[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Translation, Scale, and Deformation Weighted Polar Active Contours

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Polar active contours have proven to be a powerful segmentation method for many medical as well as other computer vision applications, such as interactive image segmentation or tracking. Inspired by recent work on Sobolev active contours we derive a Sobolev-type function space for polar curves, which is endowed with a metric that allows us to favor origin translations and scale changes over smooth deformations of the curve. The resulting translation, scale, and deformation weighted polar active contours inherit the coarse-to-fine behavior of Sobolev active contours as well as their robustness to local minima and are thus very useful for many medical applications, such as cross-sectional vessel segmentation, aneurysm analysis, or cell tracking.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. This article is an extended version of our previous conference paper [2].

References

  1. Baust, M., Navab, N.: A spherical harmonics shape model for level set segmentation. In: 11th European Conference on Computer Vision (ECCV) (2010)

    Google Scholar 

  2. Baust, M., Yezzi, A., Unal, G., Navab, N.: A Sobolev-type metric for polar active contours. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2011)

    Google Scholar 

  3. Bodur, O., Grady, L., Stillman, A., Setser, R., Funka-Lea, G., O’Donnell, T.: Semi-automatic aortic aneurysm analysis. In: SPIE Medical Imaging (2007)

    Google Scholar 

  4. Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007)

    Article  MathSciNet  Google Scholar 

  5. Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  6. Charpiat, G., Maurel, P., Pons, J.P., Keriven, R., Faugeras, O.: Generalized gradients: Priors on minimization flows. Int. J. Comput. Vis. 73(3), 325–344 (2007)

    Article  Google Scholar 

  7. Collewet, C.: Polar snakes: a fast and robust parametric active contour model. In: 16th IEEE International Conference on Image Processing (ICIP), pp. 3013–3016 (2009)

    Chapter  Google Scholar 

  8. Denzler, J., Niemann, H.: Active rays: polar-transformed active contours for real-time contour tracking. Real-Time Imaging 5, 203–213 (1999)

    Article  Google Scholar 

  9. Dias, J., Leitao, J.: Wall position and thickness estimation from sequences of echocardiographic images. IEEE Trans. Med. Imaging 15(1), 25–38 (1996)

    Article  Google Scholar 

  10. Floreby, L.: A multiscale algorithm for closed contour matching in image sequence. In: Proceedings of the 13th International Conference on Pattern Recognition, pp. 884–888, vol. 1 (1996)

    Chapter  Google Scholar 

  11. Friedland, N., Adam, D.: Automatic ventricular cavity boundary detection from sequential ultrasound images using simulated annealing. IEEE Trans. Med. Imaging 8(4), 344–353 (1989)

    Article  Google Scholar 

  12. Hu, M., Ping, X., Ding, Y.: A new active contour model and its application on cell segmentation. In: 8th Control, Automation, Robotics and Vision Conference (ICARCV 2004), pp. 1104–1107, vol. 2 (2004)

    Chapter  Google Scholar 

  13. Lax, P.: Functional Analysis. Wiley-Interscience, New York (2002)

    MATH  Google Scholar 

  14. Liang, J., Ding, G., Wu, Y.: Segmentation of the left ventricle from cardiac mr images based on radial gvf snake. In: International Conference on BioMedical Engineering and Informatics (BMEI 2008), vol. 2, pp. 238–242 (2008)

    Chapter  Google Scholar 

  15. Martín-Fernández, M., Alberola-López, C.: An approach for contour detection of human kidneys from ultrasound images using Markov random fields and active contours. Med. Image Anal. 9(1), 1–23 (2005)

    Article  Google Scholar 

  16. Michor, P., Mumford, D.: Riemannian geometries on the space of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Radau, P.: Evaluation of cardiac mr segmentation (2010). http://sourceforge.net/projects/cardiac-mr/files/

  18. Ravhon, R., Adam, D., Zelmanovitch, L.: Validation of ultrasonic image boundary recognition in abdominal aortic aneurysm. IEEE Trans. Med. Imaging (2001)

  19. Ray, N., Acton, S.: Motion gradient vector flow: an external force for tracking rolling leukocytes with shape and size constrained active contours. IEEE Trans. Med. Imaging 23(12), 1466–1478 (2004)

    Article  Google Scholar 

  20. Sacan, A., Ferhatosmanoglu, H., Coskun, H.: Celltrack: an open-source software for cell tracking and motility analysis. Bioinformatics 24(14), 1647–1649 (2008)

    Article  Google Scholar 

  21. Schudy, R.R.: Harmonic surfaces and parametric image operators: Their use in locating the moving endocardial surface from threedimensional cardiac ultrasound data. Tech. rep University of Rochester (1981)

  22. Sundaramoorthi, G., Mennucci, A., Soatto, S., Yezzi, A.J.: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM J. Imaging Sci. 4(1), 109–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Coarse-to-fine segmentation and tracking using Sobolev active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(5), 851–864 (2008)

    Article  Google Scholar 

  24. Sundaramoorthi, G., Yezzi, A., Mennucci, A., Sapiro, G.: New possibilities with Sobolev active contours. Int. J. Comput. Vis. (2008)

  25. Sundaramoorthi, G., Yezzi, A.J., Mennucci, A.: Sobolev active contours. Int. J. Comput. Vis. 73(3), 345–366 (2007)

    Article  Google Scholar 

  26. Teles de Figueiredo, M., Leitaa, J.: Bayesian estimation of ventricular contours in angiographic images. IEEE Trans. Med. Imaging 11(3), 416–429 (1992)

    Article  Google Scholar 

  27. Veksler, O.: Star shape prior for graph-cut image segmentation. In: ECCV (2008)

    Google Scholar 

  28. Yezzi, A., Mennucci, A.: Conformal metrics and true “gradient flows” for curves. In: Tenth IEEE International Conference on Computer Vision (ICCV 2005), vol. 1, pp. 913–919 (2005)

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Darko Zikic for many fruitful discussions and his suggestions on the preparation of the manuscript.

The first author is fully supported by the international graduate school of science and engineering (IGSSE) at Technische Universität München (TUM).

Author information

Authors and Affiliations

  1. Computer Aided Medical Procedures and Augmented Reality, Technische Universität München, 85748, Munich, Germany

    Maximilian Baust & Nassir Navab

  2. School of Electrical and Computer Engineering, Georgia Institue of Technology, Atlanta, GA, 30332, USA

    Anthony Yezzi

  3. Faculty of Engineering and Natural Sciences, Sabanci University, Instanbul, Turkey

    Gozde Unal

Authors
  1. Maximilian Baust
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anthony Yezzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gozde Unal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nassir Navab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maximilian Baust.

Appendices

Appendix A: Solving the ODE

The computation of the gradient \(\nabla _{\!E}\mathcal{H}\) requires the solution of the following ordinary differential equation, cf. also [25]:

$$g''(s) = \frac{1}{\lambda L^2}\bigl(f(s)-\bar{f}\bigr),\quad g(0)=g(L),\quad g'(0)=g'(L),$$
(34)

where we set g=k r and f=h r in order to keep the notation simple. Further, we denote the differentiation with respect to s is by ⋅′. At first, we integrate twice and obtain:

$$g(s) = g(0) + sg'(0) + \frac{1}{\lambda L^2}\int^s_0 \int^{\hat{s}}_0 f(\xi)-\bar{f}\,d\xi\,d\hat{s}.$$
(35)

Second, we perform integration by parts for the rightmost term:

(36)
(37)
(38)
(39)

Thus we end up with

$$g(s) = g(0) + sg'(0) + \frac{1}{\lambda L^2}\int^s_0(s-\hat{s})\bigl(f(\hat{s})-\bar{f}\bigr)\,d\hat{s}. $$
(40)

Now we set s=L and use the boundary condition g(0)=g(L) and obtain:

(41)
(42)
(43)

due to the definition of \(\bar{f}\). Changing \(\hat{s}\) to s eventually yields

$$g'(0) = -\frac{1}{\lambda L^3} \int^L_0s\bigl(f(s)-\bar{f}\bigr)\,ds.$$
(44)

Plugging this formula into (40), integrating both sides form 0 to L, and using again that \(\bar{g}=\bar{f}\) we finally obtain

(45)

where K γ (s) is the kernel defined in (26). Since the definition of the arc length allows us to choose the starting point arbitrarily we can conclude from (45) that g may also be obtained by g=K γ f.

Appendix B: Deriving a First Order Correction for ϕ

Now we want to derive a first-order accurate correction ψ such that \(\hat{\phi}(x)=\psi(x)\phi(x)\), where ϕ defined in (32), is close to a signed distance representation of c, which we denote by Φ(x). At first, we note that the level lines of a signed distance representation of c are solutions to c t =n. Supposing that ϵ∈ℝ is sufficiently small one might say that the level line L ϵ ={x∈Ω:Φ(x)=ϵ} can be approximated by evolving c with a forward Euler discretization of the flow c t =n:

$$c_{0+\tau} = c_0+\tau \mathbf {n},$$
(46)

where c 0=c and τ=ϵ—see also sketch below:

As we are only interested in the geometry of c we can add a tangential component to the flow c t =n:

$$c_t = \mathbf {n}+ \frac{\mathbf {s}\cdot \mathbf {t}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {t}= \frac{\mathbf {s}\cdot \mathbf {n}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {n}+\frac{\mathbf {s}\cdot \mathbf {t}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {t}=\frac{1}{\mathbf {s}\cdot \mathbf {n}}\mathbf {s},$$
(47)

since t and n are an orthonormal basis. From this we may conclude that the c t =n and c t =1/(s⋅n)s yield the same geometric curve. This means that the level line L ϵ can also be approximated by performing a forward Euler step of the flow c t =1/(s⋅n)s:

$$c_{0+\tau}=c_0+\frac{\tau}{ \mathbf {s}\cdot \mathbf {n}}\mathbf {s},$$
(48)

where again c 0=c and τ=ϵ. As ϕ(x) gives us the signed distance ϵ/(s⋅n) we thus know that the correction ψ is given by

$$\psi(x)=\mathbf {s}\bigl(\theta(x)\bigr) \cdot \mathbf {n}\bigl(\theta(x)\bigr).$$
(49)

Of course, this correction is only first-order accurate in ϵ, because we derived it via a forward Euler discretization.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baust, M., Yezzi, A., Unal, G. et al. Translation, Scale, and Deformation Weighted Polar Active Contours. J Math Imaging Vis 44, 354–365 (2012). https://doi.org/10.1007/s10851-012-0331-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-012-0331-5

Keywords

Search

Navigation