Highly Accurate PDE-Based Morphology for General Structuring Elements

Abstract

Abstract. Modelling the morphological processes of dilation and erosion with convex structuring elements with partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high rotational invariance. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology.

Highly Accurate Schemes for PDE-Based Morphology with General Convex Structuring Elements

“…Typically, these key issues need to be specified by the user of the algorithm. In detail, there are the following points: (i) The order of the interpolation formula, (ii) the We now elaborate on exactly these topics by discussing a series of numerical experiments, thereby extending the work Breuß and Weickert (2009). For following the structure of the discussion, let us note that it makes sense to consider not only the issues (i)-(iii) individually, but also relationships between them.…”

“…Compared to that work, and in addition to the contents just stated above, we have added a detailed numerical study of important properties of the method: For instance, we investigate its accuracy depending on the anisotropy of the structuring element, and the influence of the interpolation formula used as part of the method on the quality of results. Also on the theoretical side we extend here the paper Breuß and Weickert (2009) considerably, as we prove that the scheme is consistent as well as stable under a mild restriction on the time step size which is very important for practical applications. Moreover, we give more details concerning the construction of our scheme.…”

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

“…To be applicable to images, PDEs are discretized, and specific numerical schemes are used: for instance, see [29][30][31] and, more recently, a variant of the flux corrected transport technique [32,33] used for tensor images and matrix morphological processing [34]. Other morphological PDEs, based on curve evolution, the level set or the Eikonal equation, are used for image segmentation, generalized distance computation or shape analysis.…”

Adaptive Continuous-Scale Morphology for Matrix Fields

“…The PDE-based approach is conceptually attractive since it allows for digital scalability and even adaptivity of the represented structuring element. This versatility was exploited, for example in [9] and [11] to create adaptive, PDE-based dilation processes for grey value images. In [15] the idea of morphological adaptivity has been transferred to the setting of matrix fields utilising the operatoralgebraic framework proposed in [18].…”