General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.
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Transcript of General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.
![Page 1: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/1.jpg)
General Orthonormal MRA
Ref: Rao & Bopardikar, Ch. 3
![Page 2: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/2.jpg)
Outline
• MRA characteristics– Nestedness, translation, dilation, …
• Properties of scaling functions
• Properties of wavelets
• Digital filter implementations
![Page 3: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/3.jpg)
Recall Formal Definition of an MRA
An MRA consists of the nested linear vector space such that
• There exists a function (t) (called scaling function) such that is a basis for V0
• If and vice versa•
• Remarks:– Does not require the set of (t) and its integer translates
to be orthogonal (in general)– No mention of wavelet
2101 VVVV
integer:)( kkt
1)2( then )( kk VtfVtf
)(lim 2 RLV jj
}0{jV
![Page 4: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/4.jpg)
Properties of Scaling Functions
1)( dtt
1)(
2dtt
0for basis
tindependenlinearly :)(
)()(),(
V
Zkkt
nntt
)(t
)1( t
Explained using Haar basis
![Page 5: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/5.jpg)
Dilation of Scaling Functions
)(2)2
1(),
2
1(
)(2
1)2(),2(
nntt
nntt
)(2)2(),2( nntt kkk
)(t
)2( t
)12( t
)2/(t
1
2
1t
k
k
V
Zllt
for basistindependenlinearly
:)2(
![Page 6: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/6.jpg)
Nested Spaces
• Every vector in V0 belongs to V1 as well
– In particular (t)
• Possible to express (t) as a linear combination of the basis for V1
ZkktV
ZkktV
VV
:)2(: of basis
:)(: of basis
1
0
10
![Page 7: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/7.jpg)
Haar may be misleading …
• One can translate an arbitrary function by integers and compress it by 2; BUT there is no reason to think that the spaces Vj created by the function and its translates and dilates will necessarily be nested in each other
V0
V1
10 VV 10 VV
Remark
![Page 8: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/8.jpg)
Two-Scale Relations(Scaling Fns)
n
ntnct )2()()(
2
1)(
)2()()(1
:sidesboth gintegratin
n
n
nc
dtntncdtt
n
nc 2)(
![Page 9: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/9.jpg)
n
ntnct )2()()(
nn
lntncnltnclt )22()())(2()()(
)()2()(2
1)(),( llmcmcltt
m
Constraints on c(n)
![Page 10: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/10.jpg)
n
kk ntnkatf )2(),()(
)(),()(),(
0)(),()(
)()()( :fn detail
01
000
010
nttfnttf
nttgVtg
tftftg
),0()(),(0 nanttf )2(),(),1()(),(1 mtntmanttfm
Orthogonal Projection in Subspaces
n
ntnatf )(),0()(0
n
ntnatf )2(),1()(1 Finer approx
Coarser approx
See next page
![Page 11: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/11.jpg)
)(2
1
)2(),2()()2(),(
mc
mtntncmttn
)2(2
1)2(),(
)2(2
1
)2(),22()(
)2(),)1(2()()2(),1(
nmcmtnt
mc
mtntnc
mtntncmtt
n
n
m
nmcmana
2
)2(),1(),0(
From previous pageFiner coefficients and coarser ones are related by c(n)
![Page 12: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/12.jpg)
0)( dtt
1)(
2dtt
0for basist independenlinearly
)()(),(
W
nntt
0)(),( ntt
Properties of Wavelets
Orthogonality
![Page 13: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/13.jpg)
Two-Scale Relations(wavelet)
0)(
)(2
1)2()()(0
:sidesboth gintegratin
)2()()()( 1
n
nn
n
nd
nddtntnddtt
ntndtVt
![Page 14: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/14.jpg)
)()2()(2
1)(),( llmcmcltt
m
We showed :
Similarly :
)()2()(2
1)(),( llmdmdltt
m
0)2()(2
1)(),(
m
lmcmdltt
Constraints on c(n) and d(n)
![Page 15: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/15.jpg)
),0()(),(
)(),0()(),0(
)()()(
1
001
nbnttf
ntnbntna
tgtftf
nn
Function Reconstruction
m
m
mtntmanttf
mtmatf
)2(),(),1()(),(
)2(),1()(
1
1
See next page
![Page 16: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/16.jpg)
m
nmdmanb
2
)2(),1(),0(
)(2
1
)2(),2()()2(),(
md
mtntndmttn
)2(2
1
)2(),22()()2(),1(
md
mtntndmttn
)2(2
1)2(),( nmdmtnt
Detail coefficients and finer representation are related by d(n)
![Page 17: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/17.jpg)
jj
kj
kjk WVWV
WWWV
WWVWVV
WVV
WVV
1
1011
100112
110
001
Nested Space
VN
VN-1 WN-1
VN-2 WN-2
VN-3 WN-3
![Page 18: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/18.jpg)
Digital Filter Implementation
Use existing methodology in signal processing for discrete wavelet comput
ation
![Page 19: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/19.jpg)
Digital Filter Implementation
)()(~2
)()(
)()(~
2
)()(
Define
ngngnd
ng
nhnhnc
nh
m
nmcmana
2
)2(),1(),0(
m
nmdmanb
2
)2(),1(),0(
Recall
m
mnhmana )2(~
),1(),0(
m
mngmanb )2(~),1(),0(
Then
![Page 20: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/20.jpg)
n
nh 1)(
n
n
n
kngnh
kkngng
kknhnh
0)2()(
)(2
1)2()(
)(2
1)2()(
n
nc 2)(
)()2()(2
1llmdmd
m
0)2()(2
1
m
lmcmd
)()2()(2
1llmcmc
m
n
ng 0)(
n
nd 0)(
![Page 21: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/21.jpg)
)0(
~)2,1()1(
~)1,1()2(
~)0,1()3(
~)1,1(
)2(~
),1()1,0(
hahahaha
mhmaam
)0(
~)0,1()1(
~)1,1()2(
~)2,1()3(
~)3,1(
)(~
),1()0,0(
hahahaha
mhmaam
)0,0(a
)0(~h)1(
~h)2(
~h)3(
~h
)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a
)1,0(a
)0(~h)1(
~h)2(
~h)3(
~h
)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a
two)offactor aby n (decimatio
samples indexed retain and ~
with ),1( Convolve
:),0( getting ...
evenhma
naCoarsening
![Page 22: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/22.jpg)
Similarly, …
two)offactor aby n (decimatio
samples indexed retain and ~ with ),1( Convolve
:),0( getting ...
evengma
nbencethe differComputing
)0,0(b
)0(~g)1(~g)2(~g)3(~g
)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a
)1,0(b
)0(~g)1(~g)2(~g)3(~g
)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a
![Page 23: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/23.jpg)
Signal Reconstruction
m lm l
ll
mltmdlbmltmcla
ltlbltla
tgtftf
)22()(),0()22()(),0(
)(),0()(),0(
)()()( 001
m
mtmct )2()()(
m
mtmdt )2()()(
n ln l
ntlndlbntlnclatf
mln
)2()2(),0()2()2(),0( )(
2 ngSubstituti
1
![Page 24: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/24.jpg)
)2()1,0()1(0)0()0,0()1(0)2()1,0(
)2()1,0()0()0,0()2()1,0()2(),0(
caccacca
cacacalclal
)1(c)0(c)1(c)2(c
0 )0,0(a 0 )1,0(a)1,0( a
)2(c
Subdivision … getting a(1,n):Zero insertion (upsampling) and convolve with 2H
n=0
)2(),1( )(1 ntnatfn
l l
l l
lnglblnhla
lndlblnclana
)2(),0(2)2(),0(2
)2(),0()2(),0(),1(
Hence
![Page 25: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/25.jpg)
)2()1,0()1(0)0()0,0()1(0)2()1,0(
)2()1,0()0()0,0()2()1,0()2(),0(
dbddbddb
dbdbdbldlbl
)1(d)0(d)1(d)2(d
0 )0,0(b 0 )1,0(b)1,0( b
)2(d
Detail part: … getting a(1,n):upsampling and convolve with 2Gn=0
Similarly, …
![Page 26: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/26.jpg)
Notations of Digital Filters
![Page 27: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/27.jpg)
Interpolator and Decimator
nnMxny for )()(
otherwise0
,for )('
kkMnM
ny
nx
![Page 28: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/28.jpg)
H~ H
G~ G
analysis filter bankperfect reconstruction pair:
Whatever goes into analysis bank isrecovered perfectly by the synthesisbank
synthesis filter bank
H~
H
G~
G
![Page 29: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/29.jpg)
Haar Revisited
3,2,1,05379)( nnx
Analysis Filters
0
9
7
3
5 0
8
5
4
2.5
2
0
1
2
-1
2.5
2
-1 0
h(-n)
0.5 0.5
0
-1g(-n)0.5
-0.5
2
1)1()0( hh
2
1)1(,
2
1)0( gg
Haar:
![Page 30: General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3.](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649e4c5503460f94b41c4f/html5/thumbnails/30.jpg)
Haar Revisited
Synthesis Filters
0 1
2 h(n)
1
0
2 g(n)
0
9
7
3
50
8
4
2
0
1
-1
2
-1
1
11