The Discrete Wavelet Transform for Image Compression

1. The Discrete Wavelet Transformfor Image Compression Speaker: Jing-De Huang Advisor: Jian-Jiun Ding Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC

2. Outline • Subband Coding • Multiresolution Analysis • Discrete Wavelet Transform • The Fast Wavelet Transform • Wavelet Transforms in Two Dimension • Image Compression • Simulation Result

3. 1. Subband Coding h0(n) 2  2  g0(n) Analysis Synthesis + h1(n) 2  2  g1(n) Low band High band   / 2 0

4. 1. Subband Coding • Cross-modulated Z-transform : For error-free reconstruction For finite impulse response (FIR) filters and ignoring the delay FIR synthesis filters are cross-modulated copies of the analysis filters  with one (and only one) being sign reversed.

5. 1. Subband Coding • Biorthogonal The analysis and synthesis filter impulse responses of all two-band, real-coefficient, perfect reconstruction filter banks are subject to the biorthogonality constraint

6. 1. Subband Coding • Orthonormal • one solution of biorthogonal • used in the fast wavelet transform • the relationship of the four filter is :

7. 2. Multiresolution Analysis • Expansion of a signal f (x) : If is an orthonormal basis for V , then If the expansion is unique, the are called basis functions. The function space of the expansion set : If are not orthonormal but are an orthogonal basis for V , then the basis funcitons and their duals are called biorthogonal.

8. 2. Multiresolution Analysis • Scaling function The subspace spanned over k for any j : The scaling functions of any subspace can be built from double-resolution copies of themselves. That is, where the coefficients are called scaling function coefficients.

9. 2. Multiresolution Analysis • Requirements of scaling function: • The scaling function is orthogonal to its integer translates. • The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. That is • The only function that is common to all is .That is • Any function can be represented with arbitrary precision. That is,

10. 2. Multiresolution Analysis • Wavelet function spans the difference between any two adjacent scaling subspaces and for all that spans the space where The wavelet function can be expressed as a weighted sum of shifted, double-resolution scaling functions. That is, where the are called the wavelet function coefficients. It can be shown that

11. 2. Multiresolution Analysis Figure 2 The relationship between scaling and wavelet function spaces. The scaling and wavelet function subspaces in Fig. 2 are related by We can express the space of all measurable, square-integrable function as or

12. 3 Discrete Wavelet Transform • Wavelet series expansion where j0 is an arbitrary starting scale called the approximation or scaling coefficients called the detail or wavelet coefficients

13. 3 Discrete Wavelet Transform • Discrete Wavelet Transform the function f(x) is a sequence of numbers where j0 is an arbitrary starting scale called the approximation or scaling coefficients called the detail or wavelet coefficients

14. 4 The Fast Wavelet Transform • Fast Wavelet Transform (FWT) • computationally efficient implementation of the DWT • the relationship between the coefficients of the DWT at adjacent scales • also called Mallat's herringbone algorithm • resembles the twoband subband coding scheme

15. 4 The Fast Wavelet Transform Scaling x by 2j, translating it by k, and letting m = 2k + n Similarity, Consider the DWT. Assume and

16. 4 The Fast Wavelet Transform Similarity,

17. 4 The Fast Wavelet Transform Figure 3 An FWT analysis filter bank.

18. 4 The Fast Wavelet Transform Figure 4 An FWT-1 synthesis filter bank. By subband coding theorem, perfect reconstrucion for two-band orthonormal filters requires for i = {0, 1}. That is, the synthesis and analysis filters must be time-reversed versions of one another. Since the FWT analysis filter are and , the required FWT-1 synthesis filtersare and .

19. Wavelet Transform vs. Fourier Transform • Fourier transform • Basis function cover the entire signal range,varying in frequency only • Wavelet transform • Basis functions vary in frequency (called “scale”)as well as spatial extend • High frequency basis covers a smaller area • Low frequency basis covers a larger area

20. Wavelet Transform vs. Fourier Transform Time-frequency distribution for (a) sampled data, (b) FFT, and (c) FWT basis

21. 5 Wavelet Transforms in Two Dimension Figure 5 The two-dimensional FWT  the analysis filter.

22. 5 Wavelet Transforms in Two Dimension two-dimensional decomposition Figure 6 Two-scale of two-dimensional decomposition

23. 5 Wavelet Transforms in Two Dimension

24. 5 Wavelet Transforms in Two Dimension Figure 7 The two-dimensional FWT  the synthesis filter bank.

25. Common Wavelet Filters • Haar: simplest, orthogonal, not very good • Daubechies 8/8: orthogonal • Daubechies 9/7: bi-orthogonalmost commonly used if numerical reconstruction errors are acceptable • LeGall 5/3: bi-orthogonal, integer operation,can be implemented with integer operations only, used for lossless image coding

26. 6 Image Compression • Quantization • uniform scalar quantization • separate quantization step-sizes for each subband • Entropy coding • Huffman coding • Arithmetic coding Wavelet coding Quantization Entropy coding image bitstream

27. encoder C decoder I2 I1 7 Simulation Result I1: Original image with width W and height H C: Encoded jpeg stream from I1 I2: Decoded image from C CR (Compression Ratio) = sizeof(I1) / sizeof(C) RMS (Root mean square error) =

28. 7 Simulation Result Wavelet-based image compression DCT-based image compression Original image CR = 11.2460 RMS = 4.1316 CR = 10.3565 RMS = 4.0104

29. 7 Simulation Result Wavelet-based image compression DCT-based image compression Original image CR = 27.7401 RMS = 6.9763 CR = 26.4098 RMS = 6.8480

30. 7 Simulation Result Wavelet-based image compression DCT-based image compression Original image CR = 53.4333 RMS = 10.9662 CR = 51.3806 RMS = 9.6947

31. Reference • R. C. Gonzolez, R. E. Woods, "Digital Image Processing second edition", Prentice Hall, 2002. • R. C. Gonzolez, R. E. Woods, S. L. Eddins, "Digital Image Processing Using Matlab", Prentice Hall, 2004. • T. Acharya, A. K. Ray, "Image Processing: Principles and Applications", John Wiley & Sons, 2005. • B. E. Usevitch, 'A Tutorial on Modern Lossy Wavelet Image Compression: Foundations of JPEG 2000', IEEE Signal Processing Magazine, vol. 18, pp. 22-35, Sept. 2001.