Role of Mathematical Tool in digital Image Processing. Dr. Shivanand S. Gornale Ph.D.FIETE,IEng . Asst. Professor and Head Dept of Computer Science Government College, Mandya (Autonomous) firstname.lastname@example.org. DIGITAL IMAGE COMPRESSION USING WAVLETS.
Dr. Shivanand S. Gornale
Asst. Professor and Head
Dept of Computer Science
Government College, Mandya (Autonomous)
What is Compression?.
Compression is the process of representing the information in compact form. It can be obtained by removing the data redundancy.
Types of Redundancy
Image Data compression can be obtained by removing any one of these redundancies
Psycho-Visual Redundancy (Contd…)
Image Data compression techniques are basically Spatial Domain and Frequency Domain. Spatial Domain operates on gray scale values of image. Where as Frequency Domain transforms the signals and convert them into another domain.
There are different compression algorithms yet developed and these are classified into
In lossless data compression the original data can be recovered exactly from the compressed data. And these techniques generally composed of relatively two independent operations.
Some lossless data compression techniques:
Normally, these techniques provides a compression ratio of 2 to 10
where the some loss of data can be acceptable
b) JPEG 2000
c) Video Compression Standard
Lossy Compression techniques gives more compression ratio compared to the lossless compression techniques. But,
Higher Compression ratio gives the lower image quality and Vice-Versa
We except the following from the transformation.
1. To create a representation for the data in which there is a less correlation among the coefficient values. i.e. decorrelating the data. (purpose is to reduce the redundancy)
2. To have a representation in which it is possible to quantize different co-ordinates with different precision
Good quantizer is
Removes the Redundancy
The Independent Variable is Time
Time Amplitude representation
Time domain representation
Four peaks corresponding to 5, 10, 20, and 50 Hz. FT cannot distinguish the two signals very well. To FT, both signals are the same, as they constitute of the same frequency components. Therefore, FT is not a suitable tool for analyzing non-stationary signals, i.e., signals with time varying spectra.
so called Short Time Fourier Transform (STFT).
A window function “ω” is chosen The width of this window must be equal to the segment of thesignal where its stationary is valid
If the window is of constant size and with this window we have sinusoids with an increasing number of cycles.
Let us assume for instant number of cycles are fixed but the size of the window keeps changing
It clearly shows that the lower frequency function covers the long interval time, while higher frequency covers the short time interval
The peaks are not well separated from each other in time, unlike the previous case, however, in frequency domain the resolution is much better.
Better time resolution;
Poor frequency resolution
Better frequency resolution;
Poor time resolution
Wavelet Decomposition of Cameraman image at Level 1
Wavelet Decomposition at Level 2 to 5 Maximum Levels of Decomposition = log2 xmax
Where xmax is the maximum size of given image
Gives high compression rate
- greater computation time for the wavelet transform.
- can create unpleasant artifacts in the compressed image
Wavelet packet decomposition at level 1 (DB1) of Woman image
Decomposition tree at First Level
Wavelet packet decomposition at level 2
(DB1) of Woman image
Decomposition tree at Second Level
bit rate =
(bits per pixel)
Compression Factor (CF)= 1/CR
Compression Gain = 100 loge* Reference size / Compressed Size
Mean Square Error (MSE)=
Peak Signal to Noise Ratio (PSNR)=
Normalized Cross –correlation (NK)
Average Difference (AD)
Structural Content (SC)
Universal Quality Index (Q)
‘i’ is the grade and
p (i) is the grade probability