A New Method for Characterizing Lace Based on a Fractal Index

The fractal geometry, issued of a synthesis of works done in Mathematics and in Physics since more of one century, proposed in the 70’s new concepts to understand some complex phenomenon ([Man77]). The notions of fractional dimension and scale invariance have been recognized applicable quickly for the description of many natural objects, of the mountainous to the maritime facades while passing by the meteorological phenomenon, the porous surroundings ([Adl92]) or the chemical catalysis ([Avn89])... These tools make it possible to understand the phenomenon of growth far from the balance that appear in a spontaneous manner in many domains, as the dielectric straining or the dendritic growth. The physicists, the chemists, the meteorologists or the astronomers could have new quantitative measures thus to characterize the objects that they study. The applications in signal processing appeared later, toward the beginning of the 80’s. A characteristic of the first attempts is the essentially descriptive vision that there was to the work: some signals were analyzed and some behaviors fractals was or non raised, the most often under the shape of a scale invariance in a certain range of resolutions. Then a " fractal dimension " was deducted, and the developments stopped there. Two important evolutions of different nature made it possible to enter in an "operational" phase in the beginning of the 90’s. The first, that appears naturally in the developments of a this young disciplines, is the enrichment of the tools of basis of the theory in view of the applications to the variety of the natural phenomenon: from the characterization of a signal by its (only) " fractal dimension ", came to be added finer measures, as the lacunarity or the multifractal analysis; the models of fractal process, first perfectly autosimilar, diversified to take into account invariances in generalized senses; finally, the statistical methods of fractal signal processing improved to provide some estimators more robust, and applicable in more general situations. The second evolution is more conceptual, and well adapted to signal processing. Instead of continuing to research fractal phenomenon (that means scale invariant) and to describe this invariance with the help of various measurements, one realized the profit that it could have there to apply some fractal tools to ordinary signals. In other words, instead of analyzing a signal to know if it is a fractal object, one makes it undergo some fractal treatments regardless of its possible scale invariance. Image processing provides a striking example of this change of point of view: fractal compression of images has been developed ([Fis98]), and not compression of fractal images! It is the same way for segmentation ([Veh96]), filtering ([Veh02]), or watermarking: any pictures are treated with fractal methods. This important evolution can be compared with the application of classic methods. The fractal measures or multifractal spectrum associated to a signal will be calculated while making some hypotheses on this one (adherence to a class of models, and "regularization" or "extension" of the signal (in the scales rather than in space)). We applied this concept for the fractal characterization of pictures of laces. These pictures (figure 1) are indeed very rich, and so very complex to analyze. This complexity can be a handicap for the constitution of a data base adapted to this particular industry. Indeed, an important effort of standardization of the objective criterion permitting the classification of the motives of laces is set currently in motion (creation of a" thesaurus " for lace industry), and the fractal treatment of the pictures of laces makes it possible to get a reliable attribute. We first present the method used for the fractal treatment fractal (boxes method), the results we obtain and the possible extensions then for this work.