Skip to main content

Introduction to Mathematical aspects of computer vision

  • Conference paper
  • 348 Accesses

Part of the Progress in Systems and Control Theory book series (PSCT,volume 25)

Abstract

The goal of the minisymposium Mathematical Aspects of Computer Vision was to introduce the audience of MTNS98 to areas of research and open problems in computer vision that are mathematical, or, more generally, theoretical in nature. A further goal was to promote cross-fertilization between system and control and vision researchers.

Keywords

  • Computer Vision
  • IEEE Transaction
  • Anisotropic Diffusion
  • Mathematical Aspect
  • Early Vision

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-0348-8970-4_17
  • Chapter length: 8 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   74.99
Price excludes VAT (USA)
  • ISBN: 978-3-0348-8970-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   99.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y. Amit, D. Geman, and K. Wilder. Joint induction of shape features and tree classifiers. IEEE Transactions on Pattern Analysis and Machine Intellegence, 19 (11): 1300–1305, 1997.

    CrossRef  Google Scholar 

  2. S. Angenent, G. Sapiro, and A. Tannenbaum. On the affine heat equation for non-convex curves. J. American Math. Soc, 11 (3): 601–634, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  3. M.J.. Black, G. Sapiro, D.H. Marimont, and D. Heeger. Robust anisotropic diffusion. IEEE Trans, on Image Processing, 7 (3): 421–432, 1998.

    CrossRef  Google Scholar 

  4. A. Blake, B. Bascle, M. Isard, and J. MacCormick. Statistical models of visual shape and motion. Phil. Trans, of the Royal Soc. of London Series A-Math. Phys. and Engr. Sciences, 356 (1740): 1283–1301, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  5. E. Calabi, P.J. Olver, C. Shakiban, A. Tannenbaum, and S. Haker. Differential and numerically invariant signature curves applied to object recognition. Internat. J. of Computer Vision, 26 (2): 107–135, 1998.

    CrossRef  Google Scholar 

  6. V. Caselles, J.M. Morel, G. Sapiro, and A. Tannenbaum. Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis. IEEE Trans, on Image Processing, 7 (3): 269–273, 1998.

    CrossRef  Google Scholar 

  7. V. Caselles, J.M. Morel, and C. Sbert. An axiomatic approach to image interpolation. IEEE Trans, on Image Processing, 7 (3): 376–386, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  8. R. Cipolla. The visual motion of curves and surfaces. Phil. Trans, of the Royal Soc. of London Series A-Math. Phys. and Engr. Sciences, 356 (1740): 1103–1118, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  9. R. Cipolla, G. Fletcher, and P. Giblin. Following cusps. Internat. J. of Computer Vision, 23 (2): 115–129, 1997.

    CrossRef  Google Scholar 

  10. I. Daubechies, S. Mallat, and A.S. Willsky. Special issue on wavelet transforms and multiresolution signal analysis - introduction. IEEE Trans, on Info. Theory, 38 (2): 529–531, 1992.

    Google Scholar 

  11. O. Faugeras and R. Keriven. Variational principles, surface evolution, pde’s, level set methods, and the stereo problem. IEEE Trans, on Image Processing, 7 (3): 336–344, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  12. O. Faugeras and T. Papadopoulo. Grassmann-cayley algebra for modelling systems of cameras and the algebraic equations of the manifold of trifocal tensors. Phil. Trans, of the Royal Soc. of London Series A-Math. Phys. and Engr. Sciences, 356 (1740): 1123–1150, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  13. O. Faugeras and L. Robert. What can two images tell us about a third one? Internat. J. of Computer Vision, 18 (1): 5–19, 1996.

    CrossRef  Google Scholar 

  14. O. Faugeras, L. Robert, S. Laveau, G. Csurka, C. Zeller, C. Gauclin, and I. Zoghlami. 3-d reconstruction of urban scenes from image sequences. Computer Vision and Image Understanding, 69(3): 292–309, 1998.

    Google Scholar 

  15. D. Geman and B. Jedynak. An active testing model for tracking roads in satellite images. IEEE Transactions on Pattern Analysis and Machine Intellegence, 18 (1): 1–14, 1996.

    CrossRef  Google Scholar 

  16. P. Giblin. Apparent contours: an outline. Phil. Trans, of the Royal Soc. of London Series A-Math. Phys. and Engr. Sciences, 356 (1740): 1087–1102, 1998.

    MathSciNet  MATH  CrossRef  Google Scholar 

  17. M. Isard and A. Blake. Condensation - conditional density propagation for visual tracking. Internat. J. of Computer Vision, 29 (l): 5–28, 1998.

    CrossRef  Google Scholar 

  18. J.J. Koenderink. The structure of images. Biological Cybernetics, 50 (5): 363–370, 1984.

    MathSciNet  MATH  CrossRef  Google Scholar 

  19. J.J. Koenderink. Optic flow. Vision Research, 26 (1): 161–179, 1986.

    CrossRef  Google Scholar 

  20. J.J. Koenderink and A.J. Vandoorn. Facts on optic flow. Biological Cybernetics, 56 (4): 247–254, 1987.

    MATH  CrossRef  Google Scholar 

  21. J.J. Koenderink and A.J. Vandoorn. Generic neighborhood operators. IEEE Transactions on Pattern Analysis and Machine Intelle-gence, 14 (6): 597–605, 1992.

    CrossRef  Google Scholar 

  22. J.J. Koenderink and A.J. Vandoorn. 2-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15 (5): 439–443, 1994.

    MATH  CrossRef  Google Scholar 

  23. G. Koepfler, C. Lopez, and J.M. Morel. A multiscale algorithm for image segmentation by variational method. SIAM Journal on Numerical Analysis, 31 (l): 282–299, 1994.

    MathSciNet  MATH  CrossRef  Google Scholar 

  24. P. Kube and P. Perona. Scale-space properties of quadratic feature detectors. IEEE Transactions on Pattern Analysis and Machine Intellegence, 18 (10): 987–999, 1996.

    CrossRef  Google Scholar 

  25. J. Malik. Interpreting line drawings of curved objects. Internat. J. of Computer Vision, 1 (1): 73–103, 1987.

    CrossRef  Google Scholar 

  26. J. Malik and P. Perona. Preattentive texture-discrimination with early vision mechanisms. J. of the Optical S oc. of America A-Optical Image Science and Vision, 7 (5): 923–932, 1990.

    CrossRef  Google Scholar 

  27. J. Malik and R. Rosenholtz. Computing local surface orientation and shape from texture for curved surfaces. Internat. J. of Computer Vision, 23 (2): 149–168, 1997.

    CrossRef  Google Scholar 

  28. S. Mallat. Wavelets for a vision. Proceddings of IEEE, 84 (4): 604–614, 1996.

    CrossRef  Google Scholar 

  29. S. Mallat and S. Zhong. Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intellegence, 14 (7): 710–732, 1992.

    CrossRef  Google Scholar 

  30. J.M. Morel. The mumford-shah conjecture in image processing. Asterisque, (241): 221–242, 1997.

    MathSciNet  Google Scholar 

  31. P. Perona. Deformable kernels for early vision. IEEE Transactions on Pattern Analysis and Machine Intellegence, 17 (5): 488–499, 1995.

    CrossRef  Google Scholar 

  32. P. Perona. Orientation diffusions. IEEE Trans, on Image Processing, 7 (3): 457–467, 1998.

    CrossRef  Google Scholar 

  33. P. Perona and J. Malik. Scale-space and edge-detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intellegence, 12 (7): 629–639, 1990.

    CrossRef  Google Scholar 

  34. J. Sato and R. Cipolla. Quasi-invariant parameterisations and matching of curves in images. Internat. J. of Computer Vision, 28 (2): 117–136, 1998.

    CrossRef  Google Scholar 

  35. K. Siddiqi, Y.B. Lauziere, A. Tannenbaum, and S.W. Zucker. Area and length minimizing flows for shape segmentation. IEEE Trans, on Image Processing, 7 (3): 433–443, 1998.

    CrossRef  Google Scholar 

  36. S. Soatto. 3-d structure from visual motion: Modeling, representation and observability. Automatica, 33 (7): 1287–1312, 1997.

    MathSciNet  MATH  CrossRef  Google Scholar 

  37. S. Soatto, R. Prezza, and P. Perona. Motion estimation via dynamic vision. IEEE Transactions on Automatic Control, 41 (3): 393–413, 1996.

    MATH  CrossRef  Google Scholar 

  38. S. Soatto and P. Perona. Recursive 3-d visual motion estimation using subspace constraints. Internat. J. of Computer Vision, 22 (3): 235–259, 1997.

    CrossRef  Google Scholar 

  39. S. Soatto and P. Perona. Reducing “structure from motion”: A general framework for dynamic vision part 1: Modeling. IEEE Transactions on Pattern Analysis and Machine Intellegence, 20 (9): 933–942, 1998.

    CrossRef  Google Scholar 

  40. P.C. Teo, G. Sapiro, and B.A. Wandell. Creating connected representations of cortical gray matter for functional mri visualization. IEEE Transactions on Medical Imaging, 16 (6): 852–863, 1997.

    CrossRef  Google Scholar 

  41. Y.L. You, W.Y. Xu, A. Tannenbaum, and M. Kaveh. Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans, on Image Processing, 5 (11): 1539–1553, 1996.

    CrossRef  Google Scholar 

  42. S.C. Zhu and D. Mumford. Prior learning and gibbs reaction-diffusion. IEEE Transactions on Pattern Analysis and Machine Intellegence, 19 (11): 1236–1250, 1997.

    CrossRef  Google Scholar 

  43. S.C. Zhu, Y.N. Wu, and D. Mumford. Filters, random fields and maximum entropy (frame): Towards a unified theory for texture modeling. Internat. J. of Computer Vision, 27 (2): 107–126, 1998.

    CrossRef  Google Scholar 

  44. Faugeras, O. (1993). Three-Dimensional Computer Vision: A Geometric Viewpoint. Cambridge: MIT Press.

    Google Scholar 

  45. Horn, B. K. P. and Brooks, M.J. (1989). Shape from Shading. Cambridge: MIT Press.

    Google Scholar 

  46. Isard, M. and Blake, A. (1996). Contour tracking by stochastic propagation of conditional density. In Proceedings of Fourth European Conference on Computer Vision. ECCV ’96, Cambridge, UK. Edited by: Buxton, B.; Cipolla, R. Berlin, Germany: Springer-Verlag, 1996. p. 343–356 vol.1.

    CrossRef  Google Scholar 

  47. Haralick, R. M. and L.G. Shapiro. (1992). Computer and Robot Vision, Volumes I and I I. Reading, MA: Addison-Wesley.

    Google Scholar 

  48. Horn, B. K. P. (1986). Robot Vision. Cambridge: MIT Press.

    Google Scholar 

  49. Koenderink, J. J. (1990). Solid Shape. Cambridge: MIT Press.

    Google Scholar 

  50. Marr, D. (1982). Vision. San Francisco: Freeman.

    Google Scholar 

  51. Nalwa, V.S. (1993). A Guided Tour of Computer Vision. Reading, MA: Addison Wesley.

    Google Scholar 

  52. E. Trucco, E. and Verri, A. (1998). Introductory Techniques for 3-D Computer Vision, NJ: Prentice-Hall, 1998.

    Google Scholar 

  53. Ullman, S. (1996). High-level Vision: Object Recognition and Visual Cognition. Cambridge: MIT Press.

    MATH  Google Scholar 

  54. Shi, J. and Malik, J (1997). Normalized cuts and image segmentation. In Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Juan, Puerto Rico, pp. 731–737.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1999 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Malik, J., Perona, P. (1999). Introduction to Mathematical aspects of computer vision. In: Picci, G., Gilliam, D.S. (eds) Dynamical Systems, Control, Coding, Computer Vision. Progress in Systems and Control Theory, vol 25. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8970-4_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8970-4_17

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9848-5

  • Online ISBN: 978-3-0348-8970-4

  • eBook Packages: Springer Book Archive