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Uniqueness Results for Image Reconstruction from Features on Curves in α-Scale Spaces

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Abstract

A sufficient condition is presented for two-dimensional images on a finite rectangular domain Ω=(−A,A)×(−B,B) to be completely determined by features on curves t↦(ξ(t),t) in the three-dimensional domain Ω×(0,∞) of an α-scale space. For any fixed finite set of points in the image, the values of the α-scale space at these points at all scales together do not provide sufficient information to reconstruct the image, even if spatial derivatives up to any order are included as well. On the other hand, the image is completely fixed by the values of the scale space and its derivative along any straight line in Ω×(0,∞) for which ξ:(0,∞)→Ω is linear but not constant. A similar result holds for curves for which ξ is of the form ξ(t)=(ξ 1(t),0) with ξ 1 periodic and not constant. If the locations at which the scale space is evaluated form a curve on a cylinder in Ω×(0,∞) with some periodic structure, like a helix, then it is sufficient to evaluate the α-scale space without spatial derivatives.

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Acknowledgements

The author would like to thank Prof. Jan de Graaf, Dr. Georg Prokert, and Dr. Bart Janssen for some fruitful discussions. The author is also grateful for the valuable comments and suggestions by the anonymous reviewers.

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    E. Vondenhoff

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Correspondence to E. Vondenhoff.

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The author is currently employed at Statistics Netherlands. The views expressed in this paper are those of the author and do not necessarily reflect the policies of Statistics Netherlands. The contents of this paper are not related to Statistics Netherlands in any sense.

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Vondenhoff, E. Uniqueness Results for Image Reconstruction from Features on Curves in α-Scale Spaces. J Math Imaging Vis 45, 1–12 (2013). https://doi.org/10.1007/s10851-012-0340-4

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