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

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.