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## First order derivative regularization

A way of forcing a function to be piecewise constant is forcing its first order derivatives to be sparse. Thus, using a first order derivative regularization operator and forcing the model residuals to follow a Cauchy distribution should make the letters blocky'' and preserve the letter edges Youzwishen (2001). Obtaining model residuals following a Cauchy distribution can be achieve posing the inverse problem as Iterative Reweighed Least Squares (IRLS) Darche (1989).

The Cauchy norm first order derivative edge-preserving regularization fitting goal was set following the nonlinear iterations: starting with , at the kth iteration the algorithm solves
 (1)
where
 (2)
is a non-stationary convolution matrix, is the result of the kth nonlinear iteration, and are the (k-1)th diagonal weighting operators, and and are the first order derivative operator in the x and y directions, is the identity matrix, the scalar is the trade-off parameter controlling the discontinuities in the solution, and the scalar balances the relative importance of the data and model residuals.

comp_images_2d
Figure 4
A) Original image, B) Deblurred image using LS with the first order derivative edge-preserving regularization

comp_graph_2d
Figure 5
Comparison between Figures A and B; A) Slice y=229 and B) Slice x=229.

We were successful in obtaining what we designed the algorithm to produce. The result is blocky in the x and y directions (Figures  and ). However, the derivatives in the x and y directions do not produce an isotropic result. We know that letters often have round shapes. Thus, the problem could benefit from using a more isotropic operator to calculate the diagonal weights.

Next: Gradient magnitude and Laplacian Up: Regularization Schemes Previous: Regularization Schemes
Stanford Exploration Project
10/14/2003