1. An Improved Version of the Inverse Hyperanalytic Wavelet Transform
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher

2. 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”

3. Shift-Invariant Wavelet Transforms
One-Dimensional DWT (1D - DWT)
Undecimated DWT (UDWT)
Dual -Tree Complex Wavelet Transform (DT-CWT)
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”

4. UDWT Advantage Disadvantages Shift-invariant High redundancy
Reduced directional selectivity
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, “An Improved Version of the Inverse Hyperanalytic Wavelet Transform”

5. DT-CWT Advantages Disadvantages 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”

6. 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”

7. IADWT The new implementation:
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, “An Improved Version of the Inverse Hyperanalytic Wavelet Transform”

8. Simulation Results ADWT
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, “An Improved Version of the Inverse Hyperanalytic Wavelet Transform”

9. 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 WT
Degree of invariance
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
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”

10. HWT
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, “An Improved Version of the Inverse Hyperanalytic Wavelet Transform”

11. Simulation Results HWT
Ioana Firoiu, Alexandru Isar, Jean-Marc Boucher, “An Improved Version of the Inverse Hyperanalytic Wavelet Transform”

12. 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”