Post on 06-Feb-2017
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
Impulse responses, analysis filters:
Lowpass Highpass
Impulse responses, synthesis filters
Lowpass Highpass
Mandatory in JPEG2000
Frequency responses:
1
2
0 0
frequency ω fr
equen
cy r
esponse
1 1 3 1 1, , , ,
4 2 2 2 4
1 1 1, ,
4 2 4
1 1 3 1 1, , , ,
4 2 2 2 4
1 1 1, ,
4 2 4
“Biorthogonal 5/3 filters”
“LeGall-Tabatabai filters” [LeGall, Tabatabai, 1988]
2
h0 ej
h1 ej
g1 ej
g0 ej
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 26
Quadrature Mirror Filters (QMF)
QMFs achieve aliasing
cancellation by choosing
Highpass band is the mirror
image of the lowpass band
in the frequency domain
Need to design only one
prototype filter
1 0
1 0
( ) ( )
( ) ( )
h z h z
g z g z
Example:
16-tap QMF filterbank
[Croisier, Esteban, Galand, 1976]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 27
Conjugate quadrature filters
Achieve aliasing cancelation by
Impulse responses
Perfect reconstruction: find power complementary prototype filter
Orthonormal subband transform!
1
0 0
1 1
1 1
( ) ( )
( )
h z g z f z
h z g z zf z
[Smith, Barnwell, 1986]
Prototype filter
0 0
1
1 1 1 1k
h k g k f k
h k g k f k
2 2
2F F
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 28
Lifting
Analysis filters
L “lifting steps”
First step can be interpreted as prediction of odd samples
from the even samples
K0
1 21L L
K1
even samples 2x n
odd samples
2 1x n
0low band y
1high band y
[Sweldens 1996]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 29
Lifting (cont.)
Synthesis filters
Perfect reconstruction (biorthogonality) is directly built into
lifting structure
Powerful for both implementation and filter/wavelet design
1 21L L
even samples 2x n
odd samples
2 1x n
0low band y
1high band y
1
0K
1
1K
-
-
-
-
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 30
Example: lifting implementation of 5/3 filters
even samples 2x n
1
2
z
1 z1
4
1/2
odd samples
2 1x n
0low band y
1high band y
Verify by considering response to unit impulse in even and
odd input channel.
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 31
Reversible subband transform
Observation: lifting operators can be nonlinear.
Incorporate the necessary rounding into lifting operator:
Used in JPEG2000 as part of 5/3 biorthogonal wavelet
transform
K0
1 21L L
K1
even samples 2x n
odd samples
2 1x n
0low band y
1high band y
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 32
Wavelet compression results
Original
512x512
8bpp
0.074 bpp 0.048 bpp
Error
images
enlarged
[Gonzalez, Woods, 2001]
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 33
Embedded zero-tree wavelet algorithm
X X X X
X X X X
X X X X
X X X X
X X
X X X
X
X X
X X
X X X X
X X X X
X X X X
X X X X
X
X X
X X
X X X X
X X X X
X X X X
X X X X
Idea: Conditional coding of all descendants (incl. children)
Coefficient magnitude > threshold: significant coefficients
Four cases
1. ZTR: zero-tree, coefficient and all descendants are not significant
2. IZ: coefficient is not significant, but some descendants are significant
3. POS: positive significant
4. NEG: negative significant
For the highest bands, ZTR and IZ symbols are merged into one symbol Z
„Parent“
„Children“
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 34
Embedded zero-tree wavelet algorithm (cont.)
Successive approximation quantization and encoding Initial „dominant“ pass
• Set initial threshold θ, determine significant coefficients
• Arithmetic coding of symbols ZTR, IZ, POS, NEG
Subordinate pass • Refine magnitude of all coefficients found significant so far by one bit
(subdivide magnitude bin by two)
• Arithmetic coding of sequence of zeros and ones.
Repeat dominant pass • Omit previously found significant coefficients
• Decrease threshold θ by factor of 2, determine new significant coefficients
• Arithmetic coding of symbols ZTR, IZ, POS, NEG
Repeat subordinate and dominant passes, until bit budget is exhausted.
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 35
Embedded zero-tree wavelet algorithm (cont.)
Decoding: bitstream can be truncated to yield a coarser
approximation: „embedded“ representation
Further details: J. M. Shapiro, „Embedded image coding using
zerotrees of wavelet coefficients,“ IEEE Transactions on Signal
Processing, vol. 41, no. 12, pp. 3445-3462, December 1993.
Enhancement SPIHT coder: A. Said, A., W. A. Pearlman, „A
new, fast, and efficient image codec based on set partitioning in
hierarchical trees,“ IEEE Transactions on Circuits and Systems
for Video Technology, vol. 63 , pp. 243-250, June 1996.
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 36
Conclusions: subband and wavelet coding
Overlapping basis functions can reduce blocking artifacts
Biorthogonal subband transforms = perfect reconstruction
DCT can be understood as a (poor) filter bank
Discrete Wavelet Transform = cascaded dyadic subband splits
Quadrature mirror filters and conjugate quadrature filters: aliasing cancellation
Lifting: powerful for implementation and wavelet construction
Lifting allows reversible (i.e., lossless) wavelet transform
Zero-trees: exploit statistical dependencies across subbands
Bernd Girod: EE398A Image and Video Compression Subband and Wavelet Coding no. 37
Reading
Taubman, Marcellin, Sections 4.2, 6.1-6.4, 7
H. S. Malvar, D. H. Staelin, “The LOT: transform coding
without blocking effects,” IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 37, no. 4, pp. 553-559,
April 1989.
D. Le Gall, A. Tabatabai, “Sub-band coding of digital
images using symmetric short kernel filters and arithmetic
coding techniques,” Proc. ICASSP-88, vol. 2, pp. 761-764,
April 1988.
J. M. Shapiro, “Embedded image coding using zerotrees of
wavelet coefficients,” IEEE Transactions on Signal
Processing,” vol. 41, no. 12, pp. 3445-3462, Dec. 1993.