Affine and projective differential geometric invariants of space curves

Alfons H. Salden; Bart M. ter Haar Romeny; Max A. Viergever

doi:10.1117/12.146646

23 June 1993 Affine and projective differential geometric invariants of space curves

Alfons H. Salden, Bart M. ter Haar Romeny, Max A. Viergever

Proceedings Volume 2031, Geometric Methods in Computer Vision II; (1993) https://doi.org/10.1117/12.146646
Event: SPIE's 1993 International Symposium on Optics, Imaging, and Instrumentation, 1993, San Diego, CA, United States

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.

Citation Download Citation

Alfons H. Salden, Bart M. ter Haar Romeny, and Max A. Viergever "Affine and projective differential geometric invariants of space curves", Proc. SPIE 2031, Geometric Methods in Computer Vision II, (23 June 1993); https://doi.org/10.1117/12.146646

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