1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -

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


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


77


HLL

HLL

HLL

HLL

HLL

HLL

HLL

HLL

88


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


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


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


Row

Column

1313


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


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


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


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

1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -

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Transcript of 1 MRA 1dim / 2dim LLLLLLHL LLLHLLHH HH HL LH + - + + -