A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

Abstract

Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation / erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum-minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well.

Highly Accurate PDE-Based Morphology for General Structuring Elements

“…As it turns out, this is feasible but involves technical difficulties, especially for the case of general ellipses as structuring elements we discuss here in detail. We validate experimentally that the attractive features discussed in [26], namely a sharp resolution of edges and high rotational invariance, do carry over to the general case. In order to compare the performance of the FCT-scheme to set-theoretical algorithms, we use a diamond-shaped structuring element.…”

“…Then, by taking into account the so-called viscosity form of the predictor scheme, the dissipation can be quantified on a discrete level and is negated in a second step using stabilised inverse diffusion [27]. For details we refer to [26]. Let us note that the basic idea to negate dissipation by a corrector step was invented by Boris and Book [3,5].…”

“…The use of a comparable high-resolution ansatz, specifically an essentially non-oscillatory (ENO) approach, was reported in [16]. In [26], Breuß and Weickert constructed a FCT-scheme for performing dilation/erosion with a disc of radius t as structuring element.…”

“…We extend the applicability of the FCT-scheme introduced in [26] from discs to general structuring elements. As it turns out, this is feasible but involves technical difficulties, especially for the case of general ellipses as structuring elements we discuss here in detail.…”

Anisotropic Continuous-Scale Morphology

“…For more information on numerical methods for hyperbolic equations and more details concerning (11), we refer the interested reader to [3,11,12,14].…”