6
STILL IMAGE S COMPRESSIO N
approximations. W e however wan t th e resultin g error s t o b e acceptable . Returnin g
to th e Malvar-Wilso n wavelets , w e wan t t o minimiz e th e error s whic h com e wit h
quantization an d w e want th e compressio n schem e to provid e u s with sketch s o f th e
signal wher e th e mos t importan t feature s ar e kept . Thi s issu e stil l lead s t o a bes t
basis search .
In speec h processing , on e i s looking fo r 'phonemes ' whic h indee d ar e th e build -
ing block s o f th e spoke n languages . Her e th e proble m i s muc h mor e difficul t sinc e
these phoneme s wil l differ fro m on e spoken languag e t o a n other . Ca n w e use thes e
phonemes a s buildin g block s an d develo p som e suitabl e algorithm s fo r extractin g
these phonemes ?
Can general algorithms possibly yield concise atomic decompositions i n a digital
signal processin g context ? Wha t i s th e rol e o f th e windowe d Fourie r transfor m o r
of th e Wigner-Vill e transforms ? Doe s th e elusiv e 'instantaneou s frequency ' exist ?
What i s th e definitio n o f frequenc y modulate d signals ? Thes e issue s wil l b e face d
in ou r thir d chapter .
Similar problem s ca n b e raise d i n imag e processing . Ar e ther e som e relevan t
scaling laws ? Shoul d th e buildin g block s b e consisten t wit h scal e invariance ? Ho w
are thes e buildin g block s defined ? I n hi s famou s boo k [65] , Davi d Mar r propose s
that th e atom s o f imag e processin g b e 'primitive s o f th e sam e genera l kin d a t
different scales' . Thes e primitives ar e called 'blobs ' and a blob has 'a rough position ,
length, widt h an d orientatio n a t whateve r scal e i t i s defined' . Howeve r a n imag e
has severa l level s o f organizatio n an d a rando m collectio n o f blob s i s not a n image .
The syntacti c structur e o f this organizatio n i s the mai n proble m i f this approac h i s
followed. Thi s issu e i s discussed i n G . Kanizsa' s treatis e [60] .
One migh t als o say tha t th e 'buildin g blocks ' of a n imag e ar e the object s whic h
can b e detecte d i n thi s imag e an d als o th e texture d component s o f thi s image .
But w e canno t expec t thes e object s o r texture s t o b e describe d b y som e explici t
mathematical formula .
The definitio n o f object s an d texture s whic h wil l b e use d i n thi s boo k wa s
proposed b y Stanle y Oshe r an d Leoni d Rudi n [80] . W e wil l retur n t o thi s mode l
in Section s 5 , 6 , 13, 14, 15 an d 16 o f thi s firs t chapter . I n thi s model , a n imag e
is decompose d int o a su m betwee n tw o piece s u an d v. Th e first piec e i s aime d a t
containing th e mai n object s containe d i n a n image . Th e secon d on e take s car e o f
the texture d component s an d o f wha t i s unorganized . I n th e Osher-Rudi n model ,
the first componen t u i s assume d t o b e a functio n wit h bounde d variation . I n
Chapter 1, Section 15, a variational formulatio n propose d b y Oshe r an d Rudi n wil l
lead t o a n interestin g modelin g o f th e texture d components . I n thi s modeling , th e
u componen t o f a n imag e i s no t furthe r decompose d int o a linea r combinatio n o f
objects. Thi s descriptio n i s a kin d o f idealizatio n abou t wha t th e Osher-Rudi n
algorithm does . Bu t lif e i s no t s o simpl e an d th e tru e stor y wil l b e unveile d i n
Section 14.
Modeling als o plays a role in studying Navier-Stoke s equation s an d turbulence .
For exampl e bot h coheren t structure s an d vorticit y filaments ar e importan t pat -
terns i n turbulen t flows. Thes e pattern s ar e observe d i n numerica l simulation s an d
also in experiments an d on e would lik e to desig n som e algorithm s fo r detectin g an d
extracting thes e pattern s fro m th e unorganize d flow. Suc h a n algorith m canno t b e
a blac k bo x an d instea d cruciall y relie s o n a bette r understandin g o f these coheren t
Previous Page Next Page