Image Processing

1. Image Compression
Done By:
Bassam Kanber
Khawla Al-Hashedy
Naglaa Fathi
Redha Qaid
Heba Al-Hakemy
Hesham Al-Aghbary
Supervisor
Ensaf Al-Zurqa

2. Goal of Image Compression
The goal of image compression is to reduce the
amount of data required to represent a digital
image.

3. Data ≠ Information
 Data and information are not synonymous terms!
 Data is the means by which information is conveyed.
 Data compression aims to reduce the amount of data
required to represent a given quantity of information
while preserving as much information as possible.

4. Data vs Information (cont’d)
 The same amount of information can be represented
by various amount of data.
 Ex1:
Your wife, Helen, will meet you at Logan Airport in
Boston at 5 minutes past 6:00 pm tomorrow night
 Ex2:
Your wife will meet you at Logan Airport at 5 minutes
past 6:00 pm tomorrow night
 Ex3:
Helen will meet you at Logan at 6:00 pm tomorrow night

5. Definitions: Compression Ratio
compression
Compression ratio:

6. Definitions: Data Redundancy
 Relative data redundancy:
Example:

7. Types of Data Redundancy
(1) Coding Redundancy
(2) Interpixel Redundancy
(3) Psychovisual Redundancy
 Compression attempts to reduce one or more of these
redundancy types.

8. Coding Redundancy
 Code: a list of symbols (letters, numbers, bits etc.)
 Code word: a sequence of symbols used to represent a
piece of information or an event (e.g., gray levels).
 Code word length: number of symbols in each code
word

9. Coding Redundancy (cont’d)
N x M image
rk: k-th gray level
P(rk): probability of rk
l(rk): # of bits for rk
Expected value:

10. Coding Redundancy (con’d)
Case 1: l(rk) = constant length
Example:

11. Case 2: l(rk) = variable length

12. Interpixel redundancy
 Interpixel redundancy implies that pixel values are
correlated (i.e., a pixel value can be reasonably
predicted by its neighbors).
autocorrelation: f(x)=g(x)

13. Interpixel redundancy (cont’d)
 To reduce interpixel redundancy, the data must be
transformed in another format (i.e., using a transformation)
 e.g., thresholding, DFT, DWT, etc.
 Example:
origin
al
thresholded

14. Psychovisual redundancy
 The human eye does not respond with equal sensitivity
to all visual information.
 It is more sensitive to the lower frequencies than to the
higher frequencies in the visual spectrum.
 Idea: discard data that is perceptually insignificant!

15. Psychovisual redundancy (cont’d)
256 gray levels 16 gray levels 16 gray levels/random noise
C=8/4 = 2:1
i.e., add to each pixel a
small pseudo-random number
prior to quantization
Example: quantization

16. Fidelity Criteria
 How close is to ?
 Criteria
 Subjective: based on human observers
 Objective: mathematically defined criteria

17. Subjective Fidelity Criteria

18. Objective Fidelity Criteria
 Root mean square error (RMS)
 Mean-square signal-to-noise ratio (SNR)

19. Objective Fidelity Criteria (cont’d)
RMSE = 5.17 RMSE = 15.67 RMSE = 14.17

20. Image Compression Models
 The image compression system is composed of 2
distinct structural blocks: an encoder & a decoder.
 Encoder performs Compression
 Decoder performs Decompression.

21.  The encoder is made up of a source encoder which removes
input redundancies, and a channel encoder, which increases the
noise immunity of the source encoder's output.
 The de-coder includes a channel decoder followed by a source
decoder.

 If the channel between the encoder and decoder is noise free (no
prone or error) the channel encoder and decoder are omitted.
 Input image f(x,y) is fed into the encoder, which creates a
compressed representation of input.
 It is stored for future for later use or transmitted for storage and
use at a remote location.
 When the compressed image is given to decoder, a reconstructed
output image f’(x,…..) is generated.
 The encoded input and decoder output are f(x, y) & f’(x, y) resp.
 In video applications, they are f(x, y, t) & f’(x, y, t) where t is time.

22.  1-The Source Encoder and Decoder::
 The source encoder is responsible for reducing or eliminating any
coding, interpixel and psychovisual redundancies in the input
image.
 Encoder is used to remove the redundancies through a series of 3
independent operations.
 Mapper:It transforms f(x,y) into a format designed to reduce
interpixel redundancies.
 It is reversible
 It may / may not reduce the amount of data to represent image.
 Ex. Run Length coding
 Quantizer: reduces the accuracy of the mapper's output in
accordance with some pre-established fidelity criterion. This stage
reduces the psychovisual redundancies of the input image.
 This operation is irreversible.

23. Encoder
 Symbol Encoder: Generates a fixed or variable length
code to represent the quantizer output and maps the
output in accordance with the code.
• In most cases, a variable-length code is used to represent
the mapped and quantized data set. It assigns the shortest
code words to the most frequently occurring output values
and thus reduces coding redundancy.
• It is reversible.
• Upon its completion, the input image has been processed
for the removal of all 3 redundancies.

24. Encoder
Mapper: transforms input data in a way
that facilitates reduction of interpixel
redundancies.

25. Encoder
Quantizer: reduces the accuracy of the
mapper’s output in accordance with some pre-established
fidelity criteria.

26. Encoder
Symbol encoder: assigns the shortest code
to the most frequently occurring output
values.

27. Decoder
• Inverse operations are performed.
• But … quantization is irreversible in
general.
• Quantization results in irreversible loss,
an inverse quantizer block is not included
in the decoder block.

29. Channel
 2-The Channel Encoder and Decoder::
 In the overall encoding-decoding process when the channel is noisy or
prone to error They are designed to reduce the impact of channel noise by
inserting a controlled form of redundancy into the source encoded data.
 As the output of the source encoder contains little redundancy, it would
be highly sensitive to transmission noise without the addition of this
"controlled redundancy."
 One of the most useful channel encoding techniques was devised by R.
W.Hamming (Hamming [1950]).
 It is based on appending enough bits to the data being encoded to ensure
that some minimum number of bits must change between valid code
words.
 Hamming(7,4) is a linear error-correcting code that encodes 4 bits of data
into 7 bits by adding 3 parity bits.
 Hamming's (7,4) algorithm can correct any single-bit error, or detect all
single-bit and two-bit errors.

30. Elements of Information Theory
 Measuring Information
 The generation of information is modeled as a probabilistic
process. Random event E occurs with probability P(E)
 I(E)=log= 1 =_ log (p(E))
 P(E)
 The base of the logarithm determines the units used to measure
the information. If the base 2 is selected the resulting
information unit is called bit. If P(E)=0.5 (two possible equally
likely events) the information is one bit
 I(E) is called the self-information of E.

31. The Information Channel
Information channel is the physical medium that
connectsthe information source to the user of information.
Self-information is transferred between an information
source and a user of the information, through the
information channel.
Information source
Generates a random sequence of symbols from a finite or
countably infinite set of possible symbols.
Output of the source is a discrete random variable

32.  The source :
Modeled as a discrete random variable
Source alphabet A={aj}
Symbols (letters) aj with probabilities P(aj)

33. A simple information system
Output of the channel is also a discrete random variable which
takes on values from a finite or countably infinite set of symbols
{b1, b2,…, b K} called the channel alphabet B

34. Entropy
Conditional entropy function
units/pixel

35. Entropy Estimation
It is not easy to estimate H reliably!
image

36. Entropy
First order estimate of H:

37. Entropy
Second order estimate of H:
Use relative frequencies of pixel blocks

38. Estimating Entropy
The first-order estimate provides only a lower-bound on
the compression that can be achieved.
Differences between higher-order estimates of entropy
and the first-order estimate indicate the presence of
interpixel redundancy!

39. Estimating Entropy
For example, consider differences:
16

40. Estimating Entropy
 Entropy of difference image:
 Better than before (i.e., H=1.81 for original image)
 However, a better transformation could be found
since:

41. Compression Types
Compression
Error-Free
Compression
(Loss-less)
Lossy
Compression

42. Error-Free Compression
 Some applications require no error in compression
(medical, business documents, etc..)
 CR=2 to 10 can be expected.
 Make use of coding redundancy and inter-pixel
redundancy.
 Ex: Huffman codes, LZW, Arithmetic coding, 1D and 2D
run-length encoding, Loss-less Predictive Coding, and
Bit-Plane Coding.

43. Run-length encoding (RLE)
 is a very simple form of data compression
 stored as a single data value and count.
 Ex: AAAABBCCCAA
Sol: 3A2B3C2A

44. Huffman Coding
 Huffman Coding
 The most popular technique for removing coding
redundancy is due to Huffman (1952)
 A variable-length coding technique.
 Optimal code (i.e., minimizes the number of code
symbols per source symbol).
 Huffman Coding yields the smallest number of code
symbols per source symbol
 Assumption: symbols are encoded one at a time!

45. Huffman Coding

46. Huffman Coding
1(0.4) 2(0.3) 3(0.1) 4(0.1) 5(0.06) 5(0.04)
bits
Lavg
2.2

     

47.  Arithmetic coding
 5 symbol message, a1a2a3a3a4 from 4 symbol
source is coded.
Source Symbol Probability Initial Subinterval
a1 0.2 [0.0, 0.2)
a2 0.2 [0.2, 0.4)
a3 0.4 [0.4, 0.8)
a4 0.2 [0.8, 1.0)

48. Arithmetic coding

49.  The final message symbol narrows to [0.06752,
0.0688).
 Any number between this interval can be used to
represent the message.
 E.g. 0.068
 3 decimal digits are used to represent the 5 symbol
message.

50. Fixed Length: LZW Coding
 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
 Included in GIF and TIFF and PDF file formats

51. 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.
 For instance sequence 255-255 can be assigned to
entry 256, the address following the locations
reserved for gray levels 0 to 255.

52. 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 -

53. 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 39-39
511 -

54. Bit-Plane Coding
 An effective technique to reduce inter pixel
redundancy is to process each bit plane
individually
 The image is decomposed into a series of binary
images.
 Each binary image is compressed using one of
well known binary compression techniques.

55. Bit-Plane Decomposition

56. Bit-Plane Encoding
 one Dimensional Run Length coding
 Two Dimensional Run Length coding

57.  Loss-less Predictive Encoding

59. Decoding LZW
 Use the dictionary for decoding the “encoded output”
sequence.
 The dictionary need not be sent with the encoded
output.
 Can be built on the “fly” by the decoder as it reads the
received code words.

60. Run-length coding (RLC)
(interpixel redundancy)
 Encodes repeating string of symbols (i.e., runs) using
a few bytes: (symbol, count)
1 1 1 1 1 0 0 0 0 0 0 1  (1,5) (0, 6) (1, 1)
a a a b b b b b b c c  (a,3) (b, 6) (c, 2)
 Can compress any type of data but cannot achieve
high compression ratios compared to other
compression methods.

61. Bit-plane coding (interpixel
redundancy)
 Process each bit plane individually.
 (1) Decompose an image into a series of binary images.
 (2) Compress each binary image (e.g., using run-length
coding)

62. Combining Huffman Coding
with Run-length Coding
 Assuming that a message has been encoded using
Huffman coding, additional compression can be
achieved using run-length coding.
e.g., (0,1)(1,1)(0,1)(1,0)(0,2)(1,4)(0,2)

63. Lossy Methods -Taxonomy
See “Image Compression Techniques”, IEEE
Potentials, February/March 2001

64. Lossy Compression
 Transform the image into a domain where
compression can be performed more efficiently (i.e.,
reduce interpixel redundancies).
~ (N/n)2 subimages

65. Transform Selection
 T(u,v) can be computed using various
transformations, for example:
DFT
DCT (Discrete Cosine Transform)
KLT (Karhunen-Loeve Transformation)

66. Example: Fourier Transform
The magnitude
of the FT
decreases, as u,
v increase! K << N

67. DCT (Discrete Cosine Transform)
Forward
Inverse

68. DCT (cont’d)
 Set of basis functions for a 4x4 image (i.e., cosines of
different frequencies).

69. DCT (cont’d)
Using
8 x 8 subimages
64 coefficients
per subimage
50% of the
coefficients
truncated
RMS error: 2.32 1.78 1.13

70. DCT (cont’d)
 DCT reduces "blocking artifacts" (i.e., boundaries
between subimages do not become very visible).

71. DCT (cont’d)
 DCT reduces "blocking artifacts" (i.e., boundaries
between subimages do not become very visible).
DFT
i.e., n-point
periodicity
gives rise to
discontinuities!
DCT
i.e., 2n-point periodicity
prevents
discontinuities!

72. DCT (cont’d)
 Subimages size selection:

73. image compression standard
 ISO & CCITT
 Binary
 Continuous-tone
 Video

74. Binary image compression
 Fax: standard
-Transmitting Docs. over Tele. Nets.
 CCITT Group 3 (Huffman) 1D & 2D
 CCITT Group 4 2D
 JBIG 1

75. Continuous-tone still image
JPEG compression standard
baseline, lossless, progressive and hierarchical

76. JPEG 2000
 wavelet and sub-band technologies
 Embedded Block Coding with Optimized
Truncation (EBCOT)

77. Features of JPEG2000
 Lossless and lossy compression.
 Protective image security.
 Region-of-interest coding.
 Robustness to bit errors.

78. JPEG-LS
 latest ISO/ITU-T standard for lossless coding of still
images.
 provides for “near-lossless” compression.
 Part-I:
the baseline system, is based on:
adaptive prediction, context modeling and Golomb
coding.
 Part-II (still under preparation).
 Designed for low-complexity.

79. Video Compression Standards
 MPEG-1
The driving focus of the standard was storage of multimedia
content on a standard CDROM, which supported data
transfer rates of 1.4 Mb/s and a total storage capability of
about 600 MB.
 MPEG-2
Designed to provide the capability for compressing, coding,
and transmitting high quality, multi-channel, multimedia
signals over broadband networks.
ex: ATM.
 MPEG-4
Digital television, interactive graphics and the World Wide
Web.