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Filterbanks Interpretation of the Discrete Wavelet Transform

Module by: Phil Schniter. E-mail the author

Summary: Introduction to the filterbanks interpretation of the DWT.

Assume that we start with a signal x⁢t∈ ℒ 2 x t ℒ 2 . Denote the best approximation at the 0 th 0 th level of coarseness by x0 ⁢t x0 t . (Recall that x0 ⁢t x0 t is the orthogonal projection of x⁢t x t onto V 0 V 0 .) Our goal, for the moment, is to decompose x0 ⁢t x0 t into scaling coefficients and wavelet coefficients at higher levels. Since x0 ⁢t∈ V 0 x0 t V 0 and V0 = V1 ⊕ W1 V0 V1 W1 , there exist coefficients c 0 ⁢n c 0 n , c 1 ⁢n c 1 n , and d 1 ⁢n d 1 n such that

x 0 ⁢t=∑nn c 0 ⁢n⁢ φ 0 , n ⁢t=∑nn c 1 ⁢n⁢ φ 1 , n ⁢t+∑nn d 1 ⁢n⁢ ψ 1 , n ⁢t x 0 t n n c 0 n φ 0 , n t n n c 1 n φ 1 , n t n n d 1 n ψ 1 , n t
(1)
Using the fact that φ 1 , n ⁢t n∈Z φ 1 , n t n is an orthonormal basis for V 1 V 1 , in conjunction with the scaling equation,
c 1 ⁢n= x 0 ⁢t ·  φ 1 , n ⁢t=∑mm c 0 ⁢m⁢ φ 0 , m ⁢t ·  φ 1 , n ⁢t=∑mm c 0 ⁢m⁢ φ 0 , m ⁢t ·  φ 1 , n ⁢t=∑mm c 0 ⁢m⁢φ⁢t−m · ∑ℓℓh⁢ℓ⁢φ⁢t−ℓ−2⁢n=∑mm c 0 ⁢m⁢∑ ℓ ℓh⁢ℓ⁢φ⁢t−m · φ⁢t−ℓ−2⁢n=∑ m m c 0 ⁢m⁢h⁢m−2⁢n c 1 n x 0 t φ 1 , n t m m c 0 m φ 0 , m t φ 1 , n t m m c 0 m φ 0 , m t φ 1 , n t m m c 0 m φ t m ℓ ℓ h ℓ φ t ℓ 2 n m m c 0 m ℓ ℓ h ℓ φ t m φ t ℓ 2 n m m c 0 m h m 2 n
(2)
where δ⁢t−ℓ−2⁢n=φ⁢t−m · φ⁢t−ℓ−2⁢n δ t ℓ 2 n φ t m φ t ℓ 2 n . The previous expression (Equation 2) indicates that c 1 ⁢n c 1 n results from convolving c 0 ⁢m c 0 m with a time-reversed version of h⁢m h m then downsampling by factor two (Figure 1).

Figure 1
Figure 1 (downsampling1.png)

Using the fact that ψ 1 , n ⁢t n∈Z ψ 1 , n t n is an orthonormal basis for W 1 W 1 , in conjunction with the wavelet scaling equation,

d 1 ⁢n= x0 ⁢t ·  ψ 1 , n ⁢t=∑mm c 0 ⁢m⁢ φ 0 , m ⁢t ·  ψ 1 , n ⁢t=∑ m m c 0 ⁢m⁢ φ 0 , m ⁢t ·  ψ 1 , n ⁢t=∑ m m c 0 ⁢m⁢φ⁢t−m · ∑ ℓ ℓg⁢ℓ⁢φ⁢t−ℓ−2⁢n=∑ m m c 0 ⁢m⁢∑ ℓ ℓg⁢ℓ⁢φ⁢t−m · φ⁢t−ℓ−2⁢n=∑ m m c 0 ⁢m⁢g⁢m−2⁢n d 1 n x0 t ψ 1 , n t m m c 0 m φ 0 , m t ψ 1 , n t m m c 0 m φ 0 , m t ψ 1 , n t m m c 0 m φ t m ℓ ℓ g ℓ φ t ℓ 2 n m m c 0 m ℓ ℓ g ℓ φ t m φ t ℓ 2 n m m c 0 m g m 2 n
(3)
where δ⁢t−ℓ−2⁢n=φ⁢t−m · φ⁢t−ℓ−2⁢n δ t ℓ 2 n φ t m φ t ℓ 2 n .

The previous expression (Equation 3) indicates that d 1 ⁢n d 1 n results from convolving c 0 ⁢m c 0 m with a time-reversed version of g⁢m g m then downsampling by factor two (Figure 2).

Figure 2
Figure 2 (downsampling3.png)

Putting these two operations together, we arrive at what looks like the analysis portion of an FIR filterbank (Figure 3):

Figure 3
Figure 3 (downsampling2.png)

We can repeat this process at the next higher level. Since V 1 = W 2 ⊕ V 2 V 1 W 2 V 2 , there exist coefficients c 2 ⁢n c 2 n and d 2 ⁢n d 2 n such that

x 1 ⁢t=∑nn c 1 ⁢n⁢ φ 1 , n ⁢t=∑nn d 2 ⁢n⁢ ψ 2 , n ⁢t+∑nn c 2 ⁢n⁢ φ 2 , n ⁢t x 1 t n n c 1 n φ 1 , n t n n d 2 n ψ 2 , n t n n c 2 n φ 2 , n t
(4)
Using the same steps as before we find that
c 2 ⁢n=∑mm c 1 ⁢m⁢h⁢m−2⁢n c 2 n m m c 1 m h m 2 n
(5)
d 2 ⁢n=∑mm c 1 ⁢m⁢g⁢m−2⁢n d 2 n m m c 1 m g m 2 n
(6)
which gives a cascaded analysis filterbank (Figure 4):

Figure 4
Figure 4 (cascade_filterbank.png)

If we use V 0 = W 1 ⊕ W 2 ⊕ W 3 ⊕⋯⊕ W k ⊕ V k V 0 W 1 W 2 W 3 ⋯ W k V k to repeat this process up to the k th k th level, we get the iterated analysis filterbank (Figure 5).

Figure 5
Figure 5 (iterated_a_filterbank.png)

As we might expect, signal reconstruction can be accomplished using cascaded two-channel synthesis filterbanks. Using the same assumptions as before, we have:

c 0 ⁢m= x 0 ⁢t ·  φ 0 , m ⁢t=∑nn c 1 ⁢n⁢ φ 1 , n ⁢t+∑nn d 1 ⁢n⁢ ψ 1 , n ⁢t ·  φ 0 , m ⁢t=∑nn c 1 ⁢n⁢ φ 1 , n ⁢t ·  φ 0 , m ⁢t+∑nn d 1 ⁢n⁢ ψ 1 , n ⁢t ·  φ 0 , m ⁢t=∑nn c 1 ⁢n⁢h⁢m−2⁢n+∑nn d 1 ⁢n⁢g⁢m−2⁢n c 0 m x 0 t φ 0 , m t n n c 1 n φ 1 , n t n n d 1 n ψ 1 , n t φ 0 , m t n n c 1 n φ 1 , n t φ 0 , m t n n d 1 n ψ 1 , n t φ 0 , m t n n c 1 n h m 2 n n n d 1 n g m 2 n
(7)
where    h⁢m−2⁢n= φ 1 , n ⁢t ·  φ 0 , m ⁢t where    h m 2 n φ 1 , n t φ 0 , m t and    g⁢m−2⁢n= ψ 1 , n ⁢t ·  φ 0 , m ⁢t and    g m 2 n ψ 1 , n t φ 0 , m t which can be implemented using the block diagram in Figure 6.

Figure 6
Figure 6 (filterbanks_block1.png)

The same procedure can be used to derive

c 1 ⁢m=∑nn c 2 ⁢n⁢h⁢m−2⁢n+∑nn d 2 ⁢n⁢g⁢m−2⁢n c 1 m n n c 2 n h m 2 n n n d 2 n g m 2 n
(8)
from which we get the diagram in Figure 7.

Figure 7
Figure 7 (filterbanks_block2.png)

To reconstruct from the k th k th level, we can use the iterated synthesis filterbank (Figure 8).

Figure 8
Figure 8 (iterated_syn_filterbank.png)

The table makes a direct comparison between wavelets and the two-channel orthogonal PR-FIR filterbanks.

Table 1
Discrete Wavelet Transform 2-Channel Orthogonal PR-FIR Filterbank
Analysis-LPF H⁢z-1 H z -1 H 0 ⁢z H 0 z
Power Symmetry H⁢z⁢H⁢z-1+H⁢−z⁢H⁢−z-1=2 H z H z -1 H z H z -1 2 H 0 ⁢z⁢ H 0 ⁢z-1+ H 0 ⁢−z⁢ H 0 ⁢−z-1=1 H 0 z H 0 z -1 H 0 z H 0 z -1 1
Analysis HPF G⁢z-1 G z -1 H 1 ⁢z H 1 z
Spectral Reverse G⁢z=±⁢z−P⁢H⁢−z-1  ,   P is odd    P P is odd G z ± z P H z -1 H 1 ⁢z=±⁢z−(N−1)⁢ H 0 ⁢−z-1  ,   N is even    N N is even H 1 z ± z N 1 H 0 z -1
Synthesis LPF H⁢z H z G 0 ⁢z=2⁢z−(N−1)⁢ H 0 ⁢z-1 G 0 z 2 z N 1 H 0 z -1
Synthesis HPF G⁢z G z G 1 ⁢z=2⁢z−(N−1)⁢ H 1 ⁢z-1 G 1 z 2 z N 1 H 1 z -1

From the table, we see that the discrete wavelet transform that we have been developing is identical to two-channel orthogonal PR-FIR filterbanks in all but a couple details.

  1. Orthogonal PR-FIR filterbanks employ synthesis filters with twice the gain of the analysis filters, whereas in the DWT the gains are equal.
  2. Orthogonal PR-FIR filterbanks employ causal filters of length NN, whereas the DWT filters are not constrained to be causal.
For convenience, however, the wavelet filters H⁢z H z and G⁢z G z are usually chosen to be causal. For both to have even impulse response length NN, we require that P=N−1 P N 1 .

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