Post on 04-Feb-2016
description
11
MRA 1dim / 2dimMRA 1dim / 2dimMRA 1dim / 2dimMRA 1dim / 2dim
LLLLLLLL
LLLLHLHL
LLLLLHLH
LLLLHHHH
HHHH
HLHL
LHLH
++--++
++--
),(11,1 yx
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
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
++ ---- ++ ++
++--
-- ++--++
--
++--
--++
++--++
++-- ++
--++
++--
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
++--
--++
55
Standard DecompositionStandard Decomposition 1dim1dimStandard DecompositionStandard Decomposition 1dim1dim
LLLLLLLL
HLLL
HLL HL H
HLL
LLLL HL H
LLLLLL HLL HL H
66
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
77
Standard DecompositionStandard Decomposition 2dim2dimStandard DecompositionStandard Decomposition 2dim2dim
HLL
HLL
HLL
HLL
HLL
HLL
HLL
HLL
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
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
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
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
1212
Standard DecompositionStandard Decomposition 2dim - V2dim - V22Standard DecompositionStandard Decomposition 2dim - V2dim - V22
Row
Column
1313
Standard DecompositionStandard Decomposition 2dim - V2dim - V33Standard DecompositionStandard Decomposition 2dim - V2dim - V33
Row
Column
1414
DecompositionDecomposition 1dim1dimDecompositionDecomposition 1dim1dim
Decomposition ( c[1..2j] )
c := c/Sqrt(2j) // normalizeg := 2j
WHILE g >= 2DecompositionStep( c[1..g] )
g := g/2ENDWHILE
END
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
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
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
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
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
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
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
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
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
2424
NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V22NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V22
Row Column
Row Column
2525
Row Row
RowColumn
Column
Column
NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V33NonStandard DecompositionNonStandard Decomposition 2dim - V2dim - V33
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
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
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
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
3030
DWTJava Implementation - DWTDWTJava Implementation - DWT
3131
DWTJava Implementation - class ImageDWTDWTJava Implementation - class ImageDWT
3232
DWTJava Implementation - set_hCoefficientDWTJava Implementation - set_hCoefficient
3333
DWTJava Implementation - getGCoeffDWTJava Implementation - getGCoeff
3434
DWTJava Implementation - fwd_DWTDWTJava Implementation - fwd_DWT
3535
DWTJava Implementation - inv_DWTDWTJava Implementation - inv_DWT
3636
DWTJava Implementation - fwd_DWT_2DDWTJava Implementation - fwd_DWT_2D
3737
DWTJava Implementation - inv_DWT_2DDWTJava Implementation - inv_DWT_2D
3838
DWTJava Implementation - decomposition_NonStandardDWTJava Implementation - decomposition_NonStandard
3939
DWTJava Implementation - reconstruction_NonStandardDWTJava Implementation - reconstruction_NonStandard
4040
DWTJava Implementation - transposeDWTJava Implementation - transpose
4141
2D Forward Wavelet Transform2D Forward Wavelet Transform 2D Forward Wavelet Transform2D Forward Wavelet Transform
4242
2D Inverse Wavelet Transform 2D Inverse Wavelet Transform 2D Inverse Wavelet Transform 2D Inverse Wavelet Transform
4343
EndEnd