A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform

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 C. Nafornita, I. Firoiu,D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

4. D04 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 C. Nafornita, I. Firoiu,D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform”

6. inter-scale and inter-band inter-scale and intra-band intra-scale and inter-band Dependencies intra-scale and intra-band 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-bandCorrelation 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 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-bandWhite Gaussian Noise • Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean: • Generally the2D DWT correlates the input 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-bandCorrelation m2 = m1+q, k1 =k2=k. Orthogonal wavelets: The intercorrelation of the wavelet coefficients depends solely of theautocorrelation of the input signal. 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-bandWhite Gaussian Noise Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean: The wavelet coefficients withdifferent resolutionsof a white Gaussian noiseare not correlated inside a sub-band. 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-bandAsymptotic Regime The intra-band coefficientsareasimptotically decorrelatedfor 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 theautocorrelation of the input signal. 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-bandVariances For k=1 or 2 or 3 : For k=4 : 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-bandWhite Gaussian Noise Input image: bi-dimensional i.i.d. white Gaussian noise with variance and zero mean: 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-bandAsymptotic 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. 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”