Bayesian 3-D Path Search and Its Applications to Focusing Seismic Data

Tomographic reconstruction from non-calibrated noisy projections in non-destructive evaluation

“…Otherwise, accurate reconstruction is not possible and it yields solutions which do not have the properties expected in the original cost function. In the field of inverse problems concerning reconstructions with small-range dependences (image restoration, image segmentation, optic flow, etc), parameter choice has been extensively studied [1,12,28,36] and several techniques are now available [2,8,16,23,43,44]. However, for tomographic reconstruction these techniques are less useful.…”

“…where C (1) is a 'fidelity term' between x and y derived from the projection model (1) and C (2) is a 'regularization term' whose parameters are θ 2 . λ 0 is a regularization parameter.…”

“…Note that simple prior distributions like the Gaussian prior (used here), the -function [15] or the generalized Gaussian prior [37] are sufficient for tomographic reconstruction, in contrast to other low-level vision tasks which often need sophisticated priors [2][3][4]. For instance, it would be possible to introduce a second prior in (9), weighted by a parameter λ 0 , in order to force the x v values to be close to 0 or 1.…”

“…However, when t varies, x = tu always represents the 'same configuration' up to a multiplicative constant. In other words, from two different configurations x(1) and x(2), we can arbitrarily make C 0 (x(1)) < C 0 (x( 2)) (or C 0 (x( 1)) > C 0 (x( 2))) by playing on the norms x(1) and x (2) . With the prior distribution P γ , we alleviate this drawback since x is concentrated in the vicinity of µ on S. It is now impossible to move along a straight line tu.…”

“…The choice of the regularization parameters remains the major drawback of the regularization approach, especially when there is more than one parameter. In our case, there is essentially one crucial parameter, that is λ 0 , which balances the weights of C (1) and C (2) . In this experiment, we chose λ 1 = • • • = λ 4 = (1 − λ 0 )/4 and we determined λ 0 experimentally.…”