%%
@article{fds200674,
Author = {W.K. Allard and Guanglian Chen and Mauro Maggioni},
Title = {W.K. Allard, G. Chen, M. Maggioni Multiscale Geometric
Methods for Data Sets II: Geometric Wavelets},
Journal = {to appear in ACHA},
Year = {2011},
Key = {fds200674}
}
@article{fds167778,
Author = {W.K. Allard},
Title = {A boundary approximation algorithm for planar
domains},
Year = {2009},
url = {http://www.math.duke.edu/~wka/bdry.pdf},
Key = {fds167778}
}
@article{fds243260,
Author = {Allard, WK},
Title = {Total variation for image denoising: III.
Examples},
Journal = {Siam Journal on Imaging Sciences},
Volume = {2},
Year = {2009},
url = {http://www.math.duke.edu/~wka/new.pdf},
Key = {fds243260}
}
@article{fds243261,
Author = {Allard, WK},
Title = {Total variation regularization for image denoising,
geometric theory},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {39},
Number = {4},
Pages = {11501190},
Year = {2007},
Month = {December},
ISSN = {00361410},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000253016600006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {Let Ω be an open subset of ℝn, where 2 < n < 7; we assume
n > 2 because the case n = 1 has been treated elsewhere (see
[S. S. Alliney, IEEE Trans. Signal Process., 40 (1992), pp.
15481562] and is quite different from the case n > 1; we
assume n ≤ 7 because we will make use of the regularity
theory for area minimizing hypersurfaces. Let ℱ(Ω.) = {f
∈ L1 (Ω)∩lL∞(Ω) : f ≥ 0} Suppose s ∈ ℱ(Ω) and
γ : ℝ → [0, ∞) is locally Lipschitzian, positive on
ℝ ∼ {0}, and zero at zero. Let F(f) = ∫Ω γ(f(x) 
s(x)) dℒn x for f ∈ ℱ(Ω); here ℒn is Lebesgue
measure on ℝn. Note that F(f) = 0 if and only if f(x)
s(x) for ℒn almost all x ∈ ℝn. In the denoising
literature F would be called a fidelity in that it measures
deviation from s, which could be a noisy grayscale image.
Let ε > 0 and let F ε(f) = εTV(f) + F(f) for f ∈
ℱ(Ω); here TV(f) is the total variation of f. A minimizer
of Fε is called a total variation regularization of s.
Rudin, Osher, and Fatemi and Chan and Esedoglu have studied
total variation regularizations where γ(y) = y 2 and γ(y)
= y, y ∈ ℝ, respectively. As these and other examples
show, the geometry of a total variation regularization is
quite sensitive to changes in γ. Let f be a total variation
regularization of s. The first main result of this paper is
that the reduced boundaries of the sets {f > y}, 0 < y <
∞, are embedded C1, μ hypersurfaces for any μ ∈ (0,1)
where n > 2 and any μ ∈ (0,1] where n = 2; moreover, the
generalized mean curvature of the sets {f ≥ y} will be
bounded in terms of y, ε and the magnitude of s near the
point in question. In fact, this result holds for a more
general class of fidelities than those described above. A
second result gives precise curvature information about the
reduced boundary of {f > y} in regions where s is smooth,
provided F is convex. This curvature information will allow
us to construct a number of interesting examples of total
variation regularizations in this and in a subsequent paper.
In addition, a number of other theorems about
regularizations are proved. © 2007 Society for Industrial
and Applied Mathematics.},
Doi = {10.1137/060662617},
Key = {fds243261}
}
@article{fds243262,
Author = {Allard, WK},
Title = {Total variation regularization for image denoising: I.
Geometric theory using total variation regularization; II
Examples.},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {39},
Number = {4},
Year = {2007},
Month = {November},
url = {http://www.math.duke.edu/~wka},
Abstract = {http://www.math.duke.edu/~wka},
Key = {fds243262}
}
@article{fds243263,
Author = {Allard, WK},
Title = {The reconstruction of surfaces in R3 by reflection},
Journal = {The Journal of Geometric Analysis},
Volume = {9},
Number = {5},
Year = {1999},
url = {http://www.math.duke.edu/faculty/allard/papers/allabst.ps},
Key = {fds243263}
}
@article{fds9254,
Author = {William K Allard},
Title = {An Introduction to the Deferred Execution
Tool},
Journal = {Proceedings of Ninth SIAM Annual Conference on Parallel
Processing for Scientific Computing, (SIAM) March
1999},
Key = {fds9254}
}
@article{fds9393,
Author = {William K. Allard and John Trangenstein},
Title = {On the Performance of a Distributed Object Oriented Adaptive
Mesh Refinement Code},
url = {http://www.math.duke.edu/~wka/papers/adaptive.ps},
Key = {fds9393}
}
@article{fds9392,
Author = {William K. Allard},
Title = {Users Guide to the Deferred Execution Tool},
url = {http://www.math.duke.edu/~wka/papers/deferred.ps},
Key = {fds9392}
}
