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Curve evolution, differential morphology, and distance transforms applied to multiscale and eikonal problems

Maragos P., Butt M.A.

Fundamenta Informaticae

2000

Vol. 41, Nr 1,2

91-129

In differential morphology, multiscale dilations and erosions are modeled via nonlinear partial differential equations (PDEs) in scale-space. Curve evolution employs methods of differential geometry to study the differential equations governing the propagation of time-evolving curves, under velocities dependent on global information or on local geometric properties of the curve. The PDEs governing multiscale morphology, and most cases of curve evolution, are of the Hamilton-Jacobi type and are related to the eikonal PDE of optics. In this paper, we explore the common theoretical concepts, tools, and numerical algorithms used in differential morphology and curve evolution, by emphasizing level set methods. Morphological operator representations of various curve evolution cases are discussed, as well as evolution laws for various morphological curve operations. We also focus on distance transforms, as the major route to connect differential morphology and curve evolution to the eikonal PDE. Furthermore, we discuss applications of differential morphology and curve evolution to various multiscale and/or eikonal problems, such as distance transform computation, ray tracing in optics, eikonal image halftoning, and watershed-based image segmentation.