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MRA 1dim / 2dim

MRA 1dim / 2dim. LL LL. LL HL. HL. LL LH. LL HH. +. +. -. LH. HH. -. +. 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.

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MRA 1dim / 2dim

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  1. MRA 1dim / 2dim LL LL LL HL HL LL LH LL HH + + - LH HH - +

  2. 2-Dim Haar Base FunctionsStandard 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 V2Standard construction y + + - + - + - x + + - + - - + - - + - + + - + + - + - - + - - + - + + - + + - + + - - + + - - + - - + +

  4. 2-Dim Haar Base Functions for V2Standard construction y x + - - + + - - +

  5. Standard Decomposition 1dim L H LL HL H LLL HLL HL H LL LL HL LL HLL HL H

  6. Standard Decomposition 2dim

  7. Standard Decomposition 2dim L H L H L H L H L H L H L H L H

  8. Standard Decomposition 2dim LL HL H LL HL H LL HL H LL HL H LL HL H LL HL H LL HL H LL HL H

  9. Standard Decomposition 2dim LLL HLL HL H LLL HLL HL H LLL HLL HL H LLL HLL HL H LLL HLL HL H LLL HLL HL H LLL HLL HL H LLL HLL HL H

  10. Standard Decomposition 2dim LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H LL LL HL LL HLL HL H

  11. Standard Decomposition 2dim - V2 L H LL HL H LLL LHL LH Row HLL HHL HH Col LL LL LL HL HLL HL LL HL HL HHH HLL HHL HH

  12. Standard Decomposition 2dim - V2 Row Column

  13. Standard Decomposition 2dim - V3 Row Column

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

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

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

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

  18. Standard Decomposition 2dim 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

  19. Standard Reconstruction 2dim 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

  20. 2-Dim Haar Base FunctionsNonstandard 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 V2Non-Standard construction y + + - + - + - x + + - - - + + - + - + + + - + - - - - + - + + + - + + - + + - - - - + +

  22. 2-Dim Haar Base Functions for V2Non-Standard construction y x + - + + - - +

  23. NonStandard Decomposition L H Row Col LL HL LLL LLH HL Row LH HH LH HH Col LL LL LL HL HL LL LH LL HH LH HH

  24. NonStandard Decomposition 2dim - V2 Row Column Row Column

  25. NonStandard Decomposition 2dim - V3 Row Row Column Column Row Column

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

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

  28. NonStandard Decomposition 2dim NonStandardDecomposition ( c[1..2j,1..2j] ) c = c/2j g = 2j 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 Reconstruction 2dim NonStandardReconstruction ( c[1..2j,1..2j] ) g = 2 WHILE g <= 2j 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 = 2jc END

  30. DWTJava Implementation - DWT

  31. DWTJava Implementation - class ImageDWT

  32. DWTJava Implementation - set_hCoefficient

  33. DWTJava Implementation - getGCoeff

  34. DWTJava Implementation - fwd_DWT

  35. DWTJava Implementation - inv_DWT

  36. DWTJava Implementation - fwd_DWT_2D

  37. DWTJava Implementation - inv_DWT_2D

  38. DWTJava Implementation - decomposition_NonStandard

  39. DWTJava Implementation - reconstruction_NonStandard

  40. DWTJava Implementation - transpose

  41. 2D Forward Wavelet Transform

  42. 2D Inverse Wavelet Transform

  43. End

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