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Homotopy-based surface reconstruction with application to acoustic signals

Abstract

This work introduces a new algorithm for surface reconstruction in ℝ3 from spatially arranged one-dimensional cross sections embedded in ℝ3. This is generally the case with acoustic signals that pierce an object non-destructively. Continuous deformations (homotopies) that smoothly reconstruct information between any pair of successive cross sections are derived. The zero level set of the resulting homotopy field generates the desired surface. Four types of homotopies are suggested that are well suited to generate a smooth surface. We also provide derivation of necessary higher order homotopies that can generate a C 2 surface. An algorithm to generate surface from acoustic sonar signals is presented with results. Reconstruction accuracies of the homotopies are compared by means of simulations performed on basic geometric primitives.

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References

  1. 1.

    Keppel, E.: Approximating complex surfaces by triangulation of contour lines. IBM J. Res. Dev. 19(1), 2–11 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Fuchs, H., Kedem, Z.M., Uselton, S.P.: Optimal surface reconstruction from planar contours. Commun. ACM 20(10), 693–702 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Boissonnat, J.D.: Shape reconstruction from planar cross sections. Comput. Vis. Graph. Image Process. 44(1), 1–29 (1988)

    Article  Google Scholar 

  4. 4.

    Bajaj, C.L., Coyle, E.J., Lin, K.N.: Arbitrary topology shape reconstruction from planar cross sections. Graph. Models Image Process. 58(6), 524–543 (1996)

    Article  Google Scholar 

  5. 5.

    Boissonnat, J.D., Memari, P.: Shape reconstruction from unorganized cross sections. In: Proceedings of the Fifth Eurographics Symposium on Geometry Processing, p. 98. Eurographics Association (2007)

  6. 6.

    Liu, L., Bajaj, C., Deasy, J.O., Low, D.A., Ju, T.: Surface reconstruction from non-parallel curve networks. In: Computer Graphics Forum, vol. 27, p. 155. (2008)

    Google Scholar 

  7. 7.

    Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. Comput. Graph. 26(2), 71–78 (1992)

    Article  Google Scholar 

  8. 8.

    Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pages, p. 76. ACM, New York (2001)

    Google Scholar 

  9. 9.

    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discrete Comput. Geom. 22(4), 481–504 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Zhang, Y., Rohling, R., Pai, D.K.: Direct surface extraction from 3D Freehand ultrasound images. In: IEEE Visualization, pp. 45–52. (2002)

    Google Scholar 

  11. 11.

    Allgower, E.L., Georg, K.: Numerical continuation methods: an introduction. Springer, New York (1990)

    MATH  Google Scholar 

  12. 12.

    Shinagawa, Y., Kunii, T.L.: The homotopy model: a generalized model for smooth surface generation from cross-sectional data. Vis. Comput. 7(2), 72–86 (1991)

    Article  Google Scholar 

  13. 13.

    Berzin, D., Hagiwara, I.: Minimal area for surface reconstruction from cross sections. Vis. Comput. 18(7), 437–444 (2002)

    Article  Google Scholar 

  14. 14.

    Fujimura, K., Kuo, E.: Shape reconstruction from contours using isotopic deformation. Graph. Models Image Process. 61(3), 127–147 (1999)

    Article  Google Scholar 

  15. 15.

    Armstrong, M.A.: Basic Topology. Springer, Berlin (1983)

    MATH  Google Scholar 

  16. 16.

    Ona, E., Andersen, L.N., Knudsen, H.P., Berg, S.: Calibrating multibeam, wideband sonar with reference targets. In: OCEANS 2007-Europe, pp. 1–5 (2007)

    Chapter  Google Scholar 

  17. 17.

    Hyman, J.M.: Accurate monotonicity preserving cubic interpolation. SIAM J. Sci. Stat. Comput. 4(4), 645–654 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Wolberg, G., Alfy, I.: Monotonic cubic spline interpolation. In: Proceedings of Computer Graphics International, 1999, pp. 188–195. (1999)

    Chapter  Google Scholar 

  19. 19.

    Späth, H.: Exponential spline interpolation. Computing 4(3), 225–233 (1969)

    Article  MATH  Google Scholar 

  20. 20.

    Pruess, S.: An algorithm for computing smoothing splines in tension. Computing 19(4), 365–373 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Beros, I., Marusic, M.: Evaluation of tension splines. Math Commun 4, 73–81 (1999)

    MATH  MathSciNet  Google Scholar 

  22. 22.

    de Boor, C., Swartz, B.: Piecewise monotone interpolation. J. Approx. Theory 21(4), 411–416 (1977)

    Article  MATH  Google Scholar 

  23. 23.

    Costantini, P., Morandi, R.: Monotone and convex cubic spline interpolation. Calcolo 21(3), 281–294 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Simmonds, E.J., MacLennan, D.N.: Fisheries Acoustics: Theory and Practice. Blackwell, Oxford (2005)

    Book  Google Scholar 

  25. 25.

    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  26. 26.

    Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, New York (2003)

    MATH  Google Scholar 

  27. 27.

    Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2 edn. Prentice Hall, New York (2002)

    Google Scholar 

  28. 28.

    Aspert, N., Santa-Cruz, D., Ebrahimi, T.: Mesh: Measuring errors between surfaces using the Hausdorff distance. In: Proceedings of the IEEE International Conference on Multimedia and Expo, vol. 1, pp. 705–708. (2002)

    Chapter  Google Scholar 

  29. 29.

    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Ojaswa Sharma.

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Sharma, O., Anton, F. Homotopy-based surface reconstruction with application to acoustic signals. Vis Comput 27, 373–386 (2011). https://doi.org/10.1007/s00371-011-0544-4

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Keywords

  • Homotopy
  • Continuous deformations
  • Surface reconstruction
  • Shape preserving
  • Acoustic signal
  • Sonar