1. A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform
Corina Nafornita1, Ioana Firoiu1,2, Dorina Isar1, Jean-Marc Boucher2, Alexandru Isar1
1 Politehnica University of Timisoara, Romania
2 Telecom Bretagne, France

2. Goal Computation of the correlation functions:
inter-scale and inter-band dependency,
inter-scale and intra-band dependency,
intra-scale and intra-band dependency.
Computation of expected value and variance of the wavelet coefficients.
Results useful for the design of different signal processing systems based on the wavelet theory.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

3. 2D-DWT 2D DWT coefficients level m, subband k where
In the DWT implementation: the lines of the input image are filtered using lowpass (m0) and a highpass m1 filter. The two images are decimated.
The columns are filtered in the same manner
The DWT of the input signal is the scalar product between input signal and the 2D wavelet
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

4. D04
For m=0, decomposition on three levels. Details on horizontal, vertical, diagonal.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

5. Expectations m-scale, k-subband
The mean of coefficients in a detail subband is zero; it depends on the mean of the signal for the approximation subband
m-scale, k-subband
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

6. Dependencies intra-scale and intra-band inter-scale and inter-band
inter-scale and intra-band
intra-scale and inter-band
We study dependencies between coefficients from different scales/subbands (3 cases): second order moments, variances
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

7. Inter-scale and Inter-band Correlation
m2 = m1+q, k1 ≠ k2
The inter-scale and inter-band dependency of the wavelet coefficients depends on the:
autocorrelation of the input signal,
intercorrelation of the mother wavelets that generate the sub-bands
For different scales and subbands: intercorrelation of wav.coefficients depends on both :
-autocorrelation of input signal Rx
-intercorrelation of mother wavelets that generate the subbands
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

8. Inter-scale and Inter-band White Gaussian Noise
Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:
Generally the 2D DWT correlates the input signal.
For different scales and subbands: When the input is White Gaussian noise, the intercorrelation depends on intercorrelation of mother wavelets that generate the subbands
So the DWT correlates the signal
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

9. Inter-scale and Intra-band Correlation
m2 = m1+q, k1 =k2=k.
Orthogonal wavelets:
The intercorrelation of the wavelet coefficients depends solely of the autocorrelation of the input signal.
For different scales same subband and orthogonal wavelets, the intercorrelation of wav.coeff.depends only on the autocorrelation of the input signal Rx
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

10. Inter-scale and Intra-band White Gaussian Noise
Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:
The wavelet coefficients with different resolutions of a white Gaussian noise are not correlated inside a sub-band.
For different scales same subband: input =WGN, coefficients are not correlated. The intercorrelation is proportional with unit impulse.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

11. Inter-scale and Intra-band Asymptotic Regime
For infinite number of iterations (asymptotic): decorrelation -> intercorrelation proportional with unit impulse
The intra-band coefficients are asimptotically decorrelated for orthogonal wavelets.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

12. Intra-scale and Intra-band Correlation
m2 = m1= m, k2 = k1= k.
The autocorrelation of the wavelet coefficients depends solely on the autocorrelation of the input signal.
For same subband&scale: autocorrelation depens only on the autocorrelation of input signal Rx
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

13. Intra-scale and Intra-band Variances
For k=1 or 2 or 3 :
For k=4 :
Second order moments for detail subimages and approximation images were computed
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

14. Intra-scale and Intra-band White Gaussian Noise
Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean:
For same band&scale: Input WGN, autocorrelation proportional with unit impulse, variance of wav.coeff is the same with the one of the noise
DWT does not correlate the input
In the same band and at the same scale, the 2D DWT does not correlate the i.i.d. white Gaussian noise.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

15. Intra-scale and Intra-band Asymptotic Regime
For k=1 or 2 or 3 :
Asymptotically the 2D DWT transforms every colored noise into a white one. Hence this transform can be regarded as a whitening system in an intra-band and intra-scale scenario.
For infinite number of iterations: autocorrelations are proportional with unit impulse and the variances have the same values.
Whitening system
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

16. Conclusions 2D DWT : sub-optimal bi-dimensional whitening system.
Contributions
formulas describing inter-scale and inter-band; inter-scale and intra-band and intra-scale and intra-band dependencies of the coefficients of the 2D DWT,
expected values and variances of the wavelet coefficients belonging to the same band and having the same scale.
Use
design of different image processing systems which apply 2D DWT for compression, denoising, watermarking, segmentation, classification…
develop a second order statistical analysis of some complex 2D WTs.
C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”