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In the previous section we changed the norm of the minimization problem to prevent the roughener from smoothing over the edges of the model. In this subsection we shift from a statistical to a more mechanical approach to attain the same goal.
The gradient magnitude () is an isotropic edgedetection operator that can be used to calculate the diagonal weights. Unfortunately, it is a nonlinear operator thus couldn't be used for regularization. Instead we used the Laplacian, which is a regularization operator used in several SEP applications Claerbout and Fomel (2001).
The gradient magnitude edgepreserving regularization fitting goal was set
following the nonlinear iterations:
starting with , at the k^{th} iteration the algorithm solves
 

 (3) 
where
 
(4) 
is a nonstationary convolution matrix, is the result of the k^{th} nonlinear iteration, is the (k1)^{th} diagonal weighting operators, is the identity matrix, is the gradient magnitude, is the Laplacian operator, the scalar is the tradeoff parameter controlling the discontinuities in the solution,
and the scalar balances the relative importance of the data and model residuals.
comp_images_laplac_2d
Figure 6 A) Original image, B) Deblurred image using LS with the gradient magnitude edgepreserving regularization () with laplacian edgepreserving regularization
comp_graph_2d_laplac
Figure 7 Comparison between Figures A and B; A) Slice y=229 and B) Slice x=229.
Figures and show a considerable improvement over Figures and . They are noise free but keep the round features of the original image. However, since we are not imposing blockiness on the model but rather on the derivative of the model (using the Laplacian as the regularization operator), the edges are not as sharp as the previous case.
Next: Gradient magnitude and smooth
Up: Regularization Schemes
Previous: First order derivative regularization
Stanford Exploration Project
10/14/2003