4
STILL IMAGE S COMPRESSIO N
reducing th e dimensionalit y o f th e proble m withou t losin g importan t informatio n
for imag e approximatio n an d modeling .
Learning an d extractin g th e mai n feature s whic h ca n b e foun d i n a give n collec -
tion C of signal s ma y b e see n a s a preliminar y tas k t o b e performe d befor e modeling .
Here w e ar e considerin g on e dimensiona l signals , bu t als o image s o r turbulen t flow s
in flui d dynamics . Thes e feature s o r primitive s wil l the n b e th e 'buildin g blocks '
which ar e aime d a t bein g use d i n compressing , denoisin g o r othe r tasks . I n man y
instances o f thi s program , th e feature s whic h ar e detecte d happe n t o b e som e 'os -
cillating patterns' . Thi s progra m lead s t o th e celebrate d 'atomi c decompositions '
on whic h w e wil l focus .
In orde r t o understan d som e intricat e collectio n o f signals , on e migh t tr y t o
break thes e signal s int o simple r 'buildin g blocks' . Analyzin g actuall y mean s de -
composing o r breakin g a comple x structur e int o simple r entitie s o r 'buildin g blocks '
which ca n b e bette r understood . The n on e i s faced wit h th e proble m o f reconstruct -
ing wha t ha s bee n broken . Thi s reconstructio n i s name d 'synthesis' . Fo r achievin g
this task , on e shoul d discove r th e assembl y (o r syntactic ) rule s whic h ar e no w pavin g
the roa d t o complexity . I n orde r t o unvei l thes e assembl y rules , Naok i Sait o i s de -
veloping a hierarch y o f featur e extractor s whic h woul d integrat e th e loca l feature s
into a globa l perception . Suc h a hierarch y o f featur e extractor s i s emulatin g th e
human brain .
To summariz e ou r discussion , modelin g shoul d star t wit h a lis t o f buildin g
blocks, bu t shoul d als o includ e som e assembl y rule s tha t tel l u s ho w t o combin e
these buildin g blocks .
When the building blocks can be modeled as elements of a vector space and
when the assembly rule is linear combination, then this model is named an 'atomic
decomposition7.
A superficia l readin g migh t lea d t o th e conclusio n tha t 'atomi c decomposition '
is a linea r methodology . Thi s i s a t th e sam e tim e righ t an d wrong . Seekin g a n
optimal collectio n B o f 'atoms ' whic h shoul d b e use d i n th e atomi c decompositio n
is a highl y nonlinea r an d controversia l issue . W e wil l retur n t o thi s proble m i n
Section 7 . Bu t a mor e subtl e nonlinearit y come s fro m th e fac t tha t th e relatio n
between a give n signa l / an d th e atom s t o b e use d i n decomposin g thi s signa l
might als o b e nonlinear . A beautifu l illustratio n i s give n b y th e famou s nonlinea r
approximation o f continuou s function s / o n [—1,1 ] b y rationa l function s g(x)
P(x)/Q(x) wher e P an d Q ar e tw o polynomial s wit h degree s no t exceedin g N an d
Q(x) doe s no t vanis h o n [0,1]. W e the n writ e P G V/v, Q G V/V . Th e correspondin g
rational function s P/Q, P G V/v , Q G V/ v ar e th e atom s t o b e use d i n a n atomi c
decomposition
oo
(3.1) / ( z ) = £ i M s ) / W * ) . PjZV^QjeVj
0
In thi s context , Vladimi r Pelle r [83 ] aske d himsel f i f (3.1) migh t yiel d som e mor e
effective approximation s t o / tha n ordinar y Fourie r serie s expansions . H e wante d
to kno w i f th e atom s Pj(x)/Qj(x) i n th e right-han d sid e o f (3.1) migh t hav e a fas t
decay a s j o o eve n i f / i s no t C°° . H e prove d a fine theore m whic h say s tha t w e
can selec t Pj G V}, Q
3
; G V/, j 1, suc h tha t
(3.2) ||P
i
/Q
J
-||
L
=o
[
_
1
,
1 ]
C',.r* , q=l,2,...
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