Abstract: Standard wavelet linear approximations (truncating high frequency coefficients) generate oscillations (Gibbs' phenomenon) near singularities in piecewise smooth functions. Nonlinear and data dependent methods are often used to overcome this problem. Recently, partial differential equation (PDE) and variational techniques have been introduced into wavelet transforms for the same purpose.
In this talk, I will present our work on two different approaches that we have been working on in this direction. One is to use PDE ideas to directly change the standard wavelet transform algorithms so as to generate wavelet coefficients which can avoid oscillations in reconstructions when the high frequency coefficients are truncated. We have designed an adaptive ENO wavelet transform by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet transforms retains the essential properties and advantages of standard wavelet transforms without any edge artifacts. We have shown the stability and a rigorous error bound which depends only on the size of the derivative of the function away from the discontinuities.
The second one is to stay with standard wavelet transforms and use variational PDE techniques to modify the coefficients in the truncation process so that the oscillations are reduced in the reconstruction processes. In particular, we use minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have fewer oscillations near edges. We will also discuss a new method for using TV for error-concealment (filling in missing wavelet coefficients).
Joint work with Jackie Shen, Math Dept, Univ. Minn., and Haomin Zhou, Math
Dept, Georgia Tech.
Future talks in Scientific Computing Seminar
March 24 : Randolph E. Bank, Department of Mathematics, UC San Diego.
March 31 : Roland Freund, Department of Mathematics, UC Davis.
April 7 : Jin-Fa Lee, The ElectroScience Laboratory, The Ohio State University.
April 28 : Gene H Golub, Department of Computer Science, Stanford University.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.