Morphological PDE and Dilation/Erosion Semigroups on Length Spaces

Abstract

This paper gives a survey of recent research on Hamilton-Jacobi partial dierential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We rst introduce the most general pair of dilation/erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups.

Morphological Counterpart of Ornstein–Uhlenbeck Semigroups and PDEs

“…For the application of the latter model to adaptive morphology, see [15]. In the most general case, this family of morphological PDEs and semigroups are formulated in the framework of length spaces [5] and a similar counterpart in ultrametric spaces [6]. A general theory of morphological counterparts of linear shift-invariant scalespaces based on the Cramér-Fourier transform has been proposed [24].…”

Morphological PDEs on Graphs for Image Processing on Surfaces and Point Clouds

“…We now briefly review some of these operators formulated as Hamilton-Jacobi equations, curve evolution level sets and morphological flows. For a detailed review and numerical algorithms, see [2,25].…”

“…Other particular forms of the Hamilton-Jacobi model Equation (1) cover the flat morphology by disks [7] (i.e., f t = ±||∇ f ||), as well as operators with more general P-power concave structuring functions (i.e., f t = ±||∇ f || p ). For the application of the latter model to adaptive morphology, see [25] and the references therein.…”

Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

“…More recently, morphological semigroups for functions on length spaces have been studied [3], whose basic ingredients are the convolution in the (max, +)-algebra (or supremal convolution), the metric distance and a convex shape function. More precisely, given a length space (X, d), a bounded function f : X → R and an increasing convex one-dimensional (shape) function L : R + → R + such that L(0) = 0, the multiscale dilation D L; t f and erosion E L; t f operators of f on (X, d) according to L at scale t > 0 are dened as D L; t f (x) = sup y∈X f (y) − tL d(x, y) t , ∀x ∈ X, E L; t f (x) = inf y∈X f (y) + tL d(x, y) t , ∀x ∈ X.…”