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Unique tomographic reconstruction of vector fields using boundary data.

S J Norton1

  • 1Nat. Inst. of Stand. and Technol., Gaithersburg, MD.

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|January 1, 1992
PubMed
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Reconstructing vector fields from line integrals is challenging. However, the solenoidal component is uniquely determined, aiding in applications like fluid velocity field analysis using vector tomography.

Area of Science:

  • Physics
  • Applied Mathematics
  • Fluid Dynamics

Background:

  • Reconstructing a vector field v(r) from line integrals is generally an undetermined problem.
  • Vector fields are defined by two component functions, leading to ambiguity in reconstruction.

Purpose of the Study:

  • To demonstrate that the solenoidal component of a vector field is uniquely determined by its line integrals.
  • To present a method for calculating the scalar potential of a divergenceless vector field.
  • To explore applications in fluid velocity field reconstruction.

Main Methods:

  • Decomposition of the vector field into irrotational and solenoidal components.
  • Utilizing the Fourier domain and a vector analog of the projection slice theorem.
  • Deriving a formula for the scalar potential using boundary conditions.

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Main Results:

  • The solenoidal part of the vector field v(r) is uniquely determined by its line integrals.
  • A formula for the scalar potential phi(r) is derived for divergenceless fields.
  • The vector tomography method is applicable to fluid velocity measurements.

Conclusions:

  • Unique determination of the solenoidal component simplifies vector field reconstruction.
  • The Fourier domain approach offers an efficient method for analysis.
  • Vector tomography provides a powerful tool for fluid dynamics research.