Nat. Inst. of Stand. and Technol., Gaithersburg, MD.
IEEE Trans Image Process. 1992;1(3):406-12. doi: 10.1109/83.148612.
The problem of reconstructing a vector field v(r) from its line integrals (through some domain D) is generally undetermined since v(r) is defined by two component functions. When v(r) is decomposed into its irrotational and solenoidal components, it is shown that the solenoidal part is uniquely determined by the line integrals of v(r). This is demonstrated in a particularly simple manner in the Fourier domain using a vector analog of the well-known projection slice theorem. In addition, under the constraint that v (r) is divergenceless in D, a formula for the scalar potential phi(r) is given in terms of the normal component of v(r) on the boundary D. An important application of vector tomography, i.e., a fluid velocity field from reciprocal acoustic travel time measurements or Doppler backscattering measurements, is considered.
从其线积分（通过某个域 D）重建向量场 v(r) 的问题通常是不确定的，因为 v(r) 由两个分量函数定义。当 v(r) 被分解为无旋和有旋分量时，可以证明有旋部分是由 v(r) 的线积分唯一确定的。这在傅里叶域中通过对著名的投影切片定理的向量类比以特别简单的方式证明。此外，在约束条件下 v(r) 在 D 中无散度，可以给出一个关于边界 D 上 v(r) 的法向分量的标量势 φ(r) 的公式。向量层析成像的一个重要应用，即从互易声传播时间测量或多普勒反向散射测量中获得的流体速度场，被考虑在内。
