Three-dimensional vector field inversion formula using first moment transverse transform in quaternionic approaches

We present the reconstruction formula for the solenoidal part of a vector field compactly supported in a unit ball of R3 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.