Communications-2010, Bucharest, June 11 A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform Corina Nafornita 1, Ioana Firoiu 1,2, Dorina Isar 1, Jean-Marc Boucher 2, Alexandru Isar 1 1 Politehnica University of Timisoara, Romania 2 Telecom Bretagne, France

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 2/16 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.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 3/16 2D-DWT 2D DWT coefficients level m, subband k where

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 4/16 D04D04

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 5/16 Expectations m-scale, k-subband

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 6/16 Dependencies intra-scale and intra-band inter-scale and inter-band inter-scale and intra-band intra-scale and inter-band

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 7/16 Inter-scale and Inter-band Correlation m 2 = m 1 +q, k 1 ≠ k 2 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

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 8/16 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.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 9/16 Inter-scale and Intra-band Correlation m 2 = m 1 +q, k 1 =k 2 =k. Orthogonal wavelets: The intercorrelation of the wavelet coefficients depends solely of the autocorrelation of the input signal.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 10/16 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.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 11/16 Inter-scale and Intra-band Asymptotic Regime The intra-band coefficients are asimptotically decorrelated for orthogonal wavelets.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 12/16 Intra-scale and Intra-band Correlation m 2 = m 1 = m, k 2 = k 1 = k. The autocorrelation of the wavelet coefficients depends solely on the autocorrelation of the input signal.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 13/16 Intra-scale and Intra-band Variances For k=1 or 2 or 3 : For k=4 :

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 14/16 Intra-scale and Intra-band White 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.

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 15/16 Intra-scale and Intra-band Asymptotic Regime 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 k=1 or 2 or 3 :

Communications-2010, Bucharest, June 11 C. Nafornita, I. Firoiu, D. Isar, J.-M. Boucher, A. Isar, “A Second Order Statistical Analysis of the 2D Discrete Wavelet Transform” 16/16 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.