An Improved Version of the Inverse Hyperanalytic Wavelet ...

An Improved Version of the Inverse Hyperanalytic Wavelet ...

An Improved Version of the Inverse Hyperanalytic Wavelet Transform Ioana Firoiu, Alexandru Isar, JeanMarc Boucher Introduction Wavelet techniques based on the Discrete Wavelet Transform (DWT) Advantages Sparsity of coefficients Disadvantages Shift-sensitivity (input signal shift unpredictable change in the output coefficients) Poor directional selectivity Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 2

Shift-Invariant Wavelet Transforms One-Dimensional DWT (1D - DWT) Undecimated DWT (UDWT) Dual -Tree Complex Wavelet Transform (DTCWT) Analytical DWT Two-Dimensional DWT (2D - DWT) 2D UDWT 2D DT-CWT Hyperanalytical Wavelet Transform (HWT) Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 3 UDWT Advantage

Shift-invariant Disadvantages High redundancy Reduced directional selectivity Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 4 DT-CWT Advantages Quasi shift-invariant Good directional selectivity Disadvantages

Redundancy Filters from the 2nd branch can be only approximated Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 5 ADWT DWT at whose entry we apply the analytical signal defined as: xa=x+iH{x} where H{x} denotes the Hilbert transform of x Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse

Hyperanalytic Wavelet Transform 6 IADWT The new implementation: Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 7 Simulation Results ADWT Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 8

Objective Comparison Degree of invariance: Grad = 1 d/m d standard deviation and m mean of the sequences of energies of a certain type of coefficients corresponding to 16 shifts Degree of invariance WT W. lev. 1 W. lev. 2 W. lev. 3 W. lev. 4 Sn.lev. 4

ADWT new 0.9967 0.9995 0.9986 0.9988 0.9990 DT CWT 1.0000 0.9811 0.9749

0.9734 0.9997 ADWT old 0.9969 0.9982 0.9981 0.9983 0.9995 DWT 0.9236

0.8265 0.7878 0.8149 0.9958 Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 9 HWT HWT f x, y f x, y , a x, y . a x, y x, y iH x x, y

j H y x, y k H x H y x, y i 2 j 2 k 2 1, and ij ji k HWT f x, y DWT f x, y iDWT H x f x, y jDWT H y f x, y

kDWT H y H x f x, y f a x, y , x, y DWT f a x, y . Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 10 Simulation Results HWT Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 11 Conclusion The new implementation of the IHWT has

a better shift-invariance Its application in image denoising slightly improves the results obtained applying the old implementation of the IHWT Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, An Improved Version of the Inverse Hyperanalytic Wavelet Transform 12

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