Sinusoidal Sensitivity Calculation for Line Segment Geometries

TL;DR

This paper derives a closed-form solution for sinusoidal coil sensitivity modeling of line segment geometries, enabling precise bias field simulations for MRI data correction.

Contribution

It introduces a general analytical formula for magnetic fields from line segments and provides a CUDA implementation for efficient bias field generation.

Findings

Derived a full generality Fourier transform of line segment magnetic fields.

Discussed sampling criteria and model equivalence.

Provided an accelerated implementation for practical use.

Abstract

Purpose: Provide a closed-form solution to the sinusoidal coil sensitivity model proposed by Kern et al. Solution allows for the precise computations of varied, simulated bias fields which can be directly applied onto raw intensity datasets. Methods: Fourier distribution theory and standard integration techniques were used to calculate the Fourier transform of measured magnetic field produced from line segment sources. Results: A $L^1_{\rm loc}(\mathbb{R}^3)$ function is derived in full generality for arbitrary line segment geometries. Sampling criteria and equivalence to the original sinusoidal model are discussed. Lastly a CUDA accelerated implementation $\texttt{biasgen}$ is provided for on-demand sensitivity and bias generation. Conclusion: Given the modeling flexibility of the simulated procedure, practitioners will now have access to a more diverse ecosystem of simulated datasets which may be used to quantitatively compare prospective debiasing methods.