Digital Geometry and Its Applications to Medical Imaging

  • Reneta P. Barneva
  • Valentin E. Brimkov
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 13)

Digital geometry is a modern discipline dealing with geometric properties of digital objects (also called digital pictures). These are usually modeled as sets of points with integer coordinates representing the pixels/voxels of the considered objects. Digital geometry is developed with the expectation that it would provide an adequate mathematical background for new advanced approaches and algorithms for various problems arising in image analysis and processing, computer graphics, medical imaging, and other areas of visual computing. In this chapter we first provide a brief discussion on the motivation, basic directions, and achievements of digital geometry. Then we consider typical examples of research problems and their solutions. We focus our attention on problems related to digital manifolds. The latter play an important role in computer graphics, 3D image analysis, volume modeling, process visualization, and so forth — in short, in all areas where discrete multidimensional data need to be represented, visualized, processed, or analyzed. The objects in these areas often represent surfaces and volumes of real objects. We discuss some applications of digital curves and surfaces to medical imaging, implied by theoretical results on digital manifolds.

Keywords

Manifold Assure Hull Agglomeration Betti 

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References

  1. 1.
    Basu S (2006) Journal of Symbolic Computation 41(10):1125–1154MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Basu S, Pollack R, Roy M (2005) Computing the first Betti number and describing the connected components of semi-algebraic sets. In: Proc. STOC'05Google Scholar
  3. 3.
    Bertrand G, Malgouyres R (1999) Journal of Mathematical Imaging Vision 11:207–221MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brimkov VE (2006) Discrete volume polyhedrization: complexity and bounds on performance. In: Tavares JM et al. (eds) CompIMAGE — Computational Modelling of Objects Represented in Images: Fundamentals, Methods and Applications. Taylor & Francis, London, Leiden, New York, Philadelphia, SingaporeGoogle Scholar
  5. 5.
    Brimkov VE, Andres E, Barneva RP (2002) Pattern Recognition Letters 23:623–636MATHCrossRefGoogle Scholar
  6. 6.
    Brimkov VE, Coeurjolly D, Klette R (2007) Discrete Applied Mathematics 155:468–495MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brimkov VE, Klette R (2004) Curves, hypersurfaces, and good pairs of adjacency relations. In: Proc. Int. Workshop Combinatorial Image Analysis, LNCS 3322. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  8. 8.
    Brimkov VE, Moroni D, Barneva R (2006) Combinatorial relations for digital pictures. In: Kuba, A et al. (eds), Discrete Geometry for Computer Imagery, LNCS 4245. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  9. 9.
    Chen L (2004) Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology. Scientific & Practical Computing, RockvilleGoogle Scholar
  10. 10.
    Chen L (2005) Gradually varied surfaces and gradually varied functions. CITR-TR 156. The University of Auckland, AucklandGoogle Scholar
  11. 11.
    Chen L, Cooley DH, Zhang J (1999) Information Sciences 115:201–220MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen L, Zhang J (1993) Digital manifolds: an intuitive definition and some properties. In: Proc. Symp. Solid Modeling Applications, ACM/SIGGRAPHGoogle Scholar
  13. 13.
    CHomP (Atlanta) & CAPD (Krak ó w). Homology algorithms and software. www.math.gatech.edu/chomp/homology/
  14. 14.
    Coeurjolly D, Guillaume A, Sivignon I (2004) Reversible discrete volume polyhedrization using Marching Cubes simplification. In: Proc. Vision Geometry XII, SPIE 5300Google Scholar
  15. 15.
    Cohen-Or D, Kaufman A, Kong TY (1996) On the soundness of surface voxelizations. In: Kong TY, Rosenfeld A (eds), Topological Algorithms for Digital Image Processing, Elsevier, AmsterdamGoogle Scholar
  16. 16.
    Daragon X, Couprie M, Bertrand G (2005) Journal of Mathematical Imaging and Vision, 23(3):379–399CrossRefMathSciNetGoogle Scholar
  17. 17.
    Desbrun M, Kanso E, Kong Y (2005) Discrete differential forms for computational modeling. In: ACM SIGGRAPH 2005 Course Notes on Discrete Differential Geometry, Chapter 7Google Scholar
  18. 18.
    De Silva, Plex V - A Mathlab library for studying simplicial homology. math.stanford.edu/comptop/programs/plex/plexintro.pdfGoogle Scholar
  19. 19.
    Fourey S, Malgouyres R (2002) Discrete Applied Mathematics 125:59–80CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kaczynski T, Mischaikow K, Mrozek M (2004) Applied Mathematical Sciences, Vol. 157. Springer, BerlinGoogle Scholar
  21. 21.
    Kaufman A (1987) An algorithm for 3D scan-conversion of polygons. In: Proc. EurographicsGoogle Scholar
  22. 22.
    Kaufman A (1993) Volume Graphics 26(7):51–64Google Scholar
  23. 23.
    Kaufman A (1987) Computer Graphics 21(4):171–179CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kaufman A, Shimony E (1986) 3D scan-conversion algorithms for voxel-based graphics. In: Proc. Workshop on Interactive 3D Graphics: ACM, New YorkGoogle Scholar
  25. 25.
    Klette G (2006) Branch voxels and junctions in 3D skeletons. In: Reulke R et al. (eds), Combinatorial Image Analysis, LNCS 4040, Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  26. 26.
    Klette G, Pan M (2004) 3D topological thinning by identifying non-simple voxels. In: Proc. Int. Workshop Combinatorial Image Analysis, LNCS 3322Google Scholar
  27. 27.
    Klette G, Pan M (2005) Characterization of curve-like structures in 3D medical images. In: Proc. Image Vision Computing New ZealandGoogle Scholar
  28. 28.
    Klette R, Rosenfeld A (2004) Digital Geometry — Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco, CAMATHGoogle Scholar
  29. 29.
    Klette R, Sun H-J (2001) Digital planar segment based polyhedrization for surface area estimation. In: Arcelli C, Cordella LP, Sanniti di Baja G (eds), Visual Form, Springer, BerlinGoogle Scholar
  30. 30.
    Kim CE (1983) IEEE Transactions on Pattern Analysis Machine Intelligence 5:231–234MATHCrossRefGoogle Scholar
  31. 31.
    Kong TY (2004) International Journal of Pattern Recognition Artificial Intelligence 9:813–844CrossRefGoogle Scholar
  32. 32.
    Kong TY, Rosenfeld A (1989) Computer Vision Graphics Image Processing 48:357–393CrossRefGoogle Scholar
  33. 33.
    Kovalevsky VA (1989) Computer Vision, Graphics, and Image Processing 46(2):141–161CrossRefGoogle Scholar
  34. 34.
    Lachaud J-O, Montanvert A (2000) Graphical Models and Image Processing 62:129–164Google Scholar
  35. 35.
    Latecki LJ (1998) Discrete Representations of Spatial Objects in Computer Vision. Kluwer, DordrechtMATHGoogle Scholar
  36. 36.
    Li F, Klette R (2006) Calculation of the number of tunnels. IMA Preprint Series 2113Google Scholar
  37. 37.
    Lohmann G (1988) Volumetric Image Analysis. Wiley & Teubner, ChichesterGoogle Scholar
  38. 38.
    Lorensen WE, Cline HE (1987) Computer Graphics 21:163–170CrossRefGoogle Scholar
  39. 39.
    Ma C-M, Wan S-Y (2000) Computer Vision Image Understanding 80:364–378MATHCrossRefGoogle Scholar
  40. 40.
    Malgouyres R (1997) Theoretical Computer Science 186:1–41MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Morgenthaler DG, Rosenfeld A (1981) Information Control 51:227–247MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Mylopoulos JP, Pavlidis T (1971) Journal of the ACM 18:239–246CrossRefMathSciNetGoogle Scholar
  43. 43.
    Nakamura A, Morita K, Imai K (2006) B-problem. CITR-TR-180. The University of Auckland, AucklandGoogle Scholar
  44. 44.
    Palagyi K, Kuba A (2003) Directional 3D thinning using 8 subiterations. In: Proc. Discrete Geometry Computational Imaging, LNCS 1568Google Scholar
  45. 45.
    Palagyi K, Kuba A (1998) Pattern Recognition Letters 19:613–627MATHCrossRefGoogle Scholar
  46. 46.
    Palagyi K, Sorantin E, Balogh E, Kuba A, Halmai C, Erdohelyi B, Hausegger K (2001) A sequential 3D thinning algorithm and its medical applications. In: Proc. Information Processing Medical Imaging, LNCS 2082Google Scholar
  47. 47.
    Peltier S, Alayrangues S, Fuchs L, Lachaud J (2005) Computation of homology groups and generators. In: Proc. DGCI, LNCS 3429Google Scholar
  48. 48.
    Rosenfeld A, Klette R (2001) Digital straightness. In: Electronic Notes in Theoretical Computer Science, Vol. 46Google Scholar
  49. 49.
    Saha PK, Chaudhuri BB (1996) Computer Vision Image Understanding 63:418–429CrossRefGoogle Scholar
  50. 50.
    Siguera M, Latecki LJ, Gallier J (2005) Making 3D binary digital images well-composed. In: Proc. Vision Geometry, SPIE 5675Google Scholar
  51. 51.
    Srihari SN (1981) ACM Computing Surveys 13:399–424CrossRefGoogle Scholar
  52. 52.
    White AT (1972) Pacific Journal of Mathematics 41:275–279MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science +Business Media B.V. 2009

Authors and Affiliations

  • Reneta P. Barneva
    • 1
  • Valentin E. Brimkov
    • 2
  1. 1.SUNY FredoniaFredonia
  2. 2.SUNY Buffalo State CollegeBuffalo

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