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

Multimedia – Image Compression

Dr. Lina A. NimriLebanese UniversityFaculty of Economic Sciences and Business Administration 1st branch

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 %

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

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.

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

Lossy Compression

•The most important methods of lossy compression are:1. The reduction of the color depth to the

most frequent colors in the image2. 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

3. Fractal compression

Lossy Compression - continued•The most important methods of lossy

compression are4. 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

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

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

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.

Fractal Compression

Compression –Transform Coding

La transformée en cosinus discrète et la transformation par ondelettes sont les transformations les plus populaires

Decoding

CodingQuanti-zation

Transfor-mation

Inverse Quatization

InverseTransfor-mation

Compressed image

Compressed image

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

JPEG

Segmentation Blocks 8x8

Transformation DCT

Quantization Quantization Matrix

Coding RLE + Huffman

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

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

DCT Formula

Inverse DCT Transformation

Discrete Cosine Transform DCT•Here is the matrix representing the image

and the basic functionsf00

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

f77Matrix representing the

imageMatrix of the basic

functions

F0,0

Basic DCT functions•How the basic matrix is defined?

▫Transformation formula!▫y=u.π/16 and x=v. π/16

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


cos13y.cos5x

cos15y.cos5x


cos13y.cos7x

cos15y.cos7x


cos13y.cos9x

cos15y.cos9x

cosy.cos11x cos3y.cos11x

cos5y.cos11x

cos7y.cos11x

cos9y.cos11x

cos11y.cos11x

cos13y.cos11x

cos15y.cos11x


cos5y.cos13x

cos7y.cos13x

cos9y.cos13x

cos11y.cos13x

cos13y.cos13x

cos15y.cos13x


cos5y.cos15x

cos7y.cos15x

cos9y.cos15x

cos11y.cos15x

cos13y.cos15x

cos15y.cos15x

A block of the basic matrix for F0,0

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 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 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

Basic DCT functions

▫F0,1 is the sum of the multiplication of elements

cos(π/16) cos(π/16) cos(π/16) cos(π/16) cos(π/16) cos(π/16) cos(π/16) cos(π/16)

cos(3π/16) cos(3π/16) cos(3π/16) cos(3π/16) cos(3π/16) cos(3π/16) cos(3π/16) cos(3π/16)




cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(11π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(13π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

cos(15π/16)

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)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)


cos(13π/16)

cos(15π/16)

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(π/8) cos(3π/8) cos(5π/8) cos(7π/8) cos(9π/8) cos(11π/8) cos(13π/86)

cos(15π/8)






cos(π/8) cos(3π/8) cos(5π/8) cos(7π/8) cos(9π/8) cos(11π/8) cos(13π/8 cos(15π/8)

Discrete Cosine Transform DCT

Choose a homogeneous block

Discrete Cosine Transform DCT

Choose a blockrich in texture

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

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

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

Quantization

•Quantization of a signal

Quantization Types

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

Quantization Types

•Uniform Quantization▫All intervals are the same size, except

sometimes the two extremes (longer)

Non Quantized Uniform Quantization

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

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

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.

Quantization of DCT coefficients•Quantization of the first block of Lena

with some quantization matrix

Quantization matrix Transformed matrix

Quantized matrix

Quantization of DCT coefficients•Quantization of the second block of Lena

with some quantization matrix•

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

Coding

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

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

Decompression – Example

Decompression – Example

Documents

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