Abstract

We present a novel technique for the estimation of global nonrigid motion of an object’s boundary without using point correspondences. A complete description of the motion of an object’s boundary involves specifying a displacement vector at each point of the boundary. Such a description provides a large amount of information, which needs to be processed further for study of the global characteristics of the deformation. Nonrigid motion can be studied hierarchically in terms of global nonrigid motion and point-by-point local nonrigid motion. The technique presented gives a method for estimating a global affine or polynomial transformation between two object boundaries. The novelty of the technique lies in the fact that it does not use any point correspondences. Our method uses hyperquadric models to model the data and estimate the global deformation. We show that affine or polynomial transformation between two datasets can be recovered from the hyperquadric parameters. The usefulness of the technique is twofold. First, it paves the way for viewing nonrigid motion hierarchically in terms of global and local motion. Second, it can be used as a front end to other motion analysis techniques that assume small motion. For instance, most nonrigid motion analysis algorithms make some assumptions on the type of nonrigid motion (conformal motion, small motion, etc.) that are not always satisfied in practice. When the motion between two datasets is large, our algorithm can be used to estimate the affine transformation (which includes scale and shear) or a polynomial transformation between the two datasets, which can then be used to warp the first dataset closer to the second so as to satisfy the small-motion assumption. We present experimental results with real and synthetic two- and three-dimensional data.

© 2000 Optical Society of America

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References

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  1. L. Brown, “A survey of image registration techniques,” ACM (Assoc. Comput. Mach.) Comput. Surv. 24, 325–376 (1992).
    [CrossRef]
  2. G. Wolberg, Digital Image Warping (IEEE Computer Society Press, Los Alamitos, Calif., 1990).
  3. C. Kambhamettu, D. Goldgof, “Curvature-based ap-proach to point correspondence recovery in conformal nonrigid motion,” Comput. Vision Graph. Image Process. Image Understand. 60, 26–43 (1994).
    [CrossRef]
  4. C. Kambhamettu, D. Goldgof, “Point correspondence recovery in nonrigid motion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1992), pp. 222–227.
  5. R. Jain, I. K. Sethi, “Establishing correspondence of non-rigid objects using smoothness of motion,” in Proceedings of the IEEE Workshop on Computer Vision and Robotics Control (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 83–87.
  6. S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.
  7. A. Hanson, “Hyperquadrics: smoothly deformable shapes with convex polyhedral bounds,” Comput. Vision Graph. Image Process. 44, 191–210 (1988).
    [CrossRef]
  8. S. Kumar, D. Goldgof, “A robust technique for the estimation of hyperquadrics parameters from range data,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. I, pp. 74–78.
  9. S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
    [CrossRef]
  10. I. Cohen, L. D. Cohen, “A hybrid hyperquadric model for 2-D and 3-D data fitting,” in (Institut National de Recherche en Informatique et d’ Automatique, Sophia Antipolis, France, 1994).
  11. I. Cohen, L. D. Cohen, “A hyperquadric model for 2-D and 3-D data fitting,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos Calif., 1994), Vol. II, pp. 403–405.
  12. C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
    [CrossRef]
  13. A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vis. 4, 107–126 (1990).
    [CrossRef]
  14. M. Sallam, K. W. Bowyer, “Registering time sequences of mammograms using a two-dimensional image unwarping technique,” in Proceedings of the Second International Workshop on Digital Mammography (Elsevier, Amsterdam, 1994), pp. 121–130.
  15. F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 567–585 (1989).
    [CrossRef]
  16. S. Han, D. Goldgof, “Using hyperquadrics for shape recovery from range data,” in Proceedings of the Fourth IEEE International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 492–496.
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).
  18. D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).
  19. S. Chaudhuri, S. Chatterjee, “Estimation of motion parameters for a deformable object from range data,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1989), pp. 291–295.
  20. L. Stark, “Recognizing object function through reasoning about 3-d shape and dynamic physical properties,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 546–553.
  21. M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
    [CrossRef]

1995 (1)

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

1994 (3)

C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
[CrossRef]

C. Kambhamettu, D. Goldgof, “Curvature-based ap-proach to point correspondence recovery in conformal nonrigid motion,” Comput. Vision Graph. Image Process. Image Understand. 60, 26–43 (1994).
[CrossRef]

M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
[CrossRef]

1992 (1)

L. Brown, “A survey of image registration techniques,” ACM (Assoc. Comput. Mach.) Comput. Surv. 24, 325–376 (1992).
[CrossRef]

1990 (1)

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vis. 4, 107–126 (1990).
[CrossRef]

1989 (1)

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 567–585 (1989).
[CrossRef]

1988 (1)

A. Hanson, “Hyperquadrics: smoothly deformable shapes with convex polyhedral bounds,” Comput. Vision Graph. Image Process. 44, 191–210 (1988).
[CrossRef]

Arrott, M.

C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
[CrossRef]

Bookstein, F. L.

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 567–585 (1989).
[CrossRef]

Bowyer, K.

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.

Bowyer, K. W.

M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
[CrossRef]

M. Sallam, K. W. Bowyer, “Registering time sequences of mammograms using a two-dimensional image unwarping technique,” in Proceedings of the Second International Workshop on Digital Mammography (Elsevier, Amsterdam, 1994), pp. 121–130.

Brown, L.

L. Brown, “A survey of image registration techniques,” ACM (Assoc. Comput. Mach.) Comput. Surv. 24, 325–376 (1992).
[CrossRef]

Chatterjee, S.

S. Chaudhuri, S. Chatterjee, “Estimation of motion parameters for a deformable object from range data,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1989), pp. 291–295.

Chaudhuri, S.

S. Chaudhuri, S. Chatterjee, “Estimation of motion parameters for a deformable object from range data,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1989), pp. 291–295.

Chen, C. W.

C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
[CrossRef]

Cohen, I.

I. Cohen, L. D. Cohen, “A hybrid hyperquadric model for 2-D and 3-D data fitting,” in (Institut National de Recherche en Informatique et d’ Automatique, Sophia Antipolis, France, 1994).

I. Cohen, L. D. Cohen, “A hyperquadric model for 2-D and 3-D data fitting,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos Calif., 1994), Vol. II, pp. 403–405.

Cohen, L. D.

I. Cohen, L. D. Cohen, “A hyperquadric model for 2-D and 3-D data fitting,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos Calif., 1994), Vol. II, pp. 403–405.

I. Cohen, L. D. Cohen, “A hybrid hyperquadric model for 2-D and 3-D data fitting,” in (Institut National de Recherche en Informatique et d’ Automatique, Sophia Antipolis, France, 1994).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Goldgof, D.

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

C. Kambhamettu, D. Goldgof, “Curvature-based ap-proach to point correspondence recovery in conformal nonrigid motion,” Comput. Vision Graph. Image Process. Image Understand. 60, 26–43 (1994).
[CrossRef]

S. Han, D. Goldgof, “Using hyperquadrics for shape recovery from range data,” in Proceedings of the Fourth IEEE International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 492–496.

S. Kumar, D. Goldgof, “A robust technique for the estimation of hyperquadrics parameters from range data,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. I, pp. 74–78.

S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.

C. Kambhamettu, D. Goldgof, “Point correspondence recovery in nonrigid motion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1992), pp. 222–227.

Han, S.

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

S. Han, D. Goldgof, “Using hyperquadrics for shape recovery from range data,” in Proceedings of the Fourth IEEE International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 492–496.

Hanson, A.

A. Hanson, “Hyperquadrics: smoothly deformable shapes with convex polyhedral bounds,” Comput. Vision Graph. Image Process. 44, 191–210 (1988).
[CrossRef]

Huang, T. S.

C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
[CrossRef]

Jain, R.

R. Jain, I. K. Sethi, “Establishing correspondence of non-rigid objects using smoothness of motion,” in Proceedings of the IEEE Workshop on Computer Vision and Robotics Control (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 83–87.

Kambhamettu, C.

C. Kambhamettu, D. Goldgof, “Curvature-based ap-proach to point correspondence recovery in conformal nonrigid motion,” Comput. Vision Graph. Image Process. Image Understand. 60, 26–43 (1994).
[CrossRef]

C. Kambhamettu, D. Goldgof, “Point correspondence recovery in nonrigid motion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1992), pp. 222–227.

Kumar, S.

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

S. Kumar, D. Goldgof, “A robust technique for the estimation of hyperquadrics parameters from range data,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. I, pp. 74–78.

S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

Pentland, A. P.

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vis. 4, 107–126 (1990).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Sallam, M.

M. Sallam, K. W. Bowyer, “Registering time sequences of mammograms using a two-dimensional image unwarping technique,” in Proceedings of the Second International Workshop on Digital Mammography (Elsevier, Amsterdam, 1994), pp. 121–130.

S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.

Sethi, I. K.

R. Jain, I. K. Sethi, “Establishing correspondence of non-rigid objects using smoothness of motion,” in Proceedings of the IEEE Workshop on Computer Vision and Robotics Control (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 83–87.

Stark, L.

M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
[CrossRef]

L. Stark, “Recognizing object function through reasoning about 3-d shape and dynamic physical properties,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 546–553.

Sutton, M.

M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

Wolberg, G.

G. Wolberg, Digital Image Warping (IEEE Computer Society Press, Los Alamitos, Calif., 1990).

ACM (Assoc. Comput. Mach.) Comput. Surv. (1)

L. Brown, “A survey of image registration techniques,” ACM (Assoc. Comput. Mach.) Comput. Surv. 24, 325–376 (1992).
[CrossRef]

Comput. Vision Graph. Image Process. (1)

A. Hanson, “Hyperquadrics: smoothly deformable shapes with convex polyhedral bounds,” Comput. Vision Graph. Image Process. 44, 191–210 (1988).
[CrossRef]

Comput. Vision Graph. Image Process. Image Understand. (1)

C. Kambhamettu, D. Goldgof, “Curvature-based ap-proach to point correspondence recovery in conformal nonrigid motion,” Comput. Vision Graph. Image Process. Image Understand. 60, 26–43 (1994).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

S. Kumar, S. Han, D. Goldgof, K. Bowyer, “On recovering hyperquadrics from range data,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1079–1083 (1995).
[CrossRef]

C. W. Chen, T. S. Huang, M. Arrott, “Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 342–356 (1994).
[CrossRef]

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 567–585 (1989).
[CrossRef]

Int. J. Comput. Vis. (1)

A. P. Pentland, “Automatic extraction of deformable part models,” Int. J. Comput. Vis. 4, 107–126 (1990).
[CrossRef]

Pattern Recogn. (1)

M. Sutton, L. Stark, K. W. Bowyer, “GRUFF-3: generalizing the domain of function-based recognition system,” Pattern Recogn. 27, 1743–1766 (1994).
[CrossRef]

Other (13)

M. Sallam, K. W. Bowyer, “Registering time sequences of mammograms using a two-dimensional image unwarping technique,” in Proceedings of the Second International Workshop on Digital Mammography (Elsevier, Amsterdam, 1994), pp. 121–130.

I. Cohen, L. D. Cohen, “A hybrid hyperquadric model for 2-D and 3-D data fitting,” in (Institut National de Recherche en Informatique et d’ Automatique, Sophia Antipolis, France, 1994).

I. Cohen, L. D. Cohen, “A hyperquadric model for 2-D and 3-D data fitting,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos Calif., 1994), Vol. II, pp. 403–405.

S. Han, D. Goldgof, “Using hyperquadrics for shape recovery from range data,” in Proceedings of the Fourth IEEE International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 492–496.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

S. Chaudhuri, S. Chatterjee, “Estimation of motion parameters for a deformable object from range data,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1989), pp. 291–295.

L. Stark, “Recognizing object function through reasoning about 3-d shape and dynamic physical properties,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 546–553.

C. Kambhamettu, D. Goldgof, “Point correspondence recovery in nonrigid motion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1992), pp. 222–227.

R. Jain, I. K. Sethi, “Establishing correspondence of non-rigid objects using smoothness of motion,” in Proceedings of the IEEE Workshop on Computer Vision and Robotics Control (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 83–87.

S. Kumar, M. Sallam, D. Goldgof, K. Bowyer, “Establishing point correspondences in unstructured nonrigid motion,” in Proceedings of the IEEE International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 289–294.

S. Kumar, D. Goldgof, “A robust technique for the estimation of hyperquadrics parameters from range data,” in Proceedings of the 12th International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. I, pp. 74–78.

G. Wolberg, Digital Image Warping (IEEE Computer Society Press, Los Alamitos, Calif., 1990).

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Figures (15)

Fig. 1
Fig. 1

Mammograms of the left and right breasts of the same patient.

Fig. 2
Fig. 2

Overlapped boundaries of the left and right breasts.

Fig. 3
Fig. 3

(a) Breast boundary and (b) hyperquadric model overlaid on the boundary.

Fig. 4
Fig. 4

2D hyperquadrics: inclined lines correspond to term 1, and horizontal lines correspond to term 2.

Fig. 5
Fig. 5

3D hyperquadrics (four terms each).

Fig. 6
Fig. 6

2D hyperquadric fits to ventricular boundaries.

Fig. 7
Fig. 7

(a) Original range data, (b) hyperquadric fit (six terms/part), (c) a different view of the duck’s body. The duck was manually segmented into five parts: beak, head, neck, body, and tail.

Fig. 8
Fig. 8

An ellipse can be transformed into itself (see Section 4).

Fig. 9
Fig. 9

(a) Overlapped breast boundaries from Fig. 1 and (b) the right breast boundary and the warped left breast boundary.

Fig. 10
Fig. 10

Mammograms of the left and right breasts of a different patient.

Fig. 11
Fig. 11

(a) Overlapped breast boundaries from Fig. 10 and (b) the right breast boundary and the warped left breast boundary.

Fig. 12
Fig. 12

Illustration of the use of our algorithm in determining whether the object being grasped by the robot arm is deforming. To asses the algorithm qualitatively, we can apply the estimated D and T to the after-grasp boundary, obtaining a new warped object boundary. This new warped boundary can then be compared with the original before-grasp boundary. In both cases above, the shapes of the warped boundaries are close to the original before-grasp boundaries, indicating qualitatively that the deformation estimates are reasonable. (a) Results of deformation analysis for a rigid object, (b) results for a deformation object.

Fig. 13
Fig. 13

Intensity images of an inflating balloon.

Fig. 14
Fig. 14

Hyperquadric models fitted to the range data of the inflating balloon.

Fig. 15
Fig. 15

Patches of the original range data: (a) original patches and (b) second balloon patch and the transformed first balloon patch.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

i=1N|Aix+Biy+Ciz+Di|γi=1.
xaγ1+ybγ2+zcγ3=1
|Aix+Biy+Ciz+Di|1,i=1, 2 ,, N.
EOF=i=1Ndata1Fi0(x, y, z)2 [1-Fi0(x, y, z)]2,
Fi0(x, y, z)=[fi0(x, y, z)]p(pis a small constant),
fi0(x, y, z)=i=1N|Aix+Biy+Ciz+D|γi.
Xi=Dxi+T+di.
Xi=Dxi+T.
inxi=0,
inXi=nT.
i=1N|Ai1x1+Ai2x2+Ai3x3+Ai4|γi=1.
i=1N|AiTx+Ai4|γi=1.
x=KX.
i=1N|AiTKX+Ai4|γi=1.
i=1N|BiTX+Ai4|γi=1.
B=AK,
i=1N|AiTx+Ai4|γi=1
i=1N|BiTx+Bi4|νi=1,
Xk=Pk(x1, x2, x3)=akx12+bkx22+ckx32+dkx1x2+ekx2x3+fkx3x1+gkx1+hkx3+ikx3+jk
k=1, 2, 3.
i=1N|Ai1X1+Ai2X2+Ai3X3+Ai4|γi=1,
i=1N|Ai1P1(x1, x2, x3)+Ai2P2(x1, x2, x3)
+Ai3P3(x1, x2, x3)+Ai4|γi=1.
i=1N|Bi1x12+Bi2x22+Bi3x32+Bi4x1x2+Bi5x2x3+Bi6x3x1+Bi7x1+Bi8x2+Bi9x3+Bi10|γi=1.
Ai1a1+Ai2a2+Ai3a3=Bi1,i=1, 2 ,, N.
xa2+yb2=1.
x2a+y2b2+x2a-y2b2=1.
D=0.9879-0.0225-0.02770.9052,
T=2.8854-51.9932.

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