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SUMMARY:Wavelets and Differential Operators: From Fractals to Marr's Prima
 l Sketch - Michael Unser (EPFL Lausanne)
DTSTART:20100114T150000Z
DTEND:20100114T160000Z
UID:TALK22342@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:Invariance is an attractive principle for specifying image pro
 cessing algorithms. In this presentation\, we promote affine invariance—
 more precisely\, invariance with respect to translation\, scaling and rota
 tion. As starting point\, we identify the corresponding class of invariant
  2D operators: these are combinations of the (fractional) Laplacian and th
 e complex gradient (or Wirtinger operator). We then specify some correspon
 ding differential equation and show that the solution in the real-valued c
 ase is either a fractional Brownian field (Mandelbrot and Van Ness\, 1968)
  or a polyharmonic spline (Duchon\, 1976)\, depending on the nature of the
  system input (driving term): stochastic (white noise) or deterministic (s
 tream of Dirac impulses). The affine invariance of the operator has two im
 portant consequences: (1) the statistical self-similarity of the fractiona
 l Brownian field\, and (2) the fact that the polyharmonic splines specify 
 a multiresolution analysis of L_2(ℝ^2) and lend themselves to the constr
 uction of wavelet bases. The other fundamental implication is that the cor
 responding wavelets behave like multi-scale versions of the operator from 
 which they are derived\; this makes them ideally suited for the analysis o
 f multidimensional signals with fractal characteristics (whitening propert
 y of the fractional Laplacian).\n\nThe complex extension of the approach y
 ields a new complex wavelet basis that replicates the behavior of the Lapl
 ace-gradient operator and is therefore adapted to edge detection. We intro
 duce the Marr wavelet pyramid which corresponds to a slightly redundant ve
 rsion of this transform with a Gaussian-like smoothing kernel that has bee
 n optimized for better steerability. We demonstrate that this multiresolut
 ion representation is well suited for a variety of image-processing tasks.
  In particular\, we use it to derive a primal wavelet sketch—a compact d
 escription of the image by a multiscale\, subsampled edge map—and provid
 e a corresponding iterative reconstruction algorithm.\n
LOCATION:MR14\, CMS
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