Geometric invariance of sampling

Haw-minn Lu; Yeshaiahu Fainman; Robert Hecht-Nielsen

doi:10.1117/12.323269

2 October 1998 Geometric invariance of sampling

Haw-minn Lu, Yeshaiahu Fainman, Robert Hecht-Nielsen

Author Affiliations +

Proceedings Volume 3454, Vision Geometry VII; (1998) https://doi.org/10.1117/12.323269
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1998, San Diego, CA, United States

Abstract

Recently, there has been growing interest in data structures in high dimensional pixel spaces. Among these structures are image manifolds. A point on an image manifold is an image; that is, a brightness function of two variables. In any in- depth geometric study of image manifolds,the impact of various sampling resolutions must be taken into account: will a manifold composed of a specific set of images sampled at one resolution differ geometrically from manifolds composed of the same image set sampled at other resolution. The main theorem of this paper states that under a specific Nyquist criterion, two image manifolds generated from the same source, but samples at differing pixel resolutions are not only diffeomorphic but isometric. This surprising new result proves the fundamental notion that geometric properties are invariant under sampling. Loosely speaking, this means that the two manifolds are of the same shape. In addition to proving this isometry theorem, an examination of fundamental image manifold properties such as curvature validates the theorem. Finally, the limitations of the Nyquist criterion are explored.

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Haw-minn Lu, Yeshaiahu Fainman, and Robert Hecht-Nielsen "Geometric invariance of sampling", Proc. SPIE 3454, Vision Geometry VII, (2 October 1998); https://doi.org/10.1117/12.323269

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