Article Title

Funding Information This work was partially supported by NIBIB of the National Institutes of Health; Award number: R21EB026086 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 Lloc(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 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.


INTRODUCTION
The success of parallel imaging methods in magnetic resonance (MR) imaging have allowed for quicker image acquisition with little or no cost to spatial aliasing 1 . Image reconstruction quality is dependent on how closely the gains of different radiofrequency (RF) coil contributions can be approximated. These gain normalization maps, also known as coil sensitivity maps or sensitivity maps for short, can be difficult to estimate at scan time. When incorrectly normalized, the different coil contributions can form a series of spatial inhomogeneities known bias fields in the final reconstructed image.
Fortunately, there is a rich literature of image postprocessing techniques which aim to amend the MR bias field problem 2,3 . These techniques may assume a smooth prior on the generated bias field or may choose a supervised approach to correcting bias fileds 4 . Currently in the bias field literature, there is a limitation in the diversity of simulated datasets   which may be used for debias benchmarking. Furthermore these simulated datasets are generally limited in the bias fields they produce, opting instead for a single bias functional form with varied tissue contrasts and noise settings.
One example of this trend can be seen in the popular BrainWeb dataset of Ref. [5]. This dataset contains groundtruth MR intensities of a known head phantom while additionally providing a finite selection of multiplicative nonuniformity intensities (INU) and additive measurement noises which may be added on top of the ground-truth intensities. It should be noted the bias fields generated by this dataset can be fairly smooth, even when applied with the extreme INU setting. To add on, the INU parametrization of BrainWeb controls the overall scale of the bias intensity not its functional form. Figure 1 showcases an example of the difference in bias fields between a 40% INU BrainWeb slice and a real patient thorax MR image slice. Although the anatomy and coil cages are not equivalent in the example provided, Figure 1 still highlights the need to have varied, simulated bias fields when developing fully general debiasing algorithms.
In an ideal setting, our debias testing environment should contain a wide range of physically-viable bias fields with empirically supported functional forms. One prospective model is mentiond in Ref. [6] states that observed sensitivity maps may be approximated by sparse Fourier representations of the initial emitted magnetic field. This is a model which is already applied by other popular medical software such as the Berkeley Advanced Reconstruction Toolbox 7 (BART). A current hang-up of the method is that computing the full required Fourier transform may be too computationally intensive or not numerically sensitive enough when using computational approximations such as the fast Fourier transform. Practical implementations of the model, like the one done by BART, use a small selection of dominant frequencies from an empirically verified source and then apply the fixed sensitivity maps to different phantoms and tissue contrasts.
The goal of this paper will be to provide a closed-form to sinusoidal sensitivity model of Ref. [6] for a restricted but expressive class of magnetic fields. In particular we are interested in the sensitivity maps produced by magnetic fields generated from segmented line geometries. The equation of focus will be where ⊂ ℝ 3 is a finite grid of points and is the measured magnetic field of source ( ) according to some readout direction ∈ ℝ 3 and phase encoding direction ∈ ℝ 3 . Notice as the sensitivity (1) and the measurement (2) are linear in emmited magnetic field and the field itself follows the superposition property, it will be sufficient for our purposes to calculate the Fourier transform for a single line segment magnetic field.

LINE SEGMENT MAGNETIC FIELD
The contribution of an infinitesimal line sement can be calculated using Biot-Savart law, Assuming a constant current , the total magnetic field is proportional to for some line segment . Any simple line segment can be parameterized as A change of variables revealŝ With shorthands = 1 − 0 and = − 0 we have This integral has the following solution With the additional shortand = − , Using identity || × || 2 2 = || || 2 2 || || 2 2 − ( • ) 2 , we arrive at the final symmetric form .
Calculating (1) simplifies to calculating the Fourier transform for functions of the form .
When integrating we will consider the change of variable where is an orthonormal rotation matrix. We are interested in rotating to the orthogonal basis with coordinate representations ′ = (0, 0, || || 2 ) and ′ = ( ′ , 0, ′ ). In particular we align will ′ with the (+ )-axis and place ′ in the (+ , )-halfspace. For this reason we will assume ′ > 0 throughout without loss of generality. Additionally we note in the case and are collinear, the Fourier transform ℱ { }( ) = 0 regardless of choice of basis. This change of variables produces the following simplificaitons where ⊺ = (0, 0, || || 2 ).
As the choice of basis  is independent to input , the ′ transformation can be adapted to ( 1 ) with ′′ = ⊺ ( − 1 ). Next defining = ( 0 ) − ( 1 ) we can combine relevant integrations as so where equality follows from ′ • ⊺ ( 1 − 0 ) = ′ || || 2 . Now we begin the task of calculating the Fourier transform of . We will suppress all prime notation as it is understood that integration will be done in the rotated coordinates. Although we see now the integrand does not lie in 1 (ℝ 3 ), the Fourier transform of does still exist in a distributional sense.
Upon confirmation that lim →0 + ℱ { } is also a tempered function of polynomial growth, we will have that ( lim →0 + ℱ { }) is absolutely integrable. The immediate consequence from Lebesgue dominated convergence theorem is Since ∈ 1 (ℝ 3 ), we may directly evaluate lim →0 + ℱ { } while maintaining equality to ℱ { } and circumventing the need to rely on identity (3).
With this in mind, we suppress the limit notation of our left-hand side equalties and carry-on with the integration Here a partial fraction decomposition was used on the integration-by-parts integral.Written in terms of , Summed together with equality 2 = || || 2 2 − 2 , Lastly expanding in terms of the original, canonical coordinates Note that as we confirm our earlier claim that the pointwise limit of ℱ { } is a tempered function of polynomial growth in ℝ 3 . When combined with the measurement contribution, we obtain the final closed-form solution to the sinusoidal sensitivity model. Examples of simulated sensitivities and bias fields for different grids can be found in Appendix B.

SAMPLING WITH THE SINUSOIDAL SENSITIVITY MODEL
Consider the sparse Fourier sampler where (⋅ − ) is the Dirac delta distribution centered at . Understood formally, the sinusoidal sensitivity model can be expressed in terms of this sampler as The issue in this sampler is that it is only well-defined for functions which have meaningful point evalutions. The framework used in section 3 worked with class ℱ {̄ } ∈ 1 loc (ℝ 3 ) whose behavior is only specified with respect to an integrating action against the Lebesgue measure on ℝ 3 .
In hopes to extend the sinusoidal model to work better with the equivalence class ℱ {̄ } we may consider the generalized sampler where ( ; ) = 1{|| − || 2 < } is the -cutoff function centered at and vol( ( , )) is the volume of the -radius sphere centered at . We see then for > 0, the generalized sinusoidal model does produce a well-defined result. Ideally we would like limit lim →0 + to be equivalent to our original sinusoidal sensitivity model. Before continuing, we introduce the following term for notational clarity. With = + i consider which lies in the space of continuous functions. That is have ( ) ∈ (ℝ 3 ⧵ {0}) and −i • ℱ {̄ }( ) ∈ 1 loc (ℝ 3 ). One can show that these objects satisfy the limit equality To see this, note that for continuity point we have for almost every less than a suitable radius from . With shorthand ⟨ , ⟩ = ∫ ( ) ( ) and -centered indicator As a consequence the equality is well-defined with = ℱ −1 { } and any grid sampler which does not contain the origin.

IMPLEMENTATION
A CUDA accelerated implementation biasgen can be found on GitHub * . This Python package takes in user-defined RF coil positions and sampling information to produce custom 3-dimensional bias fields. A coil positioning and visualization tool is provided for setup as well as various examples to help the user get started. As the bias generation is done independently from image intensity, biasgen can be used both to produce simulated dataset and augment existing ones. An application of biasgen on the BrainWeb phantom dataset can be found in Appendix B. * https://github.com/lucianoAvinas/biasgen

CONCLUSIONS
In this paper, we have derived the Fourier transform of measured magnetic field emitted by line segment geometries. This closed-form can be used to evaluate the sinusoidal sensitivity model to arbitrary accuracy and function smoothness. Special care was taken to discuss the distributional nature of the solved Fourier transform and settings were identified where this closed-form agreed with the sparse sampled model introduced by Ref. [6]. As next steps, further work can be done to solve (1) for smooth line curves, such as for the case of circular or cylindrical geometries.

ACKNOWLEDGMENTS
Research reported in this manuscript was partially supported by the NIBIB of the National Institutes of Health under award number R21EB026086. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
How to cite this article: Vinas L, and Sudyadhom A (2022), Sinusoidal Sensitivity Calculation for Segment Geometries, Magn Reson Med., .