1.3. ATOMI C D E C O M P O S I T I O N S AN D MODELIN G
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We now detail a n instance where this specifi c 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 , 1987 b y 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 th e larg e Magellani c cloud .
The nam e 'supernova ' come s fro m a tim e whe n peopl e though t a ne w sta r wa s
born. Toda y w e kno w i t i s th e opposit e sinc e w e ar e watchin g th e deat h o f a sta r
under th e for m o f a violen t stella r explosio n whic h destroy s thi s sta r whil e ejectin g
the product s o f nuclea r burnin g int o space .
The Hubbl e spac e telescope was launched i n April 1990. Th e primar y mirro r of
the Ritchey-Cretie n telescop e 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 coache d b y Sylvi e Roque s a t
Observatoire Midi-Pyrenees . Thi s algorith m wa s grounde d 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 base d 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 imag e is not yet the reconstructed image .
Instead i t i s a first ste p insid e a sequence o f iterations. I n th e nex t 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 ne w picture reveale d tw o faint ring s exhibitin g som e intriguin g geometry . Th e
rings ar e tangent , symmetri c an d the y intersec t a t th e cente r o f S7V1987A 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 whic h migh t diffe r fro m on e astrophysicis t 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 makin g algorithm . Befor e movin g furthe r an d
unveiling the specifi c mode l on which JPEG-2000 i s grounded, i t ma y be interestin g
to sa y a fe w genera l word s o n modelin g a s a scientifi c 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 commo n 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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