Wavelet transform in image compression
- 1. WAVELET TRANSFORM IN IMAGE COMPRESSION Presented By, E . JEEVITHA 16MMAT05 M.Phil Mathematics
- 3. INTRODUCTION Wavelets 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 ”.
- 4. HISTORICAL DEVELOPMENTS 1909 : 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.
- 5. 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.
- 7. 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.
- 9. 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).
- 10. METHODS / STEPS
- 11. 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.
- 12. 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
- 13. LL1 LH1 HL1 HH1 HL1 HH1LH1 LH1 HH1 HL1 HH2 HL2 LH2 HH2LH2 LL2 HL2 LL3 HL3 LH3 HH3 LEVEL 1 LEVEL 2 LEVEL 3
- 14. 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
- 15. APPLICATION
- 16. LL2 HL2 LH2 HH2 LH HL HH LH HL HH HL2 LH2 HH2 LL3 HL3 LH3 HH3
- 17. OTHER APPLICATIONS Wavelets 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.
- 18. 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.
- 19. ADVANTAGES The 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.
- 20. CONCLUSION Image 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.
- 21. THANK YOU