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, som e fundamen -
tal issue s abou t stil l imag e compressio n wil l b e discussed . W e will mostl y focu s o n
the specifi c exampl e o f th e image s provide d b y th e Hubbl e telescope . I n th e thir d
section, modelin g wil l be introduce d wit h a n emphasi s 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 imag e compressio n 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 genera l methodology , name d principa l componen t analysi s o r
Karhunen-Loeve expansion . Karhunen-Loev e analysi s wil l b e define d i n Sectio n 9 .
In Sectio n 10, th e limitation s o f suc h a Karhunen-Loev e approac h t o compressio n
will b e 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 (writte n BV fo r short) wil l be listed for the reader's convenience .
In Sectio n 13, thi s 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 th e 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 15 an d pave s th e
road t o Donoho' s denoisin g algorithm s whic h wil l b e describe d i n Sectio n 16.
A survey o f wavelet analysi s wil l be provided i n Sectio n 18. Wavele t analysi s i s
a varian t o n the Littlewood-Pale y analysi s whic h is described i n Sectio n 17. Th e re-
lations betwee n wavele t analysi s an d th e Littlewood-Pale y analysi s wil l 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-200 0 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 thi s boo k wil l b e motivate d b y som e aspect s o f stil l imag e
processing an d w e will try t o analyz e som e mathematical problem s whic h ar e 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 whic h ar e a s relevan t
http://dx.doi.org/10.1090/ulect/022/01
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