Chin-Cheng Chan^{1} and Justin P. Haldar^{1}

^{1}Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, United States

As MR image reconstruction algorithms become increasingly nonlinear, data-driven, and difficult to understand intuitively, it becomes more important that tools are available to assess the confidence that users should have about image reconstruction results. In this work, we suggest that a quantity known as the “local perturbation response” (LPR) provides useful information that can be used for this purpose. The LPR is analogous to a conventional point-spread function, but is well-suited to general image reconstruction methods that may have nonlinear and/or shift-varying characteristics. We illustrate the LPR in the context of several common image reconstruction techniques.

In this work, we suggest to use the local perturbation response (LPR), originally proposed to characterize spatial resolution in regularized tomography

Instead, the LPR attempts to assess image quality by applying a small perturbation to the image, and then observing how well that small perturbation is reconstructed by the image reconstruction method. Mathematically, if $$$\delta\mathbf{x}$$$ represents the small perturbation to the image in the image domain, the LPR is calculated by generating perturbed k-space data $$$\tilde{\mathbf{d}} = \mathbf{d}+\mathbf{E}\delta\mathbf{x}$$$, reconstructing the perturbed data using $$$f(\cdot)$$$, and comparing the result against the reconstruction obtained without a perturbation: $$\mathrm{LPR}=f(\mathbf{d}+\mathbf{E}\delta\mathbf{x})-f(\mathbf{d}).$$ This procedure is illustrated in Fig. 1. Importantly, this procedure can be implemented for arbitrary undersampled datasets, and does not require knowledge of any fully-sampled datasets. Ideally, if the reconstruction algorithm does a good job, the LPR will be a faithful reconstruction of the original perturbation. On the other hand, a substantial discrepancy between the LPR and the original perturbation would be a reason to doubt the results produced by the reconstruction method.

Fig. 4 shows the results of computing LPRs of the same perturbation at different spatial locations. We can observe from this result that the LPRs are spatially-varying for all four methods, although spatial variation is most extreme for the U-Net and Sparse-MRI methods, while GRAPPA and AC-LORAKS have the least amount of spatial variation.

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Figure 1. Schematic illustration of the LPR
computation process.

Figure 2. (a) The top and the middle rows show images obtained
with and without a Gaussian perturbation, respectively. Results are shown both for fully-sampled data
and for undersampled data reconstructed with different reconstruction methods. The
bottom row shows the original perturbation and the LPRs obtained with different
reconstruction methods. (b) A 1D plot depicting the variation of the LPRs along
a horizontal line passing through the center of the perturbation. Each curve is normalized to have value 1 at
the center of the perturbation.

Figure 3. The top and the middle rows show images with and
without a striped resolution test-pattern perturbation, respectively. Results
are shown both for fully-sampled data and for undersampled data reconstructed with
different reconstruction methods. The bottom row shows the original perturbation
and the LPRs obtained with different reconstruction methods.

Figure 4. LPRs computed at different spatial locations to
characterize the degree to which different reconstruction methods have
shift-varying characteristics. (a)
Illustration of four different spatial positions where Gaussian blob
perturbations were added. (b) LPRs for different reconstruction methods at each
of the spatial locations from (a).