DEPARTMENT OF COMPUTER SCIENCE

DOCTORAL DISSERTATION DEFENSE

Candidate: Wen-Liang Hwang

Advisor: Stephane Mallat

DOCTORAL DISSERTATION DEFENSE

Candidate: Wen-Liang Hwang

Advisor: Stephane Mallat

**Singularity Detection, Denoising and
Multifractal Characterization with Wavelet**

11:00 a.m., Friday, July 30, 1993

12th floor conference room, 719 Broadway

Abstract

Most of a signal information is often carried by singularities. We study the characterization of the singularities with the wavelet transform and its modulus maxima. We introduce numerical algorithm to detect and characterize pointwise singularities from the behavior of the wavelet transform maxima across scales. As an application, we develop a denoising algorithm which discriminates the signal information from noise through an analysis of local singularities. In one dimension, we recover a piecewise smooth signal, where the sharp transitions are preserved. In two dimensions, the wavelet maxima algorithm detects and characterizes the edges. The geometrical properties of edges are used to discriminate the noise from the image information and the denoising algorithm restores sharp images even at low SNR.

Multifractals are singular signals having some self-similarity properties. We develop a robust algorithm to extract the fractal parameter of fractional Brownian motion embedded in white noise. Fractal parameters are estimated from the evolution of the variance of the wavelet coefficients across scales with a modified penalty method. Self-similar multifractals have a wavelet transform whose maxima define self-similar curves in the scale-space plane. We introduce an algorithm to recover the affine self-similar parameters with a voting procedure. This voting strategy is robust with respect to renormalization noise. We describe the numerical applications to Cantor measures, dyadique multifractals and to the study of diffusion limited aggregates.