1 / 16

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

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 , A lexandru Isar 1. 1 Politehnica University of Timisoara, Romania 2 Telecom Bretagne, France. Goal.

Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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”

More Related