1 / 23

Multi-Resolution Analysis

Multi-Resolution Analysis. Non-stationary Property of Natural Image. Pyramidal Image Structure. Z-transform. The z-transform is the discrete time version of Laplace transform. Given a sequence {x(n)}, its z-transform is: In particular,. Z-Transform and Fourier Transform.

zariel
Télécharger la présentation

Multi-Resolution Analysis

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. Multi-Resolution Analysis

  2. Non-stationary Property of Natural Image

  3. Pyramidal Image Structure

  4. Z-transform The z-transform is the discrete time version of Laplace transform. Given a sequence {x(n)}, its z-transform is: In particular,

  5. Z-Transform and Fourier Transform • Discrete time Fourier transform (DTFT): • Discrete Fourier Transform:

  6. Discrete time Fourier transform (DTFT) Discrete Fourier Transform (DFT) Frequency Domain Representation Im{z} Re{z} Z-plane

  7. Sub-sequence of a Finite Sequence Let x(n) = x(0) x(1) x(2) x(3) x(4) x(5) x(6) … x0(n) = x(0) 0 x(2) 0 x(4) 0 x(6) … x1(n) = 0 x(1) 0 x(3) 0 x(5) 0 … Then, clearly, x0(n) + x1(n) = x(n), and x0(n) = [x(n) + (-1)nx(n)]/2, x1(n) = [x(n) - (-1)nx(n)]/2 Denote X(z) to be the Z-transform of x(n), then

  8. Define Let WM =exp(j2/M), then One may write Z transform of a Sub-sequence

  9. Decimation (down-sample) M-fold decimator yk(n) = x(Mn+k) = xk(Mn+k) , 0  k  M-1 Example. M = 2. y0(n) = x(2n), y1(n) = x(2n+1),

  10. Interpolation (up-sample) L-fold Expander Example. L = 2. {zL(n)} ={x(0), 0, x(1), 0, x(2), 0, …} and

  11. Frequency Scaling X(jw) -4p -2p 0 2p 4p X(jw/2) -4p -2p 0 2p 4p X(j2w) -4p -2p 0 2p 4p

  12. For M = 2, with decimation Note that For L = 2, with interpolation, In general, M-fold down-samples will stretch the spectrum M-times followed by a weighted sum. This may cause the aliasing effect. L-fold up-sample will compress the spectrum L times Frequency domain Interpretation

  13. Two-band Sub-band Filter

  14. x(n) v0(n) y0(n) H0(z) 2 z-1 v1(n) y1(n) 2 H1(z) u0(n) v0(n) 2 + G0(z) u1(n) 2 G1(z) v1(n) Filter-banks

  15. x(n) v0(n) y0(n) H0(z) 2 z-1 v1(n) y1(n) 2 H1(z) Frequency Response

  16. Frequency Domain Interpretation |X(jw)|= |X(ejw)| w -p p 2p -2p 0 |X(jw)Ho(jw)|=|X(j(w+2p))Ho(j(w+2p))|=|Y0(jw)| -p p 2p -2p w 0 |X(jw/2)Ho(jw/2)|=|X(j(w+p)/2)Ho(j(w+p)/2)|=|V0(jw)| -p p 2p -2p 0 w

  17. Frequency Domain Interpretation |X(jw)|= |X(ejw)| w -p p 2p -2p 0 |X(jw)H1(jw)|=|X(j(w+2p))H1(j(w+2p))|=|Y1(jw)| -p p 2p -2p w 0 |X(jw/2)H1(jw/2)|=|X(j(w+p)/2)H1(j(w+p)/2)|=|V1(jw)| -p p 2p -2p 0 w

  18. Perfect Reconstruction Desired PR (perfect reconstruction) condition: Implies: It can be shown that H0(z): low pass filter, H1(z): high pass filter Usually, both are chosen to be FIR filters

  19. Perfect Reconstruction Filter Families

  20. 2D Sub-band Filter 2-D four-band filter bank for sub-band image coding

  21. Daubechie’s Orthogonal Filters

  22. Sub-band Decomposition Example A 4-band split of the vase in fig.7.1 using sub-band coding system of Fig. 7.5

  23. 3-stage Forward DWT

More Related