Application to tomography

Integral (126) describes the situation in the linear tomography when rays can be parametricized as a function of $x:y=\th (x; \xi ,\eta )$ and  

$$w=(x; \xi ,\eta )=\sqrt{1+{\vert\th ^{'}_{x}\vert}^{2}}.$$ (129)

The general problem of tomography is the reconstruction of the field u₀(x,y) using the given field $u(\xi , \eta)$. We shall reduce this problem to a simpler one: to reconstruct the discontinuities of the field u₀(x,y). The possibility of this reduction follows from the fact that discontinuities are a model for the most vivid particularities of the object.

So our task is to find such an operator ${\bf P}^{(-)}$, that

$${\bf P}^{(-)}u(\xi ,\eta ) \stackrel{q}{\sim } u_{0} (x,t).$$

Let us introduce a dual IG-operator

$$u_{1}(x,y)=\hat{{\bf P}}_{w^{'}} u(\xi ,\eta ),$$

which has the family of dual lines $\{ {\hat {\th }}_{N}(\xi ) \}$ as a system of stacking lines and an arbitrary kernel w^'=w^'(M,N). It follows from the results of Chapter 12 that for any kernel w^', the operator $\hat{{\bf P}}_{w^{'}}$ gives geometrical reconstruction of the original discontinuity of the field u₀(x,y) since this operator realizes the mapping $\tau = K^{-1}\mu$.It is clear that the operator $\hat{{\bf P}}_{w^{'}}$transforms the dynamics of a discontinuity of the field $u(\xi , \eta)$ in the same way as the operator ${\bf P}_{w}$did. Then, according to formula (126)  

$$U_{1} \sim \hat{{\bf P}}_{w^{'}} u \sim \kappa^{'} {\left[ \... ...A^{'} \right] }_{\omega}R^{(+)}_{q^{''}, \nu ^{''}}[y-\tau (x)]$$ (130)

where $q^{''}=q^{'}+ {1 \over 2}, {\:}\nu^{''}=\nu^{'} - {\kappa^{'}-1 \over 2}$

$$d^{'}= { \left( {d^{2} \mu \over d\xi^{2}} - {d^{2}\hat{\th ... ...i^{2}} \right)}_{\xi =\xi^{\ast }}, {\:}\kappa^{'}=sgn(d^{'}).$$

We have taken into account that

$$\hat{\th }_{y}={1 \over \th_{y}} \gt o.$$

Now, combining formulae (128) and (130), we have:

$$u_{1}(x,y) \sim \kappa \kappa^{'} { \left[ \sqrt{{2\pi \over... ...0} \right] }_{{\omega}} R^{(+)}_{q^{''},\nu ^{''}} [y-\tau (x)]$$

where

 

q^''=q+1, (131)

$$\nu ^{''}=\nu - {\kappa -1 \over 2} - {\kappa^{'} -1 \over 2}.$$

The operator ${\bf P}_{w}$ acts like ${\bf D}^{- {1 \over 2}}_{(\kappa )}$and the operator $\hat{{\bf P}}_{w^{'}}$ acts like ${\bf D}^{- {1 \over 2}}_{(\kappa^{'})}$. This means that in order to reconstruct the order and the index of discontinuities we shall apply the correction filter ${\bf D}^{ {1 \over 2}}_{(\kappa )} {\bf D}^{- {1 \over 2}}_{(\kappa^{'})}$. It can be shown that independently on the curve $y=\tau (x)$ 

$$\kappa ^{'}=-\kappa,$$ (132)

consequently, the correction filter is ${\bf D}^{{1 \over 2}}_{(+)} {\bf D}^{{1 \over 2}}_{(-)}=\vert{\bf D}\vert \equiv {\bf D}{\bf H}$. It follows from equations (131) and (132) that

$$\nu ^{''}=\nu + 1 {\:}\rm{and} {\:}\kappa \kappa^{'}=-1.$$

But

$${\bf D}{\bf H}[-R_{q+1,\nu +1}] \sim -R_{q,\nu +2} \sim - {\bf{H}}^{2}R_{q,\nu } \sim R_{q,\nu}$$

since ${\bf{H}}^{2} =-{\bf E}$.

The kernal w^' must be chosen from the condition

$${2\pi ww^{'} \over \sqrt{\th _{\eta }\vert dd^{'}\vert}} =1.$$

I have proven that

dd^'=FG

where

$$F=\th_{x\xi }\th_{\eta } - \th_{x\eta }\th_{\xi }, {\:}G={\h... ...{x\xi } {\hat{\th }}_{y} - {\hat{\th }}_{y\xi}{\hat{\th }}_{x}.$$

Consequently,  

$$w^{'}={\sqrt{\th_{\eta }\vert FG\vert \over 2\pi w}}.$$ (133)

So we have shown that  

$${\bf P}^{(-)}={\hat{{\bf P}}}_{w^{'}}\vert{\bf D}\vert$$ (134)

where w^' is expressed by Formula (133).

For example, let  

$$\th = \eta + {a \over 2} {(\xi -x)}^{2},$$ (135)

then

$$\hat{\th }= \eta-{a \over 2}{(\xi -x)}^{2}, {\:}ww^{'}={a \over 2\pi }.$$

If equation (135) is an equation of rays in ray tomography, then according to equation (129)

$$w = \sqrt{1+a^{2}{(\xi -x)}^{2}}$$

and

$$w^{'}={a \over 2\pi \sqrt{1+a^{2}{(x-\xi )}^{2}}}.$$

Figures - give an illustration of reconstruction of the step-function

$$u_{0}(x,y)= H (y-{1 \over 2})$$

given at the square $-{1 \over 2} \leq x \leq {1 \over 2}), {\:}0 \leq y \leq 1$ (equation (126)). Figure a shows the field $u(\xi , \eta)$. Figure b is a result of application of the simplest operator $\hat{{\bf P}}_{w^{'}}$ at $w^{'} \equiv 1$. Figure c is a result of application of the operator $\hat{{\bf P}}_{w^{'}}$ with w^' chosen according to equation(135), And Figure d gives the field

$$u_{1}(x,y)= {\hat{{\bf P}}}_{w^{'}}\vert{\bf D}\vert u(\xi,\eta ).$$

It is interesting that Kirchhoff's operators of the forward and reverse wave-field continuation originate a symmetrical pair of asymptotically inverse operators ${\bf P}_{w}{\bf D}_{(+)}^{{1 \over 2}}$ and ${\hat{{\bf P}}}_{\hat{w}}{\bf D}^{{1 \over 2}}_{(-)}$ with

$$w= \sqrt{{\vert F\vert\over 2 \pi}} \equiv w_{E} {\:}{\rm and}{\:}\hat{w}= \sqrt{{\vert G\vert \over 2\pi }}=w_{E}.$$

In the n-dimensional case we have the following analog of formula (134)  

$${\bf P}^{(-)}={\hat{{\bf P}}}_{w^{'}}{\vert{\bf D}\vert}^{{m \over 2}}, {\:}m=n-1$$ (136)

where the kernel w^' is expressed by the same formula (133) but F and G are determinants of the correspondent matrices.

The formula (111) and (113) were published first by Gr. Beylkin (1984) who applied a very different approach connected with the theory of pseudodifferential operators.