1. Multimedia – Image Compression
Dr. Lina A. Nimri
Lebanese University
Faculty of Economic Sciences and Business Administration 1st branch

2. Image Compression
To store, treat or print an image, it is necessary to reduce the size of the file of the image while keeping the information applicable
It is necessary to compress the image
JPEG quality 100% JPEG quality 50 %
JPEG quality 25 % JPEG quality 0 %

3. Image Compression
Image compression is applying data compression methods on digital images
This compression is to reduce the size of an image to be able to store it using little storage space or to transmit it quickly

4. Image Compression Image compression can be lossless or lossy
Lossless compression is sometimes preferred for images:
Art drawings, technical designs, icons or comics
Requiring a high precision, such as medical scans
Lossy methods are appropriate for images:
Natural photos, such as photos in an applications or when a minor loss of exactness is acceptable while acheiving a substantial reduction in the binary code.

5. Lossy Compression
Lossless compression algorithms do not deliver compression ratios that are high enough. Hence, most multimedia compression algorithms are lossy.
The compressed data is not the same as the original data, but a close approximation of it.
Yields a much higher compression ratio than that of lossless compression

6. Lossy Compression The most important methods of lossy compression are:
The reduction of the color depth to the most frequent colors in the image
Sub-sampling of chrominance
This method takes advantage of the fact that the eye perceives brightness more vigorously than color, dropping at least half of chrominance information in the image
Fractal compression

7. Lossy Compression - continued
The most important methods of lossy compression are
Compression –Transform Coding
This is the most commonly used method
Discrete Cosine Transform (DCT) and wavelet transforms are the most popular
Transform Coding includes the application of the transformation in the image, followed by quantization and entropy coding

8. Distortion Measure Distortion is a mathematical measure of error
Where xn, yn and N are the input data sequence, reconstructed data sequence, and length of the data sequence respectively.
The visual distortion (perceptual distortion) measures the perceived error
It is possible to have a very clear visual distortion with a poor measure of distortion
An image shifted to the right of a vertical line

9. The Rate-Distortion Theory
Rate: the average number of bits needed to represent a data source
Typical Rate Distortion Function: R(D)
If D is a tolerable distortion, then R (D) is the minimum rate with which the data source can be coded while having an upper limit of distortion equals to D
Provides a framework for the study of tradeoffs between Rate and Distortion.

10. Fractal Compression
Fractal Compression is an image compression little used today
It is based on the detection of recurrence of the grounds, and tends to eliminate redundancy of information in the image.
Fractal compression is a destructive compression method as all baselines are not reflected in the final image.

11. Fractal Compression

12. Compression –Transform Coding
Decoding
Coding
Quanti-
zation
Transfor-
mation
Inverse Quatization
Inverse
Compressed image
La transformée en cosinus discrète et la transformation par ondelettes sont les transformations les plus populaires
Compressed image

13. Compression –Transform Coding: JPEG example
JPEG is a free code, thus, allowing it to be widely distributed on the Internet
It allows you to handle color images in grayscale
It has many uses such as medical scanning because it offers a powerful method that is the reason of its widespread distribution

14. JPEG Segmentation Blocks 8x8 Transformation DCT Quantization
Quantization Matrix
Coding
RLE + Huffman

15. Discrete Cosine Transform DCT
The experts are based on the fact that relevant information of an image, characterized by a two- dimensional signal Img (x, y), are only the components of the low frequency of the signal
The eye is less sensitive to high frequencies.
The DCT transforms a signal amplitude (each value of the image represents the "magnitude" of the image, and so, its intensity) into a frequency information
When working with the matrix Img (x, y), the X and Y axes represent the vertical and horizontal dimensions of the image
On the matrix DCT(i, j), the axes represent the frequency of the signal in two dimensions

16. Discrete Cosine Transform DCT
Given
An image segmented into blocks 8 × 8 pixels
Aim
Obtain a new representation of each block:
containing the same information
Information is concentrated on a few elements
Principle
Using a matrix of transformation

17. DCT Formula
Inverse DCT Transformation

18. Discrete Cosine Transform DCT
Here is the matrix representing the image and the basic functions
F0,0
f00
f01
f02
f03
f04
f05
f06
f07
f10
f11
f12
f13
f14
f15
f16
f17
f20
f21
f22
f23
f24
f25
f26
f27
f30
f31
f32
f33
f34
f35
f36
f37
f40
f41
f42
f43
f44
f45
f46
f47
f50
f51
f52
f53
f54
f55
f56
f57
f60
f61
f62
f63
f64
f65
f66
f67
f70
f71
f72
f73
f74
f75
f76
f77
Matrix representing the image
Matrix of the basic functions

19. Basic DCT functions How the basic matrix is defined?
Transformation formula!
y=u.π/16 and x=v. π/16
A block of the basic matrix for F0,0
cosy.cosx
cos3y.cosx
cos5y.cosx
cos7y.cosx
cos9y.cosx
cos11y.cosx
cos13y.cosx
cos15y.cosx
cosy.cos3x
cos3y.cos3x
cos5y.cos3x
cos7y.cos3x
cos9y.cos3x
cos11y.cos3x
cos13y.cos3x
cos15y.cos3x
cosy.cos5x
cos3y.cos5x
cos5y.cos5x
cos7y.cos5x
cos9y.cos5x
cos11y.cos5x
cos13y.cos5x
cos15y.cos5x
cosy.cos7x
cos3y.cos7x
cos5y.cos7x
cos7y.cos7x
cos9y.cos7x
cos11y.cos7x
cos13y.cos7x
cos15y.cos7x
cosy.cos9x
cos3y.cos9x
cos5y.cos9x
cos7y.cos9x
cos9y.cos9x
cos11y.cos9x
cos13y.cos9x
cos15y.cos9x
cosy.cos11x
cos3y.cos11x
cos5y.cos11x
cos7y.cos11x
cos9y.cos11x
cos11y.cos11x
cos13y.cos11x
cos15y.cos11x
cosy.cos13x
cos3y.cos13x
cos5y.cos13x
cos7y.cos13x
cos9y.cos13x
cos11y.cos13x
cos13y.cos13x
cos15y.cos13x
cosy.cos15x
cos3y.cos15x
cos5y.cos15x
cos7y.cos15x
cos9y.cos15x
cos11y.cos15x
cos13y.cos15x
cos15y.cos15x
Remark: transpose the matrix (exchange lines and columns)
This is a typing error

20. Basic DCT functions y=u.π/16 and x=v. π/16
F0,0 is the sum of the multiplication of elements
Do not forget the constant 1
f00
f01
f02
f03
f04
f05
f06
f07
f10
f11
f12
f13
f14
f15
f16
f17
f20
f21
f22
f23
f24
f25
f26
f27
f30
f31
f32
f33
f34
f35
f36
f37
f40
f41
f42
f43
f44
f45
f46
f47
f50
f51
f52
f53
f54
f55
f56
f57
f60
f61
f62
f63
f64
f65
f66
f67
f70
f71
f72
f73
f74
f75
f76
f77
Mask

21. Basic DCT functions F0,1 is the sum of the multiplication of elements
cos(π/16)
cos(3π/16)
cos(5π/16)
cos(7π/16)
cos(9π/16)
cos(11π/16)
cos(13π/16)
cos(15π/16)
Remark: transpose the matrix (exchange lines and columns)
This is a typing error

22. Basic DCT functions F1,0 is the sum of the multiplication of elements
cos(π/16)
cos(3π/16)
cos(5π/16)
cos(7π/16)
cos(9π/16)
cos(11π/16)
cos(13π/16)
cos(15π/16)
Remark: transpose the matrix (exchange lines and columns)
This is a typing error

23. Basic DCT functions
F2,0 is the sum of the multiplication of elements y=2π/16=π/8 and x=0
cos(π/8)
cos(3π/8)
cos(5π/8)
cos(7π/8)
cos(9π/8)
cos(11π/8)
cos(13π/8)
cos(15π/8)
cos(13π/86)
cos(13π/8
Remark: transpose the matrix (exchange lines and columns)
This is a typing error

24. Discrete Cosine Transform DCT
Choose a homogeneous block

25. Discrete Cosine Transform DCT
Choose a block
rich in texture

26. Discrete Cosine Transform DCT
It is observed that the coefficients of high absolute value are located on top and left
The importance of the coefficients for the reconstruction of the image decreases as you move diagonally from top left to bottom right that corresponds to high frequencies
To which the eye is the least sensitive

27. Inverse Discrete Cosine Transform IDCT
It is possible to restore the image by the inverse process IDCT Note that in theory the DCT does not introduce any loss of information on the image
The original block can be reconstructed without loss by using the Inverse DCT
In fact, the DCT coefficients, i.e. those seen in the matrix of arrival are rounded to the nearest
By making the transformation using Inverse DCT, we can observe slight differences between the treated matrix and the initial matrix

28. Quantization
In signal processing, quantization can approximate a continuous signal (or values in a discrete set of large size) by the values of a discrete set of relatively small size
Quantization: reducing the number of distinct values in the source data must undergo a compression
Quantization algorithms allow the partitioning of data values
The quantization is a primary source for loss of data in lossy compression methods

29. Quantization
Quantization of a signal

30. Quantization Types
Uniform Scalar Quantizer is the simplest type of quantizer, where the intervals are constant length.
Quantization step is fixed

31. Quantization Types Uniform Quantization
All intervals are the same size, except sometimes the two extremes (longer)
Non Quantized
Uniform Quantization

32. Quantization Types Non-uniform Adaptive Quantization
Non-uniform: intervals may have different sizes (ex: Laplacien, Gaussian)
Adaptive: the sets of intervals are calculated for each block of coefficients
Non Quantized
Adaptive Laplacien Quantization

33. Quantization of DCT coefficients
Quantification is a simple Euclidean division of DCT coefficients by a divider (step), which replaces the original coefficients by the quotient of division
this is the stage at which we will lose some of the information
DCT ranges the coefficients in order of importance
Those coding for the low frequencies towards the upper left corner, while less important towards the lower right corner of the matrix
It takes relatively small steps for the important values and increasingly large steps as we descend to the bottom right of the matrix

34. Quantization of DCT coefficients
The set of steps that is used for quantization is called a quantization matrix
Some of these matrices were built according to psycho-visual criteria, but most follow this little formula: Q (i, j) = 1 + (1 + i + j) x Fq
We will consider Fq = 5
It is a quality factor, it can be little changed in some software such as Photoshop
The quantization matrix can be given as a result of psychophysical studies, with the goal of maximizing the compression ratio while minimizing perceptual losses in JPEG images.
The quantization matrix which is created from this formula, does not need to be stored with the image: this allows a considerable gain in weight simply because it is sufficient to add the formula to the file header. During decompression, de-quantization is required using simple calculation prior to recreating the matrix from the formula.

35. Quantization of DCT coefficients
Quantization of the first block of Lena with some quantization matrix
Quantization matrix
Transformed matrix
Quantized matrix

36. Quantization of DCT coefficients
Quantization of the second block of Lena with some quantization matrix

37. Coding
The third step is to compress, this time without loss, the quantized matrix that we have
JPEG uses a VLC (Variable Length Code) coding type: Huffman coding
The Experts Group JPEG has chosen to treat zeros in a special way because of their big number
It uses a special scanning for larger possible sequences of zeros
This method is called the zig-zag sequence

38. Coding
It uses the zig-zag method, by which you count the longest possible sequence of zeros.

39. Decompression
Decompression is performed by doing the opposite: all operations are reversible, but the quantization that causes the loss
We finally found a matrix very close to the first

40. Decompression – Example
1: Matrix after division of F(u,v) by the quantization matrix
2: Matrix after Multiplication of 1 by the quantization matrix
3: Matrix after applying the Inverse Discrete Cosine Transform (IDCT)
4: Error measure of the reconstructed matrix with the original one.

41. Decompression – Example
1: Matrix after division of F(u,v) by the quantization matrix
2: Matrix after Multiplication of 1 by the quantization matrix
3: Matrix after applying the Inverse Discrete Cosine Transform (IDCT)
4: Error measure of the reconstructed matrix with the original one.