Iterative self-consistent parallel imaging (PI) reconstruction (SPIRiT) [1] has been extended for dynamic imaging (k-t SPIRiT) by additionally exploiting temporal correlations in k-t space resulting in improved magnitude and phase reconstruction accuracy of velocity encoded data [2]. Using eigen-decomposition of a modified SPIRiT operator, computationally optimized reconstruction formally translates into auto-calibrated SENSE (ESPIRiT) [3]. In this work, this principle is applied to a k-t SPIRiT operator resulting in SENSE-like reconstruction of a coil-combined x-f space object. The L₁-regularized variant of the proposed method, termed L₁ k-t ESPIRiT, is tested on dynamic cardiac short-axis view and aortic 4D flow data, and compared to standard L₁ k-t SPIRiT in terms of reconstruction accuracy and time.

k-t SPIRiT reconstructs a multi-coil x-f image series $$$\boldsymbol{\rho}$$$ by solving the optimization problem

$$\operatorname*{arg\,max}_{\boldsymbol{\rho}} \|\mathbf{d}-(\mathbf{I}_{N_c}\otimes\mathbf{I}_u\mathbf{F}_{x,f})\boldsymbol{\rho}\|_2^2+\mu\|(\color{red}{\mathbf{G}}-\mathbf{I})\boldsymbol{\rho}\|_2^2+\lambda R(\boldsymbol{\rho}) , \qquad(1)$$

where the x-f domain PI-operator $$$\mathbf{G}$$$ is composed of diagonal matrices implementing the coil-wise k-t space interpolations ($$$\mathbf{d}$$$: k-t space data, $$$\mathbf{F}_{x,f}$$$: Fourier transform from x-f to k-t space, $$$\mathbf{I}_u$$$: undersampling matrix, $$$R(\cdot)$$$: further regularization terms, $$$\mu$$$/$$$\lambda$$$: regularization parameters). More specifically, for each x-f voxel, $$$\mathbf{G}$$$ reduces to a matrix-vector multiplication resulting in a computational complexity of $$$O(N_c^2)$$$ ($$$N_c$$$: number of coils). Following [3], $$$\mathbf{G}$$$ can be reformulated, assuming multi-coil k-t space blocks are projected onto a subspace spanned by an orthonormal vector set, generating the fully sampled k-t space calibration data blocks. After voxel-wise eigen-decomposition of $$$\mathbf{G}$$$, the eigenvectors corresponding to eigenvalue one assemble the matrix $$$\mathbf{S}_{x,f}$$$ (stacked diagonal matrices) directly transforming a x-f object into its multi-coil sensitivity-weighted x-f representation. Thereby, $$$\mathbf{G}$$$ in (1) is replaced with $$$\mathbf{S}_{x,f}$$$ in the modified data-consistency term:

$$\operatorname*{arg\,max}_{\boldsymbol{\rho}} \|\mathbf{d}-(\mathbf{I}_{N_c}\otimes\mathbf{I}_u\mathbf{F}_{x,f})\color{red}{\mathbf{S}_{x,f}}\boldsymbol{\rho}\|_2^2+\lambda R(\boldsymbol{\rho}) . \qquad(2)$$

Solving (2) results in a computationally optimized equivalent of (1) with an $$$O(N_c)$$$ PI operator and regularization terms acting on a coil-combined object $$$\boldsymbol{\rho}$$$, and not on its multi-coil depiction.

Breath-held fully sampled 2D cine balanced SSFP short-axis view data (32 coils, 270x270x8 mm³ FOV, 1.4x1.4x8 mm³ voxel size, 23 heart phases, 36 ms temporal resolution) and navigator-gated 3D cine 4-point phase-contrast (PC) data (28 coils, 240x206x46 mm³ imaging volume, 2x2x2 mm³ voxel size, 25 heart phases, 33.6 ms temporal resolution, 200 cm/s isotropic encoding velocity) were acquired from two healthy subjects on a 3T scanner (Philips Ingenia, Philips Healthcare, Best, The Netherlands). 2D and 3D k-t spaces were retrospectively undersampled (5-fold) using time-interleaved Cartesian variable-density sampling along the phase encodes [4, 5]. $$$\mathbf{G}$$$ and $$$\mathbf{S}_{x,f}$$$ were derived from the fully sampled central k-t space region. Due to the sparse x-f support, L₁-regularization, $$$R(\boldsymbol{\rho}) = \|\boldsymbol{\rho}\|_1$$$, was added. k-t SPIRiT and k-t ESPIRiT reconstructions were both performed using POCS-like algorithms as described in [1] and Fig. 1a, respectively ($$$K = 30$$$ iterations each). 32- and 28-channel data were compressed to smaller number of virtual channels using [6]. All algorithms were implemented in Matlab (The Mathworks, Natick, MA, USA), and streamline visualization was performed using dedicated software (GTFlow, GyroTools, Zurich, Switzerland).

Fig. 1b shows reconstruction speed-up factors of k-t ESPIRiT relative to k-t SPIRiT depending on the number of virtual channels. Fig. 2 illustrates fully sampled reference x-t image slice series (top) and $$$\mathbf{G}$$$ operator x-f eigenvalue images (bottom) derived from the corresponding 5-fold undersampled data sets of the dynamic 2D a) and 3D b) experiments. In Fig. 3, direct Fourier transformed (IFT), k-t SPIRiT and k-t ESPIRiT reconstructed frames and temporal profiles are compared relative to the reference short-axis view data. The sampling pattern in the phase encode-time plane is also depicted. Arrows mark suppressed image artifacts present in the systolic reference and k-t SPIRiT image. Fig. 4 illustrates velocity vector field streamlines of the proposed method relative to the fully sampled reference and k-t SPIRiT. Numbers in frames depict the reconstruction speed-up factor relative k-t SPIRiT.

Eigen-decomposition of the k-t SPIRiT operator has been proposed and implemented to reduce computational cost in image reconstruction. Dynamic 2D and 3D in-vivo data revealed equivalence of k-t SPIRiT and k-t ESPIRiT, and significant reconstruction time savings of the proposed relative to the standard method. The inherent x-f support masking might come at the expense of reduced temporal resolution and this aspect deserves further study. Overall, this work has provided further advances towards feasible reconstruction times for combined PI and compressed sensing.