A Method for Detection of Occlusal Position Using a Robust Estimator

Transcript

1 Vol. 45 No. 9 Sep ,, 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 2208 Sep ) 7) 3 Character Surface Character Surface 1 Fig. 1 System overview ) 3) ) 5) 1(B) Character Surface Character Surface 8) 10) CAD/CAM F.G.P 11) 14) 15) 1 M 16) 2 3 M Roland DG Modela MDX- 15 x y

3 Vol. 45 No Fig. 2 Range image. Fig. 3 3 Process to estimate occlusal position mm 0.05 mm z mm x y z 2.5 x y 0.25 mm 2 34,051 4 Fig. 4 Initial state z z z 2 xy 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,

4 2210 Sep Fig 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 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

5 Vol. 45 No Geman McClure ρ σ =1.0 Fig. 7 ρ function of Geman and McClure (σ =1.0). M ρ (5) 1 M {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

6 2212 Sep Table 1 Parameters. M σ N γ 1 γ 2 γ 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 ) γ 1 (R, t) γ 1 t k 1 n m θ x θ y θ z 2 2 π/30 π/30 π/ x y 0.25 mm xy γ 1 x y γ 1 x y Pentium III 500 MHz PC Windows2000 C ,765 25, z z z (a) z mm (a1)(a2)(a3) (a) z mm mm mm 6 2

7 Vol. 45 No Fig. 10 Estimated state (e) x 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 (a) (b)(c)(d) (e)(f)(g) 2 (d)(g) 11 (a) (1) x y 0.25 mm z mm 0.01 mm (2) (3) (1)

8 2214 Sep (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 (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)

9 Vol. 45 No KD 2),18) 1) Besl,P.andMcKay,N.:AMethodforRegistration of 3-D Shapes, IEEE Trans. PAMI, Vol.14, No.2, pp (1992). 2) Vol.J85- D-II, No.9, pp (2002). 3) Lucchese, L., Doretto, G. and Cortelazzo, G.M.: A Frequency Domain Technique for Range Data Registration, IEEE Trans. PAMI, Vol.24, No.11, pp (2002). 4) Ucoluk, G. and Toroslu, I.H.: Reconstruction of 3-D Surface Object from its Pieces, 9th Canadian Conference on Computational Geometry, pp (1997). 5) Ucoluk, G. and Toroslu, I.H.: Automatic reconstruction of broken 3-D surface objects, Computers & Graphics, Vol.23, No.4, pp (1999). 6) Hashimura, K., Kato, S. and Itoh, H.: Finding a Quasi-Optimal Joining Position of Potsherds in 3D Multi-resolution Mesh Representation, 2nd International Conference on Software Engineering, Artificial Intelligence, Networking & Parallel/Distributed Computing, pp (2001). 7) Hashimura, K., Kato, S. and Itoh, H.: A Parametric Search Strategy for a Quasi-Optimal Joining Position of 3D Potsherds, International Journal of Computer & Information Science, Vol.3, No.2, pp (2002). 8) Papaioannou, G., Karabassi, E.-A. and Theoharis, T.: Automatic Reconstruction of Archaeological Finds A Graphics Approach, International Conference Computer Graphics and Artificial Intelligence, pp (2000). 9) Papaioannou, G., Karabassi, E.-A. and Theoharis, T.: Reconstruction of Three-Dimensional Objects Through Matching of Their Parts, IEEE Trans. PAMI, Vol.24, No.1, pp (2002). 10) Papaioannou, G., Karabassi, E.-A. and Theoharis, T.: Virtual Archaeologist: Assembling the Past, IEEE Computer Graphics and Applications, Vol.21, No.2, pp (2001). 11) 6 F.G.P Vol.28, pp (1994). 12) CAD/CAM F.G.P Vol.13, pp (1994). 13) CAD/CAM Vol.27, pp (1993). 14) FGP CAD/CAM Vol.8, pp (1991). 15) Vol.44, No.6, pp (2000). 16) Kanoh,M.,Hashimura,K.,Kato,S.andItoh, H.: A Robust Method for Detection of Occlusal Position, Asian Conference on Computer Vision, pp (2004). 17) Black, M.J. and Anandan, P.: The Robust Estimation of Multiple Motions Parametric and Piecewise-Smooth Flow Fields, Computer Vision and Image Understanding, Vol.63, No.1, pp (1996). 18) Friedman, J., Bentley, J. and Finkel, R.: An Algorithm for Finding Best Matches in Logarithmic Expected Time, ACM Trans. Mathematical Software, Vol.3, No.3, pp (1977). ( ) ( )

10 2216 Sep OS