Regularity and localized representations of random-phase textures

Abstract : This dissertation deals with random-phase texture analysis and synthesis – i.e. textures without patterns, and with random Fourier phase. The first part studies properties of a special representant of each class of micro-textures, that we name texton. We prove an optimality result with respect to the spatial concentration around the origin. We take advantage of this concentration phenomenon to propose sparse representations of micro-textures, approximate and exact under some hypothesis. We discuss generalizations of the texton to color images and extend the sparse approximations developped for gray-scale images. We interpret the optimality of concentration as a projection result, and discuss several other projection experiments on different image spaces. These numerical experiments show that the hypothesis, widely believed in signal processing, claming that “the geometry of images is encoded in their phase” deserves further inquiry. In the last part of this dissertation, we study some asymptotical properties of the random-phase texture model. We proved the convergence to a Gaussian field while extending random-phase textures towards the whole (non-periodic) discrete plane in the first part of the dissertation, and we focus here on convergence and local properties (continuity and regularity) of multi-dimensional infinite random Fourier sums. We extend to the multi-dimensional case a theorem of Billard and Kahane showing the equivalence, for the random sums considered, between a.s. uniform convergence, a.s. pointwise convergence everywhere, and a.s. continuity everywhere. We also extend to the multi-dimensional case, sufficient conditions and necessary conditions for continuity and Hölder regularity of these sums, with an anisotropic framework.
Keywords : Mathematics
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Theses
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Samuel Ronsin. Regularity and localized representations of random-phase textures. General Mathematics [math.GM]. Université René Descartes - Paris V, 2014. English. ⟨NNT : 2014PA05S024⟩. ⟨tel-01160321⟩

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