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CATEGORIES:Applied and Computational Analysis
SUMMARY:Wavelets and Differential Operators: From Fractals
to Marr's Primal Sketch - Michael Unser (EPFL Lau
sanne)
DTSTART;TZID=Europe/London:20100114T150000
DTEND;TZID=Europe/London:20100114T160000
UID:TALK22342AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/22342
DESCRIPTION:Invariance is an attractive principle for specifyi
ng image processing algorithms. In this presentati
on\, we promote affine invariance—more precisely\,
invariance with respect to translation\, scaling
and rotation. As starting point\, we identify the
corresponding class of invariant 2D operators: the
se are combinations of the (fractional) Laplacian
and the complex gradient (or Wirtinger operator).
We then specify some corresponding differential eq
uation and show that the solution in the real-valu
ed case is either a fractional Brownian field (Man
delbrot and Van Ness\, 1968) or a polyharmonic spl
ine (Duchon\, 1976)\, depending on the nature of t
he system input (driving term): stochastic (white
noise) or deterministic (stream of Dirac impulses)
. The affine invariance of the operator has two im
portant consequences: (1) the statistical self-sim
ilarity of the fractional Brownian field\, and (2)
the fact that the polyharmonic splines specify a
multiresolution analysis of L_2(ℝ^2) and lend them
selves to the construction of wavelet bases. The o
ther fundamental implication is that the correspon
ding wavelets behave like multi-scale versions of
the operator from which they are derived\; this ma
kes them ideally suited for the analysis of multid
imensional signals with fractal characteristics (w
hitening property of the fractional Laplacian).\n\
nThe complex extension of the approach yields a ne
w complex wavelet basis that replicates the behavi
or of the Laplace-gradient operator and is therefo
re adapted to edge detection. We introduce the Mar
r wavelet pyramid which corresponds to a slightly
redundant version of this transform with a Gaussia
n-like smoothing kernel that has been optimized fo
r better steerability. We demonstrate that this mu
ltiresolution representation is well suited for a
variety of image-processing tasks. In particular\,
we use it to derive a primal wavelet sketch—a com
pact description of the image by a multiscale\, su
bsampled edge map—and provide a corresponding iter
ative reconstruction algorithm.\n
LOCATION:MR14\, CMS
CONTACT:
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