A scale-space approach with wavelets to singularity estimation

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, "the structural intensity", that computes the "density" of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed

Activation detection in fMRI data via multi-scale singularity detection

Detection of active areas in the brain by functional magnetic resonance imaging (fMRI) is a challenging problem in medical imaging. Moreover, determining the onset and end of activation signal at specific locations in 3-space can determined networks of temporal relationships required for brain mapping. We introduce a method for activation detection in fMRI data via wavelet analysis of singular features. We pose the problem of determining activated areas as singularity detection in the temporal domain. Overcomplete wavelet expansion at integer scales are used to trace wavelet modulus maxima across scales to construct maxima lines. Form these maxima lines, singularities in the signal are located corresponding to the onset and end of an activation signal. We present result for simulated phantom waveforms and clinical fMRI dat from human finger tapping experiments. Different levels of noise were added to two waveforms of phantom data. No assumptions about specific frequency and amplitude of an activation signal were made prior to analysis. Detection was reliable for modest levels of random noise, but less precise at higher levels. For clinical fMRI data, activation maps were comparable to those of existing standard techniques

Tight wavelet frame packet

Ph.DDOCTOR OF PHILOSOPH

Vector extension of monogenic wavelets for geometric representation of color images

14 pagesInternational audienceMonogenic wavelets offer a geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, we propose a non-trivial extension of the monogenic framework to vector-valued signals to carry out a non marginal color monogenic wavelet transform. We also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction