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3D Edge Detection by Selection of Level Surface Patches

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Abstract

We propose a new edge detector for 3D gray-scale images, extending the 2D edge detector of Desolneux et al. (J. Math. Imaging Vis. 14(3):271–284, 2001). While the edges of a planar image are pieces of curve, the edges of a volumetric image are pieces of surface, which are more delicate to manage. The proposed edge detector works by selecting those pieces of level surface which are well-contrasted according to a statistical test, called Helmholtz principle. As it is infeasible to treat all the possible pieces of each level surface, we restrict the search to the regions that result of optimizing the Mumford-Shah functional of the gradient over the surface, throughout all scales. We assert that this selection device results in a good edge detector for a wide class of images, including several types of medical images from X-ray computed tomography and magnetic resonance.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  2. Alvarez, L., Gousseau, Y., Morel, J.-M.: The size of objects in natural and artificial images. Adv. Imaging Electron. Phys. 111, 167–242 (1999)

    Google Scholar 

  3. Ayache, N., Faverjon, B.: Efficient registration of stereo images by matching graph descriptions of edge segments. Int. J. Comput. Vis. 1(2), 107–131 (1987)

    Article  Google Scholar 

  4. Ballester, C., Caselles, V., Monasse, P.: The tree of shapes of an image. ESAIM: Control, Optim. Calc. Var. 9, 1–18 (2003)

    MATH  MathSciNet  Google Scholar 

  5. Banchoff, T.: Critical points and curvature for embedded polyhedra. J. Differ. Geom. 1, 245–256 (1967)

    MATH  MathSciNet  Google Scholar 

  6. Beucher, S., Lantuejoul, C.: Use of watersheds in contour detection. In: Proc. Int. Workshop Image Processing (1979)

  7. Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proc. 9th Int. Conf. Comput. Vis., pp. 26–33 (2003)

  8. Brown, L.G.: A survey of image registration techniques. ACM Comput. Surv. 24, 325–376 (1992)

    Article  Google Scholar 

  9. Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 679–698 (1986)

    Article  Google Scholar 

  10. Cao, F., Musé, P., Sur, F.: Extracting meaningful curves from images. J. Math. Imaging Vis. 22(2), 159–181 (2005)

    Article  Google Scholar 

  11. Carr, H., Duffy, B., Denby, B.: On histograms and isosurface statistics. IEEE Trans. Vis. Comput. Graph. 12(5), 1259–1266 (2006)

    Article  Google Scholar 

  12. Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom.: Theory Appl. 24(2), 75–94 (2003)

    MATH  MathSciNet  Google Scholar 

  13. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)

    Article  MATH  Google Scholar 

  14. Caselles, V., Meinhardt, E., Monasse, P.: Constructing the tree of shapes of an image by fusion of the trees of connected components of upper and lower level sets. Positivity 12(1), 55–73 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Clark, J.H.: Hierarchical geometric models for visible surface algorithms. Commun. ACM 19(10), 547–554 (1976)

    Article  MATH  Google Scholar 

  16. Cohen, I., Cohen, L.D., Ayache, N.: Using deformable surfaces to segment 3-D images and infer differential structures. CVGIP: Image Underst. 56(2), 242–263 (1992)

    Article  MATH  Google Scholar 

  17. Cox, J., Karron, D.B., Ferdous, N.: Topological zone organization of scalar volume data. J. Math. Imaging Vis. 18(2), 95–117 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Davis, L.S.: A survey of edge detection techniques. Comput. Graph. Image Process. 4, 248–270 (1975)

    Article  Google Scholar 

  19. Deriche, R.: Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Comput. Vis. 1(2), 167–187 (1987)

    Article  Google Scholar 

  20. Desolneux, A., Moisan, L., Morel, J.-M.: Edge detection by Helmholtz principle. J. Math. Imaging Vis. 14(3), 271–284 (2001)

    Article  MATH  Google Scholar 

  21. Desolneux, A., Moisan, L., Morel, J.-M.: Gestalt theory and computer vision. In: Seeing, Thinking and Knowing, pp. 71–101. Springer, Berlin (2004)

    Chapter  Google Scholar 

  22. Desolneux, A., Moisan, L., Morel, J.M.: Meaningful alignments. Int. J. Comput. Vis. 40(1), 7–23 (2000)

    Article  MATH  Google Scholar 

  23. Desolneux, A., Moisan, L., Morel, J.M.: Variational Snake Theory. Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, Berlin (2003)

    Google Scholar 

  24. Forsyth, D.A., Ponce, J.: Computer Vision: A Modern Approach. Prentice Hall, New York (2003)

    Google Scholar 

  25. Galassi, M. et al.: GNU Scientific Library. Network Theory Ltd. (2002)

  26. Grimson, W.E.L., Huttenlocher, D.P.: On the verification of hypothesized matches in model-based recognition. IEEE Trans. Pattern Anal. Mach. Intell. 13(12), 1201–1213 (1991)

    Article  Google Scholar 

  27. Haralick, R.M.: Digital step edges from zero crossing of second directional derivatives. IEEE Trans. Pattern Anal. Mach. Intell. 6(1), 58–68 (1984)

    Article  Google Scholar 

  28. Haralick, R.M., Shapiro, L.G.: Image segmentation techniques. Comput. Vis. Graph. Image Process. 29, 100–132 (1985)

    Article  Google Scholar 

  29. Haralick, R.M., Watson, L.T., Laffey, T.J.: The topographic primal sketch. Int. J. Robot. Res. 2(1), 50–72 (1983)

    Article  Google Scholar 

  30. Hernandez, M., Frangi, A.F.: Non-parametric geodesic active regions: Method and evaluation for cerebral aneurysms segmentation in 3DRA and CTA. Med. Image Anal. 11(3), 224–241 (2007)

    Article  Google Scholar 

  31. Hsieh, J.W., Liao, H.Y.M., Fan, K.C., Ko, M.T., Hung, Y.P.: Image registration using a new edge-based approach. Comput. Vis. Image Underst. 67(2), 112–130 (1997)

    Article  Google Scholar 

  32. Hsu, L.Y., Loew, M.H., Ostuni, J.: Automated registration of brain images using edge and surface features. IEEE Eng. Med. Biol. Mag. 18(6), 40–47 (1999)

    Article  Google Scholar 

  33. Julesz, B.: A method of coding TV signals based on edge detection. Bell Syst. Technol. 38(4), 1001–1020 (1959)

    Google Scholar 

  34. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)

    Article  Google Scholar 

  35. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: From phase transitions to active vision. Arch. Ration. Mech. Anal. 134(3), 275–301 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kirsch, R.A.: Computer determination of the constituent structure of biological images. Comput. Biomed. Res. 4(3), 315–328 (1971)

    Article  Google Scholar 

  37. Koepfler, G., Lopez, C., Morel, J.-M.: A multiscale algorithm for image segmentation by variational method. SIAM J. Numer. Anal. 31(1), 282–299 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lisani, J.L., Moisan, L., Monasse, P., Morel, J.-M.: On the theory of planar shape. Multiscale Model. Simul. 1, 1 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Lorensen, W.E., Cline, H.E.: Marching cubes: A high resolution 3D surface construction algorithm. In: Proc. 14th Annu. Conf. Comput. Graph. Interact. Tech., pp. 163–169 (1987)

  40. Lowe, D.G.: Perceptual Organization and Visual Recognition. Kluwer Academic, Dordrecht (1985)

    Google Scholar 

  41. Malladi, R., Sethian, J.A., Vemuri, B.C.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)

    Article  Google Scholar 

  42. Marr, D., Hildreth, E.: Theory of edge detection. Proc. R. Soc. Lond. 207(1167), 187–217 (1980)

    Article  Google Scholar 

  43. Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. Vis. Math. 3, 35–57 (2002)

    Google Scholar 

  44. Mokhtarian, F., Abbasi, S., Kittler, J.: Robust and efficient shape indexing through curvature scale space. In: Proc. Br. Mach. Vis. Conf., pp. 53–62 (1996)

  45. Monasse, P.: Contrast invariant representation of digital images and application to registration. PhD thesis, Université Paris IX-Dauphine, June 2000

  46. Monasse, P., Guichard, F.: Fast computation of a contrast-invariant image representation. IEEE Trans. Image Process. 9, 860–872 (2000)

    Article  Google Scholar 

  47. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1988)

    Article  MathSciNet  Google Scholar 

  48. Pascucci, V., Cole-McLaughlin, K.: Parallel computation of the topology of level sets. Algorithmica 38(2), 249–268 (2003)

    Article  MathSciNet  Google Scholar 

  49. Pielot, R., Scholz, M., Obermayer, K., Gundelfinger, E.D., Hess, A.: 3D edge detection to define landmarks for point-based warping in brain imaging. In: Proc. Int. Workshop Image Process., vol. 2 (2001)

  50. Prewitt, J.: Object enhancement and extraction. In: Picture Processing and Psychopictorics, pp. 75–149. Academic Press, New York (1970)

    Google Scholar 

  51. Roberts, L.: Machine perception of 3D solids. In: Opt. Electro-Opt. Inf. Process., pp. 159–197 (1965)

  52. Sarioz, D., Kong, T.Y., Herman, G.T.: History trees as descriptors of macromolecular structures. In: Lect. Notes Comput. Sci., vol. 4291, p. 263. Springer, Berlin (2006)

    Google Scholar 

  53. Sijbers, J., Scheunders, P., Verhoye, M., Van der Linden, A., Van Dyck, D., Raman, E.: Watershed-based segmentation of 3D MR data for volume quantization. Magn. Reson. Imaging 15, 679–688 (1997)

    Article  Google Scholar 

  54. Song, Y., Zhang, A.: Monotonic tree. In: Proc. 10th Int. Conf. Discrete Geom. Comput. Imagery (2002)

  55. Vincent, L., Soille, P.: Watersheds in digital spaces: an efficient algorithm based onimmersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991)

    Article  Google Scholar 

  56. Zhao, H.K., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: 1st IEEE Workshop Var. Lev. Set Methods, vol. 80(3), pp. 194–202 (2001)

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Authors and Affiliations

  1. Department of Information and Communication Technologies, Universitat Pompeu Fabra, Passeig de Circumval⋅lació 8, 08003, Barcelona, Spain

    Enric Meinhardt & Vicent Caselles

  2. GTC, Aragon Institute for Engineering Research (I3A), Universidad de Zaragoza, María de Luna 3, 50018, Zaragoza, Spain

    Ernesto Zacur

  3. Center for Computational Imaging & Simulation Technologies in Biomedicine, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain

    Alejandro F. Frangi

  4. Networking Research Center on Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN), Barcelona, Spain

    Alejandro F. Frangi

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  1. Enric Meinhardt
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  2. Ernesto Zacur
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  3. Alejandro F. Frangi
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  4. Vicent Caselles
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Correspondence to Enric Meinhardt.

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Meinhardt, E., Zacur, E., Frangi, A.F. et al. 3D Edge Detection by Selection of Level Surface Patches. J Math Imaging Vis 34, 1–16 (2009). https://doi.org/10.1007/s10851-008-0118-x

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