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### 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)

shivanand_gornale@yahoo.com

Contents

- Introduction
- Principles Behind Compression
- Image Compression Methodologies
- Wavelets, Wavelet Packets and their limitations
- Performance Evaluation and Results
- Discussion and Interpretation
- Conclusion
- Future Work
- Demo of MATLAB

Introduction

- In Ocean of information, storing and retrieval of information computer and Communication Systems will take a crucial role .
- Increasing use of multimedia data, digital technology requires more storage area and long transmission time for processing.
- To save the storage area and transmission time, we often need compression of multimedia data.

Introduction

What is Compression?.

Compression is the process of representing the information in compact form. It can be obtained by removing the data redundancy.

Introduction

- Redundancy
- Various data
- same information
- Examples
- Neighboring pixels are correlated

Introduction

- Fundamental Components of Compression

Removal of

- Redundancy
- removing duplication
- Irrelevancy
- omits not noticeable parts

Introduction

Types of Redundancy

- Spatial redundancy
- Correlation between neighboring pixels
- Spectral redundancy
- Correlation between different color planes or spectral bands.
- Temporal redundancy
- Correlation between adjacent frames i.e. Video

Introduction

- Digital Image Compression
- Coding redundancy
- Inter-pixel redundancy and
- Psycho visual redundancy

Image Data compression can be obtained by removing any one of these redundancies

Introduction

- Coding redundancy
- Can be Remove by
- Variable length coding

Introduction

- Inter-pixel Redundancy
- co-relation a structural or geometric relationship between the objects in the image
- Identical histogram
- reality different structure and geometry

Introduction

Psycho-Visual Redundancy

- Brightness
- perceived by human eyes
- Depends upon
- Factors other than Light reflected by the region
- Intensity Variation
- Constant intensity
- Eye does not respond with equal sensitivity

Introduction

Psycho-Visual Redundancy (Contd…)

- Certain information
- less relative importance
- In visual processing
- Psycho-visual redundant
- Human perception
- does not involvequantitative analysis
- Eliminated
- Without significantly loss

Image Compression Methodologies

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

- Lossless Algorithms
- Lossy Algorithms

Image Compression Methodologies

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.

- An representation in which its inter pixel redundancies are reduced.
- Coding the representation to eliminate coding redundancies.

Image Compression Methodologies

Some lossless data compression techniques:

- Variable Length Coding
- Huffman Coding
- Arithmetic Coding
- LZW Coding
- Bit Plane Coding
- Lossless Predictive Coding
- Integer-to-Integer Wavelet Transform

Normally, these techniques provides a compression ratio of 2 to 10

Image Compression Methodologies

- Lossy Compression

where the some loss of data can be acceptable

- Lossy Predictive Coding
- Transform Coding
- Zonal Coding
- Wavelet Coding
- Image compression Standard
- Continuous tone still image Compression Standards

a) JPEG

b) JPEG 2000

c) Video Compression Standard

Image Compression Methodologies

- Lossy Compression

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

Source Encoder

Quantizer

Entropy Encoder

Output Compressed

image

Original Image

Compressed Image

Image Compression Methodologies- Lossy image compression methods

Image Compression Methodologies

- Source Encoder (Linear Transformer)

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

- DFT
- DCT
- DWT
- CWT and
- Generalized Lapped Orthogonal Transform (Gen LOT)

Image Compression Methodologies

- Quantizer
- Variable Length Coding
- Scalar Quantization
- Vector Quantization

Good quantizer is

- Depends on the Transform and vice-versa
- Quantization methods
- wavelet transforms
- Embedded and
- Non Embedded quantizer

Image Compression Methodologies

- Entropy coding

Removes the Redundancy

- Huffman Coding
- Arithmetic Coding
- Run Length Encoding (RLE) and
- Lempel-Ziv (LZ)

Image Compression Methodologies

- Effect of Spatial and gray level resolution on compression
- Subjective process
- Hardware consideration
- number of gray levels
- integer power of 2.

Image Compression Methodologies

- Effect of varying number of samples

Image Compression Methodologies

- Varying resolution

Image Compression Methodologies

- Effect of varying the number of gray levels

Image Compression Methodologies

- Transform Coding
- Mathematical tool (Our aim is to highlight the mathematical tool )
- Changes one group of data into another group
- FT
- STFT
- DCT
- Laplace Transform (LT),
- Z Transform,
- Hilbert Transform and
- Wavelets

Wavelets

- Mathematical Transformation
- Why
- To obtain a further information from the signal that is not readily available in the raw signal.
- Raw Signal
- Normally the time-domain signal
- Processed Signal
- A signal that has been "transformed" by any of the available mathematical transformations
- Fourier Transformation
- The most popular transformation

Wavelets

- Time-Domain Signal

The Independent Variable is Time

- The Dependent Variable is the Amplitude
- Most of the Information is Hidden in the Frequency Content

Wavelets

- Frequency Transforms
- Why Frequency Information is Needed
- Be able to see any information that is not obtained in time-domain
- Types of Frequency Transformation
- Fourier Transform, Hilbert Transform, Short-time Fourier Transform, Wigner Distributions, the Radon Transform, the Wavelet Transform …

Wavelets

- Frequency Analysis

Frequency Spectrum

- Be basically the frequency components (spectral components) of that signal
- Show what frequencies exists in the signal
- Fourier Transform (FT)
- One way to find the frequency content
- Tells how much of each frequency exists in a signal

Wavelets

Time Amplitude representation

x( t)=cos(2*pi*10*t)+cos(2*pi*25*t)

+cos(2*pi*50*t)+cos(2*pi*100*t)

Wavelets

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.

Wavelets

- FT Only Gives what Frequency Components Exist in the Signal
- The Time and Frequency Information can not be Seen at the Same Time (Through FT)
- Time-frequency Representation of the Signal is Needed- Which is possible ? Yes:
- A revised version of Fourier Transform (FT)

so called Short Time Fourier Transform (STFT).

Wavelets

- Short Time Fourier Transform (STFT)
- The signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary.

A window function “ω” is chosen The width of this window must be equal to the segment of thesignal where its stationary is valid

Wavelets

- Time Frequency Plane for STFT

- The time-frequency plane of a windowed Fourier transform, where the window is a square wave. Because the same window is used for all frequencies, the resolution of the analysis at every point in the plane is identical.

Wavelets

- Problems with STFT TOO!
- Unchanged Window
- Dilemma of Resolution
- Narrow window -> poor frequency resolution
- Wide window -> poor time resolution
- Heisenberg Uncertainty Principle
- Cannot know what frequency exists at what time intervals

Wavelets

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

Wavelets

- STFT with Narrow Window

- First let's look at the first most narrow window. The STFT has a very good time resolution, but relatively poor frequency resolution:

Wavelets

- STFT with Wider Window

The peaks are not well separated from each other in time, unlike the previous case, however, in frequency domain the resolution is much better.

Wavelets

- Multiresolution Analysis (MRA)
- Wavelet Transform
- An alternative approach to the short time Fourier transform to overcome the resolution problem
- Similar to STFT: signal is multiplied with a function
- Multiresolution Analysis
- Analyze the signal at different frequencies with different resolutions

Wavelets

- Multiresolution Analysis (MRA)
- Good time resolution and poor frequency resolution at high frequencies
- Good frequency resolution and poor time resolution at low frequencies
- More suitable for short duration of higher frequency; and longer duration of lower frequency components

Wavelets

- Resolution of Time & Frequency

Better time resolution;

Poor frequency resolution

Frequency

Better frequency resolution;

Poor time resolution

Time

- Each box represents a equal portion
- Resolution in STFT is selected once for entire analysis

Wavelets

- Comparison of Transforms

Wavelets

- Wavelet Transform
- Structure of Wavelet
- Pair of Filters
- Low pass filter
- High pass filter
- Filter bank
- recursively averaging and
- differentiating coefficients

Wavelets

- Analysis of 2D DWT shows First Level of Decomposition

Wavelets

Wavelet Decomposition of Cameraman image at Level 1

Wavelets

Wavelet Decomposition at Level 2 to 5 Maximum Levels of Decomposition = log2 xmax

Where xmax is the maximum size of given image

Third Level

Second Level

Fourth Level

Fifth Level

Wavelets

- Synthesis of 2D DWT shows First Level of Reconstruction

Wavelets

- Wavelet Properties

Gives high compression rate

- Best wavelet filter bank and
- Decomposition level
- Biorthogonal wavelet filters instead
- Orthogonal
- Energy preservation
- MSE=0
- Whereas, biorthogonal filters lack it

Wavelets

- Compact Support
- Filter Length
- Vanishing order or moment
- Smoothness
- Filter magnitude response
- Group delay
- Decomposition level
- Regularity

Wavelets

- Compact Support
- It is effective for locating jump discontinuities and also for the efficient representation of signals with small support.
- The fact that they have jump discontinuities, may result in blockiness (blocking artifacts) in reconstructed images.

Wavelets

- Size of Filters (Filter Length)
- Higher order does not imply better image quality because of the length of the wavelet filter
- Long analysis filters

- greater computation time for the wavelet transform.

- can create unpleasant artifacts in the compressed image

Wavelets

- Symmetry
- Symmetric filters are good for minimizing the edge effects in the wavelet representation of discrete wavelet transform (DWT) of a function.
- Large coefficients resulting from false edges due to periodization can be avoided.

Wavelets

Vanishing Moments

Wavelets

- Vanishing Moments Cont..
- Artifacts such as wave-shaped irregularities in the reconstructed image could occur with the number of vanishing moments.
- Results in lots of zero values for the wavelet coefficients, which leads to efficient coding

Wavelet Family Properties

Wavelet Forms

Wavelets

- Wavelet Transform
- Successive iterations are performed on the low pass coefficients (approximation) from the previous stage to further reduce the number of low pass coefficients.
- Yields better energy compaction- gives good compression rate.
- Wavelet Transform (WT) often fails to accurately capture high frequency information especially at low bit rates where such information is lost in quantization noise.

Wavelets

- Wavelet Packet Transform (WP)
- It provide more flexible decomposition at any node
- Wavelet packet transform is applied to both low pass results (approximations) and high pass results (details)
- performs significantly better than wavelet for compression of images with large amount of textures

Wavelets

- Wavelet Packet (WP) presenting on extension of the octave band wavelet decomposition to full tree decomposition

Wavelets

Wavelet packet decomposition at level 1 (DB1) of Woman image

Decomposition tree at First Level

Original Image

First Level

Wavelets

Wavelet packet decomposition at level 2

(DB1) of Woman image

Original Image

Second Level

Decomposition tree at Second Level

Wavelets

- Some images have high frequency information (which is how noise may appear in an image) that is not preserved well in standard Wavelet based compression algorithms.
- Scalar wavelets do not possess all the desirable properties simultaneously such as orthogonality, symmetry, Vanishing moments and short support. And these properties are badly needed for better performance compression

Performance Criteria

- Performance criteria in Image Compression
- Compression Efficiency and
- Distortion
- Caused by the compression algorithms

Performance Criteria

- Compression Efficiency
- Bit Rate
- average number of bits per pixel

bit rate =

(bits per pixel)

Performance Criteria

- Low Bit Rate
- More Compression Ratio

CR=

Compression Factor (CF)= 1/CR

Compression Gain = 100 loge* Reference size / Compressed Size

- Distortion
- Subjective Quality Measures
- Quality is evaluated by collecting opinion of humans
- Objective Quality Measures
- Difference between original and reconstructed
- Measures
- Image Differencing
- Feature Extraction

Performance Criteria

- Subjective Evaluation of Image Quality
- Mean Opinion Score (MOS)

Where,

‘i’ is the grade and

p (i) is the grade probability

Performance Criteria

- Quality Metrics

Performance Evaluation and Results

- Wavelet Transform
- Noisy and Noiseless Fingerprint Image Compression
- Performance Analysis of Biorthogonal Wavelet Filters for Lossy Fingerprint Image Compression
- Wavelet Packet Transform
- Noisy and Noiseless Fingerprint Image
- SAR image Compression through Wavelet and Wavelet Packet

Experiments Results

- Wavelet Transform
- Wavelet transform applied on Fingerprint Image [374x388] taken from FVC 2002 Database in two forms with and without noise
- Haar, Daubechies 1 and Symlet transforms at different levels.
- At every level of these transform for both the type of images different Compression ratio is achieved
- Observation
- Haar transform gives more compression ratio for the noisy as well as noiseless fingerprint images.
- The same is achieved by Daubechies (db1) as well as Symlet (sym2) Wavelet transform
- Conclusion:
- More compression ratio we achieve in noiseless fingerprint images rather than noisy fingerprint images.

Experiments Results

- Wavelet Packet Transform
- Wavelet Packet transform applied on Fingerprint Image [374x388] taken from FVC 2002 Database in two forms with and without noise
- Haar, Daubechies1 and Symlet transforms at third level with varying threshold value is applied.
- Observation
- It seems that when there is an increase in the threshold value, it results more compression ratio for both noisy and noiseless fingerprint image
- Conclusion:
- Hence there is always necessary to select the optimal threshold value to get higher compression and to avoid minimum losses of images.

ExperimentsResults

- From above experiments it is conclude that the performance of the compression and decompression generally depends on the image characteristics. For the images mostly low frequency contents (ordinary still images) scalar wavelets give better performance.
- Scope for Multiwavelets appear to excel at preserving high frequency contents, as medical images are normally high frequency patterns.

Conclusion

Conclusion:

- It is concluded that the compression ratio depends on the type of image and type of transforms because;
- there is no filter that performs the best for all images pertaining to different applications

Future Work

- This work may further extended to do compression and decompression of color and video images using complex wavelets, Multiwavelets in presence of noise
- The proposed hybrid model of this research work may be extended to do the compression of medical images, biometrics, and bioinformatics images.

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