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                Department of Mathematical Sciences
                   The Johns Hopkins University

                             SEMINAR

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Thomas Yu                                              March 28, 2002
Department of Mathematical Sciences                    304 Whitehall Hall 
Rennselaer Polytechnic Institute                       Refreshments: 3:30 PM 
                                                       Seminar: 4:00 PM 

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    ROBUST NONLINEAR PYRAMID TRANSFORMS AND NONLINEAR SUBDIVISION 
              SCHEMES BASED ON MEDIAN-INTERPOLATION


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                               ABSTRACT

We introduce a nonlinear refinement subdivision scheme based on 
median-interpolation.  The scheme constructs a polynomial interpolating
adjacent block medians of an underlying object. The interpolating
polynomial is then used to impute block medians at the next finer
triadic scale. In general the computation of such a refinement operator
(which is shown to exist using topological arguments) requires a skillful
use of numerical methods.  Perhaps surprisingly, expressions for the
refinement operator can be obtained in closed-form for the scheme 
interpolating by polynomials of degree $D=2$.  Despite the nonlinearity 
of this scheme, convergence and regularity can be established using 
techniques reminiscent of those developed in analysis of linear 
refinement schemes.

The refinement scheme can be deployed in multiresolution fashion to
construct a nonlinear pyramid and an associated forward and inverse
transform.  We discuss the basic properties of these transforms and 
their possible use in removing badly non-Gaussian noise.  Analytic 
and computational results are presented to show that in the presence 
of highly non-Gaussian noise, the coefficients of the nonlinear 
transform have much better properties than traditional wavelet 
coefficients.

We shall also discuss some of the open problems from this project, as 
well as our recent attempts in attacking them.

This is Joint work with David Donoho and Tim Goodman.