Last time I gave an start for **Radon transformation**.

where refers to the projection of , acquired as a function of , the distance along the projection, and , the rotation angle. The simplest projection is a collection of parallel ray integrals as given by for a constant .

3. **Helmholtz’s decomposition**

**Helmholtz’s Theorem**: Let be a vector field on a bounded domain *V* in which is twice continuously differentiable. Then can be decomposed into a curl-free component and a divergence-free component.

- More precisely, we can assume that , when . And this is unique for the choices of 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*, . To obtain the information on the model parameters, measurements of some observable parameters are needed. The experimental measurements are called the* data*, . In order to compute the model parameters, first the **forward problem** is defined by devising a mapping :

while the inverse mapping is:

5. **Method**

- Linearization
- Discretization

6.**Inversion Method**

(*to be continued and completed*)

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## Published by YiMin

Traveller, in math.
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