Ye Tian^{1,2}, Apoorva Pedgaonkar^{1}, Jason Mendes^{1}, Mark Ibrahim^{3}, Brent Wilson^{3}, Edward DiBella^{1}, and Ganesh Adluru^{1}

We present a novel motion compensation reconstruction method based on spatiotemporal constrained reconstruction (STCR) by tracking the movements of every pixel in each time frame, and constrain the temporal total variation along the pixel tracks. The proposed method can handle both respiratory and cardiac motion, and has comparable reconstruction speed but offers better image quality compared with STCR.

In STCR, undersampled dynamic k-space data is reconstructed by exploiting the sparsity with spatial and temporal total variation constraints. Temporal total variation assumes pixel intensities in the object are temporally varying smoothly or in a piecewise smooth fashion. However, when inter-frame motion is present, this assumption breaks down since motion misaligns pixels representing the same feature along the time direction. To mitigate this problem, we propose to calculate the temporal total variation along the motion track of each pixel and term this pixel tracking temporal total variation (PTTTV). The cost function can be written as

$$ \left\|Am-d\right\|^2_2 + \lambda_t \left\|\sqrt{\nabla_tPm^2 + \epsilon}\right\|_1 + \lambda_s \left\|\sqrt{\nabla_xm^2 + \nabla_ym^2 + \epsilon}\right\|_1,$$

where $$$A$$$ is sampling matrix, $$$m$$$ is image to be reconstructed, $$$d$$$ is acquired data, $$$\lambda_{t,s}$$$ are constraint parameters, $$$\nabla_{t,x,y}$$$ are temporal and spatial gradient operators, $$$\epsilon$$$ is a small term to avoid singularity and $$$P$$$ is a reordering matrix produced from motion information that spatially aligns each pixel along time direction. The 1D case of pixel tracking total variation is shown in Figure 1.

To acquire motion information, an
artifact reduced reference image was first reconstructed by using STCR with $$$\lambda_t = 0.1C$$$, $$$\lambda_s = 0.03C$$$ and 20 conjugate iterations, where $$$C$$$ is the average intensity of the initial
estimation, then denoised using a Gaussian low pass filter. Motion maps were
estimated from this reference using large deformation kinematics with a greedy
algorithm^{10}. Similar to the MASTeR approach^{7}, two sets of
motion maps were estimated: forward motion maps that map each time frame to its
later time frame and backward motion maps. We then detected the tracks of every
pixel in the current frame, by finding its nearest moving position in adjacent
frames from the motion maps. Pixels along the track were aligned using $$$P$$$.
Figure 2 shows the pixel reordering and the improvement in temporal sparsity
when aligning pixels along their tracks. Further reconstruction iterations when $$$P$$$ is applied used constraint parameters $$$\lambda_t = 0.3C$$$ and $$$\lambda_s = 0.03C$$$.

One gated and one ungated golden angle radial SMS datasets were used to test the PTTTV. Both datasets were acquired under regadenoson stress without breath holding using a turboFLASH sequence on a 3T Prisma (Siemens) scanner with TR/TE = 2.7/1.6ms, FOV = 260mm, voxel size 1.8mmx1.8mmx8mm. The reconstructions using PTTTV were compared with reconstructions using STCR.

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