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

2 2-Dim Haar Base Functions Standard construction The standard construction of a two-dimensional wavelet basis consists of all possible tensor products of one-dimensional basis functions.

3 2-Dim Haar Base Functions for V 2 Standard construction x y

4 x y

5 Standard Decomposition1dim LLLL HL LL HLLHLH HL LL H LLLHLLHLH

6 Standard Decomposition2dim

7 HL HL HL HL HL HL HL HL

8 LLHLH LL H LL H LL H LL H LL H LL H LL H

9 Standard Decomposition2dim LLLHLLHLH LLLHLLHLH LLLHLLHLH LLLHLLHLH LLLHLLHLH LLLHLLHLH LLLHLLHLH LLLHLLHLH

10 Standard Decomposition2dim LLLL HL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH LLLL LL HLLHLH

11 LHLLHLH LLL HLL LHL HHL LH HH LLLLLLHL HLLLHLHL HLLHHLHH Row Col HHH HLL Standard Decomposition2dim - V 2

12 Standard Decomposition2dim - V 2 Row Column

13 Standard Decomposition2dim - V 3 Row Column

14 Decomposition1dim Decomposition ( c[1..2 j ] ) c := c/Sqrt(2 j ) // normalize g := 2 j WHILE g >= 2 DecompositionStep( c[1..g] ) g := g/2 ENDWHILE END

15 DecompositionStep1dim DecompositionStep ( c[1..2 j ] ) FOR i := 1 TO 2 j / 2 c’[i] := ( c[2i-1] + c[2i] ) / Sqrt(2) c’[2 j /2 + i] := ( c[2i-1] - c[2i] ) / Sqrt(2) ENDFOR c := c’ END

16 Reconstruction1dim Reconstruction ( c[1..2 j ] ) g := 2 WHILE g <= 2 j ReconstructionStep( c[1..g] ) g := 2 * g ENDWHILE c := c * Sqrt(2 j )// undo normalization END

17 ReconstructionStep1dim ReconstructionStep ( c[1..2 j ] ) FOR i := 1 TO 2 j / 2 c’[2i - 1] := ( c[i] + c[2 j / 2 + i] ) / Sqrt(2) c’[2i] := ( c[i] - c[2 j / 2 + i] ) / Sqrt(2) ENDFOR c := c’ END

18 Standard Decomposition2dim StandardDecomposition ( c[1..2 j,1..2 k ] ) FOR row := 1 TO 2 j Decomposition ( c[row,1..2 k ] ) ENDFOR FOR col := 1 TO 2 k Decomposition ( c[1..2 j,col] ) ENDFOR END

19 Standard Reconstruction2dim StandardReconstruction ( c[1..2 j,1..2 k ] ) FOR col := 1 TO 2 k Reconstruction ( c[1..2 j,col] ) ENDFOR FOR row := 1 TO 2 j Reconstruction ( c[row,1..2 k ] ) ENDFOR END

20 2-Dim Haar Base Functions Nonstandard construction The nonstandard construction of a two-dimensional wavelet basis consists of a single coarse scaling function along with scales and translates of three wavelet functions

21 2-Dim Haar Base Functions for V 2 Non-Standard construction ++- x y

22 2-Dim Haar Base Functions for V 2 Non-Standard construction x y

23 NonStandard Decomposition LH LLLLLHHL HH LLLLLLHL LLLHLLHH HH Row Col LLHL LHHHLH HL LH Row Col

24 NonStandard Decomposition2dim - V 2 RowColumn RowColumn

25 Row Column NonStandard Decomposition2dim - V 3

26 DecompositionStep1dim DecompositionStep ( c[1..2 j ] ) FOR i := 1 TO 2 j / 2 c’[i] := ( c[2i-1] + c[2i] ) / Sqrt(2) c’[2 j /2 + i] := ( c[2i-1] - c[2i] ) / Sqrt(2) ENDFOR c := c’ END

27 ReconstructionStep1dim ReconstructionStep ( c[1..2 j ] ) FOR i := 1 TO 2 j / 2 c’[2i - 1] := ( c[i] + c[2 j / 2 + i] ) / Sqrt(2) c’[2i] := ( c[i] - c[2 j / 2 + i] ) / Sqrt(2) ENDFOR c := c’ END

28 NonStandard Decomposition2dim NonStandardDecomposition ( c[1..2 j,1..2 j ] ) c = c/2 j g = 2 j WHILE g >= 2 FOR row = 1 TO g DecompositionStep(c[row,1..g]) ENDFOR FOR col = 1 TO g DecompositionStep(c[1..g,col]) ENDFOR g = g / 2 ENDWHILE END

29 NonStandard Reconstruction2dim NonStandardReconstruction ( c[1..2 j,1..2 j ] ) g = 2 WHILE g <= 2 j FOR col = 1 TO g ReconstructionStep(c[1..g,col]) ENDFOR FOR row = 1 TO g ReconstructionStep(c[row,1..g]) ENDFOR g = 2g ENDWHILE c = 2 j c END

30 DWT Java Implementation - DWT

31 DWT Java Implementation - class ImageDWT

32 DWT Java Implementation - set_hCoefficient

33 DWT Java Implementation - getGCoeff

34 DWT Java Implementation - fwd_DWT

35 DWT Java Implementation - inv_DWT

36 DWT Java Implementation - fwd_DWT_2D

37 DWT Java Implementation - inv_DWT_2D

38 DWT Java Implementation - decomposition_NonStandard

39 DWT Java Implementation - reconstruction_NonStandard

40 DWT Java Implementation - transpose

41 2D Forward Wavelet Transform

42 2D Inverse Wavelet Transform

43 End