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 b e considered . Th e firs t chapte r i s devoted t o i m a g e p r o c e s s i n 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 an d i s know n a s JPEG-2000 . I n th e secon d chapter , a fe w ne w
results o n th e N a v i e r - S t o k e 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 m o d u l a t e 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
tha t a 'wavele t thresholding ' wipe s awa y thi s v component .
Th e 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 14 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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