proc_miru2005_3.dvi
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- ともあき かたづ
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1 (MIRU2005) D-3D E-mal: yum@rvs.s.kyushu-u.ac.jp Fast Marchng Method 2 3 Fast Marchng Method M 2D-3D Regstraton usng 2D Dstance Maps Yum IWASHITA,RyoKURZUME, Kenj HARA, Naok ABURAYA, Masahko NAKAMOTO,KozoKONISHI, Makoto HASHIZUME, and Tsutomu HASEGAWA Graduate School of Informaton Scence and Electrcal Engneerng, Kyushu Unversty Hakozak , Hgash-ku, Fukuoka, Japan Graduate School of Desgn, Kyushu Unversty Shobaru 4 9 1, Mnam-ku, Fukuoka Japan Graduate School of Medcne, Osaka Unversty Yamadagaoka 2 2, Suta Japan Graduate School of Medcal Scence, Kyushu Unversty Madash 3 1 1, Hgash-ku, Fukuoka Japan E-mal: yum@rvs.s.kyushu-u.ac.jp Abstract Ths paper presents a fast pose estmaton algorthm of a 3D free form object n 2D mages usng 2D dstance maps. Some regstraton technques usng a slhouette mage or a contour lne have been proposed so far, but the calculaton cost for determnng pont correspondences s consderably expensve. To overcome ths problem, the proposed method utlzes a dstance map on the 2D mage plane, whch s constructed qute rapdly by the Fast Marchng Method. For pose estmaton of the object, contour lnes of the 2D mage and the projecton of the 3D object are algned usng the dstance map teratvely by the robust M-estmator. Some expermental results wth smulated models and actual mages of the endoscopc operaton are successfully carred out. Key words Texture mappng, 2D-3D regstraton, Fast Marchng Method, Dstance map, Robust M-estmater
2 Vola [16] 2 3 Allen [14] 2 albedo 3 2 [5] M Elstrom [3] 2 [15] 2 Lensch [7], [8] 2 3 Downhll Smplex octree [1], [6] Zuff [17] 1 X 3 Levenberg-Marquardt [9] [2], [10], [11] Fast Marchng Method M 2. Fast Marchng Method Sethan Fast Marchng Method [13] Fast Marchng Method 3 Level Set Method 2 Fast Marchng Method Ekonal T (p) F =1 T (p) p. Fast Marchng Method F Ekonal Fast Marchng Method Ekonal (max(d x j T, D +x j T,0) 2 + max(d y j T, D+y j T,0)2 ) 1 2 =1/F j (1) 1 Step.1 3 (known, tral, far) 1 known 0 T=0 2 known 4 known tral T j =1/F j Heap 3 far T= Step.2 Heap tral T ( mn,j mn) tral Heap known Heap Heap downheap Step.3 ( mn,j mn) 4 (( mn 1,j mn),( mn+1,j mn), ( mn,j mn 1),( mn,j mn+ 1)) far tral
![1 2 Level Set Method [13] [4] Fast Marchng Method boudary (a) Object (b) Obtaned dstance map 2 Fg.](/docs-images/101/151128309/images/3-2.jpg "2 Detected boundary and dstance map. 3. 2 3 3 3 2 3 3 OpenGL 3 3 3 OpenGL 2 2 2 3 3 detected contour 3 3 Fg.")
3 Step.4 ( mn,j mn) 4 tral 1 Heap upheap Step.5 tral Step.2 Fast Marchng Method (1) F j 1 T F 1 T 1 boundary +20 pxels contour lnes of dstance map +20 pxels +100 pxels 1 Fast Marchng Method Fg. 1 An example of the dstance map calculaton usng the Fast Marchng Method (Snakes Level Set Method [4]) 2 Fast Marchng Method OpenGL M Level Set Method [13] [4] Fast Marchng Method boudary (a) Object (b) Obtaned dstance map 2 Fg. 2 Detected boundary and dstance map OpenGL OpenGL detected contour 3 3 Fg. 3 Contour detecton of 3D geometrc model M f ( 4) f v f D ( 5) D s,t f D = D s,t D s,t D s,t (s, t) f 3 F = ψ(f ) (3) (2) M = ψ(r f ) (4)
4 r 3 ψ(z) M M 3 f r f z E(P )= ρ(z ) (5) ρ(z) ρ(z) = ψ(z) (6) z P 3 E(P ) P E P = ρ(z ) z =0 (7) z P w(z) w(z) = 1 z ψ(z) = 1 ρ(z) (8) z z E P = z w(z )z =0 (9) P ψ(z) ρ(z) w(z) z P = const. (5) E z ψ(z) = (10) 1+(z/σ) 2 ρ(z) = σ2 2 log(1 + (z/σ)2 ) (11) 1 w(z) = (12) 1+(z/σ) 2 P 3 Fg. 4 f 4 3 f Applyng the force f to all the trangular patches of the contour ponts. Object Image plane f D Steepest descent drecton Fg. 5 v f f D f Focul pont (a) 5 v Projected contour pont (s, t) f r M F (b) Force and moment around COG. y x z 3. 4 Coarse-to-fne dstance band 2 Fast Marchng Method coarse-to-fne dstance band coarse-to-fne dstance band dstance band 10 pxels dstance band , , dstance band Table 1 Computaton tme of the dstance band by the Fast Marchng Method. Image sze Whole regon Dstance band [ms] (10 pxels) [ms] 160x x x Fast Marchng Method 3 2 3
5 4. 3 Pentum IV, 3.06GHz (a)-(c) 6.(d) (e) 6.(f) 7 2 (2) f D D s,t f D = D s,t D s,t + k sgn(n D s,t D s,t )(1 n )n (13) D s,t D s,t n 3 k sgn( ) 7.(b) (13) Color mage 3D object 200mm 640x480pxel 500mm 200x250pxel 3 219,576 73, x, y, z 40mm 40mm 20mm 53 x, y, z x,y z A,B,C 3 A A 0.1mm 1 z 95.0% A 3 z y z Boundary Intal poston of 3D model (a) Frst term only (b) Frst + second terms 7 (13) 2 Fg. 7 Effect of the second term of Eq mm y Slhouette mage z x 8 Fg. 8 An example of ntal poston M 11(a) 1/4 11(b)
![2 Table 2 Accuracy of 2D-3D algnment after convergence Poston x y z Orentaton roll ptch yaw Average [mm] -0.028 0.185-4.115 Average [deg.] -0.867-0.212-0.100 SD. [mm] 0.506 0.940 2.121 SD. [deg.] 6.](/docs-images/101/151128309/images/6-3.jpg "209 3.311 1.188 Regstraton result Table 3 3 A Accuracy of 2D-3D algnment after convergence on orentaton A Poston x y z Average [mm] 0.033 0.030-3.763 SD. [mm] 0.017 0.022 0.")
![484 Orentaton roll ptch yaw Average [deg.] 0.063 0.1550-0.042 SD. [deg.] 0.146 0.131 0.016 40 Orentaton error around ptch axs [deg.] 30 20 10 0-10 -20-30 Fg.](/docs-images/101/151128309/images/6-9.jpg "9 B -40-40 -30-20 -10 0 10 20 30 40 Orentaton error around roll axs [deg.] 9 A C Dstrbuton of orentaton error M 11(c),(d) 4 M M 4. 2. 3 3 12 652 3 3 1 f 0.")
6 2D contour lne 3D object 2D contour lne 3D object 2D mage Intal poston (a) Cone Regstraton result 2D mage Intal poston (d) Rng Regstraton result 2D mage Intal poston (b) Cube Regstraton result 2D mage Intal poston (e) Tea Pot Regstraton result 2D mage Intal poston (c) Prsm Regstraton result 2D mage Intal poston (f) Pllar 6 Fg. 6 Examples of 2D-3D regstraton usng 3D models wth varous shapes. 2 Table 2 Accuracy of 2D-3D algnment after convergence Poston x y z Orentaton roll ptch yaw Average [mm] Average [deg.] SD. [mm] SD. [deg.] Regstraton result Table 3 3 A Accuracy of 2D-3D algnment after convergence on orentaton A Poston x y z Average [mm] SD. [mm] Orentaton roll ptch yaw Average [deg.] SD. [deg.] Orentaton error around ptch axs [deg.] Fg. 9 B Orentaton error around roll axs [deg.] 9 A C Dstrbuton of orentaton error M 11(c),(d) 4 M M f 0.30[us] 3 1 (a) Contour lne (b) Pose A (c) Pose B (d) Pose C 10 Fg. 10 Pose at three local mnmum solutons 9.6[ms] [pxel] 0.90[pxel 2 ] [5] f 1.15[us] k-d tree 5 f 3 7
![79-26.21 Wthout M-est. [deg.] -2.24 15.14 9.60 (a) Slhouette (occluded) (b) Dstance map and ntal pose Fg.](/docs-images/101/151128309/images/7-1.jpg "13 13 3 An example of mappng the camera mage on the 3D model. (c) Fnal pose wth M-estmator Fg.")
![02[pxel] 16 (a) Intal poston (b) 1 teraton (0.01s) Oblque-vewng endoscope Lver model (c) 3 teratons (0.](/docs-images/101/151128309/images/7-7.jpg "03s) (d) 60 teratons (0.6s) 12 3 Fg. 12 2D-3D regstraton of smulaton mages. Fg. 14 14 Expermental setup for evaluaton of the surgcal navgaton system.")
7 4 M Table 4 Comparson of accuracy of 2D-3D algnment wth and wthout M-estmator Poston x y z Orentaton roll ptch yaw M-estmator [mm] M-estmator [deg.] Wthout M-est. [mm] Wthout M-est. [deg.] (a) Slhouette (occluded) (b) Dstance map and ntal pose Fg An example of mappng the camera mage on the 3D model. (c) Fnal pose wth M-estmator Fg (d) Wthout M-estmator M Effect of robust M-estmator D mage 3D model , 14 CT 3 15 OTV-S5(OLYMPUS ) 16.(a) 16.(b) 16.(b) 16.(b1)-(b3) 1.15[pxel] 1.02[pxel] 16 (a) Intal poston (b) 1 teraton (0.01s) Oblque-vewng endoscope Lver model (c) 3 teratons (0.03s) (d) 60 teratons (0.6s) 12 3 Fg. 12 2D-3D regstraton of smulaton mages. Fg Expermental setup for evaluaton of the surgcal navgaton system. 5 1 f Table 5 Comparson of processng tme for one patch Number of ponts Proposed Pont-based on boundary method [µs] method [5] [µs] Fg. 15 3D lver model. 3 CT (
![3 Fast Marchng Method 2 2 3 2 M 3 2 Fast Marchng Method [1] L. Brune, S. Lavallee, and R. Szelsk, Usng force felds derved from 3D dstance maps for nferrng the atttude of a 3D rgd object, Proc.](/docs-images/101/151128309/images/8-1.jpg "the Second European Conference on Computer Vson, pp. 670-675, 1992 [2] Q. Delamarre and O. Faugeras, 3D Artculated Models and Mult-Vew Trackng wth Slhouettes, Proc.")
![of the Internatonal Conference on Computer Vson, Vol. 2, pp. 716-721, 1999 [3] M.D. Elstrom and P.W.](/docs-images/101/151128309/images/8-2.jpg "Smth, Stereo-Based Regstraton of Mult-Sensor Imagery for Enhanced Vsualzaton of Remote Envronments, Proc. of the 1999 IEEE Internatonal Conference on Robotcs and Automaton, pp. 1948-1953, 1999 [4] Y.")
![of Ffth Asan Conference on Computer Vson (ACCV), pp. 99-106, 2002 [6] S. Lavallee and R.](/docs-images/101/151128309/images/8-5.jpg "Szelsk, Recoverng the Poston and Orentaton of Free-Form Objects from Image Contours Usng 3D Dstance Maps, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 17, No. 4, pp.")
![378-390, 1995 [7] H. Lensch, W. Hedrch, and H.-P. Sedel, Automated Texture Regstraton and Sttchng for Real World Models, In Pacfc Graphcs 00, pp. 317-326, 2000 [8] H. Lensch, W. Hedrch, and H.-P. Sedel, Hardwareaccelerated slhouette matchng, In SIGGRAPH Sketches, 2000 [9] J.](/docs-images/101/151128309/images/8-6.jpg "Mantz and M. Vergever, A survey of medcal mage regstraton, Medcal Image Analyss, vol. 2, No. 1, pp. 1-36, 1998 [10] K. Matsushta and T. Kaneko, Effcent and handy texture mappng on 3D surfaces, Comput.")
8 Fg. 16 (a) An example of endoscopc mages for ecperment (b2) 16 (b1) (b3) Supermposton of the 3D lver model on the endoscopc mages. 17(a)) 3 ( 17(b)) 17 3 Fg. 17 (a) Endoscopc mage of the gallbladder (c1) Intal poston (b) 3D model of the gallbladder (c2) (c3) (c4) 17 2D-3D regstraton usng actual mages of the gallbladder Fast Marchng Method M 3 2 Fast Marchng Method [1] L. Brune, S. Lavallee, and R. Szelsk, Usng force felds derved from 3D dstance maps for nferrng the atttude of a 3D rgd object, Proc. the Second European Conference on Computer Vson, pp , 1992 [2] Q. Delamarre and O. Faugeras, 3D Artculated Models and Mult-Vew Trackng wth Slhouettes, Proc. of the Internatonal Conference on Computer Vson, Vol. 2, pp , 1999 [3] M.D. Elstrom and P.W. Smth, Stereo-Based Regstraton of Mult-Sensor Imagery for Enhanced Vsualzaton of Remote Envronments, Proc. of the 1999 IEEE Internatonal Conference on Robotcs and Automaton, pp , 1999 [4] Y. Iwashta, R. Kurazume, T. Tsuj, K. Hara, and T. Hasegawa, Fast Implementaton of Level Set Method and Its Realtme Applcatons, IEEE Internatonal Conference on Systems, Man and Cybernetcs, pp , 2004 [5] R. Kurazume, K. Noshno, Z. Zhang, and K. Ikeuch, Smultaneous 2D mages and 3D geometrc model regstraton for texture mappng utlzng reflectance attrbute, Proc. of Ffth Asan Conference on Computer Vson (ACCV), pp , 2002 [6] S. Lavallee and R. Szelsk, Recoverng the Poston and Orentaton of Free-Form Objects from Image Contours Usng 3D Dstance Maps, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 17, No. 4, pp , 1995 [7] H. Lensch, W. Hedrch, and H.-P. Sedel, Automated Texture Regstraton and Sttchng for Real World Models, In Pacfc Graphcs 00, pp , 2000 [8] H. Lensch, W. Hedrch, and H.-P. Sedel, Hardwareaccelerated slhouette matchng, In SIGGRAPH Sketches, 2000 [9] J. Mantz and M. Vergever, A survey of medcal mage regstraton, Medcal Image Analyss, vol. 2, No. 1, pp. 1-36, 1998 [10] K. Matsushta and T. Kaneko, Effcent and handy texture mappng on 3D surfaces, Comput. Graphcs Forum 18, pp , 1999 [11] P.J. Neugebauer and K. Klen, Texturng 3d models of real world objects from multple unregstered photographc vews, Comput. Graphcs Forum 18, pp , 1999 [12] J. Sethan, A fast marchng level set method for monotoncally advancng fronts, Proc. of the Natonal Academy of Scence, vol. 93, pp , 1996 [13] J. Sethan, Level Set Methods and Fast Marchng Methods, second edton, Cambrdge Unversty Press, UK, 1999 [14] I. Stamos and P.K. Allen, Integraton of Range and Image Sensng for Photorealstc 3D Modelng, Proc. of the 2000 IEEE Internatonal Conference on Robotcs and Automaton, pp , 2000 [15] K. Umeda, G. Godn, and M. Roux, Regstraton of Range and Color Images Usng Gradent Constrants and Range Intensty Images, Proc. of 17th Internatonal Conference on Pattern Recognton, pp , 2004 [16] P. Vola and W.M. Wells III, Algnment by maxmzaton of mutual nformaton, Int. J. of Computer Vson, vol. 24, No. 2, pp , 1997 [17] S. Zuff, A. Leardn, F. Catan, S. Fantozz, and A. Cappello, A Model-Based Method for the Reconstructon of Total Knee Replacement Knematcs, IEEE Trans. on Medcal Imagng, vol. 18, No. 10, pp ,1999
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