Abstract

Information loss may occur frequently in the imaging of living tissues by using two-photon fluorescence microscopy due to the intensive deformation of the tissue. A landmark-based optical flow interpolation scheme is proposed for image reconstruction of living aorta walls in two-photon autofluorescence image sequences. Landmarks are extracted and evaluated by an active contour-based aorta model, and are aligned and reconstructed by use of a hierarchical algorithm. The accuracy of the calculation of optical flow is improved by applying landmark-based image warping. Experimental results show that the proposed scheme outperforms commonly used optical flow interpolation techniques for the reconstruction of intensively deforming tissues.

© 2003 Optical Society of America

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References

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  1. W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
    [CrossRef] [PubMed]
  2. B. Masters, “Confocal microscopy and multi-photon excitation microscopy of human skin in vivo,” Opt. Express 8, 2–11 (2001).
    [CrossRef] [PubMed]
  3. R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
    [CrossRef] [PubMed]
  4. D. Arkas, D. Becker, “Applications of spectral imaging: detection and analysis of human melanoma and its precursors,” Pigm. Cell. 14, 2–8 (2001).
    [CrossRef]
  5. X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
    [CrossRef]
  6. J. Wang, L. Ji, H. Ma, “A combined approach for a quantitative evaluation of aorta-related drugs,” in Proceedings of the 2nd Joint Conf. of the IEEE Medicine and Biology Society and Biomedical Engineering Society, USA, Oct.2002.
  7. Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer, New York, 1993).
  8. B. K. P. Horn, B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
    [CrossRef]
  9. B. K. P. Horn, Robot Vision (MIT, Cambridge, Mass., 1990).
  10. B. Galvin, B. McCane, K. Novins, D. Mason, S. Mills, “Recovering motion fields: An evaluation of eight optical flow algorithms,” in Proceedings of the Ninth British Machine Vision Conference, P. H. Lewis, M. S. Nixon, eds. (ECS Publications Database, University of Southampton, UK). (1998). http://www.eprints.org .
  11. B. McCane, B. Galvin, K. Novins, “On the evaluation of optical flow algorithms,” in Proceedings of the Fifth Int. Conf. Control, Automation, Robotics and Vision (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1563–1567.
  12. M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
    [CrossRef]
  13. T. McInerney, D. Terzopoulos, “Deformable models in medical image analysis: A survey,” Medical Image Analysis 1, 91–108 (1996).
    [CrossRef] [PubMed]
  14. A. Singh, D. Goldgof, D. Terzopoulos, Deformable Models in Medical Image Analysis, (IEEE Press, Los Alamitos, Calif., 1998).
  15. D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
    [CrossRef] [PubMed]
  16. R. Handa, H. H. S. Ip, H. Y. Tang, “Computerised analysis of aortic distensibility on computed tomography,” in Proceedings of the Current Perspectives in Healthcare Computing 1997 Conference, Harrogate (1997), pp. 157–165.
  17. C. Davatzikos, J. Prince, “Brain image registration based on curve mapping,” in Proceedings of IEEE Workshop Biomedical Image Anal., (IEEE, Los Alamitos, Calif.1994), pp. 245–254.
  18. J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
    [CrossRef]
  19. H. Chui, A. Rangarajan, “A new algorithm for non-rigid point matching,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 44–51.
  20. R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).

2001

D. Arkas, D. Becker, “Applications of spectral imaging: detection and analysis of human melanoma and its precursors,” Pigm. Cell. 14, 2–8 (2001).
[CrossRef]

B. Masters, “Confocal microscopy and multi-photon excitation microscopy of human skin in vivo,” Opt. Express 8, 2–11 (2001).
[CrossRef] [PubMed]

2000

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
[CrossRef] [PubMed]

1997

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

1996

T. McInerney, D. Terzopoulos, “Deformable models in medical image analysis: A survey,” Medical Image Analysis 1, 91–108 (1996).
[CrossRef] [PubMed]

1990

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1988

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
[CrossRef]

1981

B. K. P. Horn, B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Arkas, D.

D. Arkas, D. Becker, “Applications of spectral imaging: detection and analysis of human melanoma and its precursors,” Pigm. Cell. 14, 2–8 (2001).
[CrossRef]

Bajcsy, R. K.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

Barillot, C.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

Becker, D.

D. Arkas, D. Becker, “Applications of spectral imaging: detection and analysis of human melanoma and its precursors,” Pigm. Cell. 14, 2–8 (2001).
[CrossRef]

Bellman, R.

R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).

Burger, P.

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Capitano, A.

R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
[CrossRef] [PubMed]

Chen, D. Y.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Chui, H.

H. Chui, A. Rangarajan, “A new algorithm for non-rigid point matching,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 44–51.

Davatzikos, C.

C. Davatzikos, J. Prince, “Brain image registration based on curve mapping,” in Proceedings of IEEE Workshop Biomedical Image Anal., (IEEE, Los Alamitos, Calif.1994), pp. 245–254.

Denk, W.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Forbat, S. M.

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Fung, Y. C.

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer, New York, 1993).

Galvin, B.

B. McCane, B. Galvin, K. Novins, “On the evaluation of optical flow algorithms,” in Proceedings of the Fifth Int. Conf. Control, Automation, Robotics and Vision (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1563–1567.

Gee, J. C.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

Gilbert, R. J.

R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
[CrossRef] [PubMed]

Goldgof, D.

A. Singh, D. Goldgof, D. Terzopoulos, Deformable Models in Medical Image Analysis, (IEEE Press, Los Alamitos, Calif., 1998).

Handa, R.

R. Handa, H. H. S. Ip, H. Y. Tang, “Computerised analysis of aortic distensibility on computed tomography,” in Proceedings of the Current Perspectives in Healthcare Computing 1997 Conference, Harrogate (1997), pp. 157–165.

Haynor, D. R.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

Hoffman, M.

R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
[CrossRef] [PubMed]

Horn, B. K. P.

B. K. P. Horn, B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

B. K. P. Horn, Robot Vision (MIT, Cambridge, Mass., 1990).

Ip, H. H. S.

R. Handa, H. H. S. Ip, H. Y. Tang, “Computerised analysis of aortic distensibility on computed tomography,” in Proceedings of the Current Perspectives in Healthcare Computing 1997 Conference, Harrogate (1997), pp. 157–165.

Ji, L.

J. Wang, L. Ji, H. Ma, “A combined approach for a quantitative evaluation of aorta-related drugs,” in Proceedings of the 2nd Joint Conf. of the IEEE Medicine and Biology Society and Biomedical Engineering Society, USA, Oct.2002.

Jin, L.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Kass, M.

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
[CrossRef]

Le Brigner, L.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

Lin, X. S.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Ma, H.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

J. Wang, L. Ji, H. Ma, “A combined approach for a quantitative evaluation of aorta-related drugs,” in Proceedings of the 2nd Joint Conf. of the IEEE Medicine and Biology Society and Biomedical Engineering Society, USA, Oct.2002.

Masters, B.

McCane, B.

B. McCane, B. Galvin, K. Novins, “On the evaluation of optical flow algorithms,” in Proceedings of the Fifth Int. Conf. Control, Automation, Robotics and Vision (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1563–1567.

McInerney, T.

T. McInerney, D. Terzopoulos, “Deformable models in medical image analysis: A survey,” Medical Image Analysis 1, 91–108 (1996).
[CrossRef] [PubMed]

Mohiaddin, R. D.

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Novins, K.

B. McCane, B. Galvin, K. Novins, “On the evaluation of optical flow algorithms,” in Proceedings of the Fifth Int. Conf. Control, Automation, Robotics and Vision (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1563–1567.

Prince, J.

C. Davatzikos, J. Prince, “Brain image registration based on curve mapping,” in Proceedings of IEEE Workshop Biomedical Image Anal., (IEEE, Los Alamitos, Calif.1994), pp. 245–254.

Rangarajan, A.

H. Chui, A. Rangarajan, “A new algorithm for non-rigid point matching,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 44–51.

Rueckert, D.

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Schunck, B. G.

B. K. P. Horn, B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Singh, A.

A. Singh, D. Goldgof, D. Terzopoulos, Deformable Models in Medical Image Analysis, (IEEE Press, Los Alamitos, Calif., 1998).

Strickler, J. H.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Sun, F.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Tang, H. Y.

R. Handa, H. H. S. Ip, H. Y. Tang, “Computerised analysis of aortic distensibility on computed tomography,” in Proceedings of the Current Perspectives in Healthcare Computing 1997 Conference, Harrogate (1997), pp. 157–165.

Terzopoulos, D.

T. McInerney, D. Terzopoulos, “Deformable models in medical image analysis: A survey,” Medical Image Analysis 1, 91–108 (1996).
[CrossRef] [PubMed]

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
[CrossRef]

A. Singh, D. Goldgof, D. Terzopoulos, Deformable Models in Medical Image Analysis, (IEEE Press, Los Alamitos, Calif., 1998).

Wang, J.

J. Wang, L. Ji, H. Ma, “A combined approach for a quantitative evaluation of aorta-related drugs,” in Proceedings of the 2nd Joint Conf. of the IEEE Medicine and Biology Society and Biomedical Engineering Society, USA, Oct.2002.

Webb, W. W.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Witkin, A.

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
[CrossRef]

Yang, G. Z.

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Zhao, J. B.

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Artif. Intell.

B. K. P. Horn, B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

IEEE Trans. Med. Imaging

D. Rueckert, P. Burger, S. M. Forbat, R. D. Mohiaddin, G. Z. Yang, “Automatic tracking of the aorta in cardiovascular mr images using deformable models,” IEEE Trans. Med. Imaging 16, 581–590 (1997).
[CrossRef] [PubMed]

Int. J. Comput. Vision

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 2, 321–331 (1988).
[CrossRef]

Medical Image Analysis

T. McInerney, D. Terzopoulos, “Deformable models in medical image analysis: A survey,” Medical Image Analysis 1, 91–108 (1996).
[CrossRef] [PubMed]

Microsc. Res. Tech.

R. J. Gilbert, M. Hoffman, A. Capitano, “Imaging of three-dimensional epithelial architecture and function in cultured caco2a monolayers with two-photon excitation microscopy,” Microsc. Res. Tech. 51, 204–210 (2000).
[CrossRef] [PubMed]

Opt. Express

Pigm. Cell.

D. Arkas, D. Becker, “Applications of spectral imaging: detection and analysis of human melanoma and its precursors,” Pigm. Cell. 14, 2–8 (2001).
[CrossRef]

Proc. SPIE

X. S. Lin, F. Sun, H. Ma, J. B. Zhao, L. Jin, D. Y. Chen, “Two-photon fluorescence imaging of rat aorta,” Proc. SPIE 4224, 7–12 (2000).
[CrossRef]

Science

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Other

A. Singh, D. Goldgof, D. Terzopoulos, Deformable Models in Medical Image Analysis, (IEEE Press, Los Alamitos, Calif., 1998).

R. Handa, H. H. S. Ip, H. Y. Tang, “Computerised analysis of aortic distensibility on computed tomography,” in Proceedings of the Current Perspectives in Healthcare Computing 1997 Conference, Harrogate (1997), pp. 157–165.

C. Davatzikos, J. Prince, “Brain image registration based on curve mapping,” in Proceedings of IEEE Workshop Biomedical Image Anal., (IEEE, Los Alamitos, Calif.1994), pp. 245–254.

J. C. Gee, L. Le Brigner, C. Barillot, D. R. Haynor, R. K. Bajcsy, “Bayesian approach to the brain image matching problem,” in Medical Imaging 1995: Image Processing, M. H. Loew, ed., Proc. SPIE2434, 145–156 (1995).
[CrossRef]

H. Chui, A. Rangarajan, “A new algorithm for non-rigid point matching,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 44–51.

R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).

J. Wang, L. Ji, H. Ma, “A combined approach for a quantitative evaluation of aorta-related drugs,” in Proceedings of the 2nd Joint Conf. of the IEEE Medicine and Biology Society and Biomedical Engineering Society, USA, Oct.2002.

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer, New York, 1993).

B. K. P. Horn, Robot Vision (MIT, Cambridge, Mass., 1990).

B. Galvin, B. McCane, K. Novins, D. Mason, S. Mills, “Recovering motion fields: An evaluation of eight optical flow algorithms,” in Proceedings of the Ninth British Machine Vision Conference, P. H. Lewis, M. S. Nixon, eds. (ECS Publications Database, University of Southampton, UK). (1998). http://www.eprints.org .

B. McCane, B. Galvin, K. Novins, “On the evaluation of optical flow algorithms,” in Proceedings of the Fifth Int. Conf. Control, Automation, Robotics and Vision (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1563–1567.

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Figures (10)

Fig. 1
Fig. 1

Illustration of the problem addressed in this paper: (a) part of the aorta body is not in the focal plane of the two-photon fluorescence microscopy due to deformation, (b) Sample image of deforming aorta with information loss.

Fig. 2
Fig. 2

(a) Illustration of the active contour-based aorta model, (b) Model fitted to a sample frame. Brighter pieces of the boundary indicate higher degrees of confidence, while darker pieces have lower degrees of confidence.

Fig. 3
Fig. 3

Framework of the landmark registration and reconstruction algorithm.

Fig. 4
Fig. 4

Examples of optical flow interpolation: (a) optical flow interpolation for a rigid object, (b) optical flow interpolation for a deformable object. From left-hand side to right-hand side: One source image, the other source image, the interpolated image. Note that optical flow interpolation can get a satisfying result on a rigid object, but its performance on an intensively deforming object is poor.

Fig. 5
Fig. 5

Framework of the proposed landmark-based optical flow interpolation scheme.

Fig. 6
Fig. 6

Mismatch error of the extracted (L e ) and the reconstructed (Lrgl) landmarks is plotted against time.

Fig. 7
Fig. 7

Landmark reconstruction results of an example frame: (a) Aorta image with information loss, (b) illustration of landmarks in different stages of processing, (c) illustrating the landmarks in detail.

Fig. 8
Fig. 8

Two groups of interpolation results of aorta wall pieces: (a) one piece of aorta at time t p , (b) same aorta piece at time t f (t p < t f ), (c) interpolated image of optical flow interpolation, (d) interpolated image of the proposed landmark-based optical flow interpolation, (e) ground-truth image. Differences can be seen between (c)s and (e)s.

Fig. 9
Fig. 9

Comparison of different interpolation schemes on a whole aorta wall: (a) Interpolated image of a global optical flow interpolation, (b) interpolated image of local optical flow interpolation, (c) interpolated image of the landmark-based optical flow interpolation, (d) ground-truth image. The greatest differences between (b), (c), and (d) are in the regions that are missing from (a).

Fig. 10
Fig. 10

Perimeters of the aorta wall calculated from the extracted, reconstructed, and ground-truth landmarks are plotted against time.

Equations (22)

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

Dli=c 1Aφipφi Ip,
Dvi=maxDli, Dli+1.
Eg=k=2NIj=1n Wjkvjk-Tk-vjk-12,
Wjk=0:minDvik-1, DvikDtminDvik-1, Dvik:minDvik-1, Dvik>Dt,
T*T2*, T3*,, TNI*=arg minT Eg.
Etvik=wtvikvik-vik-1-vik-1-vik-2,
wtvik=1-DvikDvik-1Dvik-2.
Esvik=wsvik(vi1kvi1k1)+(vi+1kvi+1k1)2(vikvik1),
wsvik=1-DvikDvi-1kDvi+1k.
Emotion=k=3NIi=1nEtvik+Esvik,
Flij=α|j-k|Dlij, j=1,, NI,
FlipFlij, j=k-1,, 1,FlifFlij, j=k+1,, NI.
Pifϕj=Pipj+uj, j=1,, Nlif
Emlip, lif=j=2Nlip Pipj-Pifϕj-Pipj-1-Pifϕj-12.
Φ*ϕ*1, ϕ*2,, ϕ*Nlip=arg minΦ Emlip, lif.
Emj=limϕj-1(Emj-1+Pipj-Pifϕj-Pipj-1-Pifϕj-12.
Pikj=Pipjδtfδtp+δtf+Pifϕjδtpδtp+δtf.
Ex+uδt, y+vδt, t+δt=Ex, y, t.
Exu+Eyv+Et=0,
u2=ux+ uy2=0, v2=vx+ vy2=0.
εx, y=xyExu+Eyv+Et2+λ2u2+v2,
El= 1Npi=1Np pri-poi2,

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