Theoretical Bounds on Parallel Imaging Implicit Data Crimes in an MRI Reproducing Kernel Hilbert Space

TL;DR

This work proposes a mathematical framework to re-characterize the implicit data crimes problem as one of error reduction during interpolation between sets of evaluation coordinates, and establishes a generalized matrix-based definition of the reconstruction error upper bound as a function of the input sampling pattern.

Abstract

Magnetic Resonance Imaging (MRI) diagnoses and manages a wide range of diseases, yet long scan times drive high costs and limit accessibility. AI methods have demonstrated substantial potential for reducing scan times, but despite rapid progress, clinical translation of AI often fails. One particular class of failure modes, referred to as implicit data crimes, are a result of hidden biases introduced when MRI datasets incompletely model the MRI physics of the acquisition. Previous work identified data crimes resulting from algorithmic completion of k-space with parallel imaging and drew on simulation to demonstrate the resulting downstream biases. This work proposes a mathematical framework to re-characterize the problem as one of error reduction during interpolation between sets of evaluation coordinates. We establish a generalized matrix-based definition of the reconstruction error upper bound as a function of the input sampling pattern. Experiments on relevant sampling pattern structures demonstrate the relevance of the framework and suggest future directions for analysis of data crimes.

Figures (5)

• Figure 1: Conceptual framework for the implicit data crime: Prospective sub-sampling occurs in clinical scanner settings. PI fills in missing k-space data before it is presented as a "fully sampled" dataset. Retrospectively sub-sampling the algorithmically completed data and comparing it to the PI reconstructed data leads to biased performance evaluation.
• Figure 2: (a) Simulated examples of PI completed k-space and (b) an example k-space completed by PI on a public dataset. Stripes are present in the log-magnitude k-space images because some points have been algorithmically synthesized.
• Figure 3: Theoretical and computational framework. (a) $T(S_{\mathrm{pro}})$ and $T(S_{\mathrm{retro}})$ are discrete sampling operators, while elements of the RKHS $H$ are continuous valued, such that $\mathbf{y}_{\mathrm{pro}} = \{f_{j_p}({\mathbf x}_p) | (j_p,{\mathbf x}_p) \in S_{\mathrm{pro}} \}$ and $\mathbf{y}_{\mathrm{retro}} = \{\hat{f}_{j_r}({\mathbf x}_r) | (j_r,{\mathbf x}_r) \in S_{\mathrm{retro}} \}$. (b) The PI data crime is then interpolation between three sets of coordinates. Interpolation weights for the experiment's error are the difference in weights for two reconstruction paths, one passing through $S_{\mathrm{retro}}$ and one skipping it.
• Figure 4: Experiment error and power functions for 2D sampling patterns. Our data-independent power function predicts k-space structure of experiment error, and its magnitude is suppressed due to the introduction of a prospective pattern.
• Figure 5: Direct interpolation of retrospective experiment error and power function calculation for 3D sampling patterns.