1.3. ATOMI C D E C O M P O S I T I O N S AN D MODELIN G 5
if an d onl y i f th e contiuou s functio n / belong s t o al l th e Beso v space s B
p
/P,p, p e
(0,1]. Thi s fo r exampl e happen s whe n f(x) = |x| 7, 7 0 . Beso v space s wil l b e
defined i n Sectio n 17, Definition 18.
The selectio n o f thes e atom s Pj/Qj i s provide d b y a nonlinea r algorithm . W e
will b e happ y t o lear n tha t i n th e n-dimensiona l version s o f Peller' s theorem , thi s
nonlinear algorith m i s a Donoho' s wavele t shrinkage . Peller' s shoul d b e viewe d a s
a compressio n algorithm .
Defining wha t i s mean t b y 'atoms ' i s a ke y issue . A n interestin g criterio n
is sparsity . W e wan t th e 'atomi c decomposition ' t o b e th e sparses t on e fo r th e
given ensembl e o f signals . Sparsit y i s pavin g th e roa d t o th e best-basi s algorith m
designed b y R. Coifman an d V. Wickerhauser [31]. On e is given a very large librar y
of orthogona l base s an d th e basi s whic h i s selecte d b y th e algorith m i s the on e fo r
which the expansion ha s a minimal length . Thi s best-basis paradig m ca n be applie d
to computationa l an d numerica l analysi s [30] . Fo r instanc e finite element s shoul d
be adapte d t o th e specifi c geometr y o f the problem . W e will retur n t o thi s issu e i n
Sections 7 and 18.
Another criterio n i s statistica l independence . Her e w e ar e treatin g a ful l col -
lection o f signals . 'Independen t componen t analysis ' (ICA ) i s aime d a t breakin g a
signal belongin g t o thi s collectio n int o a linea r combinatio n o f nearl y independen t
components. I n the ICA approach , thes e components ar e the 'atoms ' we are lookin g
for. Thi s approac h i s discussed i n a serie s o f paper s b y Jean-Frangoi s Cardos o an d
Naoki Sait o [21], [86] , [87 ] an d [88] .
In this book , a first exampl e o f 'building blocks ' i s given by the 'time-frequenc y
atoms' whic h ar e use d fo r processin g audi o signals . Th e 'time-frequency ' atom s
which hav e bee n use d b y Denni s Gabo r an d Joh n vo n Neuman n wer e a kind o f ide-
alization of the musical notes. Mor e realistic atoms were introduced b y Jean-Sylvai n
Lienard an d Xavie r Rode t [62 ] in their wor k on audio signals. Thes e Lienard-Rode t
atoms sharpl y contras t wit h wave s o r time-scal e wavelets , sinc e the y trul y mimi c
the sounds which are heard an d processed . Lienard-Rode t atom s mimi c the musica l
notes b y incorporatin g fou r importan t characteristics : th e duration , th e frequency ,
the shapin g o f th e attac k an d th e shapin g o f th e attenuation .
Expanding a given audi o signa l int o a serie s of Lienard-Rodet atom s wa s no t a
simple task . Indee d ther e ar e to o man y parameter s t o b e fixed an d ther e ar e infin -
itely many expansions . Importan t progres s was made by Henrique Malvar an d Ken -
neth Wilso n i n th e lat e eighties . Workin g independently , the y discovere d infinitel y
many orthonorma l base s consistin g o f 'time-frequenc y atoms' . Th e Malvar-Wilso n
atoms hav e a simpl e an d explici t algorithmi c definitio n whic h wil l b e unveile d i n
Chapter 1, Sectio n 18. Expandin g a n audi o signa l int o a serie s o f time-frequenc y
atoms become s a feasibl e tas k whic h shoul d b e compare d t o writin g a musica l
score whil e listenin g t o th e music . Le t u s howeve r stres s tha t thes e Ma i var-Wilson
wavelets d o no t solv e th e mai n issu e raise d b y Jean-Sylvai n Lienar d an d Xavie r
Rodet. Indee d a Ma i var-Wilso n wavele t basi s i s labelle d b y a n infinit e numbe r o f
free parameters . Thes e parameters shoul d b e fixed before expandin g a given signal .
What orthonorma l Malvar-Wilso n basi s shoul d on e use ? Thi s fundamenta l issu e
leads t o a bes t basi s searc h whic h wil l b e addresse d i n Sectio n 7 .
For a mathematician, expandin g a function int o a n orthonorma l basi s doe s no t
raise an y problem . I n signa l o r imag e processing , th e error s wil l com e fro m tw o
limitations. Firs t on e shoul d b e usin g finite sum s instea d o f infinit e serie s an d
secondly, rea l o r comple x coefficient s ar e quantized . The y ar e replace d b y digita l
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