WAVELET TRANSFORM IN

IMAGE COMPRESSIONPresented By,

E . JEEVITHA16MMAT05

M.Phil Mathematics

OVERVIEWIntroductionHistorical developmentsTechniquesMethodologyApplicationsAdvantagesConclusion

INTRODUCTIONWavelets are mathematical functions that

splits up data into different frequency components,

and then study each component with a resolution

matched to its scale.

Wavelet transform decomposes a signal into

a set of basis functions. These basis functions are

called as “ wavelets ”.

HISTORICAL DEVELOPMENTS1909 : Alfred Haar – Dissertation “On the orthogonal

function systems” for his doctoral degree. The first wavelet

related theory.

1910 : Alfred Haar : Development of a set of rectangular

basis functions.

1930 : Paul Levy investigated “ The brownian motion”.

Littlewood and Paley worked on localizing the

contributing energies of a functon.

1946 : Dennis Gabor : Used short time fourier transform.

1975 : George zweig - The first continuous wavelet

transform.

1985 : Meyer - Construction of orthogonal wavelet basis

functions with very good time and frequency localization.

1986 : Stephen Mallet – Developing the idea of Multi-

resolution analysis for DWT.

1988 : Daubechies and Mallet – The modern wavelet

theory.

1992 : Albert cohen and Daubechies constructed the

compactly supported biorthogonal wavelets.

TYPES OF

WAVELET TRANSFOR

M

WHY IMAGE COMPRESSION?Digital images usually require a

very large number of bits, this causes

critical problem for digital image data

transmission and storage.

It is the art & science of reducing

the amount of data required to represent

an image.

It is one of the most useful and

commercially successful technologies in

the field of digital image processing.

WHY WAVELETS ?Good approximation properties.

Efficient way to compress the

smooth data except in localized

region.

Easy to control wavelet properties.

( Example : Smoothness,

better accuracy near sharp

gradients).

METHODS / STEPS

Digitize the source image to a signal s, which is a

string of numbers.

Decompose the signal into a sequence of wavelet

coefficients.

Use thresholding to modify the wavelet compression

from w to another sequence w’.

Use quantization to convert w’ to a sequence q.

Apply entropy coding to compress q into a sequence e.

A B C D A+B C+D A-B C-D

L H

STEP 1

STEP 2

A

B

C

D

L H

A+B

A-B

C+D

C-D

LL

LH

HL

HH

LL1

LH1

HL1

HH1

HL1

HH1LH1

LH1 HH1

HL1HH2

HL2

LH2

HH2LH2

LL2 HL2

LL3 HL3

LH3 HH3

LEVEL 1 LEVEL 2

LEVEL 3

ORIGINAL IMAGE

20 15 30 20

17 16 31 22

15 18 17 25

21 22 19 18

35 50 5 10

33 53 1 9

33 42 - 3 - 8

43 37 - 1 1

68 103 6 19

76 79 - 4 - 7

2 - 3 4 1

- 10 5 - 2 - 9

1st HORIZONTAL SEPERATION 1st VERTICAL SEPERATION

APPLICATION

LL2 HL2

LH2 HH2

LH

HL

HH

LH

HL

HH

HL2

LH2 HH2

LL3 HL3

LH3 HH3

OTHER APPLICATIONSWavelets are a powerful statistical tool which can

be used for a wide range of applications, namely

Signal processing.

Image processing.

Smoothing and image denoising.

Fingerprint verification.

Biology for cell membrane recognition, to

distinguish the normal from the pathological

membranes.

DNA analysis, protein analysis.

Blood-pressure, heart-rate and ECG analysis.

Finance (which is more surprising), for

detecting the properties of quick variation of values.

In Internet traffic description, for designing

the services size.

Speech recognition.

Computer graphics and multi-fractal analysis.

ADVANTAGESThe advantage of wavelet compression is that, in contrast

to JPEG, wavelet algorithm does not divide image into

blocks, but analyze the whole image.

Wavelet transform is applied to sub images, so it produces

no blocking artifacts.

Wavelets have the great advantage of being able to

separate the fine details in a signal.

Wavelet allows getting best compression ratio, while

maintaining the quality of the images.

CONCLUSIONImage compression using wavelet transforms

results in an improved compression ratio as well as image

quality. Wavelet transform is the only method that

provides both spatial and frequency domain information.

These properties of wavelet transform greatly help in

identification and selection of significant and non-

significant coefficient. Wavelet transform techniques

currently provide the most promising approach to high

quality image compression, which is essential for many

real world applications.

THANK YOU