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11
MRA 1dim / 2dimMRA 1dim / 2dimMRA 1dim / 2dimMRA 1dim / 2dim
LLLLLLLL
LLLLHLHL
LLLLLHLH
LLLLHHHH
HHHH
HLHL
LHLH
++--++
++--
),(11,1 yx
![Page 2: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/2.jpg)
22
2-Dim Haar Base Functions2-Dim Haar Base FunctionsStandard constructionStandard construction2-Dim Haar Base Functions2-Dim Haar Base FunctionsStandard constructionStandard construction
The standard construction of a two-dimensional wavelet basisconsists of all possible tensor products of one-dimensional basis functions.
)()(
)()(
)()(
)()(
2
2
1
1
2
2
1
1
2
2
1
1
00
00
yx
yx
yx
yx
jk
jk
jk
jk
jk
jk
![Page 3: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/3.jpg)
33
2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Standard constructionStandard construction2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Standard constructionStandard construction
)()( 00
00 yx
++
)()( 00
00 yx )()( 1
000 yx )()( 1
100 yx
++ -- ++ -- ++ --x
y
++
--
)()( 00
00 yx
++--
)()( 00
10 yx
)()( 00
11 yx
++--
)()( 00
00 yx
++ --
-- ++
++ --
++--
)()( 10
00 yx
++
--
--
++
)()( 11
00 yx
)()( 00
10 yx )()( 1
010 yx )()( 1
110 yx
)()( 00
11 yx )()( 1
011 yx )()( 1
111 yx
++ ---- ++ ++
++--
-- ++--++
--
++--
--++
++--++
++-- ++
--++
++--
![Page 4: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/4.jpg)
44
2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Standard constructionStandard construction2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Standard constructionStandard construction
x
y
)()( 00
00 yx
++ --
-- ++
)()( 00
11 yx
++--
--++
![Page 5: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/5.jpg)
55
Standard DecompositionStandard Decomposition 1dim1dimStandard DecompositionStandard Decomposition 1dim1dim
LLLLLLLL
HLLL
HLL HL H
HLL
LLLL HL H
LLLLLL HLL HL H
![Page 6: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/6.jpg)
66
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
![Page 7: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/7.jpg)
77
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
HLL
HLL
HLL
HLL
HLL
HLL
HLL
HLL
![Page 8: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/8.jpg)
88
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
LLLL HL H
LLLL HL H
LLLL HL H
LLLL HL H
LLLL HL H
LLLL HL H
LLLL HL H
LLLL HL H
![Page 9: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/9.jpg)
99
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
LLLLLL HLL HL H
![Page 10: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/10.jpg)
1010
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
LLLLLLLL
HLLL
HLL HL H
![Page 11: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/11.jpg)
1111
LL HH LLLL HLHL HH
LLLLLL
HLLHLL
LHLLHL
HHLHHL
LHLH
HHHH
LLLLLLLL
LLLLHLHL
HLHLLLLL
HLHLHLHL
HLLHLL HHLHHL HHHH
Row
Col
HHHHHH
HLLHLL
Standard DecompositionStandard Decomposition 2dim - V2dim - V22Standard DecompositionStandard Decomposition 2dim - V2dim - V22
![Page 12: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/12.jpg)
1212
Standard DecompositionStandard Decomposition 2dim - V2dim - V22Standard DecompositionStandard Decomposition 2dim - V2dim - V22
Row
Column
![Page 13: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/13.jpg)
1313
Standard DecompositionStandard Decomposition 2dim - V2dim - V33Standard DecompositionStandard Decomposition 2dim - V2dim - V33
Row
Column
![Page 14: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/14.jpg)
1414
DecompositionDecomposition 1dim1dimDecompositionDecomposition 1dim1dim
Decomposition ( c[1..2j] )
c := c/Sqrt(2j) // normalizeg := 2j
WHILE g >= 2DecompositionStep( c[1..g] )
g := g/2ENDWHILE
END
![Page 15: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/15.jpg)
1515
DecompositionStepDecompositionStep 1dim1dimDecompositionStepDecompositionStep 1dim1dim
DecompositionStep ( c[1..2j] )
FOR i := 1 TO 2j / 2c’[i] := ( c[2i-1] + c[2i] ) / Sqrt(2)c’[2j/2 + i] := ( c[2i-1] - c[2i] ) / Sqrt(2)
ENDFOR
c := c’
END
![Page 16: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/16.jpg)
1616
ReconstructionReconstruction 1dim1dimReconstructionReconstruction 1dim1dim
Reconstruction ( c[1..2j] )
g := 2
WHILE g <= 2j
ReconstructionStep( c[1..g] ) g := 2 * g
ENDWHILE
c := c * Sqrt(2j) // undo normalizationEND
![Page 17: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/17.jpg)
1717
ReconstructionStep 1dimReconstructionStep 1dim
ReconstructionStep ( c[1..2j] )
FOR i := 1 TO 2j / 2c’[2i - 1] := ( c[i] + c[2j / 2 + i] ) / Sqrt(2)c’[2i] := ( c[i] - c[2j / 2 + i] ) / Sqrt(2)
ENDFOR
c := c’
END
![Page 18: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/18.jpg)
1818
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
StandardDecomposition ( c[1..2j,1..2k] )
FOR row := 1 TO 2j
Decomposition ( c[row,1..2k] )ENDFOR
FOR col := 1 TO 2k
Decomposition ( c[1..2j,col] )ENDFOR
END
![Page 19: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/19.jpg)
1919
Standard ReconstructionStandard Reconstruction 2dim2dimStandard ReconstructionStandard Reconstruction 2dim2dim
StandardReconstruction ( c[1..2j,1..2k] )
FOR col := 1 TO 2k
Reconstruction ( c[1..2j,col] )ENDFOR
FOR row := 1 TO 2j
Reconstruction ( c[row,1..2k] )ENDFOR
END
![Page 20: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/20.jpg)
2020
2-Dim Haar Base Functions2-Dim Haar Base FunctionsNonstandard constructionNonstandard construction2-Dim Haar Base Functions2-Dim Haar Base FunctionsNonstandard constructionNonstandard construction
The nonstandard construction of a two-dimensional wavelet basisconsists of a single coarse scaling function along with scales and translatesof three wavelet functions
)2,2(2),(
)2,2(2),(
)2,2(2),(
),(),(
,
,
,
00,0
lykxyx
lykxyx
lykxyx
yxyx
jjjjlk
jjjjlk
jjjjlk
)()(),(
)()(),(
)()(),(
)()(),(
yxyx
yxyx
yxyx
yxyx
![Page 21: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/21.jpg)
2121
2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Non-Standard constructionNon-Standard construction2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Non-Standard constructionNon-Standard construction
),(00,0 yx
++
),(00,0 yx
++ --x
y
++
--
++ --
-- ++ ++ -- --++
++-- ++
++--
--
++--++
++--
),(00,0 yx
++ -- ++ --
),(10,0 yx
++--
--++
),(10,1 yx
),(00,0 yx
),(10,0 yx ),(1
1,0 yx
),(11,0 yx
--++
),(11,1 yx
),(10,1 yx ),(11,1 yx
),(10,0 yx
++++--
--
),(10,1 yx
++--++
++--
),(11,0 yx
),(11,1 yx
![Page 22: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/22.jpg)
2222
2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Non-Standard constructionNon-Standard construction2-Dim Haar Base Functions for V2-Dim Haar Base Functions for V22
Non-Standard constructionNon-Standard construction
x
y
),(10,0 yx
++--
++--++
++--
),(11,1 yx
![Page 23: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/23.jpg)
2323
NonStandard DecompositionNonStandard DecompositionNonStandard DecompositionNonStandard Decomposition
LL HH
LLLLLL LLHLLH HLHL
HHHH
LLLLLLLL
LLLLHLHL
LLLLLHLH
LLLLHHHH
HHHH
Row
Col
LLLL HLHL
LHLH HHHH LHLH
HLHL
LHLH
Row
Col
![Page 24: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/24.jpg)
2424
NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V22NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V22
Row Column
Row Column
![Page 25: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/25.jpg)
2525
Row Row
RowColumn
Column
Column
NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V33NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V33
![Page 26: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/26.jpg)
2626
DecompositionStepDecompositionStep 1dim1dimDecompositionStepDecompositionStep 1dim1dim
DecompositionStep ( c[1..2j] )
FOR i := 1 TO 2j / 2c’[i] := ( c[2i-1] + c[2i] ) / Sqrt(2)c’[2j/2 + i] := ( c[2i-1] - c[2i] ) / Sqrt(2)
ENDFOR
c := c’
END
![Page 27: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/27.jpg)
2727
ReconstructionStep 1dimReconstructionStep 1dim
ReconstructionStep ( c[1..2j] )
FOR i := 1 TO 2j / 2c’[2i - 1] := ( c[i] + c[2j / 2 + i] ) / Sqrt(2)c’[2i] := ( c[i] - c[2j / 2 + i] ) / Sqrt(2)
ENDFOR
c := c’
END
![Page 28: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/28.jpg)
2828
NonStandard DecompositionNonStandard Decomposition 2dim2dimNonStandard DecompositionNonStandard Decomposition 2dim2dim
NonStandardDecomposition ( c[1..2j,1..2j] )
c = c/2j
g = 2j
WHILE g >= 2FOR row = 1 TO g
DecompositionStep(c[row,1..g])ENDFOR
FOR col = 1 TO gDecompositionStep(c[1..g,col])
ENDFORg = g / 2
ENDWHILE
END
![Page 29: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/29.jpg)
2929
NonStandard ReconstructionNonStandard Reconstruction 2dim2dimNonStandard ReconstructionNonStandard Reconstruction 2dim2dim
NonStandardReconstruction ( c[1..2j,1..2j] )
g = 2WHILE g <= 2j
FOR col = 1 TO gReconstructionStep(c[1..g,col])
ENDFOR FOR row = 1 TO g
ReconstructionStep(c[row,1..g])ENDFORg = 2g
ENDWHILE c = 2jc
END
![Page 30: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/30.jpg)
3030
DWTJava Implementation - DWTDWTJava Implementation - DWT
![Page 31: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/31.jpg)
3131
DWTJava Implementation - class ImageDWTDWTJava Implementation - class ImageDWT
![Page 32: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/32.jpg)
3232
DWTJava Implementation - set_hCoefficientDWTJava Implementation - set_hCoefficient
![Page 33: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/33.jpg)
3333
DWTJava Implementation - getGCoeffDWTJava Implementation - getGCoeff
![Page 34: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/34.jpg)
3434
DWTJava Implementation - fwd_DWTDWTJava Implementation - fwd_DWT
![Page 35: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/35.jpg)
3535
DWTJava Implementation - inv_DWTDWTJava Implementation - inv_DWT
![Page 36: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/36.jpg)
3636
DWTJava Implementation - fwd_DWT_2DDWTJava Implementation - fwd_DWT_2D
![Page 37: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/37.jpg)
3737
DWTJava Implementation - inv_DWT_2DDWTJava Implementation - inv_DWT_2D
![Page 38: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/38.jpg)
3838
DWTJava Implementation - decomposition_NonStandardDWTJava Implementation - decomposition_NonStandard
![Page 39: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/39.jpg)
3939
DWTJava Implementation - reconstruction_NonStandardDWTJava Implementation - reconstruction_NonStandard
![Page 40: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/40.jpg)
4040
DWTJava Implementation - transposeDWTJava Implementation - transpose
![Page 41: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/41.jpg)
4141
2D Forward Wavelet Transform2D Forward Wavelet Transform 2D Forward Wavelet Transform2D Forward Wavelet Transform
![Page 42: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/42.jpg)
4242
2D Inverse Wavelet Transform 2D Inverse Wavelet Transform 2D Inverse Wavelet Transform 2D Inverse Wavelet Transform
![Page 43: 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -](https://reader036.vdocument.in/reader036/viewer/2022062806/5697bfd11a28abf838caaebc/html5/thumbnails/43.jpg)
4343
EndEnd