1. From Spatial Regularization to Anatomical Priors in fMRI Analysis
Fragmented Map
Regularized Map
Wanmei Ou William Wells Polina Golland
Core 1 Meeting – May 23rd, 2006

2. Main Contributions
Propose new spatial regularization method: Incorporating Anatomical Information
Empirically Compare proposed methods with traditional methods
Say that the comparison is done in both synthetic and read fMRI data + traditional methods: we mean guassian smoothing. Also briefly explain what gaussian smoothing is

3. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluation
Conclusions

4. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluation
Conclusions

5. From fMRI Images to fMRI Analysis
Image Acquisition
MRI
fMRI
Task protocol: Auditory, Vision, etc.
Functional Magnetic Resonance imaging has provided researchers with a non-invasive dynamic method for studying brain activation by capturing the change in blood oxygen levels. fMRI is a 4-D image. It encodes both space (x, y, z) and time. It is taken using a conventional MR machine with special imaging parameter setting. A subject’s MR scans are usually obtained prior to his/her fMRI scans. MR images have much higher resolution than the fMRI images. i.e. standard MR images have 1mm/voxell; standard fMRI images have 4 mm/voxel. When a subject undergoes fMRI, he/she is usually ask to perform certain task according to the pre-decided experimental protocol. Auditory, vision, or memory task is typical task presented in a fMRI study. It is also common to combine different types of tasks in one experiment. Detection is usually performed on the signal of each voxel separately. These detectors are referred to as the voxel-by-voxel detectors. Arbitrary threshold is usually applied the statistic obtained from the detectors and generate the activation map. Because fMRI has a lot signal-to-noise ratio, the activation map are usually scatter and contain a lot of false positive detections, creating a need for spatial regularization. Sometimes, spatial regularization is preformed prior to detection.
Spatial
Regularization
Threshold
Voxel-by-voxel detector
Activation Map
Scatter activation

6. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluation
Conclusions

7. Goal and Approaches
Goal: Recover True Activation through Spatial Regularization
Our Approach:
Incorporate MRF into General Linear Model (GLM) Statistic
Include Anatomical Information into MRF
Similar to previous approaches, our goal is to recover truth activation from fragmented activation maps. Different from Gaussian smoothing, Decomes’s, and Woolrich’s approach, we do not use a smooth filter to model the signal instead we use regularize the sufficient statistics obtain from the GLM detector using a MRF model. We further refine our MRF by incorporating anatomical information. By incorporating anatomical information, we can capture the fact that activation is more likely occurs in gray matter, and that spatial coherency of activation is strong within each tissue and not across tissue boundaries. Our ides of this refined MRF model is inspired by the atlas-based segmentation approach.

8. Synthetic Ground Truth
Detailed Approaches
Our Approach:
1. Incorporate MRF into GLM Statistic
Capture spatial dependency
Overcome over-smoothing effect
2. Include Anatomical Information into MRF
Activation is more likely in gray matter
Spatial dependency is strong within tissue type
Activation Maps
Similar to previous approaches, our goal is to recover truth activation from fragmented activation maps. Different from Gaussian smoothing, Decomes’s, and Woolrich’s approach, we do not use a smooth filter to model the signal instead we use regularize the sufficient statistics obtain from the GLM detector using a MRF model. We further refine our MRF by incorporating anatomical information. By incorporating anatomical information, we can capture the fact that activation is more likely occurs in gray matter, and that spatial coherency of activation is strong within each tissue and not across tissue boundaries. Our ides of this refined MRF model is inspired by the atlas-based segmentation approach.
MRF
MRI
Synthetic Ground Truth
Segmentation
Gray, White, Other

9. General Linear Model (GLM)
Protocol-Independent Signal
Two Hypotheses
Not-active voxel
Active voxel
Protocol-Dependent Signal
GLM
ML Estimate
F or T statistic
P-value
Traditional Approach
The following is some background on the general linear model and the statistic we use for spatial regularization. GLM models the fMRI signal as a linear combination of the protocol-dependent and protocol-independent component. A present of the protocol-dependent component indicates the voxel is active. In a conventional GLM detection, we first estimate the amplitudes of the protocol-dependent response, and then convert it into F or T statistics from which we obtain the p-value presenting significance. Our GLM statistic is the maximum log likelihood ratio of the two hypotheses, Z_i. Cosmas has proven that z_i is a monotonically increasing function of F-statistic.
General Log Likelihood Ratio (Cosman, 04)
Our Approach

10. Markov Random Field Spatial Priors: Likelihood: MAP Estimate:
-- Hidden Activation State
-- Noisy Observation/Statistic
Again, different from Decomes’s work, our MRF prior does not model temporal signal directly, but the maximum log likelihood statistic, Z_i. In this MRF prior, Z_i is the noisy observation. X_i represents true activation state. X_i=1 indicating a voxel is activation, zero otherwise. We have not incorporate the anatomical information into this MRF prior. The distribution if the hidden nodes, which form an MRF, can be written as the product of the pairwise potential and singleton potentials. Likelihood model is the probability of observing noisy statistic given true activation state. Here, we assume that the noisy observations are independent given the activation states. Therefore we can factor this distribution into product a format. We’d like to obtain the MAP estimate of the activation state given the noisy statistic. Using the Bayes rule, we can get the following. We will discuss a approximation algorithm for this MAP estimate later, even though under the current binary activation MRF, the exact solver cannot achieve by a polynomial time algorithm because this binary MRF problem cannot reduced to the Min-flow-Max-cut problem.
MAP Estimate:

11. Incorporating Anatomical Information
Combine activation state & tissue type
MRI
Segmentation
There is only min change in the MRF formulations after we incorporate the anatomical information. In the MRF prior introduced in the last slice only encode the activation state, but the hidden node here, u_i, models both activation state and tissue type. We obtain and an extra observation from the segmentation result of MR scan. We do not assign the segmentation of each node explicitly, instead we only consider them as noisy observation. That is because there is defect in the segmentation process. Imperfect registration and mismatch resolution between fMRI image and MR image are two other reasons. We’d like to obtain the MAP estimate of the hidden nodes given the noisy activation statistic and the noisy segmentation result. Similar as before, we can formulate the MAP estimate using Bayes rule. Compare with the MAP estimate of the previous, the only different here is the extra log likely hood for the segmentation result. Again, we assume that the noisy segmentation result of different voxels are independent given the true tissue type of the voxels.
-- Hidden Activation State
-- Tissue Type
MAP Estimate:
-- Segmentation Label
-- Noise Statistic

12. Markov Random Fields Solvers
Binary MRF Min-Cut/Max-Flow (Min-Max) Binary MRF Only
Gibbs Sampling Slow
Simulated Annealing Slow
Belief Propagation Fast, Approximation
Mean Field Fast, Approximation
It can be shown that when each hidden node of the MRF only has two possible states, the MAP problem cannot reduced to the Min-Max problem which can be solved by polynomial time algorithm. However, in general, a direct search of the MAP estimate of the MRF problem involves exponential computation time. Alternative solvers include Markov chain Monte Carlo (MCMC), Simulation Annealing (note to wanmei: more research put one sentence for each algorithm). There is another branch called variation method to give reasonable approximating solution, but require much shorter computation time. There is variational methods include family of BP and meanfield. Since BP algorithm may not converge in a graph with loop and it obtains similar result as mean field in our synthetic example, we focus on the mean field approximation through out this work.

13. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluations
Conclusion and Future work

14. Mean Field Approximate by Iterative up-date rule Approximated MAP
Belief: Prob. Of voxel is active
We will explain the mean field algorithm on the MRF prior without incorporating anatomical information. The Mean Field algorithm approximates P_x|z by a product distribution Q_X through minimization of the KL-Divergence between the two distributions. The form Q_X assumes that all voxels are independent. KL-divergence is a non-negative quantity, and it is zero when the two distribution are equal. To minimize the KL-divergent w.r.t. Q_X is equivalent to minimize the free energy because the last two term does not involve Q_X. So our problem is reduced to finding the minimum of the free energy given the constrain that Q_X is a valid distribution. Through substitution & simple algebra, we can get the iterative update rule for the distribution of each voxel, b_i. When the algorithm converges, we can obtain the approximation MAP solution X_hat.
Approximated MAP

15. Mean Field
Similar up-date rule while incorporating anatomical information
We follow similar procedure as describe in the previous slice to the obtain the approximated MAP solution of the hidden node for the MRF prior with anatomical information. Compare with the previous update rule, the only different is the extra likelihood here, and the compatibility matrix is larger. Since we are interested in the MAP estimate of the activation state, we first marginalize Q w.r.t. v, and obtain the MAP solution.
Approximated MAP

16. Alternative Anatomically Guided Filters
No smoothing with Anatomical Information
Suppress all activation in the non-gray matter.
Anatomically Guided Gaussian Filter
Adjust weights based on segmentation labels.
In the previous slice, we introduced the anatomically guided MRF filter. The simplest way to utilize anatomical information is to suppress all the activation occurs at the non-gray matter. As you can see, this filter ignores any activation at the gray matter. We can also incorporate the anatomical information into the Gaussian filter. In practice, a Gaussian filter is an averaging filer with equal weight for all its neighbors. With help from the anatomical information, we can assign lower weight for the voxels who share different segment labels. This emphasis the assumption that spatial coherency of activation is strong within each tissue type and not across tissue boundaries. In our experiments we weight the voxels sharing the same segmentation label twice as heavy as the voxels that share different segmentation labels.

17. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluation
Synthetic Data
Real fMRI Data
Conclusions

18. Low SNR Fragmented Activation Maps
Experiments – Synthetic Data Sets
Low SNR Fragmented Activation Maps
Noise
SNR = -9.3dB
Noise
SNR = -6.3dB
Activation Maps
Threshold:
False positive = 0.05%
Forward
Model
GLM
Detector +
Synthetic Ground Truth
To quantitatively evaluate the performance of the method, we generated realistic phantom data by applying EM segmentation to an anatomical MRI scan and placing activation areas of variable size randomly in the gray matter. We then down-sampled the scan to an fMRI resolution. We then created simulated fMRI scans based on a fixed parametric hemodynamic response function, an event-related protocol, and varying levels of noise. The noise level is compatible to the one in real fMRI data. We created 15 data sets following the procedure. Here is a slice of one of the data sets. As we can see that without spatial regularization, the basic GLM produces a fragmented activation map that contains an umber of false detection islands at high SNR and shows very little of the original activation at low SNR. Given either of these maps, the users would have troubles inferring the true activation areas and disambiguating them from spurious false detections. We again see that spatial regularization is necessary.
We used the estimated SNR, $\widehat{\mbox{SNR}}=- 10\log_{10}(|B\hat{\beta}|^2)/\hat{\sigma^2}$, to determine a realistic level of the simulated noise as the true SNR is unaccessible for real fMRI scans. Since the signal and the noise overlap in some frequency bands, part of the noise energy is assigned to the estimated signal during detection. The estimated SNR is therefore an optimistic approximation of the true SNR, which can still be used as a monotonic upper bound of the true SNR. In our real fMRI studies, the estimated SNR is about -5dB. Here, we illustrate the results for two levels of true SNR, -6dB and -9dB, which correspond to estimated SNR of -4.3dBand -6.2dB respectively.
The gray matter voxels represent $10$\% of the total number of voxels in the volume, and the active voxels represent about $10$\% of the gray matter voxels in these images.
Real
N/A
-6.5
Synthetic1
-6.3
-5.9
Synthetic2
-9.3
-8.8

19. Experiments – Synthetic Data Sets
Noise
Forward
Model
GLM with different
smoothing methods +
Synthetic Ground Truth
No Smoothing
No Smoothing w/ Anatomical Info
Gaussian Smoothing w/o Anatomical Info
Gaussian Smoothing w/ Anatomical Info
MRF w/o Anatomical Info
MRF w/ Anatomical Info
We compare the reconstructed activation map using six spatial regularization methods: No smoothing, Gaussian smoothing and MRF model, and their corresponding anatomical guided versions.

20. ROC Analysis Without Anatomical Information
Min-Max (Exact Solver) vs. Mean Field (Approximation)
SNR = -6dB
SNR = -9dB

21. ROC Analysis Without Anatomical Information
MRF (Mean Field) vs. Gaussian Smoothing
SNR = -6dB
SNR = -9dB
Because of using the synthetic data set, we can obtain ROC analysis. The two ROC figures compare the detection performance of the GLM detectors combine with no smoothing, Gaussian smoothing, and MRF prior. Without anatomical information, it is a binary MRF prior, so we can obtain the MAP estimate using the exact solver (Min-Max) and the approximation solver (Mean Field). The following ROC curves, shown in log scale, are the average ROC performance of the detectors over 15 individually generated and processed data sets. The error bars indicate the standard deviation of the true detection rate over 15 different, independently created and processed, data sets. The Min-Max ROC curve does not have the error bars, as the estimation takes too long.
The Mean Field detection accuracy is very close to the exact Min-Max solution, providing a reasonable approximation to the exact solution that also takes much less time to compute.
%The Min-Max accuracy is sometimes lower than the Mean Field accuracy, which appears to contradict the optimality of Min-Max. However, we %note that both algorithms solve a particular estimation problem that does not necessarily describe the ground truth precisely but rather %approximates it using a Markov model. Thus, the lowest energy state under this model might not be the best detector in practice. It is still %reassuring to see that the approximate solver performs as well as the exact algorithm. It also suggests that more realistic spatial priors could %further improve the detection accuracy.
As expected, the accuracy of all methods improves with increasing SNR. At high noise levels (low SNR), Gaussian smoothing out performs MRFs. As the simplest smoothing technique, Gaussian smoothing is more robust to noise. We also believe that our current way of constructing the likelihood term in the MRF model over-emphasizes the data evidence over the prior. We are investigating ways to compensate for this in the estimation of the model. As the SNR increases, MRFs provide better regularization of the activation state (for example, at SNR=-6dB, at the false positive rate of 0.01\%, the MRF outperforms the Gaussian smoothing by about 15\% in true detection accuracy; at 70\% true detection, the MRF approximately halves the false detections compared to the Gaussian smoothing). With the improving scanning technology, we believe MRFs will become even more helpful in reducing spurious false detection islands.

22. ROC Analysis With Anatomical Information
MRF (Mean Field) vs. Gaussian Smoothing
SNR = -6dB
SNR = -9dB
The two ROC figures compare the detection performance of the GLM detectors combined with no smoothing, Gaussian smoothing, and MRF prior and their anatomically guided versions.
In addition to the trends observed before, we note that the anatomical information significantly boosts the performance of all detectors at all noise levels. At high noise levels (SNR = -9dB) and false positive rates between 0.01\% and 0.1\%, all methods gain at least 10\% in true detection rate when using the anatomical information. The MRF model benefits more than the Gaussian smoothing, but its detection accuracy is still lower. At the lower noise level (SNR = -6dB), the basic GLM detector augmented with anatomical information approaches the performance of the Gaussian smoothing. At 0.01\% false positive rate, the anatomically-guided MRF outperforms the anatomically-guided Gaussian smoothing by about 15\% in true detection rate, achieving over 90\% detection accuracy. The large boost experienced by the basic GLM when augmented with anatomical information is easy to understand: since false detections occur relatively uniformly throughout the volume, masking the gray matter improves the performance substantially.

23. Road Map Background and Motivation
Markov Random Field (MRF) and Anatomical Guided MRF Model
Mean Field Approximation Solver
Experimental Evaluation
Synthetic Data
Real fMRI Data
Conclusions

24. Evaluation on Real Data
“Ground Truth”
GLM
Majority Voting
GLM
8 task epochs
comparisons
GLM with various spatial regularizers
2 task epochs

25. Activation Maps Comparison
Anat No Smoothing Gaussian MRF
“Ground Truth” No Smoothing+Anat Gaussian+Anat MRF+Anat
Three Epochs
sm007ep3

26. Activation Maps Comparison
Anat No Smoothing Gaussian MRF
“Ground Truth” No Smoothing+Anat Gaussian+Anat MRF+Anat
Three Epochs
sm007ep3

27. Activation Maps Comparison
Anat No Smoothing Gaussian MRF
“Ground Truth” No Smoothing+Anat Gaussian+Anat MRF+Anat
Three Epochs
sm007ep3

28. Conclusions New Spatial Regularization method Empirical Evaluation
Anatomical Bias
Empirical Evaluation
ROC analysis
Activation maps
MRF + Anatomical Information
Increase detection accuracy with reduced-length signal