Bluman and Kumei's Nonlinear Scale-Space Theory

A nonlinear scale-space theory discovered by G. Bluman and S. Kumei is revisited and brought to the attention of computer vision community. Bluman and Kumei found that there exist a unique class of di usivity functions depending only on the local image grey-values that allows the related nonlinear scale-space of an image to be brought in a one-to-one correspondence with a linear scale-space of an associated image by applying Lie group theory. We follow their article and discuss the importance of this Lie-theoretic approach in the eld of image analysis and processing. Moreover, we indicate how to unravel the di erences and connections between the linear and nonlinear paradigms by applying modern geometry and Lie group theory. It is thus also shown, as we claimed some years ago, that an associated linear scale-space can be read out in a nonlinear fashion. We demonstrate that the nonlinear ltering scheme dynamically respects the localisation, ordering and grouping of regions of homogeneous grey-values and singularity sets such as edges, junctions, ridges and ruts. An explicit full discretisation of the nonlinear and linear ltering scheme illustrates that both the nonlinear and linear scale-spaces of identical input images have their own interpretation in terms of topology, geometry and dynamics. Usually one tries to incorporate the same dynamic priciples into nonlinear scale-space theories derived by means of variational and anisotropic ltering techniques. But in the derivations of latter theories the di usivity functions are always depending on the image gradient and higher order image structures. Moreover, it's not sure that these ltering schemes permit an in nite solution space nor that they can be associated with linear ltering schemes. Bluman and Kumei's approach, nevertheless, could give a strong Lie-theoretic support in order to derive variational, anisotropic and dynamic scale-space theories that are adjusted to a supervised computer vision task. This article has been submitted September 1998 to the Journal of Mathematical Vision and Imaging