Introduction

Dictionary matching is key to many cutting edge quantitative MRI sequences, but its discrete nature presents a challenge to the application of common spatial regularisation techniques. We propose a formulation of spatially regularised dictionary matching as an optimisation on a discrete Markov random field, permitting an efficient iterative optimisation using primal-dual strategies¹.

Methods

Dictionary matching assigns parameter estimates to voxels by matching a series of measurements of each voxel—its ‘fingerprint’—against a dictionary of simulated fingerprints. For a volume $$$V$$$ of voxels $$$x$$$, normalised fingerprints $$$\hat{\bf f}(x)$$$, and parameter maps $$$\theta_1(x), ..., \theta_n(x)$$$ containing parameters from discrete sets $$$\Theta_1, ..., \Theta_n$$$, the corresponding objective function is

$$\mathop{\mathrm{argmin}}_{\theta_1, ..., \theta_n}\sum_{x \in V}D\left( \hat{\bf f}(x), \hat{\bf s}\left(\theta_1(x), ..., \theta_n(x) \right) \right)\quad\text{s.t.}\quad \forall x \in V \left( \theta_1(x)\in \Theta_1 \land ...\land \theta_n(x)\in \Theta_n \right)\,,$$

where $$$\hat{\bf s}$$$ models normalised fingerprints from the relevant sequence, and $$$D$$$ defines a distance over fingerprints. We propose to include spatial regularisation of the parameter maps via a penalty term on pairs of values assigned to adjacent voxels, giving the new objective function

$$\mathop{\mathrm{argmin}}_{\theta_1, ..., \theta_n}\left[\sum_{x \in V}D\left( \hat{\bf f}(x), \hat{\bf s}\left(\theta_1(x), ..., \theta_n(x) \right) \right)+\sum_{(x,x')\in A} \sum_{i=1}^n d_i \left( \theta_i(x),\theta_i(x')\right)\right]\quad\text{s.t.}\quad \forall x \in V \left( \theta_1(x)\in \Theta_1 \land ...\land \theta_n(x)\in \Theta_n \right)\,,$$

where the set $$$A$$$ comprises all pairs of adjacent voxels, and $$$d_i$$$ defines a distance over parameter $$$\theta_i$$$. Coupling between voxels demands a global optimisation, but the data consistency term via dictionary matching is not differentiable, hence common iterative methods using gradient descent cannot be applied here.

Instead, the problem can be cast as an optimisation on a discrete Markov random field, where each vertex $$$v$$$ of a graph $$$G$$$ with edges $$$E$$$ is assigned a label $$$l(v)$$$ from the discrete set $$$L$$$. A subset of discrete Markov random field problems is described by the objective function

$$\mathop{\mathrm{argmin}}_l\left[\sum_{v \in G}q\left( v, l(v) \right)+\sum_{(v,v')\in E} r \left( l(v),l(v')\right)\right]\quad\text{s.t.}\quad \forall v \in G\left( l(v) \in L \right)\,,$$

where $$$q$$$ penalises the assignment of a label to a given vertex, and $$$r$$$ penalises the assignment of a pair of labels along an edge. Problems of this form permit efficient iterative optimisation using primal-dual methods¹.

For application to dictionary matching, a vertex is created for each voxel, with edges joining the vertices corresponding to adjacent voxels. Labels correspond to combinations of parameters, and the cost of assigning a label to a vertex is the distance between the fingerprint simulated for that combination of parameters, and the fingerprint measured in the voxel belonging to that vertex. Similarly, the cost of assigning a pair of labels along an edge is the desired regularisation penalty evaluated at the parameter combinations corresponding to those vertices' labels. Letting $$$l_i$$$ denote the value of the $$$i$$$th parameter for label $$$l$$$, the overall transformation is
$$G \leftarrow V\,,\quad E \leftarrow A\,,\quad L \leftarrow \Theta_1 \times ... \times \Theta_n\,,\quad q(v,l) \leftarrow D\left( \hat{\bf f}(v), \hat{\bf s}(l_i,...,l_n) \right)\,,\quad \text{and} \quad r(l,l') \leftarrow \sum_{i=1}^n d_i(l_i,l'_i)\,.$$
In this work, total variation (TV) regularisation was applied to the parameter maps, hence
$$d_i = w_i | l_i - l'_i |$$
for weighting parameters $$$w_i$$$. This technique was applied to data from a 3D joint T₁ and T₂ mapping sequence² acquiring four volumes with different T­₁ and T₂ weightings introduced by inversion recovery and T₂ preparation pulses. Relevant acquisition parameters included: FOV = 240 mm × 372 mm × 240 mm, isotropic 3 mm resolution, undersampling factor = 4. Measured volumes were reconstructed with motion-corrected iterative SENSE without additional regularisation. 1D Bloch simulations were used to generate a dictionary of 103 × 103 combinations of T₁ and T_­2, logarithmically spaced in the intervals [80, 1600] and [16, 320] respectively. Regularised parameter maps were estimated for 240 mm × 240 mm axial slices with the iterative optimiser FastPD^1,3, using the cosine distance for matching of fingerprints, and with TV regularisation applied to the T₁ and T₂ maps with weights $$$w_1$$$ = 5e-6 and $$$w_2$$$ = 1.5e-5 respectively. These were compared to unregularised maps from matching with the global optimum of the cosine distance.

Results

TV regularised maps in phantoms show excellent qualitative (Fig. 1) and quantitative (Fig. 2) improvement in the consistency of the estimated parameters within vials. Similar improvements are seen in-vivo (Fig. 3), with additional improvements to the appearance of respiratory motion artefacts.

Discussion and Conclusion

TV regularisation improved the apparent quality of parameters maps in all test cases, when compared to unregularised dictionary matching. Future work will involve the investigation of alternative regularisation terms, and application to other data with incoherent sampling between volumes, such as those from MR fingerprinting.

Acknowledgements

The authors acknowledge financial support from: (1) BHF RG/20/1/34802 (2) EPSRC EP/V044087/1 (3) Wellcome EPSRC Centre for Medical Engineering (NS/A000049/1), (4) ANID Millennium Institute iHEALTH, ICN2021_004; Fondecyt 1210637 and 1210638; Basal Funding, IMPACT, FB210024 and (5) the Technical University of Munich – Institute for Advanced Study.