Fast Algorithms for
Discrete Wavelet Transform
Review and Implementation
by Dan Li
F2000
DWT and FWT: Significance
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DWT
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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
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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
FWT Algorithm: An Outline
Mallat Straightfoward Filter Bank
“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
FWT Algorithm: An Overview (I)
2
H(z)
Mallat Filter Bank
D1
x[n]
2
H(z)
2
G(z)
A1
A2
Transversal Filters
x0[n]
G1(z)
2
x1[n-1]
2
Short-length Filters
H0(z)
x[n]
z-1
x[n]
A1

H1(z)
- 
+
2

2
D1
H0(z)
D1
- 
+
H0(z)+H1(z)
z-1
z-1
+

H1(z)
H(z)
2
D3
G(z)
2
A3
2
G(z)
G0(z)
D2
-
A1
+
FWT Algorithm: An Overview (II)
c0

c1

c2

-c2

c1

-c0

(1+z-1)2
2
Binomial QMF
x[n]
(1+z-1)3
(1+z-1)(1-z-1)
x0[n]
(1-z-1)2
(1+z-1)2
z-1
2
x1[n-1]
2
Classical Lattice
x[n]
z-1
2
-2-4
CORDIC Lattice
(1-z-1)2
cos  0



K’
sin0
-sin 0
2
2-16
x[n]
z-1
s(0)
2
x0[n]
x[n]
x1[n-1]
z-1
2
d(0)

z-1
q1(z)
d(1)
p2(z)



cos L
1



-1 
-2-2 
K’ 1
2-2 

2





1
1
-0
-2
s(1)
s(2)


p1(z)

1

D1
sinL
-sin L
...


cos 1
z-1

cos L


cos 1


sin1
-sin 1


cos 0

2-8
Lifting Ladder
(1+z-1)(1-z-1)
(1-z-1)3
A1

q2(z) ...
d(2)
pM(z)

1
A1
-1
D1



1
2

A1
2-2 
-2-2 
z-1

s(M)
D1
K1
A1
K2
D1
qM(z)
d(M)
Comput. Complexity: A Comparison
Arithmetic Complexity (per input point & per decomposition cell)
Mallat FB
Tran.Filter (FFT)
Tran.Filter(Short-length)
35
# of
mults
30
25
# of
20
adds
15
10
5
10
5
0
0
10
20
Filter length L
Tran.Filter (FFT)
30
25
20
15
0
Mallat FB
35
30
40
Tran.Filter(Short-length)
0
10
20
Filter length L
30
Computational Structure Complexity (per filter coefficient)
Tran.Filter (FFT)
50
40
# of
adders 30
needed 20
Tran.Filter(Short-length)
Binomial QMF filters
10
Classical Lattice
0
5
10
15
Word length w (bits)
20 CORDIC Lattice
40
Comments, Implementations, etc.
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“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.
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Implementation focused on the
following:
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Fast filtering for short and long
filters
Various formats of polyphase
structures
Reformulation of polyphase
transversal filter with the
consideration of reduced interchannel communication
Integer filter and IWT
implementations
Simulations