Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 1 Subband and wavelet coding Vector convolution, convolutional transforms Filter banks vs. vector space interpretation Orthogonal and biorthogonal subband transforms DCT as a filter bank Lapped Orthogonal Transform (LOT) Discrete Wavelet Transform (DWT) Quadrature mirror filters and conjugate quadrature filters Lifting implementation/design of the DWT Embedded zero-tree coding of wavelet coefficients
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Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 1
Subband and wavelet coding
Vector convolution, convolutional transforms
Filter banks vs. vector space interpretation
Orthogonal and biorthogonal subband transforms
DCT as a filter bank
Lapped Orthogonal Transform (LOT)
Discrete Wavelet Transform (DWT)
Quadrature mirror filters and conjugate quadrature filters
Lifting implementation/design of the DWT
Embedded zero-tree coding of wavelet coefficients
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 2
Subband coding: motivation
Coding with block-wise transform introduces visible
blocking artifacts, as bit-rate decreases.
Can we, somehow, overlap adjacent blocks,
thereby smoothing block boundaries,
but without increasing the number of transform
coefficients?
Solution: subband transform.
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 3
+
Vector convolution
+
1A 1A
y k
x k
m
m
nx 1n x 2n x 1n x
ny 1n y 2n y 1n y
i
n A i n i
y x
0A
i
n S i n i
x y
Forward transform
Inverse transform
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 4
Perfect reconstruction condition
Original domain
z-transform: “polyphase matrices”
Perfect reconstruction condition in the z-domain
Example, m=2
i i
S n i A i A n i S i I n
i
i
z A i z
H i
i
z S i z
G
1
z z z z I z z
G H H G G H
11 01
10 0000 01
10 11 00 11 01 10
H z H z
H z H zG z G z
G z G z H z H z H z H z
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 5
Filter bank interpretation of convolutional transform
0h
Analysis filterbank
Synthesis filterbank
+
1h
1mh
m
m
m
m
m
m
0g
1g
1mg
x k x k 0y n
1y n
1my n
,
; 0 ,q q jh mi j A i j q m
,; 0 ,q j q
g mi j S i j q m
subband
signals
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 6
Frequency domain perspective
0H 1H 2H
2
3
3
3
2
3
3 1G
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 7
Vector space interpretation
Subband decomposition is the projection of the input onto a
set of “analysis vectors” in the Hilbert space of square
summable sequences
Consider signal in channel q
Synthesis filterbank is linear combination of synthesis
“basis vectors”
,
n
q q q q
k k
y n h k x mn k h mn k x k x a
“Analysis vector”
n denotes shift
1
0
1
0
m
q q
q n
mn
q q
q n
x k y n g k mn
y n
x s“Synthesis vector”
n denotes shift
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 8
Orthonormal subband transforms
Orthonormal expansion
Analysis and synthesis vectors are identical!
0
, m
n n
q q
q n
x x s s
n n T
q q q qA i S i h k g k a s
Vector space Convolutional
transform Filter bank
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 9
Biorthogonal transforms
Analysis vectors and synthesis vectors are not necessarily each
orthogonal, but each analysis vector must be orthogonal to all but
one synthesis vectors (and vice versa)
Equivalent to perfect reconstruction
Important for linear-phase FIR filters, since lapped orthogonal
transforms with linear phase do not exist.
1 2
1 2 1 2 1 2 1 2 1 2, , 0 , , ,n n
q q q q n n q q m n n s a
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 10
Subbands vs. block-wise transform
Blockwise transforms are a special case of subband
decompositions with:
Number of bands m = order of transform N
Length of impulse responses of analysis/synthesis filters ≤ m
Filters used in subband coders are not in general
orthogonal.
Linear phase is desirable for images.
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 11
Subbands vs. block-wise transform (cont.)
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 12
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 4
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 5
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 6
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 7
Frequency response of a DCT of order N=8
0 30
20
10
0
10
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B] i = 0
0 /2 30
20
10
0
10
Frequency
i = 1
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 2
0 /2 30
20
10
0
10
Frequency
Fre
quency
response [
dB
]
i = 3
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
/2
Frequency
Fre
qu
en
cy r
esp
on
se [d
B]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 13
Lapped Orthogonal Transform
Orthonormal convolutional transform with perfect reconstruction
Lapped orthogonal transform (LOT): only A[0] and A[1] non-zero,
hence
TA i S i T
i
A i A n i I n
0 00 0 1 1
1 1
T
T TA A
A A A A IA A
0 1 0TA A
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 14
Example LOT basis functions, m=8
[G. Levinsky, EE392C class project, 1997]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 15
LOT vs. DCT coding
LOT quantizer step size 70
entropy 0.426 bpp
DCT quantizer step size 70
entropy 0.453 bpp
[G. Levinsky, EE392C class project, 1997]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 16
Discrete Wavelet Transform
Recursive application of a two-band filter bank to the
lowpass band of the previous stage yields octave band
splitting:
Same concept can be derived from wavelet theory:
Discrete Wavelet Transform (DWT)
frequency
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 17
Cascaded analysis / synthesis filterbanks
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 18
2-d Discrete Wavelet Transform
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 19
2-d Discrete Wavelet Transform example
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 20
2-d Discrete Wavelet Transform example
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 21
2-d Discrete Wavelet Transform example
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 22
2-d Discrete Wavelet Transform example
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 23
2-d Discrete Wavelet Transform example
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 24
Two-channel filterbank
Aliasing cancellation if : 0 1
1 0
( ) ( )
( ) ( )
g z h z
g z h z
0 0 1 1
0 0 1 1
1ˆ( ) ( ) ( ) ( ) ( ) ( )
2
1( ) ( ) ( ) ( )
2
x z h z g z h z g z x z
h z g z h z g z x z
Aliasing
2 2
2 2
0h
1h
0g
1g
x z x̂ z∑
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 25
Example: two-channel filter bank with perfect reconstruction