Image Compression, Introduction Data Compression/ Data compression, modelling and coding,Image Compression

IMAGE COMPRESSION
Under Guidance of: Presented by:-
Jayash Sharma Sir Shivangi Saxena
M.tech,Semester-II

• The problem of reducing the amount of data
required to represent a digital image.
• From a mathematical viewpoint: transforming
a 2-D pixel array into a statistically
uncorrelated data set.
Image Compression?

• For data STORAGE and data TRANSMISSION
• DVD
• Remote Sensing
• Video conference
• FAX
• Control of remotely piloted vehicle
• The bit rate of uncompressed digital cinema
data exceeds one Gbps.
Why do We Need Compression?

REDUNDANT
DATA
INFORMATION
DATA = INFORMATION + REDUNDANT DATA
Information vs Data

• Spatial redundancy
• Neighboring pixels are not independent but
correlated
 Temporal redundancy
Why Can We Compress?

• Basic data redundancies:
1. Coding redundancy
2. Inter-pixel redundancy
3. Psycho-visual redundancy
Fundamentals

Let us assume, that a discrete random variable rk in the interval [0,1]
represent the gray level of an image:
If the number of bits used to represent each value of rk is l(rk), then
the average number of bits required to represent each pixel:
The total number bits required to code an MxN image:
1
,
,
2
,
1
,
0
)
( −
=
= L
k
n
n
r
p k
k
r 
avg
L
N
M .
.
Coding Redundancy
∑
−
=
=
1
0
)
(
)
(
L
k
k
r
k
avg r
p
r
l
L

Coding Redundancy
bits
r
p
r
l
L
k
k
r
k
avg
7
.
2
)
02
.
0
(
6
)
03
.
0
(
6
)
06
.
0
(
5
)
08
.
0
(
4
)
16
.
0
(
3
)
21
.
0
(
2
)
25
.
0
(
2
)
19
.
0
(
2
)
(
)
(
7
0
2
=
+
+
+
+
+
+
+
=
= ∑
=
2
1
n
n
CR =
R
d
C
R
1
1−
=
Compression
ratio:
Relative data
redundancy:
11
.
1
7
.
2
3
=
=
R
C 099
.
0
11
.
1
1
1 =
−
=
d
R

Inter-pixel Redundancy
Spatial Redundancy
Geometric Redundancy
Inter-frame Redundancy
Here the two pictures have
Approximately the same
Histogram.
We must exploit Pixel
Dependencies.
Each pixel can be estimated
From its neighbors

Psycho-visual Redundancy
Elimination of psych-visual redundant data results in a loss of
quantitative information ,it is commonly referred as quantization.
Improved Gray-
Scale

Psycho-visual Redundancy
IGS Quantization

Fidelity Criteria
The general classes of criteria :
1. Objective fidelity criteria
2. Subjective fidelity criteria

Fidelity Criteria
Objective fidelity:
Level of information loss can be expressed as a function
of the original and the compressed and subsequently
decompressed image.
2
/
1
2
1
0
1
0
]
)
,
(
)
,
(
ˆ
[
1






−
= ∑∑
−
=
−
=
M
x
N
y
rms y
x
f
y
x
f
MN
e
∑∑
∑∑
−
=
−
=
−
=
−
=
−
= 1
0
1
0
2
1
0
1
0
2
)]
,
(
)
,
(
ˆ
[
)
,
(
ˆ
M
x
N
y
M
x
N
y
ms
y
x
f
y
x
f
y
x
f
SNR
Root-mean-square
error
Mean-square
signal-to-noise ratio

Fidelity Criteria
25
.
10
93
.
6
=
=
rm
rms
SNR
e
39
.
10
78
.
6
=
=
rm
rms
SNR
e

Fidelity Criteria
Subjective fidelity (Viewed by Human):
• By absolute rating
• By means of side-by-side comparison of and
)
,
( y
x
f )
,
(
ˆ y
x
f

Image Compression Model
Remove Input
Redundancies
Increase the
Noise Immunity
The source encoder is responsible for removing redundancy
(coding, inter-pixel, psycho-visual)
The channel encoder ensures robustness against channel noise.

Classification
 Lossless compression
 lossless compression for legal and medical documents,
computer programs
 exploit only code and inter-pixel redundancy
 Lossy compression
 digital image and video where some errors or loss can be
tolerated
 exploit both code and inter-pixel redundancy and sycho-
visual perception properties

Error-Free Compression
Applications:
• Archive of medical or business documents
• Satellite imaging
• Digital radiography
They provide: Compression ratio of 2 to 10.

Huffman
coding
Variable-length Coding
The most popular technique for removing coding redundancy is due
to Huffman (1952)
Huffman Coding yields the smallest number of code symbols per
source symbol
The resulting code is optimal

Huffman
coding (optimal
code)

Huffman
coding
symbol
bits
entropy
symbol
bits
Lavg
/
14
.
2
/
2
.
2
)
5
)(
04
.
0
(
)
5
)(
06
.
0
(
)
4
)(
1
.
0
(
)
3
)(
1
.
0
(
)
2
)(
3
.
0
(
)
1
)(
4
.
0
(
=
=
+
+
+
+
+
=

Error Free Compression Technique.
Remove Inter-pixel redundancy.
Requires no priori knowledge of probability distribution of pixels.
Assigns fixed length code words to variable length sequences.
Patented Algorithm US 4,558,302.
LZWCoding

Coding Technique
A codebook or a dictionary has to be constructed
For an 8-bit monochrome image, the first 256 entries are
assigned to the gray levels 0,1,2,..,255.
As the encoder examines image pixels, gray level sequences
that are not in the dictionary are assigned to a new entry.
LZW Coding

LZW Coding
Example
Consider the following 4 x 4 8 bit image
39 39 126 126
39 39 126 126
39 39 126 126
39 39 126 126
Dictionary Location Entry
0 0
1 1
. .
255 255
256 -
511 -
Initial Dictionary

LZW Coding
39 39 126 126
39 39 126 126
39 39 126 126
39 39 126 126
•Is 39 in the dictionary……..Yes
•What about 39-39………….No
•Then add 39-39 in entry 256
•And output the last recognized symbol…39
Dictionary Location Entry
0 0
1 1
. .
255 255
256 -
511 -
39-39

H.R. Pourreza
Bit-plane coding
Bit-plane coding is based on
decomposing a multilevel
image into a series of binary
images and compressing
each binary image .
a b c d e f
f
e
d
c
b
a

Bit-plane coding
Binary Bit-planes Gray Bit-planes
1
1
1
0
1
2
1
0
0
2
2
1
1
2
0
:
2
2
2
:
−
−
+
−
−
−
−
−
−
=
−
≤
≤
⊕
=
+
+
+
−
m
m
l
l
l
m
m
m
m
m
m
a
g
m
l
a
a
g
g
g
g
g
code
Gray
a
a
a
scale
gray
bit
m


Bit-plane
decomposition

Bit-plane coding
Binary Bit-planes Gray Bit-planes
Bit-plane
decomposition

Bit-plane coding
• Constant area Coding
• One-dimensional run-length coding
1
0
1
0
L
L
H
H
HRL
+
+
=
Average values of
black and white run
lengths
Relative Address Coding (RAC) is based on tracking the
binary transitions.
• Two-dimensional RLC

Loss-less Predictive Coding







= ∑
=
−
m
l
l
n
l
n f
round
f
1
ˆ α
In most cases, the prediction is formed by a linear combination
of m previous pixels. That is:
1-D Linear Predictive coding:






−
= ∑
=
m
i
i
n i
y
x
f
round
y
x
f
1
)
,
(
)
,
(
ˆ α
m is the order of linear predictor

H.R. Pourreza
[ ]
)
1
,
(
)
,
(
ˆ −
= y
x
f
round
y
x
f α First-order linear predictor

Lossy Compression
Lossy encoding is based on the concept of
compromising the accuracy of the reconstructed
image in exchange for increased compression.
Lossy encoding techniques are capable of
reproducing recognizable mono-chrome images
from data that have been compressed by more than
100:1 and images that are virtually
indistinguishable from the original at 10:1 to 50:1 .
Lossy Compression:
1. Spatial domain methods
2. Transform coding

Lossy Compression
Lossy Predictive Coding
• Predictive Coding : transmit the difference
between estimate of future sample & the sample itself.
• Delta modulation
• DPCM
• Adaptive predictive coding
• Differential frame coding

Lossy Compression
Lossy Predictive Coding

Lossy Compression
Lossy Predictive Coding – Delta Modulation
(DM)



−
>
+
=
= −
otherwise
e
for
e
f
f
n
n
n
n
ζ
ξ
α
0
ˆ
1



Lossy Compression
Optimal Prediction
∑
∑
∑
=
=
−
=
−
≤








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

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−
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m
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}
{
ˆ
ˆ
ˆ
}
]
ˆ
{[
}
{
α
α
α


Differential Pulse Code
Modulation (DPCM)

Lossy Compression
Optimal Prediction
)
1
,
1
(
)
1
,
(
)
1
,
1
(
)
,
1
(
)
,
1
(
97
.
0
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1
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(
97
.
0
)
,
(
ˆ
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1
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1
(
5
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,
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(
75
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−
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
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
−
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≤
∆
−
=
−
−
−
−
+
−
=
−
+
−
=
−
=
y
x
f
y
x
f
v
and
y
x
f
y
x
f
h
otherwise
y
x
f
v
h
if
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
Prediction Error
Pred. #1
Pred. #2
Pred. #3
Pred. #4

Lossy Compression
Optimal Prediction

H.R. Pourreza
Lossy Compression
Prediction Error for different predictors
Pred. #1
Pred. #3
Pred. #2
Pred. #4

Lossy Compression
Optimal Quantization
)
(
)
( function
odd
an
is
q
s
q
t =
Decision
Levels
Reconstruction
Levels

Lossy Compression
)
(
)
( function
odd
an
is
q
s
q
t =
Minimization of the mean-square quantization error
i
i
i
i
i
i
i
s
s
i
t
t
s
s
L
i
L
i
t
t
i
s
L
i
ds
s
p
t
s
i
i
−
=
−
=









=
∞
−
=
+
=
=
=
−
−
−
+
∫
−
2
1
2
,
,
2
,
1
2
0
0
2
/
,
,
2
,
1
)
(
)
(
1
1


Additional constraint for optimum uniform quantizer:
θ
=
−
=
− −
− 1
1 i
i
i
i s
s
t
t

Lossy Compression
Unit variance Laplacian probability density function
As this table constructed for a unit variance
distribution, the reconstruction and decision levels
for the case of are obtained by multiplying
the tabulated values by the standard deviation 0of
the probability density function
1
≠
σ

Lossy Compression
DPCM RMSE
The best of four possible
quantizers is selected for
each block of 16 pixels.
Scaling factors: 0.5, 1.0,
1.75 and 2.5

Lossy Compression
DPCM result images
2-level Lloyd-
Max quantizer
1.0 bits/pixel
4-level Lloyd-
Max quantizer
2.0 bits/pixel
8-level Lloyd-
Max quantizer
3.0 bits/pixel
2-level adaptive
quantizer
1.125 bits/pixel
4-level adaptive
quantizer
2.125 bits/pixel
8-leveladaptive
quantizer
3.125 bits/pixel

Lossy Compression
DPCM Prediction Error
2-level Lloyd-
Max quantizer
1.0 bits/pixel
4-level Lloyd-
Max quantizer
2.0 bits/pixel
8-level Lloyd-
Max quantizer
3.0 bits/pixel
2-level adaptive
quantizer
1.125 bits/pixel
4-level adaptive
quantizer
2.125 bits/pixel
8-leveladaptive
quantizer
3.125 bits/pixel

Lossy Compression
Transform Coding
The goal of the transformation process is to decorrelate the
pixels of each sub-image, or to pack as much information as
possible into the smallest number of transform coefficients

1
,
,
1
,
0
,
)
,
,
,
(
)
,
(
)
,
(
1
,
,
1
,
0
,
)
,
,
,
(
)
,
(
)
,
(
1
0
1
0
1
0
1
0
−
=
=
−
=
=
∑∑
∑∑
−
=
−
=
−
=
−
=
N
y
x
v
u
y
x
h
v
u
T
y
x
f
N
v
u
v
u
y
x
g
y
x
f
v
u
T
N
u
N
v
N
u
N
v


Forward kernel is Separable
if:
Lossy Compression
Transform Coding
)
,
(
).
,
(
)
,
,
,
( 2
1 v
y
g
u
x
g
v
u
y
x
g =
Forward kernel is Symmetric if:
)
,
(
).
,
(
)
,
,
,
( 1
1
2
1 v
y
g
u
x
g
v
u
y
x
g
g
g =
⇒
=

Discrete Fourier Transform (DFT):
Lossy Compression
Transform Coding
N
vy
ux
j
N
vy
ux
j
e
v
u
y
x
h
e
N
v
u
y
x
g
/
)
(
2
/
)
(
2
)
,
,
,
(
1
)
,
,
,
(
+
+
−
=
=
π
π
Walsh-Hadamard Transform (WHT):
)
2
(
)
1
(
1
)
,
,
,
(
)
,
,
,
(
1
0
)]
(
)
(
)
(
)
(
[
m
v
p
y
b
u
p
x
b
N
N
v
u
y
x
h
v
u
y
x
g
m
i
i
i
i
i
=
∑
−
=
=
−
=
+
bk(z) is the kth bit (from right to left) in the binary
representation of z.

)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
1
1
3
2
2
2
1
1
1
0
u
b
u
b
u
p
u
b
u
b
u
p
u
b
u
b
u
p
u
b
u
p
m
m
m
m
m
m
+
=
+
=
+
=
=
−
−
−
−
−
−

Lossy Compression
Transform Coding

Discrete Cosin Transform (DCT):
Lossy Compression
Transform Coding







−
=
=
=





 +





 +
=
=
1
,
,
2
,
1
2
0
1
)
(
2
)
1
2
(
cos
2
)
1
2
(
cos
)
(
)
(
)
,
,
,
(
)
,
,
,
(
N
u
for
N
u
for
N
u
where
N
v
y
N
u
x
v
u
v
u
y
x
h
v
u
y
x
g

α
π
π
α
α

Lossy Compression
Transform Coding

Lossy Compression
Transform Coding
1. Dividing the image into
sub-images of size 8x8
2. Representing each sub-
image using one of the
transforms
3. Truncating 50% of the
resulting coefficients
4. Taking the inverse
Transform of the truncated
coefficients
DFT
WHT
DCT
rmse=1.28
rmse=0.86
rmse=0.68

H.R. Pourreza
Lossy Compression
Transform Coding
• DCT Advantages:
1. Implemented in a single integrated circuit (IC)
2. Packing the most information into the fewest
coefficients
3. Minimizing the block-like appearance (blocking
artifact)

Lossy Compression
Transform Coding
Sub-image size selection
Truncating 75% of the resulting coefficients

Lossy Compression
Transform Coding
Truncating 75% of the
resulting coefficients.
Sub-images size:
8x8
4x4
2x2

Lossy Compression
Transform Coding
Bit
allocation
87.5% of the DCT coeff.
Of each 8x8 subimage.
Threshold coding (8
coef)
Zonal coding

H.R. Pourreza
Lossy Compression
Transform Coding- Bit allocation

Lossy Compression
• Zonal coding
1. Fixed number of bits / coefficient
- Coefficients are normalized by their standard
deviations and uniformly quantized
2. Fixed number of bits is distributed among the coefficients
unequally.
- A quantizer such as an optimal Lloyed-Max is designed
for each coeff.:
- DC coeff. Is modeled by Rayleigh density func.
- The remaining coeff. Are modeled by Laplcian
or
Gaussian

Lossy Compression
• Threshold coding
1. Single global threshold
2. Different threshold for each subimage (N-Largest
coding)
3. Threshold can be varied as a function of the location of
each coeff.

H
Lossy Compression
Transform Coding
Bit
allocation
67:1
rmse:
6.33
34:1
rmse:
3.42

Lossy Compression
Transform Coding
Wavelet Coding

Lossy Compression
Transform Coding
Wavelet Coding
34:1
rmse:
2.29
67:1
rmse:
2.96
No blocking

Lossy Compression
Transform Coding
108:1
rmse:
3.72
167:1
rmse:
4.73
Wavelet Coding

Lossy Compression
Transform Coding
Quantizer
selection
Effectiveness of the quantization can be improved by:
• introducing an enlarge quantization interval around
zero
• Adapting the size of the quantization interval from
scale to scale

Why Do We Need International Standards?
 International standardization is conducted to achieve
inter-operability .
 Only syntax and decoder are specified.
 Encoder is not standardized and its optimization is left to the
manufacturer.
 Standards provide state-of-the-art technology that is
developed by a group of experts in the field.
 Not only solve current problems, but also anticipate the future
application requirements.
 Most of the standards are sanction by the International
Standardization Organization (ISO) and the
Consultative Committee of the International
Telephone and Telegraph (CCITT)
Image Compression Standards

CCITT Group 3 and 4
 They are designed as FAX coding methods.
 The Group 3 applies a non-adaptive 1-D run length
coding and optionally 2-D manner.
 Both standards use the same non-adaptive 2-D coding
approach, similar to RAC technique.
 They sometime result in data expansion. Therefore, the
Joint Bilevel Imaging Group (JBIG), has adopted several
other binary compression standards, JBIG1 and JBIG2.
Binary Image Compression Standards

What Is JPEG?
 "Joint Photographic Expert Group". Voted as
international standard in 1992.
 Works with color and grayscale images, e.g., satellite,
medical, ...
 Lossy and lossless
Continues Tone Still Image Comp.

 First generation JPEG uses DCT+Run length Huffman
entropy coding.
 Second generation JPEG (JPEG2000) uses wavelet
transform + bit plane coding + Arithmetic entropy
coding.
Continues Tone Still Image Comp. - JPEG

 Still-image compression standard
 Has 3 lossless modes and 1 lossy mode
 sequential baseline encoding
 encode in one scan
 input & output data precision is limited to 8 bits, while quantized
DCT values are restricted to 11 bits
 progressive encoding
 hierarchical encoding
 lossless encoding
 Can achieve compression ratios of up-to 20 to 1 without
noticeable reduction in image quality

 Work well for continuous tone images, but not good for
cartoons or computer generated images.
 Tend to filter out high frequency data.
 Can specify a quality level (Q)
 with too low Q, resulting images may contain blocky, contouring
and ringing structures.
 5 steps of sequential baseline encoding
 transform image to luminance/chrominance space (YCbCr)
 reduce the color components (optional)
 partition image into 8x8 pixel blocks and perform DCT on each
block
 quantize resulting DCT coefficients
 variable length code the quantized coefficients

Original JPEG 27:1
JPEG Encoding

Video compression standards:
1. Video teleconferencing standards
 H.261 (Px64)
 H.262
 H.263 (10 to 30 kbit/s)
 H.320 (ISDN bandwidth)
2. Multimedia standards
 MPEG-1 (1.5 Mbit/s)
 MPEG-2 (2-10 Mbit/s)
 MPEG-4 (5 to 64 kbit/s for mobile and PSTN and uo to 4
Mbit/s for TV and film application)
Video Compression Standards

Image Compression, Introduction Data Compression/ Data compression, modelling and coding,Image Compression

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