Advanced Digital Signal Processing final report NAME : YI-WEI CHEN TEACHER : JIAN-JIUN DING

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

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

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

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:

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

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

Discrete HLT  Discrete HLT:  H[p]:

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

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

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)

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 b is small, the SRHLT has a long impulse response.  When b is large, the SRHLT has a short impulse response.

Discrete SRHLT  Analogous to discrete HLT  Discrete SRHLT:  H[p]:

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

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

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

Improvement & other image Using adaptive threshold and overlapped sectionExperiment on Tiffany image

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 h b (x) :  (i)odd function  (ii)strictly decreases with |x|  (iii)

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

Thank you.