Automatic Detection of Circular Objects by Ellipse Growing Mitsuo OKABE, Kenichi KANATANI, and Naoya OHTA 1. [4], [5], [18], [19] [14], [17] [28], [32

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Automatic Detection of Circular Objects by Ellipse Growing Mitsuo OKABE, Kenichi KANATANI, and Naoya OHTA 1. [4], [5], [18], [19] [14], [17] [28], [32] [6], [21], [27] Fujitsu Cadtech, Ltd., Oyama-shi, 323-0807 Japan Department of Information Technology, Okayama University, Okayama-shi, 700-8530 Japan Department of Computer Science, Gunma University, Kiryushi, 376-8515 Japan 5 5 5 3 2 3 2 [2], [3], [7] [10], [22] [26], [31], [33] [39] D II Vol. J85 D II No. 12 pp. 1823 1831 2002 12 1823

2002/12 Vol. J85 D II No. 12 edge detection voting for the center of the circle voting for the radius of the circle ellipse growth outlier detection ellipse fitting Does the edge segment mostly cover the ellipse? no yes search for the other segment end Fig. 1 1 Flowchart of the procedure. [2] 2 [30] 3 2 1 LMedS [29] 1 2. 2. 1 Sobel 1 8 8 Fig. 2 2 The evolute of an ellipse and its singularities. 61 2. 2 [11] 4 2 [1] 2 4 3 [16] P k 2 P C r 3(a) C γr k k = 30, γ = 1/10 1/ r 1 4 5 3. 2. 3 C 1 3 1824

3 Fig. 3 (a) (b) (c) (a) P k 2. (b) C R e R2 /2s 2 cos φ R (c) δ (a) Voting around the center of the circle passing through P and the two points away from it by k pixels on both sides. (b) A pixel away from the estimated center C by distance R votes the value R with weight e R2 /2s 2 cos φ. (c) Fitting an ellipse to the longest edge segment inside the region within distance δ from the estimated ellipse. C P R [11] R 1 e R2 /2s 2 cos φ 2 φ P Sobel CP 3(b) s C R s 1/4 R ±1 e 1/2 R 1 ) e R2 /2s 2 C s cos φ 0 C 0 6 3. 2. 4 δ 2 e x2 3(c) [12], [13], [15] 3 [18] Ax 2 +2Bxy+Cy 2 +2f(Dx+Ey)+f 2 F =0 (1) f AC B 2 > 0 (2) x Q x/f A B D x = y/f, Q = B C E (3) 1 D E F (1) [14], [16] (x, Qx) = 0 (4) (a, b) a, b 3 http://www.ail.cs.gunma-u.ac.jp/labo/program.html 1825

2002/12 Vol. J85 D II No. 12 δ Q (+), Q ( ) 1 S c c ( A S = B B C ), c = c = (c, S 1 c) F (5) 2 S λ 1, λ 2 ( D E u 1, u 2 3 1, 2 1 = 2 = c ( c/λ 1 ± δ/f) 2 c (6) ( c/λ 2 ± δ/f) 2 4 S (+), S ( ) ( ) S (±) = U 1 2 ) U (7) U u 1, u 2 2 2 5 Q (+), Q ( ) Q (±) = ( S (±) S (±) S 1 c (S (±) S 1 c) F +(c, S 1 (S (±) S)S 1 c) (8) (4) δ (x, Q (+) x)(x, Q ( ) x) < 0 (9) (1) 2 δ = 4 (2) δ 1 δ = 10 2. 5 LMedS [29] ) 4 Fig. 4 Judging if the edge segment covers more than half of the ellipse. 4 (3) {x α }, α = 1,..., N Q m = O, S m = 1 {x α } 5 2 5 Q (1) A, B,..., F (2) 3 S = med N (x α, Qx α ) 2 α=1 (10) P k Qx α 2 P k = diag(1, 1, 0) 1, 1, 0 Q x α [15] 4 S < S m Q m Q, S m S 10 S > = S m x α (x α, Qx α) 2 < 10Sm (11) P k Qx α 2 [29] σ σ 1.4826 S m Q 2.13σ e 2. 6 4 1826

(a) (b) (c) 5 (a) (b) (c) Fig. 5 Estimation of the center of the initial circle. (a) The detected circle and the fitted ellipse. (b) Voting for the center. (c) Simple Hough transform. 0 100 200 300 400 R (a) 0 100 200 300 400 R 6 R (a) (b) Fig. 6 The number of votes (ordinate) for the radius R of the initial circle (abscissa). (a) Proposed method. (b) Simple voting. (b) e 4 1 + γ (6) 1 = λ 1 (1 ± γ) 2, λ(±) 2 = λ 2 (1 ± γ) 2 (12) 1 ± γ γ = 1.2 20 e e 4, e 1 e, e (10) 100 4 e, e (11) 3. 5(a) 5(b) 5(c) 1 [11] 6(a) R 6(b) ±1 1 [11] 6(a) 0 7(a) 1827

2002/12 Vol. J85 D II No. 12 (a) (b) 7 Fig. 7 (a) (b)(1) (2) (3) (4) (a) The edge image used. (b) (1) The initial circle, (2) the hyperbola resulting from the ellipse growing, (3) the ellipse fitted by LMedS, and (4) the ellipse fitted after detecting another segment. δ = 10 7(b) (1) (2) (3) (4) 8 1, 2 4, 5 500 333 18 CPU Pentium III 600MHz OS Linux [32] 9 768 512 700 6.45cm 59.4 27.3cm [17], [28], [32] 4. 1 2 (2) No. 13680432 [1],,, 1976. [2],,,, PRMU98-123, Nov. 1998. [3] Y. C. Cheng and S. C. Lee, A new method for quadratic curve detection using K-RANSAC with acceleration techniques, Pattern Recognit., vol.28, no.5, pp.663 682, May 1995. [4] W. Chojnacki, M. J. Brooks, and A. van den Hen- 1828

論文 楕円成長法による円形物体の自動検出 図 8 当てはめ例 上段はエッジ画像 下段は初期円と当てはめた楕円を原画像に重ねた もの Fig. 8 Examples of ellipse fitting: the edge images (upper rows); the initial circles and the fitted ellipses superimposed on the original images (lower rows). 1829

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