Affine and projective differential geometric invariants of space curves

A.H. Salden, B.M. Haar Romenij, ter, M.A. Viergever

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

    1 Citation (Scopus)
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    Abstract

    A space curve, e.g., a parabolic line on a 2-dimensional surface in 3-dimensional Euclidean space, induces a plane curve under projective mapping. But 2-dimensional scalar input images of such an object are, normally, spatio-temporal slices through a luminance field caused by the interaction of an external field and that object. Consequently, the question arises how to obtain from those input images a consistent description of the space curve under projective transformations. By means of classical scale space theory, algebraic invariance theory, and classical differential geometry a new method of shape description for space curves from one or multiple views is proposed in terms of complete and irreducible sets of affine and projective differential geometric invariants. The method is based on defining implicitly connections for the observed curves that are highly correlated to the projected space curves. These projected curves are assumed to reveal themselves as coherent structures in the scale space representation of the differential structure of the input images. Several applications to stereo, optic flow, texture analysis, and image matching are briefly indicated.
    Original languageEnglish
    Title of host publicationGeometric Methods in Computer Vision II : San Diego, CA, July 11, 1993
    EditorsB.C. Vemuri
    Place of PublicationBellingham
    PublisherSPIE
    Pages64-74
    Publication statusPublished - 1993

    Publication series

    NameProceedings of SPIE
    Volume2031
    ISSN (Print)0277-786X

    Fingerprint

    Differential Invariants
    Geometric Invariants
    Space Curve
    Scale Space
    Projective Transformation
    Texture Analysis
    Curve
    Coherent Structures
    Image Matching
    Algebraic Theory
    Luminance
    Plane Curve
    Differential Geometry
    Slice
    External Field
    Euclidean space
    Optics
    Invariance
    Scalar
    Line

    Cite this

    Salden, A. H., Haar Romenij, ter, B. M., & Viergever, M. A. (1993). Affine and projective differential geometric invariants of space curves. In B. C. Vemuri (Ed.), Geometric Methods in Computer Vision II : San Diego, CA, July 11, 1993 (pp. 64-74). (Proceedings of SPIE; Vol. 2031). Bellingham: SPIE.
    Salden, A.H. ; Haar Romenij, ter, B.M. ; Viergever, M.A. / Affine and projective differential geometric invariants of space curves. Geometric Methods in Computer Vision II : San Diego, CA, July 11, 1993. editor / B.C. Vemuri. Bellingham : SPIE, 1993. pp. 64-74 (Proceedings of SPIE).
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    abstract = "A space curve, e.g., a parabolic line on a 2-dimensional surface in 3-dimensional Euclidean space, induces a plane curve under projective mapping. But 2-dimensional scalar input images of such an object are, normally, spatio-temporal slices through a luminance field caused by the interaction of an external field and that object. Consequently, the question arises how to obtain from those input images a consistent description of the space curve under projective transformations. By means of classical scale space theory, algebraic invariance theory, and classical differential geometry a new method of shape description for space curves from one or multiple views is proposed in terms of complete and irreducible sets of affine and projective differential geometric invariants. The method is based on defining implicitly connections for the observed curves that are highly correlated to the projected space curves. These projected curves are assumed to reveal themselves as coherent structures in the scale space representation of the differential structure of the input images. Several applications to stereo, optic flow, texture analysis, and image matching are briefly indicated.",
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    Salden, AH, Haar Romenij, ter, BM & Viergever, MA 1993, Affine and projective differential geometric invariants of space curves. in BC Vemuri (ed.), Geometric Methods in Computer Vision II : San Diego, CA, July 11, 1993. Proceedings of SPIE, vol. 2031, SPIE, Bellingham, pp. 64-74.

    Affine and projective differential geometric invariants of space curves. / Salden, A.H.; Haar Romenij, ter, B.M.; Viergever, M.A.

    Geometric Methods in Computer Vision II : San Diego, CA, July 11, 1993. ed. / B.C. Vemuri. Bellingham : SPIE, 1993. p. 64-74 (Proceedings of SPIE; Vol. 2031).

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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    AB - A space curve, e.g., a parabolic line on a 2-dimensional surface in 3-dimensional Euclidean space, induces a plane curve under projective mapping. But 2-dimensional scalar input images of such an object are, normally, spatio-temporal slices through a luminance field caused by the interaction of an external field and that object. Consequently, the question arises how to obtain from those input images a consistent description of the space curve under projective transformations. By means of classical scale space theory, algebraic invariance theory, and classical differential geometry a new method of shape description for space curves from one or multiple views is proposed in terms of complete and irreducible sets of affine and projective differential geometric invariants. The method is based on defining implicitly connections for the observed curves that are highly correlated to the projected space curves. These projected curves are assumed to reveal themselves as coherent structures in the scale space representation of the differential structure of the input images. Several applications to stereo, optic flow, texture analysis, and image matching are briefly indicated.

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    Salden AH, Haar Romenij, ter BM, Viergever MA. Affine and projective differential geometric invariants of space curves. In Vemuri BC, editor, Geometric Methods in Computer Vision II : San Diego, CA, July 11, 1993. Bellingham: SPIE. 1993. p. 64-74. (Proceedings of SPIE).