Sinusoidal Sensitivity Calculation for Line Segment Geometries

Purpose: Provide a closed-form solution to the sinusoidal coil sensitivity model proposed by Kern et al. This closed-form allows for precise computations of varied, simulated bias fields for ground-truth debias datasets. Methods: Fourier distribution theory and standard integration techniques were used to calculate the Fourier transform for line segment magnetic fields. 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 also discussed. Lastly a CUDA accelerated implementation $\texttt{biasgen}$ is provided by authors. Conclusion: As the derived result is influenced by coil positioning and geometry, practitioners will have access to a more diverse ecosystem of simulated datasets which may be used to compare prospective debiasing methods.