Tight Frames of Curvelets
Translated title
Tætte Rammer af Curvelets
Authors
Rasmussen, Mikkel Funch ; Hansen, Rasmus Hallgren ; Le, Jenny Thuong
Term
4. term
Education
Publication year
2018
Submitted on
2018-06-07
Pages
81
Abstract
This thesis studies the theory and applications of curvelets, a mathematical tool for efficiently representing edges in images and signals that improves on wavelets by handling curved discontinuities. We develop the geometric construction of continuous curvelets via polar windows and prove that they form a tight frame. The theory is then adapted to a discrete transform based on the USFFT to enable practical use. We analyze the sparsity of curvelet coefficients for smooth objects with a C2 edge (including star-shaped sets), showing that most coefficients are negligible while the remaining ones are uniformly bounded; this yields fast coefficient decay comparable to the smooth case without singularities and improved approximation rates over wavelets. To understand localization and decay, we examine the Fourier transform of edge fragments and use the Radon transform as a key analytical tool; we also treat coefficients in smooth regions and at corners. Finally, we demonstrate curvelets in Matlab using CurveLab for visualization, image reconstruction, and denoising, and provide supporting definitions and code in the appendices.
Dette speciale undersøger teori og anvendelser af curvelets, et matematikværktøj til effektivt at repræsentere kanter i billeder og signaler, som forbedrer wavelets ved håndtering af kurvede diskontinuiteter. Vi gennemgår den geometriske konstruktion af kontinuerte curvelets, defineret via vinduer i polære koordinater, og beviser at de udgør en tæt ramme. Dernæst tilpasses teorien til en diskret transform baseret på USFFT, så curvelets kan anvendes i praksis. Vi analyserer sparsiteten af curveletkoefficienter for glatte objekter med en C2-kant (bl.a. stjerneformede mængder) og argumenterer for, at de fleste koefficienter er negligeable, mens de resterende er ensidigt begrænsede; dette giver en hurtig koefficientafmatning, sammenlignelig med tilfældet uden singularitet, og en bedre approksimationsrate end med wavelets. For at forstå lokalisering og afklingning undersøger vi Fouriertransformen af kantfragmenter og anvender Radontransformen som analytisk værktøj; vi behandler også koefficienter i glatte områder og ved hjørner. Endelig demonstrerer vi curvelets i Matlab via CurveLab til visualisering, bildegengivelse og støjreduktion, og vedlægger nødvendige definitioner samt kode i appendikserne.
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