CHAPTER 1 Still image s compressio n 1.1. Introductio n The first chapte r i s organized a s follows. I n the second section, some fundamen - tal issue s abou t stil l imag e compressio n wil l be discussed . W e will mostly focu s o n the specifi c exampl e o f the image s provide d b y th e Hubbl e telescope . I n th e thir d section, modelin g will be introduced wit h a n emphasis o n 'atomi c decompositions' . In Sectio n 4 , a fe w experimenta l fact s illustrat e th e efficienc y o f wavelet s base d algorithms i n stil l imag e compressio n technology . Thi s sectio n shoul d b e viewe d as a kin d o f advertisemen t o r motivatio n fo r thi s firs t chapter . I n Sectio n 5 , som e terminology use d i n image compression wil l be clarified. I n Sectio n 6 , u -f v model s for stil l image s ar e introduced . Best-basis algorithm s ar e define d i n Sectio n 7 . Sectio n 8 i s devote d t o th e already obsolet e JPE G standard . Thi s standar d i s ofte n justifie d a s bein g a par - ticular exampl e o f a general methodology , name d principa l componen t analysi s o r Karhunen-Loeve expansion . Karhunen-Loev e analysi s wil l be define d i n Sectio n 9 . In Sectio n 10 , the limitation s o f suc h a Karhunen-Loev e approac h t o compressio n will be illustrate d wit h a counterexample . In Sectio n 11 , w e wil l retur n t o th e u + v model s fo r stil l image s o n whic h this wor k i s based . I n Sectio n 12 , som e wel l know n propertie s o f function s wit h bounded variations (written BV fo r short) will be listed for the reader's convenience. In Sectio n 13 , this spac e BV i s use d fo r modelin g th e geometri c content s i n stil l images. I n Sectio n 14 , the mathematica l propertie s o f the Osher-Rudi n mode l wil l be unveiled . W e als o nee d a mode l fo r th e texture s an d th e nois e tha t migh t b e contained i n ou r images . Suc h a mode l i s propose d i n Sectio n 1 5 an d pave s th e road t o Donoho' s denoisin g algorithm s whic h wil l be describe d i n Sectio n 16 . A survey of wavelet analysi s will be provided i n Section 18 . Wavele t analysi s is a variant o n the Littlewood-Paley analysi s which is described i n Section 17 . Th e re- lations betwee n wavelet analysi s an d th e Littlewood-Pale y analysi s will be clarifie d in Sectio n 19 . The n quantizatio n issue s wil l b e addresse d i n Sectio n 20 . Fourie r series expansion s o f BV function s wil l b e compare d t o thei r wavele t expansion s in Sectio n 21 . I f th e Osher-Rudi n mode l i s accepted , the n wavele t analysi s wins , which migh t explai n th e succes s o f JPEG-2000 ove r th e conventiona l JPEG . 1.2. A firs t glanc e a t compressio n an d denoising . The supernov a SM987 A The firs t chapte r o f this boo k wil l be motivate d b y som e aspect s o f still imag e processing and we will try t o analyze some mathematical problem s which are raise d by th e digita l revolution . A mai n challeng e consist s i n optimizin g compressio n while minimizin g distortion . Bu t ther e ar e man y mor e issue s which ar e a s relevan t
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