Lecture 5
Image Compression
Dr. Mohsen NASRI
College of Computer and Information Sciences,
Majmaah University, Al Majmaah
m.nasri@mu.edu.sa
Introduction
Definition
Reduce size of Image
(number of bits needed to represent an image)
Benefits
Reduce storage needed
Reduce transmission cost / latency / bandwidth
Applications
• Wireless image transmission
• image storage in multimedia/image databases
difficult to send real-time
uncompressed image over
current network
2
Need for image compression
 Huge sizes of the data files, in uncompressed form
 The storage devices have relatively slow access
=>impossible to render/index image info in real time
 limited bandwidth of the communication channels
=> Multimedia data (esp. image and video) have large data volume
The main advantage/need of/for image compression is that it
reduces the data storage requirement. It also offers an
alternative approach to reduce the communication cost
3
Need for image compression (cont…)
• Reduce the size of data.
– Reduces storage space and hence storage cost.
• Compression ratio = original data size/compressed
data size
– Reduces time to retrieve and transmit data.
4
Basic concept of a compresssion
5
Compression categories
• Compression = Image coding
Image compression
Lossless

Lossy
Lossless compression
The information content is not modified
Lossy compression
The information content is reduced and it is not recoverable

6
Image compression model
Original
Image
o Statistical Redundancy
o Psychovisual Redundancy
Reduction of data
redundancy
Reduction of
Entropy
Entropy encoding
Decompressed
Image
7
Statistical Redundancy
Statistical redundancy occurs due to the fact that pixels within an image tend
to have very similar intensities as those of its neighborhood, except at the
object boundaries or illumination changes. For still images, statistical
redundancies are essentially spatial in nature. For natural two-dimensional
images, redundancies are present along both the x- and y-dimensions. Video
signals exhibit yet another form of statistical redundancy and that is temporal.
For video, intensities of same pixel positions across successive frames tend
to be very similar, unless there is large amount of motion present. In this
lesson however, we focus our attention to only one form of statistical
redundancy and that is spatial redundancies of images. Presence of
statistical redundancies in natural images allows efficient representation in
the transformed output.
8
Psychovisual Redundancy
Psychovisual redundancy arises due to the problem of perception. Our eyes are
more responsive to slow and gradual changes of illumination than perceiving
finer details and rapid changes of intensities. Hence, to what extent we should
preserve the details for our perception and to what extent we can compromise
on the quality of reconstructed image that we perceive is essentially carried out
by exploiting the psychovisual redundancy. As we shall see later, psychovisual
redundancy has been well studied and its exploitation has been included within
the multimedia standards.
9
Measuring the quality of reconstructed images
To evaluate the image quality, we should use the peak signal-to-noise ratio
metric, which is defined (in decibels) by:
where q is the number of bits per pixel (bpp) of the original image, and MSE is
the mean-square-error which is defined by:
10
Elements of Image Compression System
A typical image compression system/image encoder consists of the following
elements
Original Image
Decoded Image
Bitstream
Encoder
0101100111...
Decoder
Compression system Compressed Decompression system
Image
The image file is converted into a series of binary data, which is called
the bit-stream
The decoder receives the encoded bit-stream and decodes it to
reconstruct the image
The total data quantity of the bit-stream is less than the total data
quantity of the original image
11
Source coding algorithms
To achieve less average length of bits per pixel of the image.
Assigns short descriptions to the more frequent outcomes and long
descriptions to the less frequent outcomes
Entropy Coding Methods
Huffman Coding
Arithmetic Coding
Lossless
Compression
Sequence of source
symbols ui
Source
Encoder
Source alphabet
U  {u1 u 2
P  { p1 p2
Sequence of code symbols ai
Code alphabet
uM}
pM}
A  {a1 a2
an}
12
Huffman code
Approach
Variable length encoding of symbols
Exploit statistical frequency of symbols
Efficient when symbol probabilities vary widely
Principle
Use fewer bits to represent frequent symbols
Use more bits to represent infrequent symbols
A
A
A
A
B
B
A
A
13
Huffman Code Data Structures
Binary (Huffman) tree
4
Represents Huffman code
Edge  code (0 or 1)
Leaf  symbol
Path to leaf  encoding
Example
A = “000”, B = “001”, C = “01”
1
0
3
To efficiently build binary tree
1
0
2
d = “1”
Priority queue
1
0
1
A
D
1
1
C
1
B
14
Huffman Code Algorithm Overview
Encoding
Calculate frequency of symbols in file
Create binary tree representing “best” encoding
Use binary tree to encode compressed file
For each symbol, output path from root to leaf
Size of encoding = length of path
Save binary tree
15
Huffman Code – Algorithm
Huffman Coding Algorithm
(1) Order the symbols according to the probabilities
Alphabet set: S1, S2,…, SN
Probabilities: P1, P2,…, PN
The symbols are arranged so that P1≧ P2 ≧ … ≧ PN
(2) Apply a contraction process to the two symbols with the smallest
probabilities. Replace the last two symbols SN and SN-1 to form a new
symbol HN-1 that has the probabilities P1 +P2.
The new set of symbols has N-1 members: S1, S2,…, SN-2 , HN-1
(3) Repeat the step 2 until the final set has only one member.
(4) The codeword for each symbol Si is obtained by traversing the binary
tree from its root to the leaf node corresponding to Si
16
Huffman Tree Construction
The message to be encoded is "ABBBBAAC"
Symbols
A
B
C
Frequency of occurrence
3
4
1
Huffman code
0
1
00
8
High frequency of occurrence : shorter bit strings
0
1
B
4
4
0
1
C
Low frequency of occurrence :Longer bit strings
1
3
A
char encoding
A
10
B
0
C
11
B
The principle is to use a lower number of
bits to encode the data that occurs more
frequently
17
Huffman Code Properties
Prefix code
No code is a prefix of another code
Example
Huffman(“I”)
 00
Huffman(“X”)
 001
// not legal prefix code
Can stop as soon as complete code found
No need for end-of-code marker
Nondeterministic
Multiple Huffman coding possible for same input
If more than two trees with same minimal weight
18
Huffman Code Properties
Greedy algorithm
Chooses best local solution at each step
Combines 2 trees with lowest frequency
Still yields overall best solution
Optimal prefix code
Based on statistical frequency
Better compression possible (depends on data)
Using other approaches (e.g., pattern dictionary)
19
Arithmetic Coding
Arithmetic Coding: a direct extension of Shannon-Fano-Elias
coding calculate the probability mass function p(xn) and the
cumulative distribution function F(xn) for the source sequence xn
Lossless compression technique
Treate multiple symbols as a single data unit
Arithmetic Coding Algorithm
Input symbol is l
Previouslow is the lower bound for the old interval
Previoushigh is the upper bound for the old interval
Range is Previoushigh - Previouslow
Let Previouslow= 0, Previoushigh = 1, Range = Previoushigh – Previouslow =1
WHILE (input symbol != EOF)
EOF = End Of File
get input symbol l
Range = Previoushigh - Previouslow
New Previouslow = Previouslow + Range* intervallow of l
New Previoushigh = Previouslow + Range* intervalhigh of l
END
Arithmetic Coding
Symbol
Probability
Sub-interval
k
0.05
[0.00,0.05)
l
0.2
[0.05,0.25)
u
0.1
[0.20,0.35)
w
0.05
[0.35,0.40)
e
0.3
[0.40,0.70)
r
0.2
[0.70,0.90)
?
0.2
[0.90,1.00)
1
Input String : k l u w r e ?
k l u wre ?
0.0713348389
0.25
0.10
0.074
0.0714
0.07136
0.071336
0.0713360
?
?
?
?
?
?
?
?
r
r
r
r
r
r
r
r
e
e
e
e
e
e
e
e
w
u
w
u
w
u
w
u
w
u
w
u
w
u
w
u
l
l
l
l
l
l
l
l
k
k
k
k
k
k
k
k
0
0.05
0.06
0.070
0.0710
0.07128
0.07132
0.0713336
1.00
0.90
0.70
0.40
0.35
0.25
0.05
0
Arithmetic Coding cont…
Input String : ESIPE
Symbol
Symbol
Probability
Sub-interval
E
0.4
[0.00,0.4[
S
0.2
[0.4,0.6[
I
0.2
[0.6,0.8[
P
0.2
[0.8,1.0[
lower bound
upper bound
0.0
1.0
E
0.0
0.4
S
0.16
0.24
I
0.208
0.224
P
0.2208
0.224
E
0.2208
0.22208
The lower bound code the Input string
Overview of Image Compression Algorithms
Embedded zero-trees Wavelet (EZW)
The EZW encoder is based on two important observations:
1. Natural images in general have a low pass spectrum. When an image
is wavelet transformed the energy in the sub-bands decreases as the
scale decreases (low scale means high resolution), so the wavelet
coefficients will, on average, be smaller in the higher sub-bands than in
the lower sub-bands. This shows that progressive encoding is a very
natural choice for compressing wavelet transformed images, since the
higher sub-bands only add detail;
2. Large wavelet coefficients are more important than smaller wavelet
coefficients.
23
Overview of Image Compression Algorithms cont…
Embedded zero-trees Wavelet (EZW)
The EZW encoder is based on DWT
– Energy is compacted into a small number of coeff.
– Significant coeff. tend to cluster at the same spatial location in
each frequency subband
LL3

Two set of info. to code:
– Where are the significant
coefficients?
– What values are the significant
coefficients?
HL3
HL2
LH3 HH3
HL1
LH2 HH2
LH1
HH1
•Three
resoluti
on level
• Two
decomp
osition
levels
24
Statistics Filters
EZW based on three basic elements
Original
Image
Reconstructed
Image
DWT
EZW
Encoder
Arithmetic
Encoder
Arithmetic
decoder
EZW
decoder
2-D IDWT
Encoded
Image
25
Overview of Image Compression Algorithms cont…
Embedded zero-trees Wavelet (EZW)
coefficients that are in the same spatial
location consist of a quad-tree.

Parent-children relation among coeff.
– Each parent coeff at level k spatially correlates with 4 coeff at
level (k-1) of same orientation
– A coeff at lowest band correlates with 3 coeff.
26
Overview of Image Compression Algorithms cont…
Structure of EZW
 Root: a
 Descendants: a1, a2, a3
27
Overview of Image Compression Algorithms cont…
Structure of EZW
 EZW scanning order of sub-bands
scan order of the transmission band
28
Overview of Image Compression Algorithms cont…
EZW algorithm
With Ti=Ti/2 and T0 = ‫׀‬Cmax‫׀‬/2
29
Overview of Image Compression Algorithms cont…
EZW algorithm
30
Overview of Image Compression Algorithms cont…
EZW algorithm
P
N
IZ
ZT
31
EZW - example
32
EZW – Example cont...
33
EZW – Example Cont...
34
EZW – Example Cont...
35
Contd..
36
JPEG
JPEG encoder
The encoding process consists of several steps:
• Color space transformation.
• Block splitting.
• Discrete Cosine Transform.
• Quantization.
• Entropy coding.
AC
Color
components
(Y, Cb, or Cr)
88
FDCT
Zig-zag
reordering
Huffman
Table
Huffman
coding
JPEG
bit-stream
Quantizer
DC
Quantization
Table
Difference
Encoding
Huffman
coding
Huffman
Table
• JPEG : Joint Photographic Experts Group
37
JPEG
JPEG decoder
Bit-stream
De-zigzag
Huffman
Decoding
De-DC
coding
Huffman
Decoding
Dequantizer
Quantization
Table
Chrominance
Upsampling
(4:2:2 or 4:2:0)
G
8X8
IDCT
YVU color
coordinate
B
R
Decoded
Image
38
Color Space Conversion
 Y   0.299000 0.587000 0.114000   R   0 
C    0.168736 0.331264 0.500002  G   128
 b 
   
Cr   0.500000 0.418688 0.081312   B  128
(a) translate from RGB to YCbCr
0.0
1.40210   Y 
 R  1.0
G   1.0 0.34414 0.71414 C  128
  
 b

 B  1.0 1.77180
0.0  Cr  128
(b) translate from YCbCr to RGB
39
DC and AC coefficients
• DC
Coefficient :
• First coefficient in every 8 x 8 block
• Represents the average value of pixels in block
• AC Coefficients : Remaining 63 coefficients in every 8 x 8 block
• DC Coefficients: treated separately from the AC Coefficients
• Differential values of DC coeffs. of all blocks are derived and encoded
40
Zig-zag ordering in an 8x8 matrix
Zig-Zag ordering: converting a 2D matrix
into a 1D array, so that the frequency
(horizontal+vertical) increases in this order,
and the coefficient variance decreases in
this order.
41
Quantization in JPEG
Quantization step is the main source of Lossy Compression
• DCT itself is not Lossy
Quantisation Tables
• In JPEG, each F[u,v] is divided by a constant q(u,v).
• Table of q(u,v) is called quantisation table.
 F (u, v) 
Quantization F (u, v)Quantization  round 

Q
(
u
,
v
)


Dequantization F (u, v) deQ  F (u, v)Quantization  Q(u, v)
 16

 12
 14

14
QY  
 18

 24
 49

 72
11 10 16
24
40
51
12 14 19
13 16 24
26
40
58
57
60
69
17 22 29
22 37 56
51 87 80
68 109 103
35 55 64 81 104 113
64 78 87 103 121 120
92 95 98 112 100 103
61 

55 
56 

62 
77 

92 
101 

99 
42
JPEG 2000
JPEG 2000 Encoder
B
G
R
Image
JPEG 2000
Bit-stream
Forward
Component
Transform
2D DWT
Context
Modeling
Quantization
EBCOT
RateDistortion
Control
Arithmetic
Coding
Tier-1
Tier-2
JPEG 2000 Decoder
JPEG 2000
Bit-stream
EBCOT
Decoder
Dequantization
2D IDWT
Inverse
Component
Transform
R
G
B
Decoded
Image
43
Quantization in JPEG 2000
 | ab(u, v) | 
q
b
(
u
,
v
)

sign
a
u
(
u
,
v
)

floor


Quantization coefficients
 b 
ab(u,v) : the wavelet coefficients of subband b
Quantization step size
b  2
Rb  b
 b 
1  11 
 2 
Rb: the nominal dynamic range of subband b
εb: number of bits alloted to the exponent of the subband’s coefficients
μb: number of bits allotted to the mantissa of the subband’s coefficients
44
Bitplane Scanning
• The decimal DWT coefficients can be converted into signed binary
format, so the DWT coefficients are decomposed into many 1-bit
planes.
• In one 1-bit-plane
– Significant
• A bit is called significant after the first bit ‘1’ is met from MSB to LSB
– Insignificant
• The bits ‘0’ before the first bit ‘1’ are insignificant
n
n
Sign
MSB
0
MSB
insignificant
0
coding
order
significant
First 1 appear
1
0
0
LSB
1
1
45
LSB
Scanning Sequence
• The scanning order of the bit-plane
• Sample
– Each element of the bit-plane is called a sample
• Column stripe
– Four vertical samples can form one column stripe
• Full stripe
– The 16 column stripes in the horizontal direction can form a full stripe
Stripe height of 4
Code block 64-bits wide
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
2
6
10
14
18
22
26
30
34
38
42
46
50
54
58
62
3
7
11
15
19
23
27
31
35
39
43
47
51
55
59
63
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
65
...
66
...
46
Comparison between JPEG and JPEG 2000
Image 1
Image 2
47
Thank You
Have a Nice Day