Uniqueness Results for Image Reconstruction from Features on Curves in alpha-Scale Spaces
A sufficient condition is presented for two-dimensional images on a finite rectangular domain Omega=(-A,A)x(-B,B) to be completely determined by features on curves ta dagger broken vertical bar(xi(t),t) in the three-dimensional domain Omega x(0,a) of an alpha-scale space. For any fixed finite set of points in the image, the values of the alpha-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 Omega x(0,a) for which xi:(0,a)-> I (c) is linear but not constant. A similar result holds for curves for which xi is of the form xi(t)=(xi (1)(t),0) with xi (1) periodic and not constant. If the locations at which the scale space is evaluated form a curve on a cylinder in Omega x(0,a) with some periodic structure, like a helix, then it is sufficient to evaluate the alpha-scale space without spatial derivatives.
alpha-scale space multiscale representation gaussian scale space poisson scale space