Reducing the effects of motion artifacts in fMRI: A structured matrix completion approach

Abstract

Functional MRI (fMRI) is widely used to study the functional organization of normal and pathological brains. However, the fMRI signal may be contaminated by subject motion artifacts that are only partially mitigated by motion correction strategies. These artifacts lead to distance-dependent biases in the inferred signal correlations. To mitigate these spurious effects, motion-corrupted volumes are censored from fMRI time series. Censoring can result in discontinuities in the fMRI signal, which may lead to substantial alterations in functional connectivity analysis. We propose a new approach to recover the missing entries from censoring based on structured low rank matrix completion. We formulated the artifact-reduction problem as the recovery of a super-resolved matrix from unprocessed fMRI measurements. We enforced a low rank prior on a large structured matrix, formed from the samples of the time series, to recover the missing entries. The recovered time series, in addition to being motion compensated, are also slice-time corrected at a fine temporal resolution. To achieve a fast and memory-efficient solution for our proposed optimization problem, we employed a variable splitting strategy. We validated the algorithm with simulations, data acquired under different motion conditions, and datasets from the ABCD study. Functional connectivity analysis showed that the proposed reconstruction resulted in connectivity matrices with lower errors in pair-wise correlation than non-censored and censored time series based on a standard processing pipeline. In addition, seed-based correlation analyses showed improved delineation of the default mode network. These demonstrate that the method can effectively reduce the adverse effects of motion in fMRI analysis.

I. INTRODUCTION

Functional magnetic resonance imaging (fMRI) provides insight into the functional organization of the brain during normal development [1], [2] and various disorders such as psychiatric illnesses, neurodegenerative diseases, and sleep disorders [3]. One application of fMRI, resting state functional MRI (rsfMRI), is of particular interest in pediatric neuroimaging as it identifies network relationships in the brain, without requiring task-performance, by assessing correlations of the blood oxygen level dependent (BOLD) signal fluctuations in disparate, but related, regions of the brain, while the subject is at rest – termed ”functional connectivity” [4]. However, separating activity-related variance in this BOLD signal from contamination by artifacts due to non-neuronal sources, including hardware instabilities, cardiac and respiratory activity, and head movement has been an area of intense research for over a decade [5]. Optimally minimizing these artifacts at the individual participant level is the largest barrier to accurate estimation of functional connectivity in clinical populations.

Head motion is particularly undesirable in fMRI [6], [7]. It changes the tissue composition in a voxel, distorts the magnetic field, and disrupts the steady state magnetization recovery of the spins in the slices that have moved [5]–[8]. These lead to disruptions in the measurement of the BOLD signal, including signal drop out and artifactual amplitude changes in many regions of the brain. Numerous strategies, including frame-by-frame spatial realignment to correct for shifts in head position, regression of motion estimates, and filtering out motion components using independent components analysis (ICA) have been employed to reduce motion artifacts [9]–[12]. These methods do not fully compensate for the intensity changes due to spin history effects, and small residual motion artifacts continue to cause distance-dependent changes in the BOLD signal correlations throughout the brain [13]–[15].

One common approach to mitigate the spurious effects in correlation is by ‘censoring’ [14], where volumes with high frame-by-frame motion, and the frames directly adjoining high-motion timepoints, are excised from the processed fMRI data, prior to functional connectivity analysis. However, this creates discontinuities in the time series and may lead to significant loss of data, resulting in unreliable estimates of correlation. To alleviate this issue, interpolation based strategies have been proposed, which replace the missing entries with synthetic data formed using signal information from neighboring time points [16], [17].

In this work, we adopt a principled approach to reduce motion artifacts from rsfMRI data based on structured low rank matrix completion. Structured low rank methods have been proposed to solve problems in numerous MRI applications [18], [19]. However, due to different signal assumptions and signal models being employed, none of the existing methods can be applied directly to fMRI time series analysis. In our approach, we excise fMRI volumes with elevated motion, and model the remaining unprocessed fMRI data using motion parameters and slice-timing information. The motion parameters are either recorded prospectively using motion tracking sensors or a camera, or estimated retrospectively from the imaging data via registration algorithms. This modeling results in a large matrix with many missing entries. We formulate the artefact-reduction problem as the recovery of this large matrix from very few measurements. To solve the ill-posed problem, we exploit the implicit structure present in every time series using a linear recurrence relation (LRR). This expresses the voxel intensity at the current time point as a linear combination of its intensities from the past. The compact representation of the LRR leads to the construction of a low rank Hankel matrix, whose entries correspond to the signal samples. The Hankel matrices from different voxels are stacked vertically to form a large structured matrix, which also has a low rank structure. We exploit this structure to recover the matrix. In addition to motion compensation, we also achieve slice-timing correction as every column of the large matrix is a volume of data at each slice acquisition time point. This enables us to downsample the time series, starting from any slice acquisition time point, to match the sampling period of the measured fMRI data.

Implementing our new algorithm mentioned above involves solving a large scale system of equations, and forming a large structured matrix. This results in high memory demand and computational complexity. To alleviate these issues, we propose a variable splitting strategy that decouples the original problem into two simpler sub-problems. We present simplifications that eliminate the need to evaluate the large structured matrix. Significant speed-up is obtained as the proposed strategy allows efficient and parallel computations in each of the sub-problems. We validated the proposed method through simulated and real motion fMRI experiments, and also tested it on twenty two datasets corresponding to eight subjects from the Adolescent Brain Cognitive Development (ABCD) study [20]. We demonstrate the improvements offered on the functional connectivity by the proposed reconstructions through functional connectivity (fcMRI) analysis.

II. MOTION COMPENSATED RECOVERY USING STRUCTURED MATRIX PRIOR

A. Forward model

We consider the motion compensated recovery of fMRI time series from unprocessed fMRI volumes. Using slice timing information and motion parameters, we relate each unprocessed volume Y_i to the desired reconstructed fMRI time series X as

$${\mathbf{Y}}_i={\mathcal{M}}_i\left({\mathcal{S}}_i(\mathbf{X})\right)+{\eta }_i,i=1,2,\dots ,n_v$$ (1)

where n_v denotes the total number of volumes. ${\mathbf{Y}}_i\in {ℝ}^{n_1\times n_2\times n_s}$ is the i^th unprocessed volume, where n₁, n₂ are the spatial dimensions of each slice and n_s is the number of slices in each volume. $\mathbf{X}\in {ℝ}^{n_r\times n_c}$ corresponds to the reconstructed fMRI time series, where n_r = n₁ × n₂ × n_s and $n_c=\frac{n_s\times n_v}{mb}$, mb is the multi-band factor associated with the echo planar imaging (EPI) acquisition. It has a higher temporal resolution (equal to the slice acquisition time) than the unprocessed fMRI volumes, which have a temporal resolution equal to the repetition time (TR). Specifically, every column of X corresponds to a full volume of data at each slice acquisition time point. We denote X as the super-resolved reconstructed fMRI time series. S_i is a linear sampling operator that samples the voxel intensities from X, which correspond to the measurement locations of the i^th unprocessed volume, i.e., ${\mathcal{S}}_i(\text{X})\in {ℝ}^{n_1\times n_2\times n_s}$. An illustration of the sampling operation on a super-resolved matrix X in the case of an interleaved EPI acquisition and mb = 1 is shown in Fig. 1. When mb > 1, entries corresponding to simultaneously acquired slices appear at the measured locations in every column of X.

Fig. 1:

Illustration of the super-resolved matrix and sampling operation: The entries of X (in green, red, and blue) correspond to the measurement locations in fMRI volumes 1, i, and n_v respectively, while the entries in black correspond to unmeasured locations. For instance, the data in red correspond to entries of different slices of a fMRI volume acquired between (i−1) and i^th TR. These slice-entries are arranged in X in the order (inter-leaved) in which they are acquired. Here, we refer to X as super-resolved as every column corresponds to a full volume of data at each slice acquisition time point. Si refers to the sampling operator that extracts the data (at i^th TR) from X, which corresponds to the measurement locations in the i^th fMRI volume (Si(X)).

The inter-volume motion occurring in the fMRI time series is captured by the six dimensional motion parameters (three rotation and three translation parameters). ${\mathcal{M}}_i$ is a motion operator that encodes these parameters for the i^th volume, and its application is equivalent to transforming the volume by the motion parameters followed by resampling. The motion parameters in ${\mathcal{M}}_i$ are estimated by spatially registering the i^th volume to the first volume, which is treated as a reference. For the model in (1), they are pre-computed once from the unprocessed fMRI volumes Y_i. We ignore the motion occurring within each slice as the scale of head movements is very small due to short slice-acquisition times in rsfMRI. Motion occurring between slices in a volume, and spin history effects contribute to the modeling error, η_i, which is the error term for the i^th volume. When SNR is sufficiently high, we can assume η_i to be additive and Gaussian [21].

Successful recovery of X from the Y_i’s not only ensures motion compensation but also slice time correction. The recovered time series can be downsampled, starting from any slice acquisition time point, to match the sampling period of the measured fMRI data. However as (1) is ill-posed, direct recovery of X is not possible without enforcing a prior on it.

B. Linear recurrence relation and structured matrix prior

We assume that the temporal signal at every voxel location r of the fMRI time series is governed by a linear recurrence relation (LRR):

$$x_n\left(r\right)=\sum _{k=1}^{L-1}{\alpha }_k\left(r\right)x_{n-k}\left(r\right)$$ (2)

Here ${\alpha }_k(\mathbf{r})\in ℝ$ are the weights and n is the signal index along the time dimension. Dropping r to ease the notation, we can compactly write (2) as

$$\underset{\text{H}}{\underbrace{\left(\begin{array}{cccc}{x}_1& x_2& \cdots & x_L\\ x_2& x_3& \cdots & x_{L+1}\\ ⋮& ⋮& \cdots & ⋮\\ x_{n_c-L+1}& x_{n_c-L+2}& \cdots & x_{n_c}\end{array}\right)}}\underset{g}{\underbrace{\left(\begin{array}{c}-{\alpha }_{L-1}\\ -{\alpha }_{L-2}\\ ⋮\\ 1\end{array}\right)}}=0$$ (3)

where $\mathbf{H}\in {ℝ}^{n_c-L+1\times L}$ is a Hankel matrix and $\text{g}\in {ℝ}^{L\times 1}$ is a vector of window coefficients. From (3), we can infer that H is rank deficient, and g is a vector in its null space. When (2) is satisfied for a window of shortest length L_min, we obtain a minimum LRR, and H will be rank deficient by one. However, in practice, L_min is not known. In such cases, the length of the window is overestimated. This results in multiple linear independent vectors g₁, g₂, ..., g_v, which satisfy (3) [22], [23], and the dimension of the null space (value of v) will depend on the rank of H.

In rsfMRI, the BOLD signals in functionally connected regions are correlated. To exploit these spatial correlations, we assume that the window coefficients corresponding to temporal signals at different voxel locations are the same. This enables us to stack the Hankel matrices from different voxel locations vertically (illustrated in Fig. 2), which leads to

$$\mathcal{H}(\mathbf{X})\left[{\text{g}}_1,{\text{g}}_2,\dots ,{\text{g}}_v\right]=0$$ (4)

Fig. 2:

Illustration of the construction of the matrices H and $\mathcal{H}(\mathbf{X})$. Top: A time series x(n) satisfying the LRR in (2) results in the construction of a Hankel matrix H. The rapidly decaying singular values of H indicate that it is low rank. Bottom:$\mathcal{H}(\mathbf{X})$ is formed by vertically stacking Hankel matrices from different voxel locations. The plot of the singular values indicate that H(X) is also low rank. We exploit this structure of $\mathcal{H}(\mathbf{X})$ to recover the missing entries of X.

Here $\mathcal{H}$ is a linear operator that map X onto a large structured matrix $\mathcal{H}(\mathbf{X})\in {ℝ}^{n_r\left(n_c-L+1\right)\times L}$. From (4), we can infer that $\mathcal{H}(\mathbf{X})$ has a large null space, and hence is low rank. The rapid decay of the singular values of $\mathcal{H}(\mathbf{X})$ is illustrated in Fig. 2.

We note that enforcing a low rank prior on $\mathcal{H}(\mathbf{X})$ to recover X results in a more constrained approach than imposing a low rank penalty on a Hankel matrix at each voxel location independently (voxel-wise penalty [24]), which is equivalent to having different spatial weights (α_k, k = 1, ... L−1) in (3) at different spatial locations. Specifically, the low rank prior on $\mathcal{H}(\mathbf{X})$ enables exploiting the spatial correlations between the time series at different voxels, in addition to the implicit temporal structure present in the time series at every voxel location. However, the voxel-wise penalty is not capable of exploiting these spatial correlations, and require additional sparsity and low rank priors for exploiting them [24].

C. Problem formulation

We formulate the recovery of X from the measurements Y_i as the following structured matrix completion problem:

$$\underset{\text{x}}{\min }\text{rank}\left[\mathcal{H}\left(\text{X}\right)\right]\text{s.t}\sum _{i=1}^{n_v}{\Vert {\text{Y}}_i-{\mathcal{M}}_i\left({\mathcal{S}}_i\left(\text{X}\right)\right)\Vert }_F^2\le \rho ,X_{ij}\ge 0$$ (5)

where ρ > 0 is an error tolerance, $\mathcal{H}(\mathbf{X})$ is a structured matrix formed from the samples of the time series. The constraint on X ensures that the solution to (5) is non-negative. This is implemented as the projection onto the non-negative orthant:

$${\mathbf{X}}^{\ast }={\mathcal{P}}_{+}(\mathbf{X})$$ (6)

where ${\left({\mathcal{P}}_{+}(\mathbf{X})\right)}_{ij}=max\left(X_{ij},0\right)$. Since (5) is NP hard, we relax the rank function using a Schatten p (0 ≤ p ≤ 1) norm and rewrite the modified objective function as

$$\underset{\text{x}}{\min }\sum _{i=1}^{n_v}{\Vert {\text{Y}}_i-{\mathcal{M}}_i\left({\mathcal{S}}_i\left(\text{X}\right)\right)\Vert }_F^2+\mu {\Vert \mathcal{H}\left(\text{X}\right)\Vert }_p^p,X_{ij}\ge 0$$ (7)

μ is a regularization parameter that balances the weight given to the data consistency term and the regularizer. The Schatten p norm acts as a regularizer, which is defined for a matrix M as ${\Vert \text{M}\Vert }_p^p:=\text{Tr}\left[{\left({\text{M}}^T\text{M}\right)}^{\frac{p}{2}}\right]={\sum }_i{\sigma }_i^p;{\sigma }_i$ are the singular values of M. When p = 1, the Schatten p norm reduces to the convex nuclear norm and for (0 ≤ p < 1), it is a non-convex penalty; When p → 0, $\parallel M{\parallel }_p^p:=\sum _i\log {\sigma }_i$[25].

Direct implementation:

The iterative re-weighted least squares (IRLS) strategy [26] can be adopted to solve (7), which decouples the problem into two sub-problems. The solution is then obtained by alternating between the estimation of a weight matrix $\mathbf{W}\in {ℝ}^{L\times L}$ and a solution to the least squares problem:

$$\mathbf{W}={\left(\underset{\mathbf{R}}{\underbrace{{[\mathcal{H}(\mathbf{X})]}^T[\mathcal{H}(\mathbf{X})]}}+ϵ\mathbf{I}\right)}^{\frac{p}{2}-1}$$ (8)

$${\text{X}}^{*}={\mathcal{P}}_{+}\left(\arg \underset{\mathbf{X}}{\min }\sum _{i=1}^{n_v}{\Vert {\text{Y}}_i-{\mathcal{M}}_i\left({\mathcal{S}}_i\left(\text{X}\right)\right)\Vert }_F^2+\mu {\Vert \mathcal{H}\left(X\right)\sqrt{\text{W}}\Vert }_F^2\right)$$ (9)

ϵ in (8) acts as a regularizer and stabilizes the inverse, and (9) can be solved using a few iterations of the Conjugate Gradient (CG) method. Note that the above implementation requires the evaluation of $\mathcal{H}(\mathbf{X})$ and solving a large scale system of equations. This results in high memory demand and computational complexity, especially in the current rsfMRI setting where the datasets are huge. To alleviate these challenges, we propose a variable splitting (VS) based optimization method.

III. VARIABLE SPLITTING BASED OPTIMIZATION ALGORITHM

We employ a variable splitting (VS) based optimization method, which decouples (7) into two sub-problems. We present simplifications that eliminate the need to form $\mathcal{H}(\mathbf{X})$, and enables efficient and parallel computations for the subproblems.

We introduce an auxiliary variable ${\mathbf{Z}}_i\in {ℝ}^{n_1\times n_2\times n_s}$ in (7) and rewrite the relaxed objective function as

$$\underset{\mathbf{X},{\mathbf{Z}}_i}{\min }\sum _{i=1}^{n_v}\left({\Vert {\mathbf{Y}}_i-{\mathcal{M}}_i\left({\mathbf{Z}}_i\right)\Vert }_F^2+\beta {\Vert {\mathbf{Z}}_i-{\mathcal{S}}_i\left(\mathbf{X}\right)\Vert }_F^2\right)+\mu {\Vert \mathcal{H}\left(\mathbf{X}\right)\Vert }_p^p,{\mathbf{X}}_{ij}\ge 0$$ (10)

where β is a penalty parameter that enforces the constraint ${\mathbf{Z}}_i={\mathcal{S}}_i\left(\mathbf{X}\right),i=1,2,\dots ,n_v$. Note that the solution to (10) tends to that of (7) as β → ∞. To solve (10), we minimize it with respect to one variable while keeping the other constant. This decouples (10) into two simpler sub-problems, which, at the n^th iteration are given by

$${\mathbf{X}}^{\left(n\right)}={\mathcal{P}}_{+}\left(\arg \underset{\mathbf{X}}{\min }\sum _{i=1}^{n_v}{\Vert {\mathbf{Z}}_i^{\left(n-1\right)}-{\mathcal{S}}_i\left(\mathbf{X}\right)\Vert }_F^2+\frac{\mu }{\beta }{\Vert \mathcal{H}\left(\mathbf{X}\right)\Vert }_p^p\right)$$ (11)

$${{\mathbf{Z}}_i}^{\left(n\right)}=\text{arg}\underset{{\mathbf{Z}}_i}{\min }\sum _{i=1}^{n_v}\left({\Vert {\mathbf{Y}}_i-{\mathcal{M}}_i\left({\mathbf{Z}}_i\right)\Vert }_F^2+\beta {\Vert {\mathbf{Z}}_i-{\mathcal{S}}_i\left({\mathbf{X}}^{\left(n\right)}\right)\Vert }_F^2\right)$$ (12)

The solution to (11) is a super-resolved time series that is also slice-time corrected, and the solution to (12) are the motion compensated fMRI volumes. The above optimization problems are solved in an iterative fashion until the cost in (7) between successive iterations is below a threshold. The final solution is then obtained by downsampling X, starting from any slice acquisition point, to match the sampling period of the measured fMRI data. In the presence of minimal motion, (11) can be solved in a standalone manner to correct the fMRI time series for slice timing effects.

A. Solution to the X sub-problem

We adopt the IRLS strategy, outlined in section II-C, to solve (11). First, we compute the weight matrix W, and then present an efficient method to solve the least squares problem.

1). Update of $\sqrt{\mathbf{W}}$:

The first step in computing the weight matrix involves forming the Gram matrix R. Instead of directly computing it, which is both a memory and computationally intensive operation, we exploit the structure present in $\mathcal{H}(\mathbf{X})$ and form R efficiently as

$$R=\sum _{r=1}^{n_r}\underset{G_r}{\underbrace{{\left[\mathcal{H}\left({\mathbf{X}}_r\right)\right]}^T\left[\mathcal{H}\left({\mathbf{X}}_r\right)\right]}}$$ (13)

where X_r is the r^th row of X and $\mathcal{H}\left({\mathbf{X}}_r\right)\in {ℝ}^{n_c-L+1\times L}$ is the corresponding Hankel matrix. To limit the influence of the time series at the background voxels on the estimation of the weight matrix, we employed a brain mask to only include the time series from the voxels inside the brain during the computation of R. To speed up the computation, G_r is computed in parallel for different values of r.

The weight matrix $\sqrt{\text{W}}$ is then computed from the eigen decomposition of R, which is given by VΛV^T; V and Λ represent the eigen vectors and eigen values, respectively. Substituting for R in (8) and simplifying further, we obtain $\sqrt{\mathbf{W}}=\mathbf{V}{(\text{Λ}+ϵ\mathbf{I})}^{-\frac{q}{2}}$, where $q=\left(1-\frac{p}{2}\right)$.

2). Least squares solution to (11):

The least squares problem corresponding to (11) is given by

$${\mathbf{X}}^{*}={\mathcal{P}}_{+}\left(\arg \underset{\mathbf{X}}{\min }\sum _{i=1}^{n_v}{\Vert {\mathbf{Z}}_i-{\mathcal{S}}_i\left(\mathbf{X}\right)\Vert }_F^2+\frac{\mu }{\beta }{\Vert \mathcal{H}\left(\mathbf{X}\right)\sqrt{\mathbf{W}}\Vert }_F^2\right)$$ (14)

Let the columns of $\sqrt{\mathbf{W}}$ be denoted as [w₁, ..., w_L]. Substituting for $\sqrt{\mathbf{W}}$ in (14), we obtain

$${\mathbf{X}}^{*}={\mathcal{P}}_{+}\left(\arg \underset{\mathbf{X}}{\min }\sum _{i=1}^{n_v}{\Vert {\mathbf{Z}}_i-{\mathcal{S}}_i\left(\mathbf{X}\right)\Vert }_F^2+\frac{\mu }{\beta }\sum _{l=1}^L{\Vert \mathcal{H}\left(\mathbf{X}\right){\mathbf{W}}_l\Vert }_2^2\right)$$ (15)

Expanding $\mathcal{H}(\mathbf{X})$ in terms of the Hankel matrix blocks, we get

$$\sum _{l=1}^L{\Vert \mathcal{H}\left(\mathbf{X}\right){\mathbf{W}}_l\Vert }_2^2=\sum _{l=1}^L\left|\right|\left(\begin{array}{c}\mathcal{H}\left({\mathbf{X}}_1\right){\mathbf{W}}_l\\ \mathcal{H}\left({\mathbf{X}}_2\right){\mathbf{W}}_l\\ ⋮\\ ⋮\\ \underset{C_l}{\underbrace{\mathcal{H}\left({\mathbf{X}}_1\right){\mathbf{W}}_l}}\end{array}\right)|{|}_2^2$$ (16)

where the k^th block of C_l, given by $\mathcal{H}\left({\mathbf{X}}_k\right){\mathbf{W}}_l$, represents a one dimensional linear convolution between X_k and ${\tilde{\mathbf{W}}}_l$:

$$\mathcal{H}\left({\mathbf{X}}_k\right){\mathbf{W}}_l={\mathbf{X}}_k\star {\tilde{\mathbf{w}}}_l={\tilde{\mathbf{w}}}_l\star {\mathbf{X}}_k={\mathbf{D}}_l{\mathbf{X}}_k^T$$ (17)

where ⋆ represents 1D convolution, the second equality is due to the commutative property of convolution, ${\tilde{\mathbf{W}}}_l$ represents a vector whose entries are flipped with respect w_l and ${\mathbf{D}}_l\in {ℝ}^{n_c-L+1\times n_c}$ is a sparse banded matrix given by

$$D_l=\left(\begin{array}{ccccccc}{w}_1^{(l)}& w_2^{(l)}& \dots & w_L^{(l)}& 0& \dots & 0\\ 0& w_1^{(l)}& \dots & w_{L-1}^{(l)}& w_L^{(l)}& \dots & 0\\ ⋮& ⋮& \dots & ⋮& ⋮& \dots & ⋮\\ 0& 0& \dots & w_1^{(l)}& w_2^{(l)}& \dots & w_L^{(l)}\end{array}\right)$$ (18)

where $w_k^{(l)}$ is the k^th entry of w_l. Rewriting each convolution block of C_l in (16) in terms of D_l we obtain,

$$C_l=\underset{B_l}{\underbrace{\left(\begin{array}{ccccc}{D}_l& 0& \cdots & \cdots & 0\\ 0& D_l& \cdots & \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& \cdots & \cdots & D_l\end{array}\right)}}\underset{V\text{}ec({\mathbf{X}}^T)}{\underbrace{\left(\begin{array}{c}{\mathbf{X}}_1^T\\ {\mathbf{X}}_2^T\\ ⋮\\ ⋮\\ {\mathbf{X}}_{n_r}^T\end{array}\right)}}$$ (19)

where $B_l\in {ℝ}^{n_r\left(n_c-L+1\right)\times n_c{n}_r}$. Using (19), we can rewrite (15) as

$${\mathbf{X}}^{\star }={\mathcal{P}}_{+}\left(\arg \underset{\mathbf{X}}{\min }{\Vert \mathbf{Z}-\mathbf{S}\text{x}\Vert }_2^2+\frac{\mu }{\beta }\sum _{l=1}^L{\Vert {\mathbf{B}}_l\text{x}\Vert }_2^2\right)$$ (20)

where z = V ec(Z^T), x = V ec(X^T) are introduced to simplify the notations and $\mathbf{S}\in {ℝ}^{n_r{n}_v\times n_r{n}_c}$ is the sampling matrix corresponding to x. Note that $\mathbf{Z}\in {ℝ}^{n_r\times n_v}$ is in the Casorati matrix form [27] such that the k^th column corresponds to the vectorized form of the volume Z_k. (20) can be solved by taking the gradient of the function inside the brackets and setting it to zero:

$$\left(\underset{\text{P}}{\underbrace{{\text{S}}^T\text{S}+\frac{\mu }{\beta }\sum _{l=1}^L{\text{B}}_l^T{\text{B}}_l}}\right)\text{x}={\text{S}}^T\text{z}$$ (21)

where $\text{P}\in {ℝ}^{n_c{n}_r\times n_c{n}_r}$ is a block diagonal matrix. The measurement locations for all the time series corresponding to a particular slice are the same. This allows us to compactly express P as

$$\text{P}=blkdiag\left({\left\{\underset{{\text{Q}}_j}{\underbrace{{\text{S}}_j^T{\text{S}}_j+\frac{\mu }{\beta }\sum _{l=1}^L{\text{D}}_l^T{\text{D}}_l}}\right\}}_{j=1}^{n_s}\right)$$ (22)

where ${\text{S}}_j^T{\text{S}}_j$ is the j^th diagonal block of S^TS and ${\text{Q}}_j\in {ℝ}^{n_1{n}_2{n}_c\times n_1{n}_2{n}_c}$ is also a block diagonal matrix. The solution to (21) is then computed as x = P⁻¹S^Tz. Instead of directly computing P⁻¹, we exploit its block diagonal structure and compute in terms of Q_j. The solution to (21) is then given by,

$${\text{x}}_{(j)}={\text{Q}}_j^{-1}{\text{S}}^T{\text{z}}_{(j)},j=1,2,\dots ,n_s$$ (23)

where x_(j) and z_(j) are the vectorized time series corresponding to the j^th slice of X and Z respectively. Finally, only the positive entries of X are retained after projection onto the non-negative orthant. To speed up the computation, (23) can be computed in parallel for different values of j.

B. Solution to the Z sub-problem

To solve (12), we rewrite the optimization problem as:

$${\mathbf{Z}}_i^{*}=\arg \underset{{\mathbf{Z}}_i}{\min }\sum _{i=1}^{n_v}\left({\Vert {\mathbf{Y}}_i-{\mathcal{M}}_i\left({\mathbf{Z}}_i\right)\Vert }_F^2+\beta {\Vert Z_i-{\tilde{\mathbf{X}}}_i\Vert }_F^2\right)$$ (24)

where $\widetilde{X_i}={\mathcal{S}}_i(X)\in {ℝ}^{n_1\times n_2\times n_s}$ represents voxel intensities of X, which correspond to the measurement locations of the i^th unprocessed fMRI volume. (24) can be decoupled into n_v independent sub-problems, which can be solved in parallel. To solve the i^th sub-problem, we compute its gradient and set it to zero:

$${\mathcal{M}}_i^T{\mathcal{M}}_i\left({\mathbf{Z}}_i\right)+\beta {\mathbf{Z}}_i=\beta \widetilde{{\mathbf{X}}_i}+{\mathcal{M}}_i^T\left({\mathbf{Y}}_i\right)$$ (25)

The above system of equations can be solved by running a few iterations of the CG algorithm.

C. Implementation details

The details of the alternating minimization algorithm to solve (11) and (12) are described in Algorithm 1. Every volume of Z is initialized (Z⁽⁰⁾) with the first volume (Y₁) of the fMRI motion time series. To set X⁽⁰⁾, first we populate the entries corresponding to the measured locations with the samples corresponding to Z, while the entries in the unmeasured locations are set to zero (See Figure. 1). The entries in each row then are set to the mean value of that row. There are a few tuning parameters associated with the proposed algorithm. They were chosen empirically.

1). Regularization parameter μ:

First, we ran the optimization algorithm on the simulation dataset (described in Section IV-A) for different values of μ, and chose the one (μ = μ_opt) that gave the lowest global mean squared error (gMSE). Then, for all other experiments (involving different datasets), we set μ = μ_opt in the algorithm.

2). Penalty parameter β:

We assessed the effect of the penalty parameter β on the convergence of the proposed variable splitting (VS) algorithm by conducting experiments on the simulation dataset. We employed a continuation strategy [28] and updated β at the end of n^th iteration as β⁽ⁿ⁾ = β⁽ⁿ⁻¹⁾ ∗ β_fac, 1.1 ≤ β_fac ≤ 2. This resulted in faster convergence (cost in (7) between successive iterates ≤ threshold) than using a fixed value of β (See Figure. S2 (a) in the supplementary material). Also, this strategy resulted in a good approximation of (7) as it converged to a solution close to the one obtained using direct implementation of (7), which was very slow. This is clearly reflected in Figure. S2 (a) and the heat map in (b). For all our experiments we set β_init = 0.25 and β_fac = 1.3, and updated β at every iteration.

3). IRLS parameters (L, ϵ, η):

The details of the structured matrix completion algorithm to solve (11) are described in Algorithm 2. The length of the window L determines the size of the Hankel matrix at each voxel. We chose a value of L such that n_s/mb < L ≤ n_c/2. In every row of X, the number of missing entries between two measurements is n_s/mb. Hence this choice of L always resulted in a few non-zero entries in every row of $\mathcal{H}(\mathbf{X})$, which aids in the completion of the matrix [29]. Note that a very large value of L is not favorable. For instance, when L = n_c (length of the time series), $\mathcal{H}(\mathbf{X})$ in (4) reduces to X, and imposing a low rank prior would result only in the exploitation of the spatial correlations; a suitable value of L would balance the weights given to both the exploitation of spatial correlations and the temporal structure at every voxel location. We denote the time series reconstructed using the proposed method with and without applying censoring to motion time series as proposed (with censoring) and proposed (no censoring) respectively. Prior to reconstruction in proposed (with censoring), volumes were censored and the locations of missing entries were encoded in the sampling operator in (1). For all experiments, we set L = 50 for proposed (no censoring), while for proposed (with censoring) we chose a larger value (L = n_c/4). This ensured that every row $\mathcal{H}(\mathbf{X})$ had few non-zero entries even for high-motion datasets. We observed that the algorithm was insensitive to different values in the neighborhood of chosen L.

We set ϵ⁽⁰⁾ = 1 and varied its value in every iteration as described in [26]. Specifically, in the m^th iteration we decreased the value of ϵ as ϵ^(m) = ϵ^(m−1)/η, η > 1. For all the experiments in this paper, we set η = 1.1; the Schatten p value was set to 0.1.

D. Memory demand and computational complexity

Compared to solving (7) using direct implementation (Section II-C), our proposed variable splitting based algorithm is both fast and memory efficient. To see this, we analyzed the memory demand and computational complexity of both algorithms. Below, we summarize the results of our analysis. For detailed descriptions, refer to the appendix.

Memory demand:

For the direct implementation, the memory demand is primarily determined by the gradient of the first term in (9), and is given by O(n_cn_r). In contrast, the maximum memory required by the proposed variable splitting algorithm is determined by the computation in (23), which is given by O(n_cn_r/n_s). This implies that the proposed algorithm provides a n_s-fold reduction in memory usage compared to the direct implementation. Typical values of n_s used in the experiments in this paper are 36, 47 and 60.

Computational complexity:

For the direct implementation, the computation complexity is determined by the evaluation of the weight matrix and the gradient of (9). The total number of operations performed is given by $O(n_c{n}_r{N}_{dir}(N_{cg}(n_v+L)+\frac{L^2}{N_{cores}})$, where N_cg and N_dir are the number of CG and IRLS (outer) iterations respectively, and N_cores is the number of available cores.

Similarly, computing the solutions to the X sub-problem and Z sub-problem determines the computational complexity for the proposed variable splitting algorithm. The total number of operations of the proposed algorithm is given by $O\left(n_c{n}_r{M}_{max}{N}_{max}{L}^2/N_{cores}\right);M_{max}$ is the total number of iterations for the X sub-problem (Algorithm 2) and

Algorithm 1:

Proposed variable splitting based algorithm for motion-compensated time series recovery.

Algorithm 2:

IRLS based structured matrix completion algorithm for solving the X sub-problem (11).

N_max refers to the number of outer iterations (Algorithm 1). From the analysis, the proposed algorithm offers a speed up of $\frac{N_{cores}{N}_{dir}\left(N_{cg}\left(n_v+L\right)+\frac{L^2}{N_{cores}}\right)}{\left(L^2{N}_{max}{M}_{max}\right)}$. Using this expression for typical values (datasets used in this paper) such as n₁, n₂ = 64, n_s = 36, n_v = 80, L = 50, N_dir = 10, N_cg = 200, N_cores = 16, N_max = 15 and M_max = 5, we expect ≈ 23 fold speed up. In practice, we observed a speed up of ≈ 27 as illustrated in Figure. S2 (a).

IV. EXPERIMENTS AND RESULTS

We tested our algorithm on fMRI data with simulated motion, data acquired under different motion conditions, and datasets from the ABCD study.

A. Datasets

1). Simulated motion fMRI experiment:

An rsfMRI dataset with TR = 3 s, voxel resolution 3.375×3.375×3.91mm³ was slice time corrected using FSL [30] and used as a reference. To generate data with motion, we retrospectively injected motion into each slice of every volume of the reference data using real, rigid head motion parameters (shown in Fig. S1 A) recorded using an in-bore Kineticor motion monitoring camera system. However, for reconstruction using the forward model in (1), the six motion parameters for every volume were obtained by registering all the volumes to the first volume of the motion data. These parameters are shown in Fig. S1 B), and the framewise displacement (FD) metric computed [14] from them are shown in Fig. 3.

Fig. 3:

Framewise displacement (FD) metric computed from motion parameters estimated with registration (see Fig. S1 for the recorded and estimated motion parameters for the data in this experiment). The FD roughly indicates the extent of motion present in the data. As part of censoring, volumes with FD > 0.5 mm are excised (along with one preceding and two succeeding volumes) from the fMRI data to account for through-plane motion artifacts as well as spin history effects.

2). Real motion fMRI experiments:

Three datasets were acquired from a single subject under different motion scenarios: 1) no motion, 2) medium motion, and 3) high motion. In the first case, the subject was instructed to remain still, while in the second and third cases, the subject was instructed to move every thirty and fifteen seconds, respectively. The datasets were collected with a protocol approved by the Institutional Review Board, on a Siemens 3T scanner using a gradient echo EPI (GRE-EPI) sequence. The scan parameters were: TR = 3 s, TE = 30 ms, flip angle = 85^o, FOV = 216 mm, voxel resolution 3 × 3 × 3 mm³, number of volumes = 160. The number of slices in each volume was 47, and the slices were acquired in an interleaved fashion. To improve the alignment of the fMRI data to the MNI template [31], a structural T1-weighted image was also acquired with the following scan parameters: TR = 1.6 s, TI = 1.1 s, flip angle = 5^o, FOV = 224 mm, voxel resolution 1 × 1 × 1 mm³, GRAPPA factor = 2.

3). Adolescent Brain Cognitive Development (ABCD) experiments:

The proposed method was tested on twenty two rsfMRI datasets from eight subjects, chosen from the ABCD study. The datasets exhibited different amounts of motion with mean FD ranging from 0.18 mm to 1.20 mm. The datasets were acquired on a Siemens 3T Prisma scanner using a GRE-EPI acquisition. The scan parameters were: TR = 800 ms, TE = 30 ms, flip angle = 52^o, FOV = 216 mm, voxel resolution 2.4 × 2.4 × 2.4 mm³, number of volumes = 383. The number of slices in each volume was 60, and the slices were acquired using simultaneous multi-slice acquisition with a multi-band factor of six. A structural T1-weighted image was also acquired with scan parameters: TR = 2.5 s, TI = 1.07 s, flip angle = 8^o and voxel resolution 1 × 1 × 1 mm³.

B. Processing for Functional connectivity (fcMRI) analysis

For the different time series, prior to fcMRI analysis, we employed the following pre-processing steps [14] to reduce variance due to non-neuronal activity in the bold signal. First, the time series were slice-time corrected using FSL [30]. Then, every volume of the fMRI time series was realigned to the first volume. This was followed by regressing multiple nuisance signals from the rsfMRI data, including white matter, cerebrospinal fluid, and whole brain signals, and six motion parameters obtained from rigid head motion correction. To reduce noise, spatial smoothing was applied using a Gaussian filter with 6 mm full width at half maximum (fwhm). Finally, the signals were detrended, standardized, and bandpass filtered (0.01 Hz < f < 0.1 Hz). For the proposed method, since the time series is already motion compensated and corrected for slice timing errors, the first two steps (motion correction and slice time correction) were excluded from the pipeline.

Processing with and without censoring resulted in different time series for fcMRI analysis. During censoring, the FD for every volume was computed from the corresponding six motion parameters [14]. Volumes with FD > 0.5 mm were identified and excised from the data along with one preceding and two succeeding volumes. We then generated multiple timecourses (for comparison to no motion (reference)) with and without applying censoring to motion time series and proposed method: motion (no censoring), motion (censoring), proposed (no censoring), and proposed (with censoring). Of note, volumes were censored from motion data after the bandpass filtering step. We also note that the time taken to process the data following the aforementioned fcMRI analysis pipeline is about 1 hour. If the proposed method is adopted, then an additional time (T_recon) to reconstruct the time series is required, which is dependent on the size of the datasets. Specifically, for datasets of size 64 × 64 × 36 × 80, T_recon = 20 mins, and size 72 × 72 × 72 × 160, T_recon = 1 hour.

C. FcMRI matrix and seed based correlation map

We generated a region-region functional connectivity matrix using an available set of 264 identified brain regions of interest (ROIs) [32]. Each ROI was modeled as a sphere of radius 10 mm and a correlation matrix was derived from the Pearson product moment correlations between the time series extracted from each pair of ROIs. A sample seed-based correlation map was also calculated between the time series of the Posterior Cingulate Cortex (PCC) and the rest of the brain.

D. Receiver Operating Characteristic (ROC) and Precision-Recall (PR) analysis

To determine the thresholds for the seed-based correlation maps, we performed ROC and PR analyses on the maps generated from different time series and reconstructions. For this purpose, a binary mask was generated by thresholding the reference correlation map at a value of 0.5. Various parameters such as sensitivity (recall), specificity, and precision were estimated using the reference binary mask and the seed-based maps at hundred thresholds between zero and one. Additional parameters such as the area under the curve (AUC) for ROC, Youden’s statistic, and the F-score were derived from the aforementioned parameters. Youden’s statistic is defined as Y := Sensitivity + Specificity − 1 and F-score is defined as the harmonic mean of precision and recall [33]. Finally, the optimal threshold for the seed based maps was chosen as the value that corresponded to maximum F-score.

E. Simulated motion fMRI

We demonstrate the effectiveness of the proposed method on reconstructing motion compensated fMRI time series from simulated-motion data in Fig. 4, which shows the voxel-wise mean squared error (MSE) maps corresponding to different fMRI time series. Brighter and darker regions in the maps correspond to higher and lower errors respectively. Slice timing and head motion correction improved the global MSE, and resulted in an improved error map (shown in (b), compared to the error map of the motion time series in (a)). However, the presence of many bright regions indicate significant deviation in signal intensities from the reference. The volumes corresponding to particularly high motion with FD > 0.5 mm, as shown in Fig. 3, were identified and excised. Interpolating the missing data in each time series using a fifth degree bspline function did not significantly improve the error map (shown in (c)). In contrast, the signal intensities corresponding to our proposed (no censoring) deviated less from the reference, except in a few regions in the brain where the error was still high. Excising the high-motion volumes and using the proposed (with censoring) to fill in the missing entries using information from the rest of the time series reduced the deviation from the reference significantly, and resulted in lower errors throughout the brain (map shown in (e)). As an example, the effectiveness of the proposed method on reconstructing the time series at a particular voxel is shown in Fig. S3 (a). The ability of the proposed method to slice-time correct a motion-free time series is also demonstrated in Fig. S3. For this, (11), which can be viewed as a regularized structured matrix completion problem, was solved in a standalone manner. The benefit of our proposed method can be appreciated from the low gMSE (0.001) and from the map shown in Fig. S3 (d), which shows lower errors throughout the brain than the uncorrected map.

Fig. 4:

Voxel-wise Mean Squared Error (MSE) maps of the time series along with the global MSE (gMSE) for (a) Motion (b) Motion corrected (motion time series corrected for slice timing & head motion) (c) Motion corrected (censor & interpolate) (d) Proposed (no censoring) and (e) Proposed (with censoring). The error map corresponding to proposed (with censoring) in (e) has significantly lower gMSE and lower errors (indicated by dark regions) than the time series in (a) to (c) and also shows improvement over proposed (no censoring). In proposed (with censoring), the high-motion volumes are censored, and the algorithm performs both motion compensation and interpolation of missing entries resulting in the lowest errors and improved pairwise correlations in the fcMRI matrix (Fig. 5).

The proposed method (with censoring) offers particular benefit for functional connectivity analysis (Fig. 5). Compared with censoring alone, the proposed method produces pair-wise correlations that more closely approximate the motion-free condition. This is illustrated by estimating linear fits (using Scikit-learn’s RANSAC algorithm) between the reference and proposed correlations, which reveals higher slope, r-value and a lower histogram spread around the line y = x, shown in blue, for our proposed (with censoring) method. Interestingly, for the motion time series censoring has a larger effect on the distribution of the histogram values around the line y = x than on the pair-wise correlations themselves.

Fig. 5:

Comparison of pair-wise fcMRI correlations generated from the simulated motion case: Here, roughly 30% of the data were censored, leaving 110 volumes of uncorrupted data for fcMRI analysis. The fcMRI correlation matrices, derived from motion time series (without and with censoring, censor+interpolate) and the proposed (without and with censoring) reconstructions, are shown from (d) to (h), and their corresponding errors with the reference correlation (c) are shown from (i) to (m). 2D histogram (heat maps) between the motion/proposed correlations (y-axis) and the reference correlations (x-axis) are shown from (n) to (r), along with the linear fit and the y = x line in red and blue respectively The higher r-value and slope associated with the fitted line (in red) in (r) indicate that the pair-wise correlation matrix from the proposed method (with censoring) deviates the least from the reference in (c).

F. Real motion fMRI

We tested our proposed algorithm on rsfMRI datasets acquired under different motion conditions as described in section IV-A. Similar to the simulated motion experiment above, we observe that the proposed method (with censoring) yields improved pair-wise correlations than the alternatives on data affected by medium motion (40% of the data marked for censoring) in Fig. 6, and high motion (60% of the data marked for censoring) in Fig. S4. As expected, even though the use of our proposed method (with censoring) outperformed the alternatives, pair-wise correlations still deviate more from the reference data in the high motion case than in the medium motion case due to high percentage of censored data. As an additional example of the benefit of the proposed method, seed based correlation maps were generated from a Posterior Cingulate Cortex (PCC) region of interest to test the ability to identify the default mode network (DMN) from the different datasets and reconstructions. Censoring alone did not allow identification of the bilateral angular gyrus regions of the DMN (see Fig. 7(b) & S5 (b), while our proposed method (with censoring) resulted in improved delineation in many regions (see Fig. 7 (d) & S5 (d)). Note that the ROC and PR analysis was used to estimate the optimal threshold to display the maps. The ROC and PR analysis also suggested that the proposed (with censoring) method was more sensitive, and had higher overlap in the activation regions with the reference than motion (censoring) (See Tables I & S1). We also demonstrate the relative performance (ROC and PR curves) of the various methods for the medium motion case in Fig. 8. Interestingly, a number of weak positive connections were seen in the high-motion case with our proposed methodology that were not observed in the medium motion case. It is possible that these represent false positive correlations due to the increased number of missing entries, and subsequent replacement by synthetic data (marked with red arrows in Figure S5 (d)).

Fig. 6:

Comparison of pair-wise fcMRI correlations generated from the medium motion case: Here, roughly 40% of the data was motion-corrupted, leaving 96 volumes of uncorrupted data for fcMRI analysis. The fcMRI matrices corresponding to motion (no censoring and censoring) and the proposed method (without and with censoring) are shown from (d) to (g). 2D histograms (heat maps) between the motion/proposed correlations (y-axis) and the reference correlations (x-axis) are shown from (h) to (k), along with linear fit and the y = x line in red and blue respectively. Higher slope, r values and lower histogram spread around y = x indicate that the proposed method (with censoring) generated lower pair-wise correlations error than the proposed (no censoring) and motion (with and without censoring) cases.

Fig. 7:

Comparison of Posterior Cingulate Cortex (PCC) seed-based functional connectivity maps from the medium motion case. The correlation map between the time series of a PCC seed (shown in green) with the rest of the brain is shown from each processing strategy, i.e., for motion (no censoring and censoring), proposed (without and with censoring), and reference shown in (a) to (e). We observe that the proposed method (with censoring) is more similar to the reference data, and is better able to identify the DMN (green arrows) than other methods.

TABLE I:

Metrics inferred from the ROC and PR analyses for the medium motion dataset: the AUC, specificity, Youden statistic (from ROC analysis); and sensitivity and F-score corresponding to the optimal threshold (from PR analysis). The threshold corresponding to the maximum F-score was chosen as optimal for the seed based maps (Fig. 7).

	AUC	Sensitivity/ Recall	Specificity	Youden statistic	F-score
Motion (No censoring)	0.94	0.40	0.78	0.76	0.50
Motion (With censoring)	0.95	0.42	0.81	0.76	0.52
Proposed (No censoring)	0.96	0.40	0.91	0.80	0.48
Proposed (With censoring)	0.97	0.49	0.92	0.85	0.56

Fig. 8:

Receiver operating characteristics (ROC) and Precision-Recall (PR) analysis to determine optimal threshold for seed based correlation maps. The ROC curve and PR curve was generated from different sensitivity and specificity & precision and recall values (estimated at hundred thresholds between zero and one), respectively. The optimal threshold was the value corresponding to maximum F-score (Table. I). As an example, the ROC and PR curves corresponding to the medium-motion case are shown in (a) and (b) respectively. The ROC curve in (a) shows that for any false positive rate, our proposed method (with censoring) achieved the highest true positive rate among all methods. Similarly from the PR curve in (b), for a high recall value, the proposed (with censoring) achieved the highest precision among all methods.

G. ABCD experiments

Pair-wise fcMRI correlations were generated for proposed (with censoring) and motion (censoring) on fourteen datasets from seven subjects (two scans per subject), and the results are shown in Fig. 9. For each subject, a dataset (third scan) with minimal motion (fewest number of volumes exceeding the FD-threshold of 0.5) was chosen as reference. The linear fits performed between the reference (x axis) and proposed/motion correlations (y axis) revealed higher median slope values, lower Interquartile Range (IQR) (distance between 25^th and 75^th percentiles) or smaller spread in the slopes and r-values for proposed (with censoring) than motion (censoring) (see Figure. 9a & 9b). Especially in high-motion scans, pair-wise correlations were significantly reduced throughout the brain for motion (censoring), and resulted in diminished slope values (See Figs. 9c to 9h). Paired t-test with a type I error rate threshold of 0.05 showed statistically significant difference in the slope values between the methods (p = 0.041), but did not show statistically significant difference in r-values.

Fig. 9:

Comparison of pair-wise fcMRI correlations generated from ABCD datasets. The fcMRI matrices, corresponding to motion (with censoring) and proposed (with censoring), were generated for fourteen datasets from seven subjects (two datasets per subject). For each subject, a dataset with minimal motion was chosen as reference. The box plots, corresponding to the slopes and r-values estimated from linear fits between the reference and motion/proposed correlations, are shown in (a) and (b) respectively. The bottom and top edges of the box correspond to the 25^th and 75^th percentiles, and the central red line indicates the median. The box plot in (a) shows that the median slope value is much lower, and the Interquartile range (distance between 25^th and 75^th percentiles) is much higher in motion (with censoring) than proposed (with censoring). Though the median r-values are similar, however, the spread of r-values is much smaller for proposed (with censoring) than motion (with censoring) (See box plot in (b)). In datasets with high motion, the pair-wise correlations were reduced throughout the brain for motion (with censoring), resulting in diminished slope values. As an example, for a dataset exhibiting high motion, (c) framewise displacement (d) reference correlation matrix (e,f) correlation matrix and heat map for motion (with censoring) and (g,h) correlation matrix and heat map for proposed (with censoring) are shown. In the heat map, the x-axis corresponds to reference and the y-axis corresponds to proposed/motion correlations. Here, roughly 57% of the data was motion-corrupted, leaving 162 volumes for fcMRI analysis. The r-values are similar between motion (censoring) and proposed (with censoring), but proposed (with censoring) has a much higher value of slope (Figure. 9f & 9h).

Out of the twenty two datasets, there were eight datasets from three subjects where the FD did not exceed the threshold of 0.5. We tested the proposed algorithm on these low-motion datasets, and demonstrated low deviation in pair-wise fcMRI correlations (see Fig. S6) from those generated using standard pipeline, as mentioned under Section IV-B. The linear fits between them revealed high r-values and slopes for all datasets (see box plots in Figs. S6 (a) & (d)). The results on a particular low-motion dataset can also be appreciated in Fig. S6.

V. DISCUSSION AND CONCLUSION

We demonstrated a principled approach to reduce motion artifacts from unprocessed rsfMRI data based on structured low rank matrix completion. To achieve this, we transformed the motion artifact-reduction problem to the recovery of a super-resolved matrix from few measurements, and relied on the low rank structure of a large structured matrix ( $\mathcal{H}(\mathbf{X})$) to recover the missing entries. Thus, recovering the missing entries is equivalent to simultaneously performing interpolation and motion compensated reconstruction. This negates the need for separate slice time correction, with the added benefit of being able to losslessly resample the data to any specified TR interval. As such, (11) could also be used to perform slice time correction and resampling of fMRI time series in the absence of motion (see Fig. S3). To keep the memory demand and computational complexity low, we proposed a variable splitting strategy. This decoupled the original optimization problem into two simpler sub-problems, which were solved in an efficient manner. We also presented simplifications that enabled us to solve the problem without storing $\mathcal{H}(\mathbf{X})$.

Experiments on a simulated dataset showed that our fast solution based on our variable splitting algorithm converged to a similar solution to that of the direct implementation of (7) (Section. II-C) in 20 mins. It offered a 27-fold speed up over the direct method (illustrated in Fig. S2(a)). The large speed up is mainly due to two factors. First, the proposed formulation resulted in reduced memory demand thus enabling efficient and parallel computations in the sub-problems. Second, to obtain a reasonable reconstructed time series, the number of CG iterations required by the direct implementation was very high (around 200). The high memory demand in every CG iteration could have also contributed to an increase in the overall computation time.

We note that the forward model in (1) and our solution to it are generic such that it can incorporate inter-or-intra-volume, and rigid or non-rigid motion. In this work, we assumed that the motion parameters are pre-recorded or pre-computed prior to reconstruction. Furthermore, we used a 3D rigid model for inter-volume head motion, and incorporated the motion within volume and spin history effects into the modeling error.

Using both simulated and real motion fMRI data, we showed that our proposed method (with censoring) closely approximates the results from the reference for both ROI-ROI fcMRI analyses and the construction of seed based correlation maps. However, in cases of very high motion (where ∼ 60% of the data needed to be censored and regenerated), we observed some deviation from the reference dataset, likely indicating the limitation of this method as currently implemented. Despite this potential degradation in performance, the resultant pairwise correlations still had lower deviation from reference than censoring alone and the seed-based fcMRI map was able to identify known DMN regions more accurately. The ROC and PR analyses also suggested that the proposed method had higher sensitivity and specificity, and the activation regions had higher overlap (F-score) with the reference than motion (censoring). Interestingly, in the high motion case, the proposed method (without censoring) was slightly more sensitive and had a higher F-score than the proposed method (with censoring) (Table SI), which could be due to the relative decimation of the available data and speaks to the limitation of censoring in general.

Using low motion ABCD datasets, we also showed that the fcMRI correlations from the proposed method had very low deviations from those generated using standard pipeline (Section IV-B), which suggests that no major artifacts were introduced in the time series by the proposed reconstruction method. We also observed that the proposed (with censoring) was more consistent and robust across the ABCD datasets with different degrees of motion than motion (censoring). This was reflected from the smaller spread in the r-values and slopes (inferred from the linear fits between the reference and motion/proposed fcMRI pair-wise correlations) for the proposed (with censoring) than those of motion (censoring) (see the box plots in Figure 9). Specifically, in high-motion cases, pair-wise fcMRI correlations were significantly reduced throughout the brain for motion (censoring), and resulted in diminished slope values.

The experiments demonstrated that in low-motion datasets, where censoring was not required, the proposed method (no censoring) can be employed to generate meaningful pairwise correlations. However, in high motion cases, the proposed method when combined with censoring was more effective in reducing the motion artifacts. This success can be attributed to the benefits of the ‘censoring and interpolating’ strategy itself. Additional benefits for the proposed methodology are derived through a) the proposed unique forward model and b) structured matrix prior that enables the exploitation of spatial correlations and temporal structure at every voxel. These result in reconstructions that have reduced motion artifacts, and thus provide us with improved functional connectivity. Despite improvements, there is some deviation between the reference and the proposed method. This error can be attributed to the artifacts arising due to through-plane motion and spin history effects, which are mitigated, but not eliminated, when the proposed method is combined with censoring. Note that even small head motion alters the magnetic field, and disrupts the steady state magnetization of the spins in the slices that have moved. Consequently, the intensities in the time series are completely altered, and it is impossible to completely eliminate these artifacts.

While there are instances where the proposed method produced higher correlations than the reference dataset, i.e., possible false positive connections, this is not a systemic finding as indicated by a slope < 1 in Fig. 9h, and as observed from the overall structure of region-region correlation matrices (Fig. 9d vs 9g ) and the pattern of seed-based connectivity maps (Fig. S5 (d) vs (e)), which are not affected. As such, the approach presented here offers great potential to restore data in motion-corrupted fMRI time series. This is critically important for applications involving high and frequent motion such as neonatal and fetal fMRI. Future work will explore the benefits of this approach in such applications.

APPENDIX

A. Memory demand and computational complexity

For the direct implementation, the update of weight matrix in (8) and the evaluation of the gradient of (9) for CG contribute to increased memory usage and computational cost. The gradient of (9) is given by

$$\sum _{i=1}^{n_v}\underset{\text{term}1}{\underbrace{S_i^T\left({\mathcal{M}}_i^T\left({\mathcal{M}}_i\left(S_i\left(\text{X}\right)\right)\right)\right)}}+\mu \text{X}\underset{S_p}{\underbrace{\left(\sum _{l=1}^L{\text{D}}_l^T{\text{D}}_l\right)}}=\sum _{i=1}^{n_v}{S}_i^T\left({\mathcal{M}}_i^T\left({\text{Y}}_i\right)\right)$$ (26)

Here the second term in (9) has been replaced with C_l from (19).

Memory demand:

The direct implementation and the variable splitting algorithm require O(n_cL) and O(L²) space to store the entries of the Hankel matrix at every voxel location and the computation of the Gram matrix R respectively.

Similarly, the sparse matrix D_l only requires O(n_cL) space to store its non-zero entries. Hence, for the direct implementation, the dominant term determining the memory demand is the space to store term 1 in (26), which is O(n_cn_r). In contrast, the proposed algorithm estimates the time series corresponding to every slice independently (23), and hence the memory required is O(n_cn_r/n_s). This implies that the proposed algorithm provides an n_s-fold reduction in the memory demand compared to the direct implementation.

Computational complexity:

For the direct implementation, the evaluation of the gram matrix requires O(n_cn_rL²/N_cores) operations (N_cores is the total number of cores), and the evaluation of S_p in (26) requires only O(n²_cL²/N_cores) operations. Observing that S_p is a sparse banded matrix with at most 2L elements in every column, the number of operations required to evaluate (XS_p) in (26) is O(n_cn_rL). Also, the number of operations required to evaluate term 1 for all volumes is O(n_cn_rn_v). Hence, the total number of operations is given by $O\left(n_c{n}_r{N}_{dir}\left(N_{cg}\left(n_v+L\right)+\frac{L^2}{N_{cores}}\right)\right)$, where N_cg and N_dir are the number of CG and IRLS (outer) iterations respectively.

Similarly, the evaluation of the solutions to the X subproblem and Z sub-problem determines the computational complexity for the proposed variable splitting algorithm. For the X sub-problem (Algorithm 2), similar to direct implementation, the computation of Gram matrix (R) and S_p requires a total of O(n_cn_rL²/N_cores) operations (assuming n_r >> n_c). To recover the time series corresponding to every slice using (23), the inverse of Q_j is not explicitly computed. An efficient solver (e.g., matlab’s mrdivide) exploiting the banded structure of the non-zero block (size n_c × n_c) in Q_j is typically used [34]. Hence, the number of operations required to estimate all slices of X is given by O(n_cn_sL²/N_cores). Assuming n_r >> n_s, the total number of operations involved in the X sub-problem is then given by O(n_cn_rM_maxL²/N_cores), where M_max is the total number of iterations in the X subproblem. Finally, the evaluation of the gradient term in (25) for CG for all volumes determines the computational complexity of the Z sub-problem. The total number of operations is given by O(n_rn_vM_cg/N_cores) where M_cg is the total number of CG iterations. Therefore, the total number of operations of the proposed algorithm is given by O(n_cn_rM_maxN_maxL²/N_cores) (assuming M_cgn_v << M_maxn_cL²), where N_max is the total number of iterations for the proposed algorithm. From the analysis, the proposed algorithm offers a speed up of $\frac{N_{cores}{N}_{dir}\left(N_{cg}\left(n_v+L\right)+\frac{L^2}{N_{cores}}\right)}{\left(N_{max}{M}_{max}{L}^2\right)}$.