Advanced Digital Signal Processing final report

1. Advanced Digital Signal Processing final report Name :Yi-weichen Teacher : Jian-Jiun Ding

2. Short Response Hilbert Transform for Edge Detection Soo-Chang Pei, Jian-JiunDing, Jiun-De Huang, Guo-CyuanGuo Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C

3. Abstract • New method : short-response Hilbert transform (SRHLT) • Edge detection • Drawbacks of general methods : • differentiation - sensitive to noise • HLT - resolution is poor • SRHLT improves drawbacks of differentiation & HLT • robust to noise • detect edges successfully

4. Differentiation • Simple • Drawbacks: • Sensitivity to noise • Not good for ramp edges • Make no difference between the significant edge and the detailed edge

5. Results of differentiation • From figure (a)&(b), the sharp edges can be detected perfectly. • From figure (c)&(d), the step edges with noise can’t be detected. • From figure (e)&(f), differentiation is not good for the ramp edges. • Edges’ form:

6. Hilbert transform (HLT) • Hilbert transform: • H(f): • longer impulse response • reduce the effect of noise • Drawback : lower resolution FT

7. Results of HLT • From figure (a)&(b), the sharp edges can’t be detected clearly. • From figure (c)&(d), the step edges with noise can be detected. • From figure (e)&(f), the ramp edges can be detected. • Due to the longer impulse responses. • Generally, HLT is better than differentiation, because general pictures

8. Discrete HLT • Discrete HLT: • H[p]:

9. Discrete radial HLT(DRHLT) • 2-D form of the discrete HLT: • H[p,q]: • Φ(θ ) is any odd symmetric function that satisfies • Example:

10. Short response HLT(SRHLT) • Combine HLT & differentiation • Canny’s criterion: • where cosechx = 2 / (ex− e−x) and tanhx = (ex− e−x ) / (ex+ e−x) • After scaling: • Then, we can define SRHLT from above criterion.

11. SRHLT • SRHLT: • Theorem: • b -> 0+ , the SRHLT becomes the HLT (H(f) = -j*sgn(f)) • b -> infinite, the SRHLT becomes the differentiation (H(f) = -j2*pi*f)

12. Results of SRHLT • In the frequency domain: • the transfer function of the SRHLT gradually changes from the step form(-j*sgn(f)) into the linear form(-j*2*pi*f)as b grows. • in the time domain: • when bis small, the SRHLT has a longimpulse response. • When bis large, the SRHLT has a short impulse response.

13. Discrete SRHLT • Analogous to discrete HLT • Discrete SRHLT: • H[p]:

14. 2-D discrete SRHLT • 2-D discrete SRHLT: • Φ(θ ) is any odd symmetric function • If • Then

15. Experiments on Lena image • (b) make no difference between the significant edge and the detailed edge • (c)lower resolution • (d)clearer

16. Experiments on Lena image+noise • (b)sensitive to noise • (c)noise robustness • (d) noise robustness& higher resolution

17. Improvement & other image Using adaptive threshold and overlapped section Experiment on Tiffany image

18. Performance measuring • From Canny’s theorem, measuring the performance of edge detection: • 1. Good detection • Higher distinction • Noise immunity • 2. Good localization • 3. Single response • Impulse response hb(x) : • (i)odd function • (ii)strictly decreases with |x| • (iii)

19. Conclusion • The SRHLT has higher robustness for noiseand can successfully detect ramp edges. • The SRHLT can avoid the pixels that near to an edge be recognized as an edge pixel. • Directionaledge detection and corner detection are also the possible applications of the SRHLT.

20. Thank you.