A Modified Infomax ICA Algorithm for fMRI Data Source Separation

This study presents a modified infomax model of Independent Component Analysis (ICA) for the source separation problem of fMRI data. Functional MRI data is processed by different blind source separation techniques including Independent Component Analysis (ICA). ICA is a statistical decomposition method used for multivariate data source separation. ICA algorithm is based on independence of extracted sources for which different techniques are used like kurtosis, negentropy, information maximization etc. The infomax method of ICA extracts unknown sources from a number of mixtures by maximizing the negentropy thus ensuring independence. In this proposed modified infomax model a higher order contrast function is used which results in fast convergence and accuracy. The Proposed algorithm is applied to general simulated signals and simulated fMRI data. Comparison of correlation results of the proposed algorithm with the conventional infomax algorithm shows better performance.


INTRODUCTION
The study of the human brain can be made by PET, SPECT, fMRI etc. fMRI is the latest technique used for measuring the functionality of the brain using blood oxygen level dependent signal (BOLD) (Ogawa et al., 1990).In fMRI the magnetic properties of the brain tissues associated with an activity are recorded in the form of 3-D images of the brain (Ogawa et al., 1990).For a specific activity a set of neurons becomes active in performing some task, thus it needs more oxygenated blood as compared to nearby idle neurons.The ratio of the oxygenated blood between active and idle neuron changes and the change in magnetic properties of the oxygenated blood is recorded by the fMRI scanner in the form of Blood-Oxygen-Level-Dependent (BOLD) signal (Ogawa et al., 1990).A number of images are taken rapidly by the fMRI scanner during the experiment.This multi image data is then statistically analyzed for active and non-active voxels and the results are presented on a statistical map.However, due to small variations of the BOLD signal, SNR values of functional MR images are too low (Friman et al., 2001) and thus it requires a de-noising step before classification.
During fMRI experiment subjects are normally performing activity and non-activity during predefined cycles of time.The BOLD signal from each brain voxel is recorded along with time thus creating time series signals which are mixtures of activity non-activity and noise and structural information etc. Location of activated voxels in this mixed data is a challenging task (McKeown and Sejnowski, 1998).
Analysis of fMRI data has been done by different approaches including Statistical Parametric Mapping (SPM), Principal Component Analysis (PCA), ICA correlation analysis, Non-negative matrix factorization etc.
fMRI data analysis approaches can be broadly divided in two categories, one is model dependent and other one is data dependent.In model dependent approaches knowledge of experimental model is required a priori.These approaches consist of SPM, time frequency analysis, correlation analysis etc.
Statistical Parametric Mapping (SPM) is a univariate fMRI data analysis approach in which the signal at each voxel is tested using t-test, f-test (Friston, 1996).In time frequency analysis approach assumption is made that task related noise and physiological activities spectra's are different and thus can be extracted from the mixed observed data (Mitra et al., 1997).In Correlation analysis approach, the voxels which are responsible for activity/non-activity are extracted using a priori knowledge of the experimental paradigm.A correlation of the experimental model and each time series is performed and the voxels are termed as active voxels in case of strong correlation results (Bandettini et al., 1993).Data driven approaches of fMRI analysis do not require any experimental model a priori like Principal Component Analysis (PCA) (Backfrieder, 1996) and Independent Component Analysis (ICA) (McKeown et al., 2003), NMF (Amir et al., 2012) etc.
Principal Component Analysis (PCA) is an important method used under data driven methods.Main assumption made when applying PCA is that the underlying sources variability.Normally the number of principal components representing the data are less than the actual dimension of the data thus the dimensionality of the data is also reduced by discarding the principal components with lowest variance.There are other multi-variate data analysis approaches, but the most promising approach is ICA, which is a data driven and powerful exploratory tool for fMRI data analysis (Qiu-Hua et al., 2010;Dyrholm et al., 2007).ICA converts multidimensional vectors into statistically independent components.Therefore this technique is broadly used in other blind source separation problems like image processing, text processing, communication and other blind source separation problems.
In this study ICA infomax algorithm is modified by making the contrast function to converge faster using its higher order cumulant.Simple fixed point and gradient descent based optimization is used for finding the unmixing matrix W. Speed and accuracy of the proposed scheme are demonstrated using general simulated data and fMRI simulated data.

ICA (INFOMAX) ALGORITHM
Independent Component Analysis (ICA) is generally used for multivariate data analysis with applications in various fields such as medical image analysis, bioinformatics, pattern recognition, digital communications (Mohammad et al., 2011), computational neuroscience and other areas where the blind source separation problem exists.ICA decomposes multivariate data into a linear sum of nonorthogonal basis vectors with coefficients being statistically independent (Dyrholm et al., 2007).Since fMRI observed data consist of sources linearly mixed by the time courses of activity, noise and physiological activity.Thus the ICA model assumes that the observed fMRI data X is a linear mixture of the source vector S and the time courses matrix T : where, X is the observed data of dimension The goal of ICA is to find the un-mixing matrix A such that: where, with the constraint that In most of the ICA algorithms, the observed data is normally centered (zero mean) and whitened before implementing the ICA algorithm.Centering of the observed data is done by subtracting the mean from the observed data X : where, X = The observed data ' X = The centered data Whitening can be done by multiplying the observed data X with some whitening matrix U so that the correlation and covariance matrix of the sources becomes identity i.e.,   I YY E T  .This can be done using the eigen-decomposition (Aapo et al., 2001) and is given as under: and D is the diagonal matrix of its eigenvalues.
Another preprocessing step for fMRI data is also required i.e. the dimension reduction due to its high dimensionality.In this case we have used Singular Value Decomposition (SVD) as a dimension reduction technique (Aapo et al., 2001).
The data is now ready for ICA algorithm in which a contrast function is maximized/minimized so that maximum independence of extracted sources is achieved.There are different flavors of ICA which consists of kurtosis maximization, negentropy, information maximization (Infomax).Now sources can be extracted sequentially i.e., one by one or symmetrically.Sequential approach is known as deflation approach and was first proposed by Delfosse and Loubaton (1995).
In the deflation based approach ICA algorithm looks for such an A which maximizes some contrast function which is basically the measure of non-Gaussianity.Different approaches are used in the literature (Aapo et al., 2001) for measuring non-Gaussianity like kurtosis, nagentropy, information maximization etc.In this study we are considering only Infomax algorithm which is discussed as under.
Infomax is a method of ICA based on information theory which aims to find independent source signals by maximizing entropy.The general strategy of Infomax algorithm begins with Eq. ( 1), where, the extracted sources Y are obtained from signal mixtures X by optimizing a un-mixing matrix A such that the extracted signals are source signals if they are mutually independent.
. It was shown that infomax contrast maximization is equivalent to the minimization of KL divergence between the distribution of and Sejnowski, 1995).Different approximations have been proposed in literature for measuring the entropy of the signals for ICA framework.Here we will consider the approximation as narrated by (Dyrholm et al., 2007): where,

PROPOSED MODIFIED INFOMAX BASED HIGHER ORDER GRADIENT FUNCTION
By looking into gradient based optimization approach, it is clear that when the gradient is stable, then it must point in that direction of A which maximizes the cost function, in this case it is the entropy of the transformed sources.Also the gradient is basically the inverse transpose of A along with the ) tanh(Y sources multiplied by the transpose of the observed data which again gives us A .Thus if we take some higher order cumulant of the gradient it will update A with a faster rate.Also all vectors of A will not change their direction as A will be normalized in each iteration and a convergence is achieved fast (Aapo et al., 2001) and ultimately independent sources will be extracted with greater accuracy in less time.Keeping these points in mind, equation ( 8) fulfills the criteria of convergence: It should be kept in mind that the value of n should be odd so that sign of elements in the matrix X and A are not lost.2a.It is quite clear that there is no one to one correspondence of the extracted sources with the actual sources even visually.Figure 2b shows the results of the extracted sources using the proposed modified infomax algorithm which are quite similar to the original sources even visually.
To check the validity of the algorithm on simulated fMRI data, proposed and conventional algorithms are also applied to extract the hidden sources in fMRI simulated data.This data is downloaded from http://mlsp.umbc.edu/simulated_fmri_data.html.This data was basically developed by Correa et al. (2005) for their algorithm testing.Figure 3 shows simulated fMRI sources and corresponding time courses while Fig. 4 shows extracted sources and time courses of simulated fMRI data using conventional infomax.The results are  not good even visually.Figure 5 shows extracted sources and time courses using the proposed infomax algorithm.By looking to the extracted sources and time courses it is quite clear that there exist one to one correspondence of sources and time courses with the actual sources of Fig. 3. Table 1 shows the execution time and correlation results of the conventional and proposed infomax algorithm for general signals of Fig. 1 and 2. Table 2 shows the execution time and correlation results of the conventional and proposed algorithm in case of fMRI data for the conventional and proposed algorithm.

CONCLUSION
In this study infomax algorithm is studied and applied on general signals and fMRI simulated signals.A higher order Infomax contrast function is also presented with simple fixed point iteration algorithm.Since the contrast function is higher order, therefore, it converges rapidly and extracted sources having a good correlation with actual sources.The proposed algorithm is tested on general signals and fMRI simulated signals and its application can be extended to other BSS problems.

Fig. 1 :
Fig. 1: Four simulated sources and eight mixturesEntropy is related to independence in such a way that maximum entropy implies independent signals.Therefore, the objective of ICA is to find the un-mixing matrix A that maximizes the entropy in the extracted signals Y .Infomax(Aapo et al., 2001) involves the maximization of the output entropy:

Fig. 2 :
Fig. 2: Four sources extracted by the conventional infomax (b) Four sources extracted by the proposed modified infomax

=
Update weight matrix A using Eq.(8) Step6 = If the change in un-mixing matrix A is less than some minimum value Stop, else go to step 5. SIMULATION RESULTS AND DISCUSSION In this study a modified version of infomax is presented.To verify the validity of the proposed algorithm, it is applied to general simulated and fMRI simulated source signals.First row of Fig. 1 shows four simulated source signals.These signals are mixed by linearly combining them with random signals thus making eight mixed signals as shown in row 2 and row 3 of Fig. 1.The conventional infomax algorithm is applied to mixed signals for extracting the underlying source signals as shown in Fig.

Table 2 :
Correlation results of extracted sources and actual simulated sources using infomax and proposed infomax