Selected Titles in This Series

Selected Title s i n Thi s Serie s

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations , 2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Ma t has, Iwahori-Heck e algebra s an d Schu r algebra s o f th e symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Frobeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/022

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Oscillating Pattern s i n Imag e Processing an d Nonlinea r

Evolution Equation s


University

LECTURE Series

Volume 2 2

Oscillating Pattern s i n Imag e Processing an d Nonlinea r

Evolution Equation s The Fifteent h

Dean Jacquelin e B . Lewi s Memorial Lecture s

Yves Meye r

American Mathematica l Societ y Providence, Rhod e Islan d

Editorial Boar d

Jerry L . Bon a (Chair ) Nige l J . Hitchi n Jean-Luc Brylinsk i Nicola i Reshetikhi n

2000 Mathematics Subject Classification. Primar y 35Q30 , 42C40 , 65T60 , 76D05 , 76D06 .

ABSTRACT. Thi s boo k i s aime d a t bridgin g a ga p betwee n signa l processing , imag e processing , and nonlinea r differentia l equations . Mor e precisel y th e rol e playe d b y oscillatin g pattern s wil l be emphasized . Thes e pattern s occu r a s th e texture d component s o f images . The y als o occu r i n frequency modulate d signal s o r chirps . Wha t i s mor e surprisin g i s th e relevanc e o f suc h pattern s in partia l differentia l equations .

The firs t hal f o f thi s essa y ca n b e use d a s a textbook , sinc e full y detaile d proof s ar e given .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Meyer, Yves . Oscillating pattern s i n imag e processin g an d nonlinea r evolutio n equation s : th e fifteenth Dea n

Jacqueline B . Lewi s memoria l lecture s / Yve s Meyer . p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 22 )

Includes bibliographica l references . ISBN 0-8218-2920- 3 (alk . paper ) 1. Evolutio n equations , Nonlinear . 2 . Image compression . 3 . Signal processing—Mathematics .

4. Oscillations . I . Universit y lectur e serie s (Providence , R.I. ) ; 22.

QA377.M495 200 1 515'.353—dc21 2001041268

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o the Assistan t t o the Publisher , America n Mathematica l Society , P. O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o [email protected].

© 200 1 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http:/ /www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 6 0 5 0 4 0 3 0 2 0 1

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 1 3 1.7. Best-basi s algorithm s i n signa l processin g 1 5 1.8. Th e ol d JPE G 1 7 1.9. Karhunen-Loev e expansion s 1 8 1.10. A n exampl e wher e th e Karhunen-Loev e approac h i s ineffective : th e

ramp functio n 1 9 1.11. A second visi t to u + v imag e model s 2 2 1.12. Th e spac e BV o f functions 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 oscillations i n som e nonlinea r PDE' s 7 1 2.1. Introductio n 7 1 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 10 1

viii C O N T E N T S

3.4. A first definitio n o f n-dimensiona l chirp s 10 3 3.5. A second definitio n o f chirp s 10 4 3.6. Jaffard' s criticis m 10 5 3.7. Chirp s an d two-microloca l space s 10 7 3.8. Wavelet s an d chirp s 10 9 3.9. A function propose d b y Rieman n contain s infinitel y man y chirp s 11 1 3.10. A generalized Rieman n functio n 11 4

Conclusion 11 7

Bibliography 119

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 i m a g e process in 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 s tandar d wil l b e un -veiled i n Marc h 200 1 an d i s known 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 , f requenc y m o d u l a t e d s ignal 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 pat tern ' (Theore m 3 , Sectio n 14 , Chapte r 1) . Her e 'oscillatin g pat terns ' 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 .

The secon d chapte r s tar t 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 mat te 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 pat tern 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

x P R E F A C E

Einstein's genera l relativit y implie s th e existenc e o f gravitationa l waves . Th e theory predicts that suc h waves might b e created by gigantic gravitational disasters . In th e cas e o f a collaps e o f a pai r o f binar y stars , thes e wave s woul d b e frequenc y modulated signals , also named chirps . Ther e is no doubt tha t detectin g such signals relies o n describin g an d analyzin g oscillatin g patterns . Thi s importan t topi c i s discussed i n Chapte r 3 .

The unity between our three chapters i s now clear. Beyon d Fourier analysi s an d its variants, new tools are available. Thes e tools help in understanding an d modelin g oscillating patterns . Suc h pattern s ar e presen t i n man y natura l images . The y help i n explainin g nonlinea r evolutions . The y ar e pivota l i n chirp s an d frequenc y modulated signals .

Let m e expres s my dee p thank s t o th e Departmen t o f Mathematics o f Rutger s University. I wa s prou d an d happ y t o giv e thes e thre e Jacquelin e Lewi s Lectures . Richard Fal k an d hi s colleague s hav e bee n kin d an d helpful .

This book is not a treatise bu t rathe r a n expanded version of my informal talks . It i s not a textbook . Instea d i t i s aimed a t describin g a scientifi c vision .

Am I th e sol e write r o f thi s book ? Thi s i s no t clea r sinc e mos t o f th e fin e results whic h wil l b e describe d o r prove d belon g t o m y student s o r m y colleagues . Moreover th e 'gran d unification ' whic h i s unveiled i n thi s essay ca n b e trace d bac k to a program whic h was launched i n the earl y fortie s b y the 'Institut e fo r th e Unit y of Science ' (se e Sectio n 2 , Chapte r 1) .

Let m e than k Alber t Cohe n an d hi s collaborator s fo r lettin g m e includ e som e still unpublished materia l an d fo r hi s encouraging remarks . Naok i Sait o read a first version o f thi s boo k an d suggeste d man y improvements . Th e anonymou s refere e was kin d an d helpful .

I woul d lik e t o dedicat e my wor k t o Jacquelin e Lewis . Le t m e als o remembe r Alberto Calderon . I mis s s o muc h ou r endles s discussion s abou t mathematics , literature an d politic s i n Argentina .


http://dx.doi.org/10.1090/ulect/022/04

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