We present the reconstruction formula for the solenoidal part of a vector field compactly supported in a unit ball of ${\mathbb{R}}^3$ by using a new type of transform called the first integral moment transverse transform. In contrast to the first integral moment transform, this transform is an integral over line components perpendicular to the rays weighted according to the length of the rays. The solenoidal part is set by the quaternionic inversion formula and consequently decomposed into two parts by using techniques where the tangential and normal parts of its Radon transform are applied. The processes to obtain the inversion formula utilize the techniques of Katsevich and Schuster's work. The main differences from this work are the approach to achieve the first integral moment transverse transforms and a more compact inversion formula in a quaternion perspective.

中文翻译：

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四元数法中采用矩矩横向变换的三维矢量场反演公式

我们给出了紧凑地支撑在一个单位球的矢量场的螺线管部分的重建公式。 ${\mathbb{[R}}^3$通过使用一种称为第一积分矩横向变换的新型变换。与第一积分矩变换相反，该变换是垂直于根据光线长度加权的光线的线分量的积分。螺线管部分由四元离子反演公式设置，因此通过使用应用了Radon变换的切线和法线部分的技术将其分解为两部分。获得反演公式的过程利用了Katsevich和Schuster的技术。这项工作的主要区别在于在四元数方面实现第一积分矩横向变换的方法和更紧凑的反演公式。