Showing papers on "Wavelet transform modulus maxima method published in 1999"
1 Jan 1999
TL;DR: In this paper, the wavelet transform modulus maxima is used to describe the scaling properties of singular measures of fractal objects, and it is shown that the generalized fractal dimensions D q and the f (α) singularity spectrum can be determined from the scaling behavior of these partition functions.
Abstract: The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions D q and the f ( α ) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.
3 citations
TL;DR: Multiresolution analysis, specifically the discrete wavelet transform modulus-maxima (mod-max) method, is utilized for the extraction of mammographic mass shape features to classify masses as round, nodular, or stellate in a classification system.
Abstract: In this article, multiresolution analysis, specifically the discrete wavelet transform modulus-maxima (mod-max) method, is utilized for the extraction of mammographic mass shape features. These shape features are used in a classification system to classify masses as round, nodular, or stellate. The multiresolution shape features are compared with traditional uniresolution shape features for their class discriminating abilities. The study involved 60 digitized mammographic images. The masses were segmented manually by radiologists, prior to introduction to the classification system. The uniresolution and multiresolution shape features were calculated using the radial distance measure of the mass boundaries. The discriminating power of the shape features were analyzed via linear discriminant analysis (LDA). The classification system utilized a simple Euclidean metric to determine class membership. The system was tested using the apparent and leave-one-out test methods. The classification system when using the multiresolution and uniresolution shape features resulted in classification rates of 83% and 80% for the apparent and leave one-out test methods, respectively. In comparison, when only the uniresolution shape features were used, the classification rates were 72 and 68% for the apparent and leave-one-out test methods, respectively.