The Discrete Wavelet Transform for 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)h0(n) h1(n)h1(n) 2  2  g0(n)g0(n) g1(n)g1(n) + AnalysisSynthesis  / 2 Low bandHigh band 0  

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

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 the expansion is unique, the are called basis functions. The function space of the expansion set : If is an orthonormal basis for V, then 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: 1.The scaling function is orthogonal to its integer translates. 2.The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. That is 3.The only function that is common to all is. That is 4.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 j 0 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 where j 0 is an arbitrary starting scale called the approximation or scaling coefficients called the detail or wavelet coefficients the function f(x) is a sequence of numbers

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 2 j, translating it by k, and letting m = 2k + n Similarity, Consider the DWT. Assume and

16 4 The Fast Wavelet Transform Similarity,

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

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 filters are 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 Figure 6 Two-scale of two-dimensional decomposition 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-orthogonal most 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 Wavelet coding Quantization Entropy coding imagebitstream Quantization –uniform scalar quantization –separate quantization step-sizes for each subband Entropy coding –Huffman coding –Arithmetic coding

27 7 Simulation Result I 1 : Original image with width W and height H C: Encoded jpeg stream from I 1 I 2 : Decoded image from C CR (Compression Ratio) = sizeof(I 1 ) / sizeof(C) RMS (Root mean square error) = I1I1 CI2I2 encoderdecoder

28 7 Simulation Result Original image DCT-based image compression Wavelet-based image compression CR = RMS = CR = RMS =

29 7 Simulation Result Original image DCT-based image compression Wavelet-based image compression CR = RMS = CR = RMS =

30 7 Simulation Result Original image DCT-based image compression Wavelet-based image compression CR = RMS = CR = RMS =

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