) Inset of panel d for GSP, EGL and GGL. Active regions using EGL are characterized by rounded edges, i.e., blobs, while for GGL the shape of bilateral response are elongated in fitting with the anatomy of pCG, (f–h) surface plots of conditional means from inset. Note vertical scale, especially for GSP, which shows large shrinkage compared to EGL and GGL, (i and h) graph plot (gplot.m) of second and third eigenvectors of EGL and GGL. Heterogeneous graph weights of GGL are easily observed compared to EGL.

Mentions: fMRI data collected from twelve subjects during a study of the visual motion system (Harrison et al., 2007) were used for our comparative analyses. The study had a 2 × 2 factorial design with motion type (coherent or incoherent) and motion speed as the two factors. Single subject analyses were performed, with no smoothing, using SPM2 (http://www.fil.ion.ucl.ac.uk/spm) to generate contrast images of the main effect of coherence. Images (one slice) of the twelve contrast images are shown in Fig. 5a. These constitute the data, Y, and the design matrix, X = 1, was a column of ones, implementing a single-sample t-test. The aim was to estimate μ(u); the conditional expectation of the main effect of coherent motion as a function of position in the brain. We calculated μ(u) under the different priors above.


Diffusion-based spatial priors for imaging.

Harrison LM, Penny W, Ashburner J, Trujillo-Barreto N, Friston KJ - Neuroimage (2007)

Real data: twelve contrast images of a slice showing bilateral response in posterior cingulated gryi (pCG) during a study of coherent motion (Harrison et al., 2007). (a) Twelve samples (b) estimated conditional means using EGL and GGL (left and right). (c) Inset of panel b with contour plot of a local diffusion kernel overlaid. Distinguishing borders between regions of high/low parameter estimates is difficult due to smoothing by the EGL. However, borders are easily seen on the right. (d) Posterior probability maps; where white regions indicate p(w > 0.5) > 0.95. (e) Inset of panel d for GSP, EGL and GGL. Active regions using EGL are characterized by rounded edges, i.e., blobs, while for GGL the shape of bilateral response are elongated in fitting with the anatomy of pCG, (f–h) surface plots of conditional means from inset. Note vertical scale, especially for GSP, which shows large shrinkage compared to EGL and GGL, (i and h) graph plot (gplot.m) of second and third eigenvectors of EGL and GGL. Heterogeneous graph weights of GGL are easily observed compared to EGL.
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Related In: Results  -  Collection

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fig5: Real data: twelve contrast images of a slice showing bilateral response in posterior cingulated gryi (pCG) during a study of coherent motion (Harrison et al., 2007). (a) Twelve samples (b) estimated conditional means using EGL and GGL (left and right). (c) Inset of panel b with contour plot of a local diffusion kernel overlaid. Distinguishing borders between regions of high/low parameter estimates is difficult due to smoothing by the EGL. However, borders are easily seen on the right. (d) Posterior probability maps; where white regions indicate p(w > 0.5) > 0.95. (e) Inset of panel d for GSP, EGL and GGL. Active regions using EGL are characterized by rounded edges, i.e., blobs, while for GGL the shape of bilateral response are elongated in fitting with the anatomy of pCG, (f–h) surface plots of conditional means from inset. Note vertical scale, especially for GSP, which shows large shrinkage compared to EGL and GGL, (i and h) graph plot (gplot.m) of second and third eigenvectors of EGL and GGL. Heterogeneous graph weights of GGL are easily observed compared to EGL.
Mentions: fMRI data collected from twelve subjects during a study of the visual motion system (Harrison et al., 2007) were used for our comparative analyses. The study had a 2 × 2 factorial design with motion type (coherent or incoherent) and motion speed as the two factors. Single subject analyses were performed, with no smoothing, using SPM2 (http://www.fil.ion.ucl.ac.uk/spm) to generate contrast images of the main effect of coherence. Images (one slice) of the twelve contrast images are shown in Fig. 5a. These constitute the data, Y, and the design matrix, X = 1, was a column of ones, implementing a single-sample t-test. The aim was to estimate μ(u); the conditional expectation of the main effect of coherent motion as a function of position in the brain. We calculated μ(u) under the different priors above.

Bottom Line: This can furnish a non-stationary smoothing process that preserves features, which would otherwise be lost with a fixed Gaussian kernel.We describe a Bayesian framework that incorporates non-stationary, adaptive smoothing into a generative model to extract spatial features in parameter estimates.Critically, this means adaptive smoothing becomes an integral part of estimation and inference.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London, WC1N 3BG, UK. l.harrison@fil.ion.ucl.ac.uk

ABSTRACT
We describe a Bayesian scheme to analyze images, which uses spatial priors encoded by a diffusion kernel, based on a weighted graph Laplacian. This provides a general framework to formulate a spatial model, whose parameters can be optimized. The application we have in mind is a spatiotemporal model for imaging data. We illustrate the method on a random effects analysis of fMRI contrast images from multiple subjects; this simplifies exposition of the model and enables a clear description of its salient features. Typically, imaging data are smoothed using a fixed Gaussian kernel as a pre-processing step before applying a mass-univariate statistical model (e.g., a general linear model) to provide images of parameter estimates. An alternative is to include smoothness in a multivariate statistical model (Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. Neuroimage 24, 350-362). The advantage of the latter is that each parameter field is smoothed automatically, according to a measure of uncertainty, given the data. In this work, we investigate the use of diffusion kernels to encode spatial correlations among parameter estimates. Nonlinear diffusion has a long history in image processing; in particular, flows that depend on local image geometry (Romeny, B.M.T., 1994. Geometry-driven Diffusion in Computer Vision. Kluwer Academic Publishers) can be used as adaptive filters. This can furnish a non-stationary smoothing process that preserves features, which would otherwise be lost with a fixed Gaussian kernel. We describe a Bayesian framework that incorporates non-stationary, adaptive smoothing into a generative model to extract spatial features in parameter estimates. Critically, this means adaptive smoothing becomes an integral part of estimation and inference. We illustrate the method using synthetic and real fMRI data.

Show MeSH