1.3. ATOMI C D E C O M P O S I T I O N S AN D MODELIN G 7
structures. Suc h a n understandin g i s base d o n a carefu l analysi s o f experimen -
tal dat a an d numerica l experiments . Thes e issue s hav e muc h t o d o wit h imag e
processing.
We then understan d tha t modelin g i s not limite d t o mathematica l description s
of som e classe s o f objects , signal s o r images . Scientifi c task s ca n als o b e modeled .
That i s th e approac h whic h ha s bee n advocate d b y Davi d Mar r i n hi s pioneerin g
treatise [65] . Davi d Mar r insiste d o n th e rol e playe d b y th e choic e o f a particula r
representation fo r achievin g som e signa l processin g tasks . H e wrote [65 ] pp. 20-21:
A representatio n i s a forma l syste m fo r makin g explici t certai n en -
tities o r type s o f information , togethe r wit h a specificatio n o f ho w
the syste m doe s this . An d I shal l cal l th e resul t o f usin g a represen -
tation t o describ e a give n entit y a descriptio n o f th e entit y i n tha t
representation. Fo r example , th e Arabic , Roma n an d binar y numer -
ical system s ar e al l forma l system s fo r representin g numbers . Th e
Arabic representatio n consist s i n a strin g o f symbols draw n fro m th e
set 0 , 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 an d th e rul e fo r constructin g th e de -
scription o f a particula r intege r n i s tha t on e decompose s n int o a
sum o f multiples o f powers o f 10... A musical scor e provide s a wa y of
representing a symphony ; th e alphabe t allow s th e constructio n o f a
written representatio n o f words... A representation, therefor e i s not a
foreign ide a a t all...w e al l us e representation s al l th e time . However ,
the notio n tha t on e ca n captur e som e aspect s o f realit y b y makin g a
description o f it usin g a symbol an d tha t t o d o so can be useful seem s
to m e a fascinating an d powerfu l idea...An y particula r representatio n
makes certai n informatio n explici t a t th e expens e o f information tha t
is pushed int o the background an d may be quite hard to recover. Thi s
issue i s important, becaus e ho w informatio n i s presented ca n greatl y
affect ho w eas y i t i s t o d o differen t thing s wit h it . Thi s i s eviden t
even fro m ou r number s example : i t i s eas y t o add , t o substrac t an d
even to multipl y i f the Arabi c o r binary representation s ar e used, bu t
it i s not a t al l easy t o d o these thing s -especiall y multiplication - wit h
Roman numerals .
Similarly i n image processing, 'zooming ' o r comparing severa l views of the sam e
image wit h increasin g resolutio n i s a n issu e whic h i s trivia l i f wavele t analysi s i s
being used. Orthonorma l wavele t base s are here playing the same role for expandin g
signals o r image s a s Arabi c numeral s fo r rea l number s an d Fourie r analysi s ca n b e
compared t o Roma n numerals .
Here th e buildin g block s ar e 'time-scal e wavelets
7
o f Sectio n 18. Thes e ne w
wavelets diffe r fro m th e 'time-frequency ' wavelet s whic h ar e use d fo r processin g
audio signals . Time-scal e wavelet s d o no t idealiz e som e pattern s whic h ca n b e
detected i n natura l images . The y ar e rathe r artificia l buildin g blocks . Thi s le d
many scientist s t o questio n thi s ne w too l sinc e ther e doe s no t exis t a uniqu e an d
universally accepte d wavele t analysis . Similarl y th e Arabi c numeral s suffe r fro m
some arbitrariness .
Wavelet analysi s i s th e simples t scientifi c too l whic h permit s u s t o achiev e a
multiresolution analysis . Th e multiresolution analysi s which is based on these 'time -
scale' wavelets is modeling the widely accepted fac t tha t image s contain informatio n
over a ful l rang e o f scales .
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