Singular integrals, image smoothness,
and the recovery of texture in image deblurring
Alfred Carasso
Mathematical and Computational Sciences Division
Tuesday, December 16, 2003 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Tuesday, December 16, 2003 13:00-14:00,
Room 4550
Boulder
Abstract:
Total variation (TV) image deblurring is a PDE-based technique that
preserves edges,
but often eliminates vital small-scale information, or texture.
This
phenomenon reflects the fact that most natural images are not of bounded
variation. The present paper reconsiders the image deblurring problem
in Lipschitz
spaces wherein a wide class of non-smooth
images can
be accomodated. A fast FFT-based deblurring method is developed that can
recover
texture in cases where TV deblurring fails completely. Singular
integrals,
such as the Poisson kernel, are used to create an effective new image
analysis tool that can
calibrate the lack of smoothness in an image.
The Poisson kernel is then used to regularize the deblurring problem by
appropriately
constraining its solutions
leading to new
bounds that substantially improve on the Tikhonov-Miller method. This
new so-called
Poisson Singular Integral or PSI method is only one of an
infinite variety of singular
integral deblurring methods that can be constructed. The method is found
to be well-behaved in
both the L1 and L2 norms, producing results closely matching those
obtained in the theoretically
optimal, but practically unrealizable, case of true Wiener filtering.
Deblurring experiments on
synthetically defocused images are used to illustrate the PSI method's
very significant improvements
over both the total variation and Tikhonov-Miller methods. In addition,
successful reconstructions with
inexact Lipschitz data highlight the robustness and practicality
of the PSI method.
Contact: A. J. KearsleyNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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