1. Graphing Scaling Functions and Wavelets (Also check out Sweldens’ Applet) Jyun-Ming Chen Spring 2001

2. Graphing the Scaling Functions Recipe for plotting Start with a sequence of values at level j The limit function f(x) is exactly

3. Example (Haar)

4. Haar (level-2)

5. Graph by Convolution Math Primer References: –Graphing scaling fns and wavelets (from Goswami, p.132) –Alternate interpretation for ease of computation (from Discrete-Time Signal Processing, Oppenheim & Schafer, pp. 22-24)

6. Ex: Convoluting Haar 0100 00100000 11 Upsampling by inserting 0 00110000 01234567 01234567 01 11 01 p sequence of Haar

7. 00110000 01234567 11 01 0000101000000000 0123456789101112131415 0000111100000000 0123456789101112131415 11 01 11 01 Upsampling by inserting 0

8. Summary (Haar scaling fns) Father scaling function Dilated and translated copy See other examples in Appendix B.

9. Graphing the Wavelets Recipe for plotting Start with a sequence of values at level j Convolve with the refinement sequence q to create a sequence of scaling function coefficient c j+1 Convolve with the sequence p to reach the wavelet’s limit function

10. Haar Wavelet (level-1)

11. Convolution 0100 00100000 1 Insert 0 0010000 01234567 01234567 01 1 01 q sequence of Haar

12. 0010000 01234567 11 01 000010 000000000 0123456789101112131415 000011 00000000 0123456789101112131415 11 01 11 01 Insert 0 p sequence of Haar

13. Graphing D4

14. wavelet scaling fn