Vol. 45 No. 9 Sep. 2004 3,, 1 1 M A Method for Detection of Occlusal Position Using a Robust Estimator Masayoshi Kanoh,, Kyoji Hashimura,, Shohei Kato and Hidenori Itoh We have developed a system that enables dental prostheses to be designed on a computer. The occlusal position for the upper and lower jaw objects should be found when designing dental prostheses on our system. However, there is often great difficulty in manually finding this position. To reduce this difficulty, we considered finding it and occluding jaw objects automatically. In positioning upper and lower jaw objects, some regions, such as teethridges and sides of teeth, do not completely close on one another, even if these objects are occluded. The evaluation value for these regions is determined to be an error, and consequently finding the occlusal position requires robust assessment. In this paper, we propose a method of evaluation using a robust estimator, the M-estimator. We also propose an algorithm applied multi-resolution representation to improve runtime. 1. Graduate School of Engineering, Nagoya Institute of Technology Presently with School of Life System Science and Technology, Chukyo University Presently with Toyota Communication Systems Co., Ltd. 1 (A) (B) (C) 2 3 2207
2208 Sep. 2004 6) 7) 3 Character Surface Character Surface 1 Fig. 1 System overview. 2 3 1) 3) 3 1 2 3 4) 5) 1(B) Character Surface Character Surface 8) 10) 2 2 2 CAD/CAM F.G.P 11) 14) 15) 1 M 16) 2 3 M 4 5 2. Roland DG Modela MDX- 15 x y
Vol. 45 No. 9 3 2209 2 Fig. 2 Range image. Fig. 3 3 Process to estimate occlusal position. 0.05 5.00 mm 0.05 mm z 0.025 mm x y z 2.5 x y 0.25 mm 2 34,051 4 Fig. 4 Initial state. 3. 2 3.1 3 2 2 3.2 z z 4 3.3 2 2.5 z 2 xy 2 5 2 2 3.1 xy γ γ 1 γ 3.2 A (x, y, z) T A A γ A γ (i, j) a γ i,j = min (x,y,z) T A z,
2210 Sep. 2004 Fig. 6 6 2 Evaluation of occlusion with 2D image. Fig. 5 5 Flow for creating 2D image (lower jaw). (γi x<γ(i +1), γj y<γ(j +1)). (1) xy a γ i,j = 3.3 R B R B R (x, y, z) T B R B R γ B R,γ (i, j) b R,γ i,j = max (x,y,z) T BR z, (γi x<γ(i +1), γj y<γ(j +1)). (2) xy b R,γ i,j = z z z z z 0 3.4 M 2 M M 3.4 γ (A γ, B R,γ ) B R,γ t =(u, v) 6 A γ (i, j) ε R,γ,t i,j =a γ i,j br,γ i+u,j+v. (3) ε A γ t B R,γ M M(A γ, B R,γ, t) = ρ(ε R,γ,t i,j i j min ε R,γ,t k,l,σ). (4) k,l σ ρ M M Geman McClure ρ 17) ρ(x, σ) = x2 σ + x. (5) 2 σ =1.0 7 x 1 1 6
Vol. 45 No. 9 3 2211 7 Geman McClure ρ σ =1.0 Fig. 7 ρ function of Geman and McClure (σ =1.0). M ρ (5) 1 M 4. 2 3 {R,t} 8 N γ k (k = 1,,N) 0 <γ k 1 <γ k correct() γ k γ k 1 t : A B : (R, t) 1 begin 2 k := N; % N 3 R := I; % I 4 t A, B γ k ; 5 repeat 6 error := ; 7 R := R 8 t := t 9 A γ k S R,γ k ; 10 for each R S R,γ k 11 begin 12 B R,γ k S t ; 13 for each t S t ; 14 begin 15 e := M(A γ k, B R,γ k, t ); 16 if (e < error) then 17 begin 18 error := e; 19 (R, t ):=(R, t ); 20 end 21 end 22 end 23 (R, t) :=(R, t ); 24 if (γ k = γ 1 ) % γ 1 25 then escape := true; 26 else 27 begin 28 t := correct(γ k,γ k 1, t); 29 k := k 1; 30 end 31 until (escape) 32 end. 8 Fig. 8 Detection algorithm for estimating occlusal posision with multi-resolution representation. correct(γ k,γ k 1, t) = γ k t. (6) γ k 1 S R,γ 4.1 x y z θ X(θ) Y (θ) Z(θ) R γ S R,γ S R,γ = {RX(δ x γθ x )Y (δ y γθ y )Z(δ z γθ z ) n δ x n, n δ y n, n δ z n}. (7) δ x δ y δ z n θ x θ y θ z
2212 Sep. 2004 1 Table 1 Parameters. M σ N γ 1 γ 2 γ 3 1.0 3 0.3 0.6 0.9 9 z Fig. 9 Rotational angle according to different of pixelization scale (z axis). x y z 9 z S t 4.2 t S t S t = {t +(δ i,δ j ) m δ i m, m δ j m}. (8) δ i δ j m γxy γ γ 1 xy γ 1 3 γ N γ 1 A γ k S R,γ k 8 9 B R,γ k S t 8 12 (R, t ) 8 10 22 γ 1 (R, t) γ 1 t k 1 n m θ x θ y θ z 2 2 π/30 π/30 π/30 8 24 30 5. 1 x y 0.25 mm xy 3.2 3.3 γ 1 x y γ 1 x y 0.3 1 Pentium III 500 MHz PC Windows2000 C++ 4 36,765 25,493 10 30 10 11 z z z (a) z 0 1.0 mm (a1)(a2)(a3) (a) z 0 0.3 mm 0.3 0.6 mm 0.6 1.0 mm 6 2
Vol. 45 No. 9 3 2213 2 10 Fig. 10 Estimated state. 12 2 13 (e) x 10 6. 1 M 2 11 Fig. 11 Occlusal state. M 2 2 M ρ { x 2 x<50 ρ(x) = (9) 2500 otherwise. 6 2,500 (9) 2,500 = 50 2 z 50 mm 12 13 (a) (b)(c)(d) (e)(f)(g) 2 (d)(g) 11 (a) (1) x y 0.25 mm z 0.025 mm 0.01 mm (2) (3) (1)
2214 Sep. 2004 12 2 (1) (a) (b)(c)(d) (e)(f)(g) 2 Fig. 12 Comparison of our algorithm and least squares method (1): (a) is initial state; (b), (c) and (d) are estimated states with our algorithm; (e), (f) and (g) are estimated state with least squares method. 13 2 (2) (a) (b)(c)(d) (e)(f)(g) 2 Fig. 13 Comparison of our algorithm and least squares method (2): (a) is initial state; (b), (c) and (d) are estimated states with our algorithm; (e), (f) and (g) are estimated state with least squares method. (2) 1 2 (3)
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