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In this short note, we test various edge-preserving regularization schemes in
the context of deblurring a text image with random noise. The blurred text
image was created by Nagy and O'Leary (2003a) as a test case. Even if the blurring
filter is known exactly, as it is in this case, sharp features are nearly in the
nullspace of the filter which we must ``invert'', or deblur. Those eigenvalues of
the filter matrix corresponding to edges may be well below the noise level, and
thus difficult or impossible to resolve.
We know that letters should be homogeneous for intervals (piecewise constant), thus it makes sense to impose model smoothness using a regularization operator. But letters also have abrupt discontinuities, thus using a regularization operator that imposes model smoothness considerably degrades our ability to discern the letter edges.
We present three separate strategies which allow the model regularization operator to preserve edges in the
deblurring process. All are implemented as regularization operators in an
unconstrained least-squares deblurring problem.