1.3. ATOMI C DECOMPOSITION S AN D MODELIN G 3 We now detail an instance where this specific difficult y appears . Th e supernov a SN1987 A wa s observe d durin g th e nigh t o f Februar y 2 3 , 198 7 by Ia n Shelton , a Canadian astronomer . Thi s discover y too k plac e a t Campana s (Chile) . Th e sta r which ble w u p i s name d Sandulea k 69-20 2 an d wa s discovere d b y th e America n astronomer Nicola s Sandulea k i n the larg e Magellani c cloud . The nam e 'supernova ' come s from a time whe n peopl e thought a new star wa s born. Toda y w e know i t i s the opposit e sinc e w e are watchin g th e deat h o f a sta r under th e for m o f a violent stella r explosio n whic h destroy s thi s sta r whil e ejectin g the product s o f nuclear burnin g int o space . The Hubble space telescope was launched i n April 1990 . Th e primary mirro r of the Ritchey-Cretien telescope had a serious spherical aberration. Therefor e th e first pictures of SN1987A take n by Hubble needed to be deblurred an d denoised. Severa l algorithms hav e bee n use d an d th e winnin g tea m wa s coached b y Sylvi e Roques a t Observatoire Midi-Pyrenees . Thi s algorith m wa s grounded o n som e previou s wor k by Andr e Lanne s [14] . S. Roques ' algorith m begin s wit h usin g Donoho' s wavele t shrinkage . Thi s de - noising algorith m wil l be analyze d durin g thi s first lectur e (Sectio n 14 , Theorem 4 , and Sectio n 16) . I t i s based o n a specifi c u + v modeling . Thi s modelin g wil l als o be describe d i n thi s first chapte r (Section s 6 , 11 , 13). Returning to astronomy, thi s denoised image is not yet the reconstructed image . Instead i t i s a first step inside a sequence of iterations. I n the next steps , S . Roque s incorporates he r knowledg e o f th e optic s o f th e Hubbl e spac e telescop e an d use s the trainin g th e astronomer s hav e develope d t o deblu r suc h fain t image s [58] . The Correctiv e Optic s Spac e Telescop e Axia l Replacemen t wa s a spectacula r repair missio n whic h too k plac e i n Decembe r 1993 . A ne w wid e field planetar y camera wa s fixed o n Hubbl e an d a ne w pictur e o f SN19S7A becam e available . This new picture revealed two faint ring s exhibiting some intriguing geometry . Th e rings ar e tangent , symmetri c an d the y intersec t a t th e cente r o f S7V1987 A Thes e rings wer e absen t fro m al l th e previou s pictures . Shoul d thes e ring s b e treate d a s artifacts? The n the y shoul d b e include d insid e th e v componen t o f th e / = u + v decomposition. D o th e ring s revea l som e ejecte d gase s a s wa s suggeste d b y Evr y Schatzman? I n thi s cas e thes e ring s belon g t o th e u component . Som e a prior i knowledge i s neede d t o distinguis h betwee n u an d v an d thi s knowledg e i s base d upon som e scientific trainin g which might diffe r fro m on e astrophysicist t o another . This exampl e tell s u s tha t extractin g a 'hig h priority ' componen t u fro m a n image / ma y b e a risk y task . A mode l i s neede d t o perfor m thi s task . Thi s model will provide us with a decision making algorithm. Befor e movin g further an d unveiling the specific mode l on which JPEG-2000 i s grounded, i t may be interestin g to sa y a few genera l word s o n modelin g a s a scientific task . 1.3. Atomi c decomposition s an d modelin g Modeling i s a multifacete d scientifi c attitude . A t th e first level , modelin g is a n attemp t a t a mathematica l descriptio n o f a collectio n o f objects , image s o r signals whic h ar e sharin g som e common features . A preliminary analysi s i s neede d to detec t thes e features . Onc e detecte d an d analyze d thes e feature s migh t hel p i n modeling th e ensembl e o f signal s unde r study . I n [86] , [87] , [88 ] an d [89] , Naok i Saito raise s th e issu e o f definin g an d buildin g a featur e extracto r I : X i- Y o n a give n collectio n X o f signal s an d images . Thi s featur e extracto r wil l hel p i n
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