Digital Image Processing

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Digital Image Processing Chapter 8 Image
Compression 11 August 2006
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Data vs Information
Information Matter (????) Data The means by
which information is conveyed
Image Compression
Reducing the amount of data required to
represent a digital image while keeping
information as much as possible
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Relative Data Redundancy and Compression Ratio
Relative Data Redundancy
Compression Ratio
Types of data redundancy
1. Coding redundancy 2. Interpixel redundancy 3.
Psychovisual redundancy
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Coding Redundancy
Different coding methods yield different
amount of data needed to represent the same
information.
Example of Coding Redundancy Variable Length
Coding vs. Fixed Length Coding
Lavg 2.7 bits/symbol
Lavg 3 bits/symbol
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Variable Length Coding
Concept assign the longest code word to the
symbol with the least probability of occurrence.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Interpixel Redundancy
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Interpixel redundancy Parts of an image
are highly correlated. In other words,we
can predict a given pixel from its neighbor.
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Run Length Coding
The gray scale image of size 343x1024 pixels
Binary image 343x1024x1 351232 bits
Line No. 100
Run length coding
Line 100 (1,63) (0,87) (1,37) (0,5) (1,4)
(0,556) (1,62) (0,210)
Total 12166 runs, each run use 11 bits ? Total
133826 Bits
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Psychovisual Redundancy
4-bit gray scale image
4-bit IGS image
8-bit gray scale image
False contours
The eye does not response with equal
sensitivity to all visual information.
(Images from Rafael C. Gonzalez and Richard E.
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Improved Gray Scale Quantization
Pixel i-1 i i1 i2 i3
Gray level N/A 0110 1100 1000 1011 1000
0111 1111 0100
Sum 0000 0000 0110 1100 1001 0111 1000 1110 1111
0100
IGS Code N/A 0110 1001 1000 1111

Algorithm 1. Add the least significant 4 bits of
the previous value of Sum to the 8-bit
current pixel. If the most significant 4 bit of
the pixel is 1111 then add 0000 instead. Keep
the result in Sum 2. Keep only the most
significant 4 bits of Sum for IGS code.
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Fidelity Criteria how good is the compression
algorithm

• Objective Fidelity Criterion
• RMSE, PSNR
• Subjective Fidelity Criterion
• Human Rating

(Images from Rafael C. Gonzalez and Richard E.
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Image Compression Models
Reduce data redundancy
Increase noise immunity
Source encoder
Channel encoder
Channel
Noise
Source decoder
Channel decoder
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Source Encoder and Decoder Models
Source encoder
Mapper
Quantizer
Symbol encoder
Reduce interpixel redundancy
Reduce psychovisual redundancy
Reduce coding redundancy
Source decoder
Inverse mapper
Symbol decoder
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Channel Encoder and Decoder
- Hamming code, Turbo code,
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Information Theory
Measuring information
Entropy or Uncertainty Average information per
symbol
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Simple Information System
Binary Symmetric Channel
Source
Destination
A a1, a2 0, 1 z P(a1), P(a2)
B b1,b2 0, 1 v P(b1), P(b2)
(1-Pe)
0
0
P(a1)
P(a1)(1-Pe)(1-P(a1))Pe
Pe
Source
Destination
Pe
1
1
1-P(a1)
(1-P(a1))(1-Pe)P(a1)Pe
(1-Pe)
Pe probability of error
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Binary Symmetric Channel
Source
Destination
A a1, a2 0, 1 z P(a1), P(a2)
B b1,b2 0, 1 v P(b1), P(b2)
H(zb1) - P(a1b1)log2P(a1b1)
- P(a2b1)log2P(a2b1) H(zb2) -
P(a1b2)log2P(a1b2) -
P(a2b2)log2P(a2b2)
H(z) - P(a1)log2P(a1) -
P(a2)log2P(a2)
H(zv) H(zb1) H(zb2)
Mutual information
I(z,v)H(z) - H(zv)
Capacity
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Binary Symmetric Channel
Let pe probability of error
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Binary Symmetric Channel
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Communication System Model
2 Cases to be considered Noiseless and noisy
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Noiseless Coding Theorem
Problem How to code data as compact as possible?
Shannons first theorem defines the minimum
average code word length per source that can be
achieved.
Let source be A, z which is zero memory source
with J symbols. (zero memory each outcome is
independent from other outcomes)
then a set of source output of n element be
Example
for n 3,
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Noiseless Coding Theorem (cont.)
Probability of each aj is
Entropy of source
Each code word length l(ai) can be
Then average code word length can be
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Noiseless Coding Theorem (cont.)
We get
from
then
or
The minimum average code word length per source
symbol cannot lower than the entropy.
Coding efficiency
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Extension Coding Example
H 0.918 Lavg 1
H 1.83 Lavg 1.89
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Noisy Coding Theorem
Problem How to code data as reliable as possible?
Example Repeat each code 3 times
Source data 1,0,0,1,1
Data to be sent 111,000,000,111,111
Shannons second theorem the maximum rate of
coded information is
j code size
r Block length
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Rate Distortion Function for BSC
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Error-Free Compression Huffman Coding
Huffman coding give the smallest possible
number of code symbols per source symbols.
Step 1 Source reduction
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Error-Free Compression Huffman Coding
Step 2 Code assignment procedure
The code is instantaneous uniquely decodable
without referencing succeeding symbols.
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Near Optimal Variable Length Codes
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Arithmetic Coding
Nonblock code one-to-one correspondence between
source symbols And code words does not
exist. Concept The entire sequences of source
symbols is assigned a single arithmetic code word
in the form of a number in an interval of real
number between 0 and 1.
(Images from Rafael C. Gonzalez and Richard E.
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Arithmetic Coding Example
0.2x0.4
0.040.8x0.04
0.0560.8x0.016
The number between 0.0688 and 0.06752 can be
used to represent the sequence a1 a2 a3 a3 a4
0.2x0.2
0.040.4x0.04
0.0560.4x0.016
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LZW Coding
Lempel-Ziv-Welch coding assign fixed length
code words to variable length sequences of
source symbols.
24 Bits
9 Bits
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LZW Coding Algorithm
0. Initialize a dictionary by all possible gray
values (0-255) 1. Input current pixel 2. If the
current pixel combined with previous pixels
form one of existing dictionary entries
Then 2.1 Move to the next pixel and
repeat Step 1 Else 2.2 Output the
dictionary location of the currently recognized
sequence (which is not include the current
pixel) 2.3 Create a new dictionary entry by
appending the currently recognized
sequence in 2.2 with the current pixel 2.4
Move to the next pixel and repeat Step 1
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LZW Coding Example
Currently recognized Sequences 39 39 126 126 39 39
-39 126 126-126 39 39-39 39-39-126 126
Dictionary Location Entry 0 0
1 1 255 255 256
39-39 257 39-126 258 126-126
259 126-39 260 39-39-126 261
126-126-39 262 39-39-126-126
Encoded Output (9 bits) 39 39 126 126 256 258
260
Input pixel 39 39 126 126 39 39 126 126 39 39 126
126
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Bit-Plane Coding
Original image
Bit 7
Binary image compression
Bit 6
Binary image compression

Bit 0
Binary image compression
Bit plane images
Example of binary image compression Run length
coding
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Bit Planes
Bit 3
Bit 7
Bit 2
Bit 6
Original gray scale image
Bit 1
Bit 5
Bit 0
Bit 4
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Gray-coded Bit Planes
Original bit planes
Gray code
a7
g7
and
a6
g6
ai Original bit planes
a5
g5
XOR
a4
g4
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Gray-coded Bit Planes (cont.)
There are less 0-1 and 1-0 transitions in
grayed code bit planes. Hence gray coded bit
planes are more efficient for coding.
a3
g3
a2
g2
a1
g1
a0
g0
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Relative Address Coding (RAC)
Concept Tracking binary transitions that begin
and end eack black and white run
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Contour tracing and Coding
Represent each contour by a set of boundary
points and directionals.
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Error-Free Bit-Plane Coding
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Lossless VS Lossy Coding
Lossless coding
Lossy coding
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Transform Coding (for fixed resolution
transforms)
Encoder
Construct nxn subimages
Forward transform
Input image (NxN)
Symbol encoder
Quantizer
Quantization process causes The transform coding
lossy
Compressed image
Decoder
Construct nxn subimages
Inverse transform
Decompressed image
Symbol decoder
Examples of transformations used for image
compression DFT and DCT
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Transform Coding (for fixed resolution
transforms)

• 3 Parameters that effect transform coding
  performance
• Type of transformation
• Size of subimage
• Quantization algorithm

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2D Discrete Transformation
Forward transform
where g(x,y,u,v) forward transformation kernel
or basis function
T(u,v) is called the transform coefficient image.
Inverse transform
where h(x,y,u,v) inverse transformation kernel
or inverse basis
function
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Transform Example Walsh-Hadamard Basis Functions
N 2m
bk(z) the kth bit of z
Advantage simple, easy to implement Disadvantage
not good packing ability
N 4
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Transform Example Discrete Cosine Basis
Functions
DCT is one of the most frequently used
transform for image compression. For example, DCT
is used in JPG files.
N 4
Advantage good packing ability, modulate
computational complexity
N 4
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Transform Coding Examples
Error
Fourier
RMS Error 1.28
Hadamard
Original image 512x512 pixels
Subimage size 8x8 pixels 64 pixels
RMS Error 0.86
DCT
Quatization by truncating 50 of coefficients
(only 32 max cofficients are kept.)
(Images from Rafael C. Gonzalez and Richard E.
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RMS Error 0.68
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DCT vs DFT Coding
DFT coefficients have abrupt changes
at boundaries of blocks
1 Block
Advantage of DCT over DFT is that the DCT
coefficients are more continuous at boundaries of
blocks.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Subimage Size and Transform Coding Performance
This experiment Quatization is made
by truncating 75 of transform coefficients
DCT is the best
Size 8x8 is enough
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
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Subimage Size and Transform Coding Performance
DCT Coefficients
Reconstructed by using 25 of coefficients (CR
41) with 8x8 sub- images
Zoomed detail Subimage size 2x2 pixels
Zoomed detail Original
Zoomed detail Subimage size 8x8 pixels
Zoomed detail Subimage size 4x4 pixels
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Quantization Process Bit Allocation
To assign different numbers of bits to
represent transform coefficients based on
importance of each coefficient - More
importance coefficeitns ? assign a large number
of bits - Less importance coefficients ?
assign a small number of bits
or not assign at all
2 Popular bit allocation methods 1. Zonal coding
allocate bits based on the basis of
maximum variance, using fixed mask for all
subimages 2. Threshold coding allocate bits
based on maximum magnitudes of coefficients
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Example Results with Different Bit Allocation
Methods
Reconstructed by using 12.5 of coefficients (8
coefficients with largest variance are
used)
Reconstructed by using 12.5 of coefficients (8
coefficients with largest magnitude are used)
Threshold coding Error
Zonal coding Error
Zoom details
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Zonal Coding Example
Zonal mask
Zonal bit allocation
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Threshold Coding Example
Threshold mask
Thresholded coefficient ordering
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Thresholding Coding Quantization
3 Popular Thresholding Methods Method 1 Global
thresholding Use a single global threshold
value for all subimages Method 2 N-largest
coding Keep only N largest coefficients Method
3 Normalized thresholding each subimage is
normalized by a normalization matrix before
rounding
Bit allocation
Restoration before decompressing
Example of Normalization Matrix Z(u,v)
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DCT Coding Example
(CR 381)
(CR 671)
Method - Normalized Thresholding, - Subimage
size 8x8 pixels
Error image RMS Error 3.42
Blocking Artifact at Subimage boundaries
Zoom details
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Wavelet Transform Coding Multiresolution approach
Unlike DFT and DCT, Wavelet transform is a
multiresolution transform.
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What is a Wavelet Transform
One up on a time, human uses a line to represent
a number. For example
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With this numerical system, we need a lot of
space to represent a number 1,000,000.
Then, after an Arabic number system is invented,
life is much easier. We can represent a number by
a digit number
X,XXX,XXX
An Arabic number is one kind of
multiresolution Representation.
The 1st digit 1x
The 2nd digit 10x
The 3rd digit 100x

Like a number, any signal can also be represented
by a multiresolution data structure, the wavelet
transform.
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What is a Wavelet Transform
Wavelet transform has its background from
multiresolution analysis and subband coding.
Other important background

• - Nyquist theorem The minimun sampling rate
  needed for sampling
• a signal without loss of information is twice the
  maximum frequency
• of the signal.
• We can perform frequency shift by multiplying a
  complex sinusiodal
• signal in time domain.

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Wavelet History Image Pyramid
If we smooth and then down sample an image
repeatedly, we will get a pyramidal image
Coarser, decrease resolution
Finer, increase resolution
Pyramidal structured image
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Image Pyramid and Multiscale Decomposition
Image NxN
Down Sampling By 2
Image N/2xN/2
Smooth
Question What Information is loss after
down Sampling?
Up Sampling By 2
Interpolate
Answer Loss Information is A prediction error
image
Predicted Image NxN
Prediction Error (loss details) NxN

S
-
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Image Pyramid and Multiscale Decomposition (cont.)
Hence we can decompose an image using the
following process
Approxi- -mation Image N/2xN/2
Approxi- -mation Image N/4xN/4
Smooth and down sampling by 2
Smooth and down sampling by 2
Image NxN
.
Up sampling by 2 and interpolate
Up sampling by 2 and interpolate
-
-

S
S

Prediction Error NxN
Prediction Error N/2xN/2
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Image Pyramid and Multiscale Decomposition (cont.)
Original Image NxN
Approximation image N/8xN/8
Prediction error N/4xN/4

Prediction error N/2xN2
Prediction error (residue) NxN
Multiresolution Representation
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Multiresolution Decomposition Process
Note that this process is not a wavelet
decomposition process !
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Example of Pyramid Images
Approximation Images (using Gaussian Smoothing)
Prediction residues
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Subband Coding
Subband decomposition process
h0(n)
Approximation
Down Sampling by 2
LPF
a(n) N/2 points
x(n) N points
h1(n)
Detail
Down Sampling by 2
d(n) N/2 points
HPF
Freq. shift by N/2
All information of x(n) is completely preserved
in a(n) and d(n).
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Subband Coding (cont.)
Subband reconstruction process
g0(n)
Up Sampling by 2
Interpolation
a(n) N/2 points
x(n) N points
S
g1(n)
Up Sampling by 2
d(n) N/2 points
Interpolation
Freq. shift by N/2
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Subband Coding (cont.)
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2D Subband Coding
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Example of 2D Subband Coding
Vertical detail filtering in x- direction using
h0(n) and in y- direction using h1(n)
Approximation filtering in both x and y
directions using h0(n)
Horizontal detail filtering in x- direction
using h1(n) and in y- direction using h0(n)
Diagonal detail filtering in both x and y
directions using h1(n)
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1D Discrete Wavelet Transformation
h y(n)
d1(n) N/2 points
x(n) N points
h j(n)
h y(n)
d2(n) N/4 points
h j(n)
Note that the number of points of x(n)
and wavelet coefficients are equal.
h y(n)
d3(n) N/8 points
h j(n)
a3(n) N/8 points
y(n) a wavelet function j(n) a scaling
function
Wavelet coefficients (N points)
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1D Discrete Wavelet Transformation
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2D Discrete Wavelet Transformation
Original image NxN
d diagonal detail h horizontal detail v
vertical detail a approximation
h1
d1
a1
v1
d2
h2
Level 1
Level 3
v2
a2
d3
h3
Level 2
a3
v3
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2D Discrete Wavelet Transformation (cont.)
h2
h1
a3
h3
Original image NxN
d3
v3
d2
v2
d1
v1
Wavelet coefficients NxN
d diagonal detail filtering in both x and y
directions using h y(n) h horizontal detail
filtering in x-direction using h y(n) and in y
direction using h
j(n) v vertical detail filtering in
x-direction using h j(n) and in y
direction using h y(n) a approximation
filtering in both x and y directions using h j(n)
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Example of 2D Wavelet Transformation
Original image
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Example of 2D Wavelet Transformation (cont.)
The first level wavelet decomposition
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Example of 2D Wavelet Transformation (cont.)
The second level wavelet decomposition
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Example of 2D Wavelet Transformation (cont.)
HL3
LL3
HL2
HL1
HH3
LH3
LH2
HH2
LH1
HH1
The third level wavelet decomposition
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Example of 2D Wavelet Transformation
Level 1
Level 2
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Example of 2D Wavelet Transformation
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Examples Types of Wavelet Transform
Daubechies wavelets
Haar wavelets
Biorthogonal wavelets
Symlets
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Wavelet Transform Coding for Image Compression
Unlike DFT and DCT, Wavelet transform is a
multiresolution transform.
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Wavelet Transform Coding Example
(CR 381)
(CR 671)
Error Image RMS Error 2.29
Error Image RMS Error 2.96
Zoom details
No blocking Artifact
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Wavelet Transform Coding Example (cont.)
(CR 1081)
(CR 1671)
Error image RMS Error 3.72
Error image RMS Error 4.73
Zoom details
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Wavelet Transform Coding vs. DCT Coding
Wavelet
DCT 8x8
(CR 671)
(CR 671)
Error image RMS Error 2.96
Error image RMS Error 6.33
Zoom details
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Type of Wavelet Transform and Performance
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No. of Wavelet Transform Level and Performance
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Threshold Level and Performance
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(No Transcript)
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Table 8.14 (Cont)
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Table 8.19 (Cont)
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Lossless Predictive Coding Model
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Lossless Predictive Coding Example
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Lossy Predictive Coding Model
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Delta Modulation
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Linear Prediction Techniques Examples
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Quantization Function
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Lloyd-Max Quantizers
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Lossy DCPM
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DCPM Result Images
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Error Images of DCPM
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