Digital Signal Processing

1. Digital Signal Processing Instructor: L. J. Wang, Dept. of Information Technology, National Pingtung Institue of CommerceReference: Gilbert Strang and Truong Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. Reference: Mark S. Drew, Simon Fraser University, Canada. (http://www.sfu.ca/) L. J. Wang

2. Transforms • The transform of a signal (a vector) is a new repre-sentation of that signal. • Three groups of transforms: • Lossless (orthogonal) transforms • Invertible (biorthogonal) transforms • Lossy transforms (not invertible) L. J. Wang

3. Lossless (orthogonal) transforms • A lossless unitary transform is like rotation. • The transformed signal has the same length as the original. • The same signal is measured along new perpendicular axes. • Example: • FFT (Fast Fourier Transform) • DCT (Discrete Cosine Transform) • DST (Discrete Sine Transform) • HT (Hartley Transform) L. J. Wang

4. Invertible (biorthogonal) transforms • For biorthogonal transforms, lengths and angles may change. • The new axes are not necessarily perpendicular, but no information is lost. • Perfect reconstruction is still available. • It just inverts. • These transforms don’t remove any information (or any noise), they just move it around – aiming to separate out the noise and decorrelate the signal. L. J. Wang

5. Lossy transforms • Orthogonal transforms give orthogonal matrices and unitary transforms • Biorthogonal transforms give invertible matrices and perfect reconstruction. • For Lossy transforms: • The irreversible step is to destroy small components, as we do below in “compression”. • The invertible is lost. L. J. Wang

6. An example of transforms • The transform from x to y is executed by a matrix : • The matrix that recovers x from y is changed only by the factor 1/2 : L. J. Wang

7. An example of transforms (II) • If x(0)=1.2, x(1)=1.0, x(2)=-1.0, x(3)=-1.2 then: (compute sums and differences) y(0)=x(0)+x(1)=1.2+1.0=2.2 y(1)=x(0)-x(1)=1.2-1.0=0.2 y(2)=x(2)+x(3)=(-1.0)+(-1.2)=-2.2 y(3)=x(2)-x(3)=(-1.0)-(-1.2)= 0.2 • y(1) and y(3) are much smaller than y(0) and y(2). If we cancel the small numbers y(1) and y(3) --- the compressed signal yc is yc(0)=2.2, yc(1)=0, yc(2)=-2.2 and yc(3)=0. L. J. Wang

8. An example of transforms (III) • Those numbers 0.2 were below our threshold. In the compressed yc they are gone. • Finally, the signal xc reconstructed from yc: xc(0)=(yc(0)+yc(1))/2=(2.2+0)/2=1.1 xc(1)=(yc(0)-yc(1))/2=(2.2-0)/2=1.1 xc(2)=(yc(2)+yc(3))/2=(-2.2+0)/2=-1.1 xc(3)=(yc(2)-yc(3))/2=(-2.2-0)/2=-1.1 • The small difference between x(0) and x(1) is lost. L. J. Wang

9. Discrete Cosine Transform (DCT) • From spatial domain to frequency domain: L. J. Wang

10. DEFINITIONS: DCT/IDCT • Discrete Cosine Transform (DCT): • Inverse Discrete Cosine Transform (IDCT): L. J. Wang

11. 64 (8 x 8) DCT basis functions L. J. Wang

12. Why DCT not FFT? • DCT is like FFT, but can approximate linear signals well with few coefficients. L. J. Wang

13. Computing the DCT • Factoring reduces problem to a series of 1D DCTs: • Most software implementations use fixed point arithmetic. Some fast implementations approximate coefficients so all multiplies are shifts and adds. L. J. Wang

14. Wavelets • A key idea for wavelets is concept of “scale”. • Sums and differences of neighbors are at the finest scale. • This is recursion --- the same transform at a new scale. • It leads to a multiresolution of the original signal. • Averages and details will appear at different scales. L. J. Wang

15. Wavelets (II) • The wavelet formulation keeps the differences y(1) and y(3) at the finest level, and iterates only on y(0) and y(2). (Following an example of transforms) • Iteration means sum and difference of the transform: z(0)=y(0)+y(2)=0 z(2)=y(0)-y(2)=4.4 • At this level there is extra compression. One component z(2) now does most of the work of the original four. L. J. Wang

16. Wavelets (III) • The wavelet transform is: z(0)=0 y(1)=0.2 z(2)=4.4 y(3)=0.2 • The compressed signal is: (after threshold) zc(0)=0 yc(1)=0 zc(2)=4.4 yc(3)=0 L. J. Wang

17. Wavelets (IV) • The multilevel transforms by recursion is clear in a flow-graph: • This is two-step wavelet transform. It is invertible. • To draw the flow-graph of the inverse, just reverse the arrows. L. J. Wang

18. Wavelet Transforms L. J. Wang

19. Wavelet Transforms (II) L. J. Wang

20. Wavelet Transforms (III) L. J. Wang

21. Wavelet Transforms (IV) L. J. Wang

22. Wavelet Transforms (V) L. J. Wang

23. Wavelet Transforms (VI) L. J. Wang

24. Wavelet Transforms (VII) L. J. Wang

25. Haar Wavelets • Scaling functions: • Haar scaling function is defined by and is shown in Fig.H-1. Some examples of its translated and scaled versions are shown in Fig.H-2~H-4. • Two-scale relation for Haar scaling function is L. J. Wang

26. Haar Wavelets (II) L. J. Wang

27. Haar Wavelets (III) • Wavelets: • Haar wavelet  (x) is given by and is shown in Fig.H-5. • Two-scale relation for Haar wavelet is L. J. Wang

28. Haar Wavelets (IV) L. J. Wang

29. Haar Wavelets (V) • Decomposition relation: • Reconstruction relation: L. J. Wang

30. Biorthogonal Filter Bank L. J. Wang

31. 5/3 Analysis Filter L. J. Wang

32. 5/3 Synthesis Filter L. J. Wang

33. 5/3 Filter Examples L. J. Wang

34. 9/7 Wavelet Filter L. J. Wang

35. 9/7 Filter Example L. J. Wang

36. One stage in a multiscale image decomposition L. J. Wang

37. One level of a wavelet decomposition in 3 steps L. J. Wang

38. Image decomposition L. J. Wang

39. Daubechies Wavelet D4 Filter L. J. Wang

40. Daubechies Wavelet D4 Filter (II) L. J. Wang

41. One stage in a multiscale image reconstruction L. J. Wang