Contents
Preface i x
Chapter 1. Stil l image s compressio n 1
1.1. Introductio n 1
1.2. A first glanc e a t compressio n an d denoising . Th e supernov a SN19S7A 1
1.3. Atomi c decomposition s an d modelin g 3
1.4. Wavelet s an d stil l imag e compression : som e succes s storie s 8
1.5. Sampling , quantization , thresholdin g an d compressio n 9
1.6. A firs t visi t t o u -f- v model s fo r stil l image s 13
1.7. Best-basi s algorithm s i n signa l processin g 15
1.8. Th e ol d JPE G 17
1.9. Karhunen-Loev e expansion s 18
1.10. A n exampl e wher e th e Karhunen-Loev e approac h i s ineffective : th e
ramp functio n 19
1.11. A secon d visi t to u + v imag e model s 2 2
1.12. Th e spac e BV o f function s wit h bounde d variatio n i n th e plan e 2 3
1.13. Th e Osher-Rudi n mode l 2 7
1.14. Th e mathematica l propertie s o f the Osher-Rudi n mode l 3 0
1.15. Modelin g texture s 4 2
1.16. Wavele t shrinkag e 4 5
1.17. Littlewood-Pale y analysi s 5 0
1.18. A survey o f wavele t analysi s 5 7
1.19. Littlewood-Pale y analysi s an d wavele t analysi s 6 5
1.20. Quantizatio n issues : Fourie r serie s vs . wavele t serie s 6 6
1.21. Fourie r serie s vs . wavele t series : expansion s o f BV function s 6 7
Chapter 2 . Th e rol e o f oscillation s i n som e nonlinea r PDE' s 7 1
2.1. Introductio n 7
2.2. Improve d Gagliardo-Nirenber g inequalitie s 7 2
2.3. Improve d Poincar e estimate s 7 9
2.4. Wavele t coefficient s o f integrabl e function s 8 0
2.5. A firs t mode l case : th e nonlinea r hea t equatio n 8 1
2.6. Th e Navier-Stoke s equation s 8 4
2.7. Modelin g coheren t structure s 8 8
2.8. Th e nonlinea r Schrodinge r equatio n 9 1
Chapter 3 . Frequenc y modulate d signals , chirp s an d th e Virg o progra m 9 3
3.1. Introductio n 9 3
3.2. Holde r classe s wit h negativ e exponent s 9 6
3.3. Infinitel y oscillatin g function s 101
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