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Role of Mathematical Tool in digital Image Processing

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. DIGITAL IMAGE COMPRESSION USING WAVLETS.

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Role of Mathematical Tool in digital Image Processing

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  1. 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

  2. DIGITAL IMAGE COMPRESSION USING WAVLETS

  3. 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

  4. 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.

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

  6. Introduction • Redundancy • Various data • same information • Examples • Neighboring pixels are correlated

  7. Introduction • Fundamental Components of Compression Removal of • Redundancy • removing duplication • Irrelevancy • omits not noticeable parts

  8. 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

  9. 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

  10. Introduction • Coding redundancy • Can be Remove by • Variable length coding

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

  12. 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

  13. 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

  14. 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

  15. 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.

  16. 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

  17. 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

  18. 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

  19. Input Image Source Encoder Quantizer Entropy Encoder Output Compressed image Original Image Compressed Image Image Compression Methodologies • Lossy image compression methods

  20. 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)

  21. 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

  22. Image Compression Methodologies • Entropy coding Removes the Redundancy • Huffman Coding • Arithmetic Coding • Run Length Encoding (RLE) and • Lempel-Ziv (LZ)

  23. Image Compression Methodologies • Effect of Spatial and gray level resolution on compression • Subjective process • Hardware consideration • number of gray levels • integer power of 2.

  24. Image Compression Methodologies • Effect of varying number of samples

  25. Image Compression Methodologies • Varying resolution

  26. Image Compression Methodologies • Effect of varying the number of gray levels

  27. 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

  28. 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

  29. 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

  30. 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 …

  31. 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

  32. 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)

  33. Wavelets Frequency Representation Time domain representation

  34. 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.

  35. 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).

  36. 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

  37. 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.

  38. 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

  39. 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

  40. 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:

  41. 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.

  42. 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

  43. 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

  44. 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

  45. Wavelets • Comparison of Transforms

  46. Wavelets • Wavelet Transform • Structure of Wavelet • Pair of Filters • Low pass filter • High pass filter • Filter bank • recursively averaging and • differentiating coefficients

  47. Wavelets • Analysis of 2D DWT shows First Level of Decomposition

  48. IMAGE LL HL Approximation. Horizontal Coefficients Details LH HH Vertical Diagonal Details Details Wavelets • Wavelet Decomposition

  49. Wavelets Wavelet Decomposition of Cameraman image at Level 1

  50. 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

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