Sparse representations for signals have been of much interest mainly for accelerating MRI acquisitions. In this work, a different methodology of utilizing sparse representations for identifying functional connectivity networks is investigated. Specifically, fMRI signals are decomposed into morphological components with sparse spatial overlap. This assumption is more physically plausible than the statistical independence assumption of the conventional Independent Component Analysis (ICA).^1,2 The proposed formulation can be related to the Morphological Component Analysis (MCA)³ and is different to that of ^4,5. Unlike ⁶, the dictionary is learned from the data using the K-Singular Value Decomposition (K-SVD) algorithm⁷.

Consider the acquired fMRI data represented as a spatial-temporal function $$$s(r,t)$$$. Conventional data-driven methods detect the patterns of connectivity between brain regions by decomposing the corresponding data matrix formed from $$$s(r,t)$$$ as $$\boldsymbol{S}=\boldsymbol{DX}+\boldsymbol{E},$$ where $$$\bf D$$$ is the dictionary with the $$$k$$$-th column $$$\bf d_k$$$ being the time series (also referred to as atoms) of the $$$k$$$-th decomposed component; the $$$k$$$-th row $$$\bf x^{(k)}$$$ of the matrix $$$\bf X$$$ represents the spatial variation of the $$$k$$$-th component from which functional connectivity of the brain is inferred; $$$\boldsymbol{E}$$$ models the noise residual component. We further assume that the acquired fMRI data can be represented as a sparse linear combination of the atoms, i.e. $$$\boldsymbol{S}=\sum_k \boldsymbol{d_{k}}\boldsymbol{x^{(k)}}$$$, resulting in the following optimization problem⁷: $$\{ \boldsymbol{\widehat{D}},\boldsymbol{\widehat{X}} \}=\arg \min_{\boldsymbol{D,X}}||\boldsymbol{S}-\boldsymbol{DX}||_F^2 \text{ subject to } ||\boldsymbol{x_{i}}||_0\leq T_{0}.$$ The sparsity constraint implies that the decomposed signal sources can spatially overlap, which is a valid assumption for commonly identified BOLD functional networks. Specifically, we assume no more than $$$T_0$$$ components are simultaneously activated at each voxel. The optimization problem can be related to the MCA method used to solve the blind source separation problem, except that $$$\boldsymbol{d_k}$$$ are the atoms of the dictionary $$$\bf D$$$ rather then the dictionaries themselves.³ The non-smooth convex optimization problem is solved using the K-SVD algorithm iteratively in two stages.⁷ In the sparse encoding stage, the matrix $$$\bf X$$$ is computed using any pursuit method.^8,9 In the second stage, each $$$\boldsymbol{d_k}$$$ is updated sequentially by finding a residual matrix $$$\boldsymbol{E_k}=\boldsymbol{S}-\sum_{j\neq k} \boldsymbol{d_j} \boldsymbol{x^{(j)}}$$$, choosing the columns of $$$\boldsymbol{E_k}$$$ that used $$$\boldsymbol{d_k}$$$ in their representation, and finding a rank-one approximation.⁷ Once the matrices $$$\boldsymbol{\widehat{D}}$$$ and $$$\boldsymbol{\widehat{X}}$$$ are estimated, the spatial components reflecting functional connectivity are the rows of $$$\boldsymbol{\widehat{X}}$$$.

MCA-KSVD analysis was applied to both task and resting-state fMRI (RS-fMRI) on human volunteers. BOLD task functional data (31 slices, 4 mm slice thickness, TR=2s, TE=30ms, 255 time frames) were obtained using auditory stimulation with spiral acquisition. RS-fMRI was acquired with 240 time frames. Figure 1 shows task activation map estimated by correlating fMRI time series with the task paradigm convolved with the hemodynamic response function and thresholded with the value $$$r=0.2$$$. Spatial ICA was performed with 20 components and the derived components were used as initialization for the MCA-KSVD with 200 iterations to ensure algorithm convergence. The MCA-KSVD results were also examined with random initializations, yielding similar results. $$$T_0$$$ was set to allow maximum 8 components to spatially overlap at each voxel. Both the ICA and MCA-KSVD components were thresholded so that the number of the displayed voxels was the same as that of the task activation map. It can be seen from Fig. 1 that the MCA- KSVD method identified task-related network structures. Figure 2 shows the resulting ICA and MCA-KSVD components in the case of the RS-fMRI. Resting-state networks such as the visual, sensori-motor, DMN, and executive control networks are resolved by the MCA-KSVD. As $$$T_0$$$ decreases, the derived networks appear sparser, as illustrated in Fig. 3, and in the extreme case when $$$T_0=1$$$, MCA-KSVD behaves similarly to the spatial clustering. The autocorrelation functions of the ICA and MCA-KSVD components were computed to quantify the spatial resolution. Figure 2 shows that the resolution of MCA-KSVD components is higher than that of the ICA in the case of the strong visual network. Figure 4 shows the pair-wise correlation between ICA components and similarly between MCA-KSVD components. In contrast to ICA, MCA-KSVD decomposition does not require its components to be independent.A data-driven method exploiting sparse representations for functional connectivity detection has been presented for task- and RS-fMRI. The method decomposes fMRI data into morphological spatial components which have sparse spatial overlap, unlike the conventional ICA analysis. Experimental results prove that the MCA-KSVD method can identify functional networks in both task- and RS-fMRI. The detected networks resemble those of the ICA, implying that MCA-KSVD can be used as an alternative method for investigating brain functional connectivity, with increased spatial resolution.