doi:10.1016/S0167-8655(99)00138-5 
Copyright © 2000 Elsevier Science B.V. All rights reserved.
A Hough transform technique for the detection of reflectional symmetry and skew-symmetry
Raymond K. K. Yip 
, 
Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong
Received 27 August 1998;
Revised 9 July 1999.
Available online 14 February 2000.
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Abstract
In this paper, a simple and yet robust Hough transform algorithm is proposed to detect and analyze reflectional symmetry and skew-symmetry (reflectional symmetry under parallel projection). It is applicable to shapes that contain global, local and slightly deformed reflectional/skew-symmetries under the presence of noise and occlusion.
Author Keywords: Author Keywords: Reflectional symmetry; Skew-symmetry; Skew-symmetry axis; Skew-transverse angle; Hough transformation
Fig. 1. (a) Properties of reflectional symmetry. (b) Under parallel projection, the reflectional symmetry of (a) becomes skew-symmetry.
Fig. 2. (a) A step-by-step example. (b) Detected result of (a).
Fig. 3. Input image and results of experiment 1: (a) Input image, (b) Edge detected image, (c) Midpoints of φ=130°, (d) Voted skew-symmetry axis (first pass) showing the four peak position with (ρs,θs)=(34, 126),(190, 60),(115, 100) and (101, 347), (e) Voted skew-transverse angle φ (second pass) for (ρs,θs)=(34, 126). Peak position indicates φt=150°, (f) Voted skew-transverse angle φ (second pass) for (ρs,θs)=(115, 100). Peak position indicates φt=77° and (g) Symmetry-axis found by the proposed algorithm.
Fig. 4. Input image and results of experiment 2: (a) Input image, (b) Edge detected image, (c) Midpoints of φ=30°, (d) Voted skew-symmetry axis (first pass) showing the peak position with (ρs,θs)= (164, 36), (e) Voted skew-transverse angle φ (second pass) for (ρs,θs)= (164, 36). Peak position indicates φt=60°, (f) Skew-symmetry axis (ρs,θs)= (21, 322) and φt=161° found by the proposed algorithm.
Fig. 5. Input image and results of experiment 3: (a) Input image, (b) Edge detected image, (c) Midpoints of φ=161°, (d) Voted skew-symmetry axis (first pass) showing the first three peak position with (ρs,θs)= (97, 107), (21, 322) and (41, 328), (e) Voted skew-transverse angle φ (second pass) for (ρs,θs)= (97,107). Peak position indicates φt=107°, (f) Recovered skew-symmetric points of (ρs,θs)= (97, 107) and φt=107°, T=1.0, (g) Recovered skew-symmetric points of (ρs,θs)= (21, 322) and φt=161°, T=1.0 and (h) Recovered skew-symmetric points of (ρs,θs)= (41, 338) and φt=164°, T=1.0.
Fig. 6. Input image and results of experiment 4: (a) Input image, (b) Skewed input image, (c) Edge detected image, (d) Voted skew-symmetry axis (first pass) showing the first two peak position with (ρs,θs)= (34, 125) and (25, 128), (e) Voted skew-transverse angle φ (second pass) for (ρs,θs)= (34,125). Peak position indicates φt=152°, (f) Detected results, (g) Recovered skew-symmetric points of (ρs,θs)= (34, 125) and φt=152°, T=1.0 and (h) Recovered skew-symmetric points of (ρs,θs)= (25, 128) and φt=152°, T=1.0.
Fig. 7. Input image and results of experiment 5: (a) Input image, (b) Edge detected image, (c) Midpoints of φ=137°, (d) Voted skew-symmetry axis (first pass) showing the first three peak position with (ρs,θs)= (68, 117), (83, 113) and (96, 108), (e) Detected results, (f) Recovered skew-symmetric points of (ρs,θs)= (68, 117) and φt=134°, T=1.0 and (g) Recovered skew-symmetric points of (ρs,θs)= (83, 113) and φt=152°, T=1.0 and (h) Recovered skew-symmetric points of (ρs,θs)= (96, 108) and φt=121°, T=1.0.
Table 1. Results of experiment 1
Table 2. Results of experiment 2
Table 3. Results of experiment 3
Table 4. Results of experiment 4
Table 5. Results of experiment 5
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