Synopsis

Mean apparent propagator (MAP) reconstructs the diffusion pdf using a dictionary based on Hermite functions. The first element corresponds to a tensor approximation; and the following elements add non-gaussian components. To improve non-gaussian accuracy, one needs to increase the size of the dictionary, but it also increases the number of q-space samples for a robust optimisation. We propose the use of compressed sensing to efficiently increase the number of atoms in the dictionary by exploiting its sparsity for a better reconstruction.

Purpose

Introduction

The mean apparent propagator (MAP) ^1,2 reconstruction optimises the weights, $$$ \mathbf{x} $$$, of its dictionary to find the best fit to the q-space signal,

$$ \min || \Phi (\mathbf{A},\mathbf{q}) \mathbf{x} - P(\mathbf{q}) ||_2^2\ \ \ [1], \\ \textrm{s.t}\ \Psi (\mathbf{A},\mathbf{r}) \mathbf{x} \geq 0 \ \ \ $$

where $$$ \Phi (\mathbf{A},\mathbf{q}) $$$ is the dictionary in q-space, $$$ \Psi (\mathbf{A},\mathbf{r}) $$$ is the dictionary in the diffusion propagator space; and the scaling tensor $$$ \mathbf{A} $$$ is used to scale the atoms of both dictionaries. Normally, the dictionaries are truncated to the first 50 atoms; however, it may be insufficient to describe complex microstructures. To overcome this, the length of the dictionary needs to be increased, though it also requires more q-space samples for a robust optimisation. We propose to use compressed sensing to find the weights from a larger dictionary while using a reduced number of samples.

Theory

Given the sparse weighting of the dictionary ², we propose a MAP reconstruction based on compressed sensing ³; hence,

$$ \min\ || \mathbf{x} ||_1 \ \ \ [2], \\ \textrm{s.t} \ || SF\Psi (\mathbf{A},\mathbf{r}) \mathbf{x} - P(\mathbf{q}) ||_2^2 \leq \epsilon$$

where $$$ SF $$$ is the undersampled Fourier operator; and the error in the data consistency term is proportional to the noise in the data.

Methods

MAP vs MAP-CS

A comparison was done between [1] and [2]. For MAP, we used 50 atoms as in ^1,2. For MAP-CS, we used an extended dictionary with 161 atoms. The CS implementation was done using SPGL1 ^4,5.

To obtain $$$ \mathbf{A} = \textrm{diag}({\mu _x}^2, {\mu _y}^2, {\mu _z}^2) $$$ for both methods, we first fitted the diffusion tensor, $$$ \textrm{D}(\mathbf{q},\Delta)=\textrm{exp}(-4\pi^{2}\Delta\mathbf{qDq}^{T}) $$$, in its eigen decomposition $$$ \mathbf{D}=Q(\vec{\mathbf{\theta}})\Lambda(\vec{\mathbf{\lambda}})Q(\vec{\mathbf{\theta}})^{T} $$$ to constrain $$$\vec{\mathbf{\lambda}}$$$ and $$$\vec{\mathbf{\theta}}$$$. Second, a dilated version of the atoms as in ^1,2 was used; thus, $$$\mathbf{\mu}_{\{x,y,z\}}=2\pi\sqrt{\beta 2 \lambda_{\{x,y,z\}} \Delta}$$$ and $$$\beta=0.5$$$ in order to generate the atoms of each dictionary.

Data

The simulated data was generated using Camino software ^6,7 for two-crossing fibres with 60° and 90° angles. We added Gaussian noise to the simulated measurements in the real and imaginary channel to simulate $$$\textrm{SNR}=10$$$.

The fully sampled q-space data was retrospectively undersampled in order to apply each reconstruction method. For MAP, we used one fourth of the data (64 samples); For MAP-CS, we used one eighth of the data (32 samples).

We also acquired real data from a 60° diffusion phantom ⁸, using $$$\delta =31~\textrm{ms}$$$ and $$$\Delta =39~\textrm{ms}$$$; thus, $$$q_{\textrm{max}} = 73.64~\textrm{mm}^{-1}$$$. The scan was done with $$$\textrm{TE} =139~\textrm{ms}, \textrm{TR}=2~\textrm{s}$$$ and resolution $$$2.0 \textrm{x} 2.0 \textrm{x} 3.0~\textrm{mm}^3$$$.

Quality Indexes

To quantitatively evaluate the reconstructions against the ground truth (fully sampled), we used the normalised-root-mean-squared-error (NRMSE) and the Pearson's correlation coefficient (PC).

Results

Figure 1 shows the indexes for the 60° and 90° crossing-angle reconstructions.

Figure 2 and 3 show the weights for each element of the dictionary for the 60° and 90° noisy simulations and their respective reconstructions. The weights are similar in both reconstructions; however, the weighting from the additional atoms makes an improvement in the reconstruction quality. There is also a reduction in the weights of the first atoms.

Figure 4 shows the NRMSE from the reconstruction for each voxel in the diffusion phantom. MAP-CS has a lower NRMSE when compared to MAP.

Figure 5 shows the PC from the reconstruction for each voxel in the diffusion phantom. The MAP-CS proposal has a higher PC when compared to MAP.

Conclusion

Given the sparse representation of the weights in the dictionary, we propose a CS-based MAP reconstruction. Quality indexes indicate that the proposed method has a comparable reconstruction, if not better, with respect to the least-squares MAP reconstruction with half the q-space sampling. As a major advantage of this method, the CS-based reconstruction increases the length of the MAP dictionary to improve reconstruction quality; additionally, a larger dictionary compensates the scaling of $$$ \mathbf{A} $$$ when it is not appropriate enough to describe the underlying microstructure.

Acknowledgements

References

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