Signal, Noise and Preprocessing

Signal, Noise and Preprocessing
* Methods and Models for fMRI Analysis September 25th, 2015 Lars Kasper, PhD TNU & MR-Technology Group Institute for Biomedical Engineering, UZH & ETHZ Generous slide support: Guillaume Flandin Ged Ridgway Klaas Enno Stephan John Ashburner *Huettel et al. After Andreea’s practical intro, this will be the first theoretical talk of the morning. start with what measuring in fMRI This is why I want to start not with preprocessing itself, but instead on a perspective of what we measure in fMRI – which is, mainly noise; and what we want to measure in fMRI, which is: BOLD This will then motivate preprocessing. The title, by the way, is borrowed from a chapter in Scott Huettel’s fabulous book. Functional Magnetic Resonance Imaging, which I can heartily recommend for reading. Especially, since we start right with the acquired data in this talk and do not shed light on the question how an MR scanner works or what is the brain physiology allowing us to do fMRI.

Overview of SPM for fMRI
Preprocessing Image time-series Kernel Design matrix Statistical parametric map (SPM) Realignment Smoothing General linear model Random field theory Statistical inference Normalisation Instead, we start right with preprocessing. You have seen this slide before, and as you can see, preproc including realigment/smoothing and normalisation is a prerequisite to our stat analysis and inference. To understand its necessity, we should quickly remind ourselves what we actually do when we measure fMRI data. p <0.05 Template Parameter estimates Lars Kasper fMRI Preprocessing

fMRI = Acquiring Movies
…of three- dimensional Blood Oxygen-Level Dependent (BOLD) contrast images typically echo- planar images (EPI) y x z Task Run/Session: Time Series of Images … scan 1 time scan N Task Functional MRI is basically acquiring a movie of a living, behaving brain. We acquire a set of 3D-images that share a BOLD contrast. In one session or run, we gain several hundreds of images (N), we call a time series. We then try to relate the dynamics, i.e. changes in that movie to putative brain processes which we induce e.g. by a block design of task and no-task phases during the run. No Task Lars Kasper fMRI Preprocessing

fMRI = Acquiring Movies
The Localized Time-series is the Fundamental Information Unit of fMRI Signal: Fluctuation through Blood oxygen level dependent (BOLD) contrast Noise: All other fluctuations Run/Session: Time Series of Images … scan 1 time scan N Since we assume different dynamics in different parts of the brain, we look at image intensity fluctuations in certain regions of interest, down to the pixel dimension, and correlate them to the task. Thus, the localized time-series is the fundamental info unit of fMRI. We consider signal all fluct due to BOLD. But they are obscured by other fluctuations that stem from various sources. Those we call noise, since our inference becomes of the putative effect (red) becomes weaker, if other fluct dominate it. Lars Kasper fMRI Preprocessing

fMRI Movie: An example Let’s give an example of how bad the situation is. Here is a time series of a behaving brain – before preprocessing, freshly squeezed from the scanner. Can anybody tell me what this subject is thinking? Or doing? Lars Kasper fMRI Preprocessing

fMRI Movie: Subtract the Mean
interest in fluctuations only OK, let’s make it a bit simpler. Since we are only interested in fluctuations, we can subtract the mean of all images from each image, to see differences only. Now, anybody? Well, we actually see a lot of fluctuation here…but most likely, only one is BOLD…e.g. movement; pulsation of the CSF; even spiking in some of the scans. Lars Kasper fMRI Preprocessing

Introducing the Dataset (MoAE)
Mother of All Experiments: Auditory Stimulation TR 7 seconds 6 TR rest 6 TR binaural stimulation (1 bi-syllabic word per second) Chapter 28 of SPM manual Lars Kasper fMRI Preprocessing

The Goal of Preprocessing
Before After So, I hope I have convinved you that the information content in BOLD fMRI is rather like this. The aim of preprocessing, on the other hand, is reaching a state like that, where statistical inference makes sense, i.e. improve sensitivity for the typically small BOLD changes on the order max a few % mean signal. Preprocessing Lars Kasper fMRI Preprocessing

Sources of Noise in fMRI
Acquisition Timing Subject Motion Anatomical Identity Inter-subject variability Thermal Noise Physiological Noise Slice-Timing Realignment Co-registration Segmentation Smoothing PhysIO Toolbox Temporal Preproc Spatial Preproc Spatial Preproc Spatial Preproc Spatial Preproc To do so, we tackle different sources of noise in different preprocessing step. A lot of them fall in the category of spatial preproc, which will be the lion share of this talk. It will be framed by remarks on temporal preprocessing and noise modelling, i.e. what to do with the noise you cannot preprocess away. Noise Modeling Lars Kasper fMRI Preprocessing

The SPM Graphical User Interface
Preprocessing Realignment Slice-Timing Correction Co-registration Unified Segmentation & Normalisation Smoothing… Noise Modeling Physiological Confound Regressors 1. 2. As Andreea pointed out, the preproc steps are all nicely aligned in the upper third of the SPM GUI. Noise modelling is part of the 1st level design, which you will hear about in the next lecture Lars Kasper fMRI Preprocessing

Acquisition Timing Subject Motion Anatomical Identity Inter-subject variability Thermal Noise Physiological Noise Slice-Timing Realignment Co-registration Segmentation Smoothing PhysIO Toolbox Temporal Preproc Let’s start with temporal preproc, i.e. slice-timing correction. Lars Kasper fMRI Preprocessing

Slice-timing correction (STC)
Slices of 1 scan volume are not acquired simultaneously (60 ms per slice) Creates shifts of up to 1 volume repetition time (TR), i.e. several seconds Reduces sensitivity for time-locked effects (smaller correlation) True 2D Acquisition Same-Timepoint Assumption As I mentioned, MR images of a whole vol take about 3 seconds, but they are typically not acquired simultaneously, but slice by slice. This creates a shift of true acq time for some voxels of up to 1 TR, i.e. 2-3 seconds. Now, if we leave it at that, and assume they all stem from the same time-point, we would lose sensitivity for time-locked effects. z time Lars Kasper fMRI Preprocessing

Slice-timing correction (STC)
Slice-timing correction: All voxel time series are aligned to acquisition time of 1 slice Missing data is sinc-interpolated (band-limited signal) Before or after realignment? before: dominant through-slice motion after: dominant within-slice motion At all? block design: for long TR (3s+) & short blocks (10s) improves estimates > 5 % event-related: for normal TRs (2s+) improves estimates > 5 % As I mentioned, MR images of a whole vol take about 3 seconds, but they are typically not acquired simultaneously, but slice by slice. This creates a shift of true acq time for some voxels of up to 1 TR, i.e. 2-3 seconds. Now, if we leave it at that, and assume they all stem from the same time-point, we would lose sensitivity for time-locked effects. Sladky et al, NeuroImage 2011 Lars Kasper fMRI Preprocessing

Interpolation - Intuition
Support Point Summing Moving Average Wolfram Mathworld, Convolution Lars Kasper fMRI Preprocessing

STC Results: Simulation
Slice-timing Correction Temporal-Derivative Modelling TR 1s 4s Block Stimulation TR 1s 4s 10s blocks 15s Event-Related Stimulation event/ 4±2 s 6±3s Sladky et al, NeuroImage 2011 Lars Kasper fMRI Preprocessing

STC Results: Experiment
STC was considered obsolete for a couple of years, since informed basis function modelling (see GLM lecture) provides an alternative, but a recent paper convincingly showed that it actually is better to do it. in particular, only SMA visible in finger tapping with STC Sladky et al, NeuroImage 2011 Lars Kasper fMRI Preprocessing

Acquisition Timing Subject Motion Anatomical Identity Inter-subject variability Thermal Noise Physiological Noise Slice-Timing Realignment Co-registration Segmentation Smoothing PhysIO Toolbox Spatial Preproc Spatial Preproc Spatial Preproc Spatial Preproc Let’s have a look at the large block of spatial preproc… I already mentioned the question of spatial identity of a voxel. This is addressed in one way or the other by all these steps. Lars Kasper fMRI Preprocessing

Finite Resolution and Voxel Identity
voxel = volume element (3D pixel) Q: What is the same voxel…in the same session (realignment) ….in which region…(coreg) ….between subjects(normalize) Lars Kasper fMRI Preprocessing

Preproc = Correct Voxel Mismatch
Voxel Mismatch Between Functional Scans/Runs Functional/Structural Images Subjects Realignment Inter-Modal Coregistration Normalisation/ Segmentation To summarise… Smoothing, on the other hand, is our last resort. If we are not too far off, we smear out remaining voxel mismatches at the end. The order of these steps is important…and somehow also reflects the generalisability and complexity of the respective problems Smoothing Lars Kasper fMRI Preprocessing

Spatial Preprocessing
REALIGN COREG SEGMENT NORM WRITE SMOOTH GLM Lars Kasper fMRI Preprocessing

Input Output fMRI time-series Structural MRI TPMs Segmentation Deformation Fields (y_struct.nii) Kernel REALIGN COREG SEGMENT NORM WRITE SMOOTH Start with realign…then coreg…then segment, and use this for normalize. Finally smoothing and off we go to the GLM (stat. analysis). MNI Space Motion corrected Mean functional (Headers changed) GLM Lars Kasper fMRI Preprocessing

General Remarks: Image Registration
Realignment, Co-Registration and Normalisation (via Unified Segmentation) are all image registration methods Goal: Manipulate one set of images to arrive in same coordinate system as a reference image Key ingredients for image registration Voxel-to-world mapping Transformation Similarity Measure Optimisation Interpolation All but smoothing part of spatial preproc steps belong to the class of image registration algorithms. That’s why, before turning to the individual algorithms, I want to give the following remarks on image registration Image reg aims at manipulating one set of images to arrive in the same coordinate system as another, ref image Key ingredients to accomplish this are…an understand how voxel data represents data in the world, a set of allowed trafos to manipulate images, a measure to quantify similarity, an algorithm to optimise this similarity, and finally, a way to interpolate, to get back voxels from the world. Lars Kasper fMRI Preprocessing

A. Voxel-to-World Mapping
3D images are made up of voxels. Voxel intensities are stored on disk as lists of numbers. Meta-information about the data: image dimensions conversion from list to 3D array “voxel-to-world mapping” Spatial transformation that maps from: data coordinates (voxel column i, row j, slice k) to: a real-world position (x,y,z mm) in a coordinate system e.g.: Scanner coordinates T&T/MNI coordinates Lars Kasper fMRI Preprocessing

A. Voxel-to-World: Standard Spaces
Talairach Atlas MNI/ICBM AVG152 Template Also DICOM scanner-based voxel-world mapping Definition of coordinate system: Origin (0,0,0): anterior commissure Right = +X; Anterior = +Y; Superior = +Z Actual brain dimensions European brains, a bit dilated (bug) Lars Kasper fMRI Preprocessing

B. Transformations Transformations describe the mapping of all image voxels from one coordinate system into another Types of transformations rigid body = translation + rotation affine = rigid body + scaling + shear non-linear = any mapping (x,y,z) to new values (x’,y’, z’) described by deformation fields Translation Rotation Scaling Shear non-linear deformation Lars Kasper fMRI Preprocessing

C. Similarity & D. Optimisation
Similarity measure summarizes resemblance of (transformed) image and reference into 1 number mean-squared difference correlation-coefficient mutual information Automatic image registration uses an optimisation algorithm to maximise/minimise an “objective function” Similarity measure is part of objective function Algorithm searches for transformation that maximises similarity of transformed image to reference Also includes constraints on allowed transformations (priors) intra-modality (same contrast) inter-modality (different contrasts possible) Lars Kasper fMRI Preprocessing

Preprocessing Step Categorisation
B. Allowed Transformations Rigid-Body Affine Non-linear REALIGN COREG SEGMENT NORM WRITE C. Similarity Measure Mean-squared Difference Mutual Information Tissue Class Probability D. Optimisation Exact Linearized Solution Conjugate Direction Line Search Iterated Conditional Modes (EM/Levenberg-Marquardt) Lars Kasper fMRI Preprocessing

E. Reslicing/Interpolation
Finally, images have to be saved as voxel intensity list on disk again After applying transformation parameters, data is re-sampled onto same grid of voxels as reference image Reoriented Resliced 1x1x3 mm voxel size 2x2x2 mm voxel size Lars Kasper fMRI Preprocessing

E. B-spline Interpolation
Lars Kasper fMRI Preprocessing

Input Output fMRI time-series Structural MRI TPMs Segmentation Deformation Fields (y_struct.nii) Kernel REALIGN COREG SEGMENT NORM WRITE SMOOTH This might have been a lengthy intro, but it helps to understand and classify the following algorithms very simply as different image registrations. So, let’s see how the individual image registration flavours work in the SPM preproc pipeline… MNI Space Motion corrected Mean functional (Headers changed) GLM Lars Kasper fMRI Preprocessing

Realignment Aligns all volumes of all runs spatially
fMRI time-series Aligns all volumes of all runs spatially Rigid-body transformation: three translations, three rotations Objective function: mean squared error of corresponding voxel intensities Voxel correspondence via Interpolation REALIGN Motion corrected Mean functional Lars Kasper fMRI Preprocessing

Realignment Output: Parameters
Lars Kasper fMRI Preprocessing

fMRI Run after Realignment
slice 21…after warping…slice 34 before Lars Kasper fMRI Preprocessing

Co-Registration Aligns structural image to mean functional image
Structural MRI Aligns structural image to mean functional image Affine transformation: translations, rotations, scaling, shearing Objective function: mutual information, since contrast different Optimisation via Powell’s method: conjugate directions, line seach along parameters Typically only trafo matrix (“header”) changed COREG Optimisation via Powell’s method: initialisation with 12 (i.e. number of parameters of function) search direction long each parameter; best parameter weighting of all parameters combined to a new search direction; Search direction with maximum contribution to new search direction is removed from search direction list Motion corrected Mean functional (Headers changed) Lars Kasper fMRI Preprocessing

Co-Registration: Output
Joint Histogram Marginal Histogram intensity bins functional Mean functional Anatomical MRI Joint and marginal Histogram Quantify how well one image predicts the other how much shared information Joint probability distribution estimated from joint histogram intensity bins structural Joint Histogram: h(if,is) Count of voxels who have intensity if in functional and is in structural image Lars Kasper fMRI Preprocessing

Co-Registration: Output
Voxels of same tissue identity should have same intensity in an MR-contrast In a second MR contrast, this intensity might be different, but still the same among all voxels of the same tissue type Therefore, aligned voxels in 2 images induce crisp peaks in joint histogram Lars Kasper fMRI Preprocessing

Acquisition Timing Subject Motion Anatomical Identity Inter-subject variability Thermal Noise Physiological Noise Slice-Timing Realignment Co-registration Segmentation Smoothing PhysIO Toolbox Spatial Preproc Lars Kasper fMRI Preprocessing

Spatial Normalisation: Reasons
Inter-Subject Variability Inter-Subject Averaging Increase sensitivity with more subjects (fixed-effects) Generalise findings to population as a whole (mixed-effects) Ensure Comparability between studies (alignment to standard space) Talairach and Tournoux (T&T) convention using the Montreal Neurological Institute (MNI) space Templates from 152/305 subjects Why Is this a problem? Lars Kasper fMRI Preprocessing

Unified Segmentation Warps structural image to standard space (MNI)
Deformation Fields Segmented Images Warps structural image to standard space (MNI) Non-linear transformation: discrete cosine transforms (~1000) Objective function: Bayes probability of voxel intensity Structural MRI TPMs NORM WRITE SEGMENT Even though its called unified segmentation, it is a 2-step procedure in the SPM pipeline…first estimation, then writing out images… MNI Space Motion corrected Motion corrected (Headers changed) Lars Kasper fMRI Preprocessing

Theory: Segmentation/Normalisation
An SPM Perspective on fMRI Preprocessing Theory: Segmentation/Normalisation Why is normalisation difficult? No simple similarity measure, a lot of possible transformations… Different Imaging Sequences (Contrasts, geometry distortion) Noise, artefacts, partial volume effects Intensity inhomogeneity (bias field) Normalisation of segmented tissues is more robust and precise than of original image Tissue segmentation benefits from spatially aligned tissue probability maps (of prior segmentation data) Motivates a unified model of segmentation/normalisation The rationale behind this method is that normalisation is harder than coreg and realign, since we deal with different brains, sessions, even scanners. It would be easier to break down the problem a bit by using a similarity measure that is robust to all these imperfections …and the anwer is: tissue probability maps Lars Kasper fMRI Preprocessing Lars Kasper

Theory: Unified Model Segmentation
An SPM Perspective on fMRI Preprocessing Theory: Unified Model Segmentation Bayesian generative model1 of voxel intensities 𝑦 𝑖 from tissue class probabilities, deformation fields and bias fields Objective function: log joint probability of all voxel intensities 𝒚 ℰ= log 𝑃(𝒚|𝝁,𝝈,𝜸, 𝒃 𝟏…𝑲 ,𝜶,𝜷) [1] Ashburner & Friston (2005), Neuroimage pixel count Gaussian Mixture Model probability of intensity in given voxel for tissue class image Intensity 𝑦 CSF WM 𝑘=1 𝐾 𝛾 𝑘⋅ ⋅𝑃 𝑦 𝑖 𝑐 𝑖 =𝑘 𝝁 𝑘 𝝈 𝑘 Prior: Tissue probability maps TPMs in MNI space 𝒃 𝟏 𝒃 𝟐 𝒃 𝟑 What is the probability of a given pixel to be of tissue class gray matter, white matter/csf Model has different ingredients… Output: Deformation fields that can be applied to both structural and functional images Bias Field Raw Corrected coil inhomo- geneities 𝝆(𝜷) Deformation Fields ~1000 discrete cosine transforms 𝒃 𝑘 (𝜶) Lars Kasper fMRI Preprocessing Lars Kasper

Summary of the unified model
SPM12 implements a generative model of voxel intensity from tissue class probabilities Principled Bayesian probabilistic formulation Gaussian mixture model: segmentation by tissue-class dependent Gaussian intensity distributions voxel-wise prior mixture proportions given by tissue probabilities maps Deformations of prior tissue probability maps also modeled Non-linear deformations are constrained by regularisation factors inverse of estimated transformation for TPMs normalises the original image Bias field correction is included within the model Lars Kasper fMRI Preprocessing

Segmentation results segmentation works irrespective of image contrast
PD Spatially normalised BrainWeb phantoms Estimated Tissue probability maps (TPMs) Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997) Lars Kasper fMRI Preprocessing

Benefits of Unified Segmentation
Affine registration Non-linear registration Lars Kasper fMRI Preprocessing

Spatial normalisation – Limitations
Seek to match functionally homologous regions, but... Challenging high-dimensional optimisation many local optima Different cortices can have different folding patterns No exact match between structure and function Interesting recent paper Amiez et al. (2013), PMID: Compromise Correct relatively large-scale variability Smooth over finer-scale residual differences Lars Kasper fMRI Preprocessing

Smoothing – Why blurring the data?
Intra-subject signal quality Suppresses thermal noise (averaging) Increases sensitivity to effects of similar scale to kernel (matched filter theorem) Single-subject statistical analysis Makes data more Gaussian (central limit theorem) Reduces the number of multiple comparisons Second-level statistical analysis Improves spatial overlap by blurring anatomical differences Kernel SMOOTH MNI Space GLM Lars Kasper fMRI Preprocessing

Smoothing – How is it implemented?
Convolution with a 3D Gaussian kernel, of specified full-width at half-maximum (FWHM) in mm mathematically equivalent to slice-timing operation or reslicing, but different kernels there (Sinc, b-spline) Gaussian kernel is separable, and we can smooth 2D data with 2 separate 1D convolutions Lars Kasper fMRI Preprocessing

fMRI Run after Smoothing
slice 21…after warping…slice 34 before Lars Kasper fMRI Preprocessing

Input Output fMRI time-series Structural MRI TPMs Segmentation Deformation Field (y_struct.nii) Kernel REALIGN COREG SEGMENT NORM WRITE SMOOTH MNI Space Motion corrected Mean functional (Headers changed) GLM Lars Kasper fMRI Preprocessing

Acquisition Timing Subject Motion Anatomical Identity Inter-subject variability Thermal Noise Physiological Noise Slice-Timing Realignment Co-registration Segmentation Smoothing PhysIO Toolbox Temporal Preproc Spatial Preproc Spatial Preproc Spatial Preproc Spatial Preproc To do so, we tackle different sources of noise in different preprocessing step. A lot of them fall in the category of spatial preproc, which will be the lion share of this talk. It will be framed by remarks on temporal preprocessing and noise modelling, i.e. what to do with the noise you cannot preprocess away. Noise Modeling Lars Kasper fMRI Preprocessing

Thank you… …and: TNU Zurich, in particular: Klaas
MR-technology Group IBT, Everyone I borrowed slides from  Lars Kasper fMRI Preprocessing

Mixture of Gaussians Classification is based on a Mixture of Gaussians model, which represents the intensity probability density by a number of Gaussian distributions. Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled e.g. partial volume effects Frequency (number of pixels) Image Intensity Lars Kasper fMRI Preprocessing

Tissue Probability Maps
Tissue probability maps (TPMs) are used as the prior, instead of the proportion of voxels in each class ICBM Tissue Probabilistic Atlases. These tissue probability maps were kindly provided by the International Consortium for Brain Mapping Lars Kasper fMRI Preprocessing

Deforming the Tissue Probability Maps
Tissue probability maps images are warped to match the subject The inverse transform warps to the TPMs Lars Kasper fMRI Preprocessing

Why regularisation? – Overfitting
Regularisation constrains deformations to realistic range (implemented as priors) Affine registration (error = ) Non-linear registration without regularisation (error = ) Template image Non-linear registration using regularisation (error = 302.7) Lars Kasper fMRI Preprocessing

Modelling inhomogeneity
A multiplicative bias field is modelled as a linear combination of basis functions. Corrupted image Bias Field Corrected image Lars Kasper fMRI Preprocessing

Unified segmentation: The maths
Mixture of Gaussians: probability of voxel i having intensity yi, given it is from a specific cluster k (e.g. tissue class gray matter) Prior probability of voxel’s tissue class (e.g. voxel proportion) 𝛾 𝑘 Joint Probability: Marginal probability of voxel intensity: Joint probability all voxels’ intensity: Lars Kasper fMRI Preprocessing

US Maths: Bias Field Implemented by adjusting the Means and Variances of the Gaussians on a pixel-by pixel basis by a function smoothly varying in space, 𝜌 𝑖 𝜷 : 𝜇 𝑘 ↦ 𝜇 𝑘 𝜌 𝑖 𝜷 , 𝜎 𝑘 2 ↦ 𝜎 𝑘 𝜌 𝑖 𝜷 2 𝜌 𝑖 is the exponential of a linear combination of low frequency basis functions Parameters to be estimated: vector 𝜷 intensity probability conditioned on cluster identitiy: Lars Kasper fMRI Preprocessing

US Maths: Spatial Priors by TPMs
Replacing stationary mixing proportions 𝛾 𝑘 by voxel-dependent proportions which are informed by the prior tissue probabilities 𝑏 𝑖𝑘 for this voxel 𝑖 and different tissue types 𝑘 𝛾 𝑘 ↦ 𝛾 𝑘 𝑖 = 𝛾 𝑘 ⋅ 𝑏 𝑖𝑘 𝑗=1 𝐾 𝛾 𝑗 𝑏 𝑖𝑗 Note: 𝐾 can be larger than the number of tissue classes, since each class can be reflected by a mixture of Gaussians, e.g. 3 Gaussians for gray matter (to allow for non-Gaussian distributions per tissue class) E.g. partial volume effects Lars Kasper fMRI Preprocessing

US Maths: Deformation Fields
Deformation (and thereby normalisation) is implemented by allowing the prior TPMs (which are in MNI-space) to be spatially transformed by a parameterised mapping b 𝑖𝑘 ↦ b 𝑖𝑘 𝛼 ⇒𝑃 𝑐 𝑖 =𝑘 𝛾,𝛼 = 𝛾 𝑘 𝑏 𝑖𝑘 (𝛼) 𝑗=0 𝐾 𝛾 𝑗 𝑏 𝑖𝑗 (𝛼) Parameter vector to be estimated: 𝜶 about 1000 discrete cosine transforms Lars Kasper fMRI Preprocessing

US Maths: Regularisation
Linear Regularisation of Bias Field and Deformation Field Estimates By including prior distributions for 𝛼 and 𝛽 as zero-mean multivariate Gaussians Covariance: 𝛼 𝑇 𝐶 𝛼 𝛼=𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑦;𝜌 𝛽 = exp ( 𝐾 70𝑚𝑚 ∗𝑁(0,𝛽)) Thus, the final objective function to be maximised is the log-joint probability of intensity, bias and deformation field parameters: Equivalenty, the negative free energy is minimised: Lars Kasper fMRI Preprocessing

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