Advertisement

Integral Invariant Signatures

  • Siddharth Manay
  • Byung-Woo Hong
  • Anthony J. Yezzi
  • Stefano Soatto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3024)

Abstract

For shapes represented as closed planar contours, we introduce a class of functionals that are invariant with respect to the Euclidean and similarity group, obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential cousins, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (in the limit), they are not as sensitive to noise in the data. We exploit the integral invariants to define a unique signature, from which the original shape can be reconstructed uniquely up to the symmetry group, and a notion of scale-space that allows analysis at multiple levels of resolution. The invariant signature can be used as a basis to define various notions of distance between shapes, and we illustrate the potential of the integral invariant representation for shape matching on real and synthetic data.

Keywords

Object Recognition Kernel Size Shape Match Moment Invariant Shape Representation 
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.

References

  1. 1.
    Alferez, R., Wang, Y.F.: Geometric and illumination invariants for object recognition. PAMI 21(6), 505–536 (1999)Google Scholar
  2. 2.
    Arbter, K., Snyder, W.E., Burkhardt, H., Hirzinger, G.: Applications of affine invariant fourier descriptors to recognition of 3-d objects. PAMI 12(7), 640–646 (1990)Google Scholar
  3. 3.
    Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. PAMI 24(4), 509–522 (2002)Google Scholar
  4. 4.
    Bengtsson, A., Eklundh, J.-O.: Shape representation by multiscale contour approximation. PAMI 13(1), 85–93 (1991)Google Scholar
  5. 5.
    Boutin, M.: Numerically invariant signature curves. IJCV 40(3), 235–248 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    Brandt, R.D., Lin, F.: Representations that uniquely characterize images modulo translation, rotation and scaling. PRL 17, 1001–1015 (1996)Google Scholar
  7. 7.
    Bruckstein, A., Katzir, N., Lindenbaum, M., Porat, M.: Similarity invariant signatures for partially occluded planar shapes. IJCV 7(3), 271–285 (1992)CrossRefGoogle Scholar
  8. 8.
    Bruckstein, M., Holt, R.J., Netravali, A.N., Richardson, T.J.: Invariant signatures for planar shape recognition under partial occlusion. CVGIP:IU 58(1), 49–65 (1993)CrossRefGoogle Scholar
  9. 9.
    Bruckstein, M., Rivlin, E., Weiss, I.: Scale-space semi-local invariants. IVC 15(5), 335–344 (1997)Google Scholar
  10. 10.
    Calabi, E., Olver, P., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. IJCV 26, 107–135 (1998)CrossRefGoogle Scholar
  11. 11.
    Chetverikov, D., Khenokh, Y.: Matching for shape defect detection. In: Solina, F., Leonardis, A. (eds.) CAIP 1999. LNCS, vol. 1689(2), pp. 367–374. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Cohignac, T., Lopez, C., Morel, J.M.: Integral and local affine invariant parameter and applicatioin to shape recognition. ICPR 1, 164–168 (1994)Google Scholar
  13. 13.
    Cole, J.B., Murase, H., Naito, S.: A lie group theoretical approach to the invariance problem in feature extraction and object recognition. PRL 12, 519–523 (1991)Google Scholar
  14. 14.
    Dickson, L.E.: Algebraic Invariants. John-Weiley & Sons, West Sussex (1914)zbMATHGoogle Scholar
  15. 15.
    Dieudonne, J., Carrell, J.: Invariant Theory: Old and New. Academic Press, London (1970)Google Scholar
  16. 16.
    Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pat. Rec. 26(1), 167–174 (1993)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Forsyth, D.A., Mundy, J.L., Zisserman, A.P., Coelho, C., Heller, A., Othwell, C.A.: Invariant descriptors for 3-d object recognition and pose. PAMI 13(10), 971–991 (1991)Google Scholar
  18. 18.
    Forsyth, D.A., Mundy, J.L., Zisserman, A., Brown, C.M.: Projectively invariant representations using implicit algebraic curves. IVC 9(2), 130–136 (1991)Google Scholar
  19. 19.
    Van Gool, L., Moons, T., Pauwels, E., Oosterlinck, A.: Semi-differential invariants. In: Mundy, J., Zisserman, A. (eds.) Geometric Invariance in Computer Vision, pp. 193–214. MIT, Cambridge (1992)Google Scholar
  20. 20.
    Van Gool, L., Moons, T., Ungureanu, D.: Affine/photometric invariants for planar intensity patterns. In: Buxton, B.F., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1064, pp. 642–651. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  21. 21.
    Grace, J.H., Young, A.: The Algebra of Invariants, Cambridge (1903)Google Scholar
  22. 22.
    Hann, C.E., Hickman, M.S.: Projective curvature and integral invariants. IJCV 40(3), 235–248 (2000)CrossRefGoogle Scholar
  23. 23.
    Hu, M.K.: Visual pattern recognition by moment invariants. IRE Trans. on IT 8, 179–187 (1961)Google Scholar
  24. 24.
    Kanatani, K.: Group Theoretical Methods in Image Understanding. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  25. 25.
    Lane, E.P.: Projective Differential Geometry of Curves and Surfaces. University of Chicago Press, Chicago (1932)Google Scholar
  26. 26.
    Lasenby, J., Bayro-Corrochano, E., Lasenby, A.N., Sommer, G.: A new framework for the formation of invariants and multiple-view constraints in computer vision. ICIP (1996)Google Scholar
  27. 27.
    Lei, G.: Recognition of planar objects in 3-d space from single perspective views using cross ratio. Robot. and Automat. 6(4), 432–437 (1990)CrossRefGoogle Scholar
  28. 28.
    Lenz, R.: Group Theoretical Methods in Image Processing. LNCS, vol. 413. Springer, Heidelberg (1990)Google Scholar
  29. 29.
    Li, S.Z.: Shape matching based on invariants. In: Omidvar, O.M. (ed.) Progress in Neural Networks: Shape Recognition, vol. 6, pp. 203–228. Intellect, Bristol (1999)Google Scholar
  30. 30.
    Liao, S., Pawlak, M.: On image analysis by moments. PAMI 18(3), 254–266 (1996)Google Scholar
  31. 31.
    Miyatake, T., Matsuyama, T., Nagao, M.: Affine transform invariant curve recognition using fourier descriptors. Inform. Processing Soc. Japan 24(1), 64–71 (1983)Google Scholar
  32. 32.
    Mokhtarian, F., Mackworth, A.K.: A theory of multi-scale, curvature-based shape representation for planar curves. PAMI 14(8), 789–805 (1992)Google Scholar
  33. 33.
    Mumford, D., Fogarty, J., Kirwan, F.C.: Geometric invariant theory, 3rd edn. Springer, Berlin (1994)Google Scholar
  34. 34.
    Mumford, D., Latto, A., Shah, J.: The representation of shape. In: IEEE Workshop on Comp. Vis., pp. 183–191 (1984)Google Scholar
  35. 35.
    Mundy, J.L., Zisserman, A. (eds.): Geometric Invariance in Computer Vision. MIT, Cambridge (1992)Google Scholar
  36. 36.
    Nielsen, L., Saprr, G.: Projective area-invariants as an extension of the crossratio. CVGIP 54(1), 145–159 (1991)zbMATHCrossRefGoogle Scholar
  37. 37.
    Olver, P.J.: Equivalence, Invariants and Symmetry, Cambridge (1995)Google Scholar
  38. 38.
    Pajdla, T., Van Gool, L.: Matching of 3-d curves using semi-differential invariants. In: ICCV, pp. 390–395 (1995)Google Scholar
  39. 39.
    Reiss, T.H.: Recognizing Planar Objects Using Invariant Image Features. LNCS, vol. 676. Springer, Heidelberg (1993)zbMATHCrossRefGoogle Scholar
  40. 40.
    Rothwell, C., Zisserman, A., Forsyth, D., Mundy, J.: Canonical frames for planar object recognition. In: Sandini, G. (ed.) ECCV 1992. LNCS, vol. 588, pp. 757–772. Springer, Heidelberg (1992)Google Scholar
  41. 41.
    Rothwell, C., Zisserman, A., Forsyth, D., Mundy, J.: Planar object recognition using projective shape representation. IJCV 16, 57–99 (1995)CrossRefGoogle Scholar
  42. 42.
    Sapiro, G., Tannenbaum, A.: Affine invariant scale space. IJCV 11(1), 25–44 (1993)CrossRefGoogle Scholar
  43. 43.
    Sapiro, G., Tannenbaum, A.: Area and length preserving geometric invariant scale-spaces. PAMI 17(1), 67–72 (1995)Google Scholar
  44. 44.
    Sato, J., Cipolla, R.: Affine integral invariants for extracting symmetry axes. IVC 15(8), 627–635 (1997)Google Scholar
  45. 45.
    Shashua, A., Navab, N.: Relative affine structure: Canonical model for 3d from 2d geometry and applications. PAMI 18(9), 873–883 (1996)Google Scholar
  46. 46.
    Springer, C.E.: Geometry and Analysis of Projective Spaces. Freeman, San Francisco (1964)zbMATHGoogle Scholar
  47. 47.
    Tieng, Q.M., Boles, W.W.: Recognition of 2d object contours using the wavelet transform zero-crossing representation. PAMI 19(8), 910–916 (1997)Google Scholar
  48. 48.
    Verestoy, J., Chetverikov, D.: Shape detect detection in ferrite cores. Machine. Graphics and Vision 6(2), 225–236 (1997)Google Scholar
  49. 49.
    Weiss, I.: Noise resistant invariants of curves. PAMI 15(9), 943–948 (1993)Google Scholar
  50. 50.
    Witkin, P.: Scale-space filtering. In: Int. Joint. Conf. AI, pp. 1019–1021 (1983)Google Scholar
  51. 51.
    Zahn, C.T., Roskies, R.Z.: Fourier descriptors for plane closed curves. Trans. Comp. 21, 269–281 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Zisserman, A., Forsyth, D.A., Mundy, J.L., Rothwell, C.A., Liu, J.S.: 3D object recognition using invariance. Art. Int. 78, 239–288 (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Siddharth Manay
    • 1
  • Byung-Woo Hong
    • 2
  • Anthony J. Yezzi
    • 3
  • Stefano Soatto
    • 1
  1. 1.University of California at Los AngelesLos AngelesUSA
  2. 2.University of OxfordOxfordUK
  3. 3.Georgia Institute of TechnologyAtlantaUSA

Personalised recommendations