1. Fast Algorithms for Discrete Wavelet Transform
Review and Implementation
by Dan Li
F2000

2. DWT and FWT: Significance
Multi-resolution mode to access the information
Extensively (and intensively) used in information processing
Advantage over other transforms
e.g. (in JPEG 2000), DWT provides 20-30% improvement in compression efficiency as oppose to DCT.
FWT
Multi-resolution mode to access the information
DWT: intensive computation and large memory requirement.
FWT makes DWT practicable in real applications
Main factors controlling the speed of DWT:
Filter length
Floating point operation vs. integer operation

3. FWT Algorithm: An Outline
Mallat Straightfoward Filter Bank
“Regular”
Structure
“Regular”
Structure
Polyphase Transversal Filters#
(* based on FFT for fast filtering)
Polyphase Short-length Filters*
(* also known as “fast-running FIR algo”)
Binomial QMF Filters
Classical Lattice Filters
“Irregular”
Structure
CORDIC Lattice Filters
Lifting Scheme and Integer WT

4. FWT Algorithm: An Overview (I)
H(z)
G(z)
x[n] D1 D2 D3 A3 A2 A1
 2
Mallat Filter Bank A1 D1
G0(z)
x[n]
 2
H0(z)
G1(z)
H1(z) 
x0[n]
x1[n-1]
z-1
Transversal Filters
x[n]
 2 -
z-1
H0(z)
H0(z)+H1(z)
H1(z)  + D1 A1
Short-length Filters

5. FWT Algorithm: An Overview (II)c0 A1 D1
x[n]
 2
x0[n]
x1[n-1]
z-1
(1+z-1)3
(1-z-1)3
(1+z-1)2
(1+z-1)(1-z-1)
(1-z-1)2   c1 c2
-c0
-c2
Binomial QMF
x[n]
 2
z-1 K’  
cos 0
sin0
-sin0
cos 1
sin1
-sin1
... A1 D1
cos L
sinL
-sinL -1
Classical Lattice A1 D1  1 
2-2
-2-2
z-1
 2 K’ -1
x[n]
-2-4
2-8
2-16
-0
-2 2
CORDIC Lattice
x[n]
 2
x0[n]
x1[n-1]
z-1
... 
p1(z)
q1(z)
p2(z)
q2(z)
pM(z)
qM(z) K1 K2 A1 D1
s(0)
d(0)
s(1)
d(1)
s(2)
d(2)
s(M)
d(M)
Lifting Ladder

6. Comput. Complexity: A Comparison
Arithmetic Complexity (per input point & per decomposition cell)
# of
mults
# of
adds
Filter length L
Filter length L
Computational Structure Complexity (per filter coefficient)
# of
adders
needed
Word length w (bits)

7. Comments, Implementations, etc.
“Efficiency” in the sense of arithmetic complexity and computational structure
Straightforward filter bank: classical and used in many commercial s/w
Polyphase structure: more efficient than direct FB. (Worthy of further exploration!)
FFT based filtering: efficient for medium or long filters
Fast running FIR filter: good for short filters
Binomial QMF: reduces the # of mults with expense of additional adds
Lattice: easier to implement with each relatively simpler stages
CORDIC: most suitable fore efficient VLSI implementation since only addition and shifts involved and least possible adders required
Lifting scheme: lead to IWT which is faster than floating-point DWT and ideal for lossless coding/compression.
Implementation focused on the following:
Fast filtering for short and long filters
Various formats of polyphase structures
Reformulation of polyphase transversal filter with the consideration of reduced inter-channel communication
Integer filter and IWT implementations
Simulations