Integral (126) describes the situation in the linear tomography when rays can be parametricized as a function of and
(129) |
So our task is to find such an operator , that
Let us introduce a dual IG-operator which has the family of dual lines 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 gives geometrical reconstruction of the original discontinuity of the field u0(x,y) since this operator realizes the mapping .It is clear that the operator transforms the dynamics of a discontinuity of the field in the same way as the operator did. Then, according to formula (126)(130) |
We have taken into account that
Now, combining formulae (128) and (130), we have: whereq''=q+1, | (131) |
The operator acts like and the operator acts like . This means that in order to reconstruct the order and the index of discontinuities we shall apply the correction filter . It can be shown that independently on the curve
(132) |
The kernal w' must be chosen from the condition
I have proven thatdd'=FG
where Consequently,(133) |
(134) |
(135) |
Figures - give an illustration of reconstruction of the step-function
given at the square (equation (126)). Figure a shows the field . Figure b is a result of application of the simplest operator at . Figure c is a result of application of the operator with w' chosen according to equation(135), And Figure d gives the fieldIt is interesting that Kirchhoff's operators of the forward and reverse wave-field continuation originate a symmetrical pair of asymptotically inverse operators and with
In the n-dimensional case we have the following analog of formula (134)
(136) |
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.