induction to wavelet transform and image compression

Introduction to Wavelet
Transform and Image
Compression
Student: Kang-Hua Hsu 徐康華
Advisor: Jian-Jiun Ding 丁建均
E-mail:
Graduate Institute of Communication Engineering
National Taiwan University, Taipei, Taiwan, ROC
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Outline (1)

Introduction

Multiresolution Analysis (MRA)
- Subband Coding
- Haar Transform
- Multiresolution Expansion

Wavelet Transform (WT)
- Continuous WT
- Discrete WT
- Fast WT
- 2-D WT

Wavelet Packets

Fundamentals of Image Compression
- Coding Redundancy
- Interpixel Redundancy

- Psychovisual Redundancy
- Image Compression Model
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Outline (2)

Lossless Compression
- Variable-Length Coding
- Bit-plane Coding
- Lossless Predictive Coding

Lossy Compression
- Lossy Predictive Coding
- Transform Coding
- Wavelet Coding

Conclusion

Reference
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Introduction(1)-WT v.s FT
Bases of the
•
FT: time-unlimited weighted sinusoids with different
frequencies. No temporal information.
•
WT: limited duration small waves with varying

frequencies, which are called wavelets. WTs contain the
temporal time information.
Thus, the WT is more adaptive.
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Introduction(2)-WT v.s TFA
•
Temporal information is related to the time-frequency
analysis.
•
The time-frequency analysis is constrained by the
Heisenberg uncertainty principal.
•
Compare tiles in a time-frequency plane (Heisenberg cell):
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Introduction(3)-MRA
•
It represents and analyzes signals at more than one
resolution.
•
2 related operations with ties to MRA:

Subband coding

Haar transform
•
MRA is just a concept, and the wavelet-based

transformation is one method to implement it.
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Introduction(4)-WT
•
The WT can be classified according to the of its input
and output.

Continuous WT (CWT)

Discrete WT (DWT)
•
1-D 2-D transform (for image processing)
•
DWT Fast WT (FWT)
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recursive relaon of the coecients
MRA-Subband Coding(1)
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•
Since the bandwidth of the resulng subbands is smaller
than that of the original image, the subbands can be
downsampled without loss of informaon.
•
We wish to select so that the

input can be perfectly reconstructed.

Biorthogonal

Orthonormal
( ) ( ) ( ) ( )
0 1 0 1
, , ,h n h n g n g n
MRA-Subband Coding(2)
•
Biorthogonal filter bank:
•
Orthonormal (it’s also biorthogonal) filet bank:
: time-reversed relation
,where 2K denotes the number of coefficients in each filter.
•
The other 3 filters can be obtained from one prototype filter.
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[ ] [ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
[ ] [ ]
0 0
0 1
1 1
1 0
, 2
, 2 0

, 2
, 2 0
g k h n k n
g k h n k
g k h n k n
g k h n k
δ
δ

− =


− =


− =


− =


1 0
( ) ( 1) (2 1 )
( ) (2 1 ), {0,1}
n
i i
g n g K n
h n g K n i

= − − −



= − − =


MRA-Subband Coding(3)
•
1-D to 2-D: 1-D two-band subband coding to the rows and
then to the columns of the original image.
•
Where a is the approximation (Its histogram is scattered, and
thus lowly compressible.) and d means detail (highly
compressible because their histogram is centralized, and thus
easily to be modeled).
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FWT can be implemented by subband coding!
Haar Transform
will put the lower frequency components of X at
the top-left corner of Y. This is similar to the
DWT.
This implies the resolution (frequency) and location (time).
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1 / 2 1 / 2 1 / 2 1 / 2
1 / 2 1 / 2 1 / 2 1 / 2
1 / 2 1 / 2 0 0
0 0 1 / 2 1 / 2

H
 
 
− −
 
=
 
−
 
 
−
 
T
Y H X H= × ×
Multiresolution Expansions(1)
•
, : the real-valued expansion coefficients.
, : the real-valued expansion functions.
•
Scaling function : span the approximation of the
signal.
•
: this is the reason of it’s name.
•
If we define , then
•
, : scaling function coefficients
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( ) ( )
k k
k
f x x
α φ
=
∑
k
α
( )
k
x
φ
( )
x
φ
/2
,
( ) 2 (2 )
j j
j k
x x k
φ φ
= −
{ }
,
( )
j j k
k
V span x

φ
=
0 1 2
V V V⊂ ⊂ ⊂ ⊂
( ) ( ) ( )
2 2
n
x h n x n
φ
φ φ
= −
∑
( )
h n
φ
Multiresolution Expansions(2)
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•
4 requirements of the scaling function:

The scaling function is orthogonal to its integer translates.

The subspaces spanned by the scaling function at low scales
are nested within those spanned at higher scales.

The only function that is common to all is .

Any function can be represented with arbitrary coarse

resolution, because the coarser portions can be represented
by the finer portions.
j
V
( ) { }
0f x V
−∞
= =
Multiresolution Expansions(3)
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•
The wavelet function : spans difference between any
two adjacent scaling subspaces, and .
•
span the subspace .
( )
x
ψ
j
V
1j
V
+
( )
( )
2
,
2 2

j
j
j k
x x k
ψ ψ
= −
j
W
Multiresolution Expansions(4)
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•
,
: wavelet function coefficients
•
Relation between the scaling coefficients and the wavelet
coefficients:
This is similar to the relation between the impulse response
of the analysis and synthesis filters in page 11. There is
time-reverse relation in both cases.
( ) ( ) 2 (2 )
n
x h n x n
ψ
ψ φ
= −
∑
( )h n
ψ

( ) ( 1) (1 )
n
h n h n
ψ φ
= − −
CWT
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•
The definition of the CWT is
•
Continuous input to a continuous output with 2 continuous
variables, translation and scaling.
•
Inverse transform:
It’s guaranteed to be reversible if the admissibility criterion is
satisfied.
•
Hard to implement!
( ) ( )
1
,
| |
x
W s f x dt
s
s
φ
τ

τ ψ
∞
−∞
−
 
=
 ÷
 
∫
( ) ( )
2
0
1
,
x
f x W s d ds
s
C s s
ψ
ψ
τ
τ ψ τ
∞ ∞
−∞
−
 
=
 ÷
 
∫ ∫

2
| ( ) |
| |
f
C df
f
ψ
Ψ
= < ∞
∫
DWT(1)
•
wavelet series expansion:
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0 0
0
, ,
( ) ( ) ( ) ( ) ( )
j j k j j k
k j j k
f x c k x d k x
φ ψ
∞
=
= +
∑ ∑∑
: arbitrary starting scale
0

j
0
( )
j
c k
( )
j
d k
: approximation or scaling coefficients
: detail or wavelet coefficients
0 0 0
, ,
( ) ( ), ( ) ( ) ( )
j j k j k
c k f x x f x x dx
φ φ
= =
∫
% %
, ,
( ) ( ), ( ) ( ) ( )
j j k j k
d k f x x f x x dx
ψ ψ
= =
∫
% %
This is still the continuous case. If we change the integral
to summation, the DWT is then developed.
DWT(2)

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0
0
0 , ,
1 1
( ) ( , ) ( ) ( , ) ( )
j k j k
k j j k
f x W j k x W j k x
M M
φ ψ
φ ψ
∞
=
= +
∑ ∑ ∑
%
%
0
1
0 ,
0
1
( , ) ( ) ( )
M
j k
x
W j k f x x

M
φ
φ
−
=
=
∑
%
1
,
0
1
( , ) ( ) ( )
M
j k
x
W j k f x x
M
ψ
ψ
−
=
=
∑
%
The coefficients measure the similarity (in linear algebra,
the orthogonal projection) of with basis functions
and .
( )
f x

0
,
( )
j k
x
φ
%
,
( )
j k
x
ψ
%
FWT(1)
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( ) ( ) ( )
2 2
n
x h n x n
φ
φ φ
= −
∑
( ) ( ) 2 (2 )
n
x h n x n
ψ
ψ φ

= −
∑
By the 2 relations we mention in subband coding,
We can then have
2 , 0
( , ) ( 2 ) ( 1, ) ( ) ( 1, )
n k k
m
W j k h m k W j m h n W j n
ψ ψ φ ψ φ
= ≥
= − + = − ∗ +
∑
2 , 0
( , ) ( 2 ) ( 1, ) ( ) ( 1, )
n k k
m
W j k h m k W j m h n W j n
φ φ φ φ φ
= ≥
= − + = − ∗ +
∑
FWT(2)
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When the input is the samples of a function or an image, we can
exploit the relation of the adjacent scale coefficients to obtain all
of the scaling and wavelet coefficients without defining the
scaling and wavelet functions.

2 , 0
( , ) ( 2 ) ( ) ( 1,, )( 1 )
n k k
m
W j k h m k W j m h n W j n
ψ ψ ψ φφ
= ≥
= − = ++ − ∗
∑
2 , 0
( , ) ( 2 ) ( ) ( 1,, )( 1 )
n k k
m
W j k h m k W j m h n W j n
φ φ φ φφ
= ≥
= − = ++ − ∗
∑
FWT(3)
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FWT resembles the two-band subband coding scheme!
1
:FWT
−
•
The constraints for perfect reconstruction is the same
as in the subband coding.
0

1
( ) ( )
( ) ( )
h n h n
h n h n
φ
ψ
= −



= −


0 0
1 1
( ) ( ) ( )
( ) ( ) ( )
g n h n h n
g n h n h n
φ
ψ
= − =



= − =


2-D WT(1)

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( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
H
V
D
x y x y
x y x y
x y y x
x y x y
φ φ φ
ψ ψ φ
ψ φ ψ
ψ ψ ψ
=


=


=


=

2-D

1-D (row)
1-D (column)
These wavelets have directional sensitivity naturally.
2-D WT(2)
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Note that the upmost-leftmost subimage is similar to the
original image due to the energy of an image is usually
distributed around lower band.
Wavelet Packets
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A wavelet packet is a more +exible decomposion.
Fundamentals of Image
Compression(1)
•
3 kinds of redundancies in an image:

Coding redundancy

Interpixel redundancy

Psychovisual redundancy
Image compression is achieved when the redundancies
were reduced or eliminated.
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Goal: To convey the same information with
least amount of data (bits).