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Analyse multifractale 2D et 3D à l'aide de la transformation en ondelettes : application en mammographie et en turbulence développée

Abstract : Since the end 80's, wavelet transform has been recognized as a privileged tool to study fractal objects, providing a unified multifractal formalism for both functions and measures. In the first part, we use the 2D WTMM (Wavelet Transform Modulus Maxima) methodology to study mammography. We illustrate the usefulness of the methodology in the study of texture segmentation of rough surfaces and in the geometric characterization of clusters of microcalcifications, which are early signs of breast cancer. In a second methodologic part, we generalize the WTMM method to provide a multifractal description of both 3D scalar and vectorial data fields, introducing the tensorial wavelet transform. We show that a recursive filter technique allows to save between 25 \% and 60 \% of computing time, as compared with FFT based filtering techniques. Then we apply the 3D WTMM method to Direct Numerical Simulations (DNS) of the Navier-Stokes equations in turbulent regime with moderate Reynolds numbers. By mesuring a significantly non-zero cancellation exponent, we bring evidence that multifractal properties of both 3D dissipation and enstrophy fields are well accounted for non-conservative multiplicative cascading processes. Moreover, we observe that the cancellation exponent decreases as the Reynolds number increases. Finally, we present the first results of a fully vectorial multifractal analysis of both velocity and vorticity fields on the same numerical simulations showing that the value of the intermittence parameter $C_2$, as measured by the tensorial 3D WTMM method, is significantly larger than the one obtained by studying 1D longitudinal velocity increments.
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Contributor : Pierre Kestener <>
Submitted on : Wednesday, December 17, 2003 - 6:38:47 PM
Last modification on : Thursday, June 27, 2019 - 1:21:23 AM
Long-term archiving on: : Friday, April 2, 2010 - 6:58:03 PM


  • HAL Id : tel-00004013, version 1


Pierre Kestener. Analyse multifractale 2D et 3D à l'aide de la transformation en ondelettes : application en mammographie et en turbulence développée. Interface homme-machine [cs.HC]. Université Sciences et Technologies - Bordeaux I, 2003. Français. ⟨tel-00004013⟩



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