Tomography Note 2

Last time I gave an start for Radon transformation.

where \mathcal{R}m(r, \phi) refers to the projection of m(x, y), acquired as a function of r, the distance along the projection, and \phi, the rotation angle. The simplest projection is a collection of parallel ray integrals as given by \mathcal{R}m(r, \phi) for a constant \phi.

3. Helmholtz’s decomposition

  • Helmholtz’s Theorem: Let \boldmath{F} be a vector field on a bounded domain V in \mathbb{R}^3 which is twice continuously differentiable. Then \boldmath{F} can be decomposed into a curl-free component and a divergence-free component.

\boldmath{F}=-\nabla \phi+\nabla\times A

  • More precisely, we can assume that \phi(x),A(x)\rightarrow 0, when |x|\rightarrow \infty. And this is unique for the choices of \phi, A when assuming the behaviors of them at infinity.

4. Forward problem and Inverse problem

  • Acoustic tomography is a type of inverse problem: specifically, the information brought by sound propagation through the field of interest is used to infer the local properties of the field. In the usual terminology of inverse problems, the set of parameters to be determined, which describes the state of the field, is called the model, \mathcal{M}. To obtain the information on the model parameters, measurements of some observable parameters are needed. The experimental measurements are called the data, \mathcal{D}. In order to compute the model parameters, first the forward problem is defined by devising a mapping \mathcal{G}:


while the inverse mapping is:


5. Method

  • Linearization
  • Discretization

6.Inversion Method

(to be continued and completed)




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