Preface This boo k consist s o f thre e chapter s i n whic h som e seemingl y unrelate d scien - tific problem s wil l be considered . Th e firs t chapte r i s devoted t o imag e processin g and mor e precisel y t o imag e compressio n an d denoising . Thi s researc h i s motivate d by th e upcomin g standar d fo r stil l imag e compression . Thi s standar d wil l b e un - veiled i n Marc h 200 1 and i s known a s JPEG-2000 . I n th e secon d chapter , a fe w ne w results o n th e Navier-Stoke s equation s an d othe r nonlinea r evolutio n equation s will b e discusse d and , i n th e thir d chapter , frequenc y modulate d signal s wil l b e analyzed. Motivatio n come s fro m th e Virg o progra m o f detectio n o f gravitationa l waves. How coul d thes e distinc t theme s possibl y b e studie d fro m th e sam e perspective ? An answe r i s foun d i n th e content s o f th e thre e chapters . Analyzing th e performance s o f a compressio n algorith m require s a mode l fo r still images . I n th e firs t chapter , ou r discussio n wil l b e base d o n th e Osher-Rudi n model. Thi s mode l originate d i n a joint pape r betwee n Stanle y Osher , Leoni d Rudi n and Ema d Fatem i [81] . I t wa s the n develope d b y Oshe r an d Rudi n an d tha t i s wh y it wil l b e name d th e Osher-Rudi n model . I t amount s t o splittin g a n imag e / int o a sum f = u + v betwee n tw o component s u an d v. Th e firs t componen t u represent s the 'objects ' whic h ar e containe d insid e th e give n imag e / . I t i s the n natura l t o as - sume tha t u belong s t o th e spac e BV o f function s wit h bounde d variation . Wavele t expansions ar e quit e effectiv e fo r analyzin g BV function s (Theore m 14 , Sectio n 21 , Chapter 1) . O n th e othe r hand , w e wil l prov e tha t th e 'textur e + noise ' componen t v i s a n 'oscillatin g pattern ' (Theore m 3 , Sectio n 14 , Chapte r 1) . Her e 'oscillatin g patterns' wil l b e define d b y som e Beso v nor m estimates . Thi s discussio n wil l impl y that a 'wavele t thresholding ' wipe s awa y thi s v component . The secon d chapte r start s wit h a n interestin g sharpenin g o f Gagliardo - Nirenberg estimate s (Theore m 16 , Sectio n 2) . I t continue s wit h a n improvemen t on Poincare' s inequalit y (Theore m 21 , Sectio n 3) . Thes e advance s rel y heavil y o n Theorem 1 4 of the firs t chapter . Th e sam e Beso v spac e whic h wa s use d fo r modelin g textures wil l b e th e hear t o f th e matte r here . This specifi c Beso v spac e an d som e relate d functiona l space s wil l agai n b e important i n thi s secon d chapter . W e ar e alludin g her e t o a ne w resul t o n th e Navier-Stokes equation s whic h wa s obtaine d b y Herber t Koc h an d Danie l Tatar u (Theorem 32 , Sectio n 6 , Chapte r 2) . Feature extractio n i s pivota l i n fluid dynamics . On e woul d lik e t o detec t an d extract th e elusiv e 'coheren t structures' . Thes e structure s ar e specifi c pattern s which appea r bot h i n experimenta l wor k i n flui d dynamic s an d i n numerica l sim - ulations. I n orde r t o stud y coheren t structures , on e need s t o investigat e localize d and oscillatin g solution s o f th e Navier-Stoke s equations . Thi s wil l als o b e don e i n Chapter 2 . ix
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