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 Discrete time Fourier transform (DTFT): Discrete Fourier Transform:

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

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

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

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),

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

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

Frequency domain Interpretation 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 For M = 2, with decimation Note that For L = 2, with interpolation,

Two-band Sub-band Filter

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

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

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

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

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

Perfect Reconstruction Filter Families

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

Daubechie’s Orthogonal Filters

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

3-stage Forward DWT