induction to wavelet transform and image compression
TRANSCRIPT
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Introduction to Wavelet
Transform and Image
Compression
Student: Kang-Hua Hsu
Advisor: Jian-Jiun Ding
E-mail: [email protected] 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 FTBases 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 relation of the coefficients
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MRA-Subband Coding(1)
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Since the bandwidth of the resulting subbands is smaller
than that of the original image, the subbands can be
downsampled without loss of information.
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
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MRA-Subband Coding(2)
Biorthogonal filter bank:
Orthonormal (its 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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? A ? A ? A
? A ? A
? A ? A ? A
? A ? A
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
H
H
! !
!
!
1 0( ) ( 1) (2 1 )
( ) (2 1 ), {0,1}
n
i i
g n g K n
h n g K n i
!
! !
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MRA-Subband Coding(3)
1-D to 2-D: 1-D two-band subband coding to the rows andthen to the columns of the original image.
Where a is the approximation (Its histogram is scattered, andthus lowly compressible.) and d means detail (highlycompressible because their histogram is centralized, and thuseasily to be modeled).
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FWT can be implemented by subband coding!
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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
!
TY H X H !
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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 its name.
If we define , then
, : scaling function coefficients
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( ) ( )k kk
f x xE J! kE
( )k xJ
xJ
/2
, ( ) 2 (2 )j j
j k x x kJ J!
_ a, ( )j j kk
V span xJ!0 1 2... ....V V V
2 2n
x h n x nJ
J J! h nJ
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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.
jV _ a0 f x V g! !
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Multiresolution Expansions(3)
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The wavelet function : spans difference between any
two adjacent scaling subspaces, and .
span the subspace .
x]
jV 1jV
2, 2 2j
j
j k x x k] ]! jW
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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 responseof the analysis and synthesis filters in page 11. There is
time-reverse relation in both cases.
( ) ( ) 2 (2 )n
x h n x n]
] J!
( )h n]
( ) ( 1) (1 )n
h n h n] J!
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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:
Its guaranteed to be reversible if the admissibility criterion is
satisfied.
Hard to implement!
1
,
| |
xW s f x dt
ss
J
XX ]
g
g
!
2
0
1,
xf x W s d ds
sC s s]
]
XX ] X
g g
g
!
2( ) |
| |
fC df
f
]
=! g
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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 xJ ]g
!
!
: arbitrary starting scale0j
0( )jc k
( )jd k
: approximation or scaling coefficients
: detail or wavelet coefficients
0 0 0, ,( ) ( ), ( ) ( ) ( )j j k j kc k f x x f x x dxJ J! ! % %
, ,( ) ( ), ( ) ( ) ( )j j k j kd k f x x f x x dx] ]! ! % %
This is still the continuous case. If we change the integral
to summation, the DWT is then developed.
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DWT(2)
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0
0
0 , ,
1 1
( ) ( , ) ( ) ( , ) ( )j k j kk j j kf x W j k x W j k xM MJ ]J ]
g
!! % %
0
1
0 ,
0
1( , ) ( ) ( )
M
j k
x
W j k f x xM
J J
!
! %
1
,0
1
( , ) ( ) ( )
M
j kx
W j k f x xM] ]
!! %
The coefficients measure the similarity (in linear algebra,
the orthogonal projection) of with basis functions
and .
f x
0 ,( )j k xJ
%, ( )j k x]%
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FWT(1)
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2 2n
x h n x nJJ J! ( ) ( ) 2 (2 )
n
x h n x n]] J!
By the 2 relations we mention in subband coding,
We can then have
2 , 0( , ) ( 2 ) ( 1, ) ( ) ( 1, )
n k km
W j k h m k W j m h n W j n] ] J ] J ! u! !
2 , 0( , ) ( 2 ) ( 1, ) ( ) ( 1, )
n k km
W j k h m k W j m h n W j nJ J J J J ! u! !
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FWT(2)
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When the input is the samples of a function or an image, we canexploit 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 km
W j k h m k W j m h n W j n] ] ] JJ ! u! ! 2 , 0
( , ) ( 2 ) ( ) ( 1,, )( 1 )n k k
m
W j k h m k W j m h n W j nJ J J J J ! u! !
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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
J
]
!
!
0 0
1 1
( ) ( ) ( )
( ) ( ) ( )
g n h n h n
g n h n h n
J
]
! !
! !
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2-DWT(1)
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( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
H
V
D
x y x y
x y x y
x y y x
x y x y
J J J
] ] J
] J ]
] ] ]
!
!
! !
2-D1-D (row)
1-D (column)
These wavelets have directional sensitivity naturally.
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2-DWT(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.
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Wavelet Packets
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A wavelet packet is a more flexible decomposition.
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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).
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Fundamentals of Image
Compression(2) Image compression can be classified to
Lossless(error-free, without distortion after
reconstructed)
Lossy
Information theory is an important tool .
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Data Information{ : information is carried by the data.
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Fundamentals of Image
Compression(3)
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1
2
R
nC
n!
11
DC
!
Evaluation of the lossless compression:
Compression ratio :
Relative data redundancy :
Evaluation of the lossy compression:
root-mean-square (rms) error
121 1
0 0
1
, ,
M N
rms
x ye f x y f x yMN
! !
!
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Coding Redundancy
We can obtain the probable information from the histogram
of the original image.
Variable-length coding: assign shorter codeword to more
probable gray level.
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If there is a set of codeword to represent theoriginal data with less bits, the original data is
said to have coding redundancy.
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Interpixel Redundancy(1)
B
ecause the value of any given pixel can bereasonably predicted from the value of its neighbors,
the information carried by individual pixels is
relatively small.
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Interpixel redundancy is resulted from the
correlation between neighboring pixels.
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Interpixel Redundancy(2)
To reduce interpixel redundancy, the original imagewill be transformed to a more efficient and nonvisual
format. This transformation is called mapping.
Run-length coding. Ex. 10000000 1,111
Difference coding.
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7 0s
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Psychovisual Redundancy
For example, the edges are more noticeable for us.
Information loss!
We truncate or coarsely quantize the gray levels (or
coefficients) that will not significantly impair the perceived
image quality.
The animation take advantage of the persistence of vision to
reduce the scanning rate.
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Humans dont respond with equal importanceto every pixel.
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Image Compression Model
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The quantizer is not necessary.
The mapper would
1.reduce the interpixel redundancy to compress directly,
such as exploiting the run-length coding.
or
2.make it more accessible for compression in the later
stage, for example, the DCT or the DWT coefficients are
good candidates for quantization stage.
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Lossless Compression
No quantizer involves in the compression procedure.
Generally, the compression ratios range from 2 to 10.
Trade-off relation between the compression ratio and the
computational complexity.
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It can be reconstructed without distortion.
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Variable-Length Coding
It merely reduces the coding redundancy.
Ex. Huffman coding
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It assigns fewer bits to the more probable gray levels than
to the less probable ones.
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Bit-plane Coding
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A monochrome or colorful image is decomposed into a series ofbinary images (that is, bit planes), and then they are compressed
by a binary compression method.
It reduces the interpixel redundancy.
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Lossless Predictive Coding
It reduces the interpixel redundancies of closely spaced
pixels.
The ability to attack the redundancy depends on the
predictor.
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It encodes the difference between the actual and predictedvalue of that pixel.
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Lossy Compression
It exploits the quantizer.
Its compression ratios range from 10 to 100 (much more
than the lossless cases).
Trade-off relation between the reconstruction accuracy and
compression performance.
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It can not be reconstructed without distortiondue to the sacrificed accuracy.
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Lossy Predictive Coding
It exploits the quantizer.
Its compression ratios range from 10 to 100 (much more
than the lossless cases).
The quantizer is designed based on the purpose for
minimizing the quantization error.
Trade-off relation between the quantizer complexity and less
quantization error.
Delta modulation (DM) is an easy example exploiting the
oversampling and 1-bit quantizer.
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It is just a lossless predictive coding containinga quantizer.
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Transform Coding(1)
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Most of the information is included among a small number
of the transformed coefficients. Thus, we truncate or coarsely
quantize the coefficients including little information.
The goal of the transformation is to pack as much informationas possible into the smallest number of transform coefficients.
Compression is achieved during the quantization of the
transformed coefficients, not during the transformation.
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Transform Coding(2)
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More truncated coefficients Higher compression ratio, butthe rms error between the reconstructed image and the original
one would also increase.
Every stage can be adapted to local image content.
Choosing the transform:
Information packing ability
Computational complexity needed
KLT WHT DCT
Information packing ability Best Not good Good
Computational complexity High Lowest Low
Practical!
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Transform Coding(3)
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Disadvantage: Blocking artifact when highly compressed(this causes errors) due to subdivision.
Size of the subimage:
Size increase: higher compression ratio, computational
complexity, and bigger block size.
How to solve the blocking artifact problem? Using the WT!
?
?
?
?
??
?
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Wavelet Coding(1)
No subdivision due to:
Computationally efficient (FWT)
Limited-duration basis functions.
Avoiding the blocking artifact!
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Wavelet coding isnot only
the transforming coding
exploiting the wavelet transform------No subdivision!
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Wavelet Coding(2)
We only truncate the detail coefficients.
The decomposition level: the initial decompositions would
draw out the majority of details. Too many decompositions is
just wasting time.
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Wavelet Coding(3)
Quantization with dead zone threshold: set a threshold to
truncate the detail coefficients that are smaller than the
threshold.
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ConclusionThe WT is a powerful tool to analyze signals. There are
many applications of the WT, such as image
compression. However, most of them are still not
adopted now due to some disadvantage. Our future
work is to improve them. For example, we could
improve the adaptive transform coding, including the
shape of the subimages, the selection of transformation,
and the quantizer design. They are all hot topics to be
studied.
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Reference[1] R.C Gonzalez, R.E Woods, Digital I age Processing,
2nd edition, Prentice Hall, 2002.[2] J.C Goswami, A.K Chan, Funda entals ofWavelets,
John Wiley & Sons, New York, 1999.
[3] Contributors of the Wikipedia, Arithmetic coding,
available inhttp://en.wikipedia.org/wiki/Arithmetic_coding.
[4] Contributors of the Wikipedia, Lempel-Ziv-Welch,available in http://en.wikipedia.org/wiki/Lempel-Ziv-
Welch.[5] S. Haykin, Co unication Syste , 4th edition, John
Wiley & Sons, New York, 2001.
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