1. Lecture 5 Image Compression
Dr. Mohsen NASRI College of Computer and Information Sciences, Majmaah University, Al Majmaah

2. Introduction Definition Benefits Applications
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

3. 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

4. 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.

5. Basic concept of a compresssion

6. 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

7. Image compression model
Original Image
Statistical Redundancy
Psychovisual Redundancy
Reduction of data redundancy
Reduction of Entropy
Entropy encoding
Decompressed Image

8. 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.

9. 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.

10. 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:

11. Elements of Image Compression System
A typical image compression system/image encoder consists of the following elements
Compressed Image
Compression system
Decompression system
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

12. 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
Source
Encoder
Sequence of source symbols ui
Sequence of code symbols ai
Source alphabet
Code alphabet

13. Huffman code Approach Principle 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 B A A A B A

14. Huffman Code Data Structures
Binary (Huffman) tree
Represents Huffman code
Edge  code (0 or 1)
Leaf  symbol
Path to leaf  encoding
Example
A = “000”, B = “001”, C = “01”
d = “1”
Priority queue
To efficiently build binary tree 4 1 3 1 1 D 2 1 1 C 1 1 A B

15. 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

16. 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

17. Huffman Tree Construction
The message to be encoded is "ABBBBAAC" C B A
Symbols 1 4 3
Frequency of occurrence 00
Huffman code 8
High frequency of occurrence : shorter bit strings
Low frequency of occurrence :Longer bit strings 1 4 B
char encoding A B C 4 1 B
 The principle is to use a lower number of bits to encode the data that occurs more frequently 1 3 C A

18. Huffman Code Properties
Prefix code
No code is a prefix of another code
Example
Huffman(“I”)  00
Huffman(“X”)  // 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

19. 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)

20. Arithmetic Coding Algorithm
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

21. Arithmetic Coding Input String : k l u w r e ? k l u w r e ?
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
[0.35,0.40) e
0.3
[0.40,0.70) r
[0.70,0.90) ?
[0.90,1.00) k l u w r e ? ? r e w u l k
1.00
0.25
0.35
0.40
0.70
0.90
0.05 1 ? r e w u l k
0.25
0.05 ? r e w u l k
0.10
0.06 ? r e w u l k
0.074
0.070 ? r e w u l k
0.0714
0.0710 ? r e w u l k ? r e w u l k ? r e w u l k

22. Arithmetic Coding cont…
Input String : ESIPE
Symbol
lower bound
upper bound
0.0
1.0 E
0.4 S
0.16
0.24 I
0.208
0.224 P
0.2208
Symbol
Probability
Sub-interval E
0.4
[0.00,0.4[ S
0.2
[0.4,0.6[ I
[0.6,0.8[ P
[0.8,1.0[
The lower bound code the Input string

23. 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.

24. 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
Two set of info. to code:
Where are the significant coefficients?
What values are the significant coefficients?
LH1
HL1
HH1
HL2
LH2
HH2
LL3
HL3
HH3
LH3
Three resolution level
Two decomposition levels

25. Statistics Filters EZW based on three basic elements DWT EZW Encoder
Arithmetic Encoder
Original Image
Encoded Image
Reconstructed
Image
Arithmetic decoder
EZW
decoder
2-D IDWT

26. 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.

27. Overview of Image Compression Algorithms cont…
Structure of EZW
Root: a
Descendants: a1, a2, a3

28. Overview of Image Compression Algorithms cont…
Structure of EZW
EZW scanning order of sub-bands
scan order of the transmission band

29. Overview of Image Compression Algorithms cont…
EZW algorithm
With Ti=Ti/2 and T0 = ׀Cmax׀/2

30. Overview of Image Compression Algorithms cont…
EZW algorithm

31. Overview of Image Compression Algorithms cont…
EZW algorithm P N IZ ZT

32. EZW - example

33. EZW – Example cont...

34. EZW – Example Cont...

35. EZW – Example Cont...

36. Contd..

37. Color components (Y, Cb, or Cr)
JPEG
JPEG encoder
The encoding process consists of several steps:
• Color space transformation.
• Block splitting.
• Discrete Cosine Transform.
• Quantization.
• Entropy coding.
Huffman Table AC
Zig-zag reordering
Huffman coding
Color components (Y, Cb, or Cr)
88 FDCT
Quantizer
JPEG bit-stream
Difference Encoding
Huffman coding DC
Quantization Table
Huffman Table
JPEG : Joint Photographic Experts Group

38. JPEG
JPEG decoder

39. Color Space Conversion
(a) translate from RGB to YCbCr
(b) translate from YCbCr to RGB

40. DC and AC coefficients First coefficient in every 8 x 8 block
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

41. 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.

42. 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.
Quantization
Dequantization

43. JPEG 2000
JPEG 2000 Encoder
JPEG 2000 Decoder

44. Quantization in JPEG 2000 Quantization coefficients
ab(u,v) : the wavelet coefficients of subband b
Quantization step size
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

45. Bitplane Scanning Significant Insignificant
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

46. 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

47. Comparison between JPEG and JPEG 2000
Image 1
Image 2

48. Thank You Have a Nice Day