Sagar Mandava^{1}, Mahesh B Keerthivasan^{1}, Diego R Martin^{2}, Maria I Altbach^{2}, and Ali Bilgin^{1,2}

Multi-contrast image acquisitions are valuable for diagnostics but the scan time scales with the number of contrast images. Accelerated acquisitions are necessary for practical scan times and require the use of constrained reconstructions. Subspace-constraints, which constrain the multi-contrast data to lie in a low-dimensional subspace, are popularly used to reconstruct these datasets. Despite yielding good quality images at most imaging contrasts, these constraints create poor image quality at certain contrasts. We demonstrate that this is due to poor recovery of higher order subspace coefficients and present a model to enable high quality recovery of these coefficients and consequently the echo-images.

The measured data can be related to $$$\mathbf{X}_{n}$$$ as follows: $$$\mathbf{Y=E(X)}$$$ where $$$\mathbf{X}_{M \times N}$$$ ($$$M$$$ spatial pixels) is a matrix with $$$\mathbf{(X}_{n})^{N}_{n=1}$$$ as its columns. A subspace-constraint for $$$\mathbf{X}$$$ takes the following form: $$$\mathbf{X} \approx \mathbf{\alpha}_{M \times K} \mathbf{\phi}_{K \times N}$$$ where $$$\mathbf{\phi}$$$ maps the $$$K$$$-dimensional subspace coefficients $$$\mathbf{\alpha}$$$ back to the $$$N$$$-dimensional contrast space. Subspace-constrained reconstruction takes the following general form: $$\min_\alpha\frac{1}{2}||\mathbf{Y}-\mathbf{E(\alpha\phi)}||_2^2+\lambda\mathbf{R(\alpha)}$$

When
no additional regularization is used, this reconstruction is the same as k-t
PCA^{3,4}. Joint-sparsity
(JS) constraints were used in^{4} while locally low rank (LLR) regularization was used in^{6} to
augment subspace constraints. The regularizer in the JS model is $$$\mathbf{R(\alpha)}=||D_{x}\mathbf{\alpha}||_{2,1}+||D_{y}\mathbf{\alpha}||_{2,1}$$$
where $$$D_{x}$$$ are $$$D_{y}$$$ are the
horizontal and vertical finite difference operators that operate on all the
subspace images and $$$||.||_{2,1}$$$ is the
mixed $$$\ell_2/\ell_1$$$ norm.
The LLR model uses the nuclear norm, $$$||.||_{*}$$$, as a surrogate for rank and is defined as: $$$\mathbf{R(\alpha)}=\sum_{j}||T_{j}{\alpha}||_{*}$$$ where $$$T_{j}$$$ is an operator that extracts the relaxation
(subspace) signals from a small spatial patch about the spatial pixel $$$j$$$ and orders the contrast signals from
the patch into a matrix.

We propose a regularization operator built on 3D block matching (BM). For a reference 3D block (3DB), the BM algorithm identifies the L-nearest 3DBs in a local neighborhood. For the ensemble of patches, the regularization operator is defined as: $$\mathbf{R(\alpha)}=\sum_{j}\sum_{k}||\Psi_{k}(\Phi(R_{j}{\alpha}))||_{*}$$

The operator $$$R_{j}({\alpha})$$$ extracts the L-nearest (in $$$\ell_2$$$ sense) 3DBs for a reference 3DB indexed by $$$j$$$ using the BM algorithm. $$$\Phi(R_{j}({\alpha}))$$$ applies a temporal transform across the extracted 3DBs. We use the SVD algorithm to allow this transform to be data driven. $$$\Psi_{k}(.)$$$ extracts the spatial patches from the $$$k^{th}$$$ coefficient dimension and orders these as columns in a matrix which are then constrained with the nuclear norm. Note that the ensemble of 3DBs and the coefficient dimension are indexed by $$$j$$$ and $$$k$$$, respectively. The BM step makes this approach a form of non-local rank (NLR) constraint on transform domain data. Therefore, the method is named NLR3D. A cartoon illustrating subspace-constraint and the working of the proposed model is presented in Figure 1.

[1] Bilgic B, Goyal VK, Adalsteinsson E. Multi-contrast reconstruction with Bayesian compressed sensing. MRM, 2011, 66, 1601– 1615.

[2] Velikina J, Alexander AL, Samsonov A. Accelerated MR parameter mapping using sparsity-promoting regularization in parametric dimension. MRM, 2013, 70, 1263-1273.

[3] Petzschner FH, Ponce IP, Blaimer M, Jakob PM, Breuer FA. Fast MR parameter mapping using k-t principal component analysis. MRM, 2011, 66, 706–716.

[4] Zhao B, Lu W, Hitchens TK, Lam F, Ho C, Liang Z-P. Accelerated MR parameter mapping with low-rank and sparsity constraints. MRM, 2014, 74, 489-498.

[5] Huang C, Graff CG, Clarkson EW, Bilgin A, Altbach MI. T2 mapping from highly undersampled data by reconstruction of principal component coefficient maps using compressed sensing. MRM, 2012, 67, 1355–1366.

[6] Tamir JI, Uecker M, Chen W, Lai P, Alley MT, Vasanawala SS, Lustig M. T2 shuffling: Sharp, multicontrast, volumetric fast spin-echo imaging. MRM, 2016, 77, 180-195.

[7] Lebel RM, Wilman AH. Transverse relaxometry with stimulated echo compensation. MRM, 2010, 64, 1005–1014.

[8] Golub G, Pereyra V. Separable nonlinear least squares: the variable projection method and its applications. Inverse Problems, 2003,19, R1–R26.

Figure 1: a) shows a multi-contrast
dataset with N
images, and b) represents the K subspace coefficient images corresponding to
the data in (a). Also shown in (b) are 3D blocks (3DB) on the subspace coefficient images. c) shows the process of extracting the
L-nearest 3DBs (in l_{2} sense) for a reference 3DB. d) collects the extracted 3DBs and applies a
temporal transform via Φ. The
transformed 3DBs are constrained one spectral frame at a time using the nuclear
norm.

Figure 2: The subspace images for
the brain dataset at R=8 for K=4.
Note that all the methods are able to
recover the first two subspace coefficients. JS shows poor performance in
recovering the third subspace coefficient and fails to recover the fourth one.
LLR shows slightly better performance in
recovering coefficients 3 and 4 but a significant amount of artifact is still
present. NLR3D shows high quality recovery of all the four coefficients.

Figure 3: Representative TE images
created from the subspace coefficients shown in Figure 2. Note the high quality
preservation
of fine detail in the TE2 (TE= 18ms)
image created using NLR3D when compared to JS and LLR. Also note the
performance at TE8 (TE= 72ms)
is comparable across the three methods
indicating that the dominant gains due to improved subspace recovery are seen in the early TEs. Relevant scan
parameters for this experiment: ESP: 9ms, ETL: 16, 3mm slice, 0.85mm
resolution, 256views/TE (subsampled to
32views/TE) and a 16 channel phased array.

Figure 4: T2 and I0 (TE=0ms) maps for the
brain dataset. NMSE values are reported on the bottom right corner of each
reconstruction.

Figure 5: TE1 (TE=11ms) images for the
knee dataset. Relevant scan parameters for this experiment: ESP: 11ms, ETL: 8,
3mm slice, 0.5mm
resolution, 512views/TE (subsampled to
21views/TE) and a 15 channel phased array. NLR3D outperforms the other
reconstructions both qualitatively and quantitatively. K=4 was used for this experiment.