Advanced Digital Signal Processing final reportNAME : 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 / (ex − e−x ) and tanh x = (ex − e−x ) / (ex + 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 section Experiment 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 hb(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.