Spatio-temporal wavelet regularization for parallel MRI reconstruction: application to functional MRI

Abstract

Background

Parallel magnetic resonance imaging (MRI) is a fast imaging technique that helps acquiring highly resolved images in space/time. Its performance depends on the reconstruction algorithm, which can proceed either in the k-space or in the image domain.

Objective and methods

To improve the performance of the widely used SENSE algorithm, 2D regularization in the wavelet domain has been investigated. In this paper, we first extend this approach to 3D-wavelet representations and the 3D sparsity-promoting regularization term, in order to address reconstruction artifacts that propagate across adjacent slices. The resulting optimality criterion is convex but nonsmooth, and we resort to the parallel proximal algorithm to minimize it. Second, to account for temporal correlation between successive scans in functional MRI (fMRI), we extend our first contribution to 3D + $t$ acquisition schemes by incorporating a prior along the time axis into the objective function.

Results

Our first method (3D-UWR-SENSE) is validated on T1-MRI anatomical data for gray/white matter segmentation. The second method (4D-UWR-SENSE) is validated for detecting evoked activity during a fast event-related functional MRI protocol.

Conclusion

We show that our algorithm outperforms the SENSE reconstruction at the subject and group levels (15 subjects) for different contrasts of interest (motor or computation tasks) and two parallel acceleration factors ($R=2$ and $R=4$) on $2\times 2\times 3\,\hbox{mm}^3$ echo planar imaging (EPI) images.

Appendices

Appendix

Optimization procedure for the 4D reconstruction

The minimization of $\mathcal {J}_{{\rm ST}}$ in Eq. (8) is performed by resorting to the concept of proximity operators [62], which was found to be fruitful in a number of recent works in convex optimization [63–65]. In what follows, we recall the definition of a proximity operator.

Definition 1

([62]) Let $\varGamma _0(\chi )$ be the class of proper lower semicontinuous convex functions from a separable real Hilbert space $\chi$ to $]\!-\infty ,+\infty ]$ and let $\varphi \in \varGamma _0(\chi )$. For every $\mathsf {x} \in \chi$, the function $\varphi +\Vert \cdot -\mathsf {x} \Vert ^2/2$ achieves its infimum at a unique point denoted by ${\rm prox}_{\varphi }\mathsf {x}$. The operator ${\rm prox}_{\varphi }\; : \; \chi \rightarrow \chi$ is the proximity operator of $\varphi$.

In this work, as the observed data are complex-valued, the definition of proximity operators is extended to a class of convex functions defined for complex-valued variables. For the function

$$\begin{aligned} \varPhi :\mathbb {C}^K&\rightarrow ]\!-\infty ,+\infty ] \;, \; x \mapsto \phi ^{{Re} }\left( {Re} (x)\right) + \phi ^{{Im}}\left( {Im}(x)\right) , \end{aligned}$$

(12)

where $\phi ^{{Re} }$ and $\phi ^{{Im}}$ are functions in $\varGamma _0({\mathbb R}^K)$ and ${Re} (x)$ [respectively ${Im}(x)$] is the vector of the real parts (respectively imaginary parts) of the components of $x\in \mathbb {C}^K$, the proximity operator is defined as

$$\begin{aligned} {{\rm prox}}_{\varPhi } :\mathbb {C}^K&\rightarrow \mathbb {C}^K \; , \; x \mapsto {{\rm prox}}_{\phi ^{{Re} }}({Re} (x))+\imath {\rm prox}_{\phi ^{{Im}}}\left( {Im}(x)\right) . \end{aligned}$$

(13)

We now provide the expressions of proximity operators involved in our reconstruction problem.

Proximity operator of the data fidelity term

According to standard rules on the calculation of proximity operators [65, Table 1.1] while denoting ${\rho ^{t}} = T^*\zeta ^{t}$, the proximity operator of the data fidelity term $\mathcal {J}_{\rm WLS}$ is given for every vector of coefficients $\zeta ^t$ (with $t\in \{1,\ldots ,N_r\}$) by ${{\rm prox}}_{\mathcal{J}_{{\rm WLS}}}(\zeta ^t) = T {u^t}$, where the image $u^t$ is such that $\forall \mathbf {r}\in \{1,\ldots ,X\}\times \{1,\ldots ,Y/R\}\times \{1,\ldots ,Z\}$,

$$\begin{aligned} {\varvec{u}}^t(\mathbf {r})= \big ({\varvec{I}}_R + 2{\varvec{S}}^{{\tiny \mathsf H }}(\mathbf {r}){\varvec{\varPsi }}^{-1}{\varvec{S}}(\mathbf {r}) \big )^{-1} \big ({\varvec{\rho }^{t}}(\mathbf {r}) + 2{\varvec{S}}^{{\tiny \mathsf H }}(\mathbf {r}){\varvec{\varPsi }}^{-1}{\varvec{d}}^t(\mathbf {r})\big ). \end{aligned}$$

(14)

Proximity operator of the spatial regularization function

According to [37], for every resolution level $j$ and orientation $o$, the proximity operator of the spatial regularization function $\varPhi _{o,j}$ is given by

$$\begin{aligned}&\forall \xi \in {\mathbb C},\qquad {{\rm prox}}_{\varPhi _{o,j}} \xi = \dfrac{{{\rm sign}}({Re} (\xi -\mu _{o,j}))}{\beta _{o,j}^{{Re} }+1}\max \{|{Re} (\xi -\mu _{o,j})|- \alpha _{o,j}^{{Re} },0\}\nonumber \\&\quad +\,\imath \dfrac{{{\rm sign}}({Im}(\xi -\mu _{o,j}))}{\beta _{o,j}^{{Im}}+1}\max \{|{Im}(\xi -\mu _{o,j})|- \alpha _{o,j}^{{Im}},0\}+\mu _{o,j} \end{aligned}$$

(15)

where the ${\rm sign}$ function is defined by ${\rm sign}(\xi )= 1$ if $\xi \ge 0$ and $-$1 otherwise.

Proximity operator of the temporal regularization function

A simple expression of the proximity operator of function $h$ is not available. We thus propose to split this regularization term as a sum of two more tractable functions $h_1$ and $h_2$:

$$\begin{aligned} h_1(\zeta ) =&\kappa \sum _{t = 1}^{N_r/2} \Vert T^*\zeta ^{2t} - T^*\zeta ^{2t-1} \Vert _p^p \end{aligned}$$

(16)

$$\begin{aligned} h_2(\zeta ) =&\kappa \sum _{t = 1}^{N_r/2-1} \Vert T^*\zeta ^{2t+1} - T^*\zeta ^{2t} \Vert _p^p. \end{aligned}$$

(17)

Since $h_1$ (respectively $h_2$) is separable with respect to the time variable $t$, its proximity operator can easily be calculated based on the proximity operator of each of the involved terms in the sum of Eq. (16) [respectively Eq. (17)]. Indeed, let us consider the following function

$$\begin{aligned} \varPsi : {\mathbb C}^K \times {\mathbb C}^K&\longrightarrow {\mathbb R}\; , \; (\zeta ^t,\zeta ^{t-1}) \mapsto \kappa \Vert T^*\zeta ^t - T^*\zeta ^{t-1} \Vert _p^p = \psi \circ H \left( \zeta ^t,\zeta ^{t-1}\right) , \end{aligned}$$

(18)

where $\psi = \kappa \Vert T^*\cdot \Vert _p^p$ and $H$ is the linear operator defined as

$$\begin{aligned} H: {\mathbb C}^K \times {\mathbb C}^K&\longrightarrow {\mathbb C}^K \; , \; (a,b) \mapsto a-b. \end{aligned}$$

(19)

Its associated adjoint operator $H^*$ is therefore given by

$$\begin{aligned} H^*: {\mathbb C}^K&\longrightarrow {\mathbb C}^K \times {\mathbb C}^K \; , \; a \mapsto (a,-a). \end{aligned}$$

(20)

Since $H H^* = 2\mathrm {Id}$, the proximity operator of $\varPsi$ can easily be calculated using [66, Prop. 11]:

$$\begin{aligned} {{\rm prox}}_{\varPsi } = {{\rm prox}}_{\psi \circ H} = {{\rm Id}} + \dfrac{1}{2}H^*\circ ( {{\rm prox}}_{2\psi } - {\rm Id}) \circ H. \end{aligned}$$

(21)

The calculation of ${\rm prox}_{2\psi }$ is discussed in [62].

Parallel proximal algorithm (PPXA)

The function to be minimized has been re-expressed as

$$\begin{aligned} \mathcal {J}_{\rm ST} (\zeta ) =&\sum _{t = 1}^{N_r} \sum _{\mathbf {r}\in \{1,\ldots ,X\} \times \{1,\ldots ,Y/R\}\times \{1,\ldots ,Z\}} \Vert {\varvec{d}}^t({\varvec{r}}) - {\varvec{S}}({\varvec{r}})(T^*\zeta ^t)({\varvec{r}}) \Vert ^2_{{\varvec{\varPsi }}^{-1}}\nonumber \\&+\,g(\zeta ) + h_1(\zeta ) + h_2(\zeta ). \end{aligned}$$

(22)

Since $\mathcal {J}_{\rm ST}$ is made up of more than two non-necessarily differentiable terms, an appropriate solution for minimizing such an optimality criterion is PPXA [47]. In particular, it is important to note that this algorithm does not require subiterations, as was the case for the constrained optimization algorithm proposed in [37]. In addition, the computations in this algorithm can be performed in a parallel manner and the convergence of the algorithm to an optimal solution to the minimization problem is guaranteed. The resulting algorithm for the minimization of the optimality criterion in Eq. (22) is given in Algorithm 1. In this algorithm, the weights $\omega _i$ have been fixed to $1/4$ for every $i\in \{1,\ldots ,4\}$. The parameter $\gamma$ has been set to 200, since this value was observed to lead to the fastest convergence in practice. The stopping parameter $\varepsilon$ has been set to $10^{-4}$ and the algorithm typically converges in less than 50 iterations.