Structurally Adaptive Mathematical Morphology Based on Nonlinear Scale-Space Decompositions

Abstract

Standard formulation of morphological operators is translation invariant in the space and in the intensity: the same processing is considered for each point of the image. A current challenging topic in mathematical morphology is the construction of adaptive operators. In previous works, the adaptive operators are based either on spatially variable neighbourhoods according to the local regularity, or on size variable neighbourhoods according to the local intensity. This paper introduces a new framework: the structurally adaptive mathematical morphology. More precisely, the rationale behind the present approach is to work on a nonlinear multi-scale image decomposition, and then to adapt intrinsically the size of the operator to the local scale of the structures. The properties of the derived operators are investigated and their practical performances are compared with respect to standard morphological operators using natural image examples.

Inner-Cheeger Opening and Applications

“…One can realize that openings and closings with square, disk or hexagon SEs, are often good enough for some filtering tasks. However, if the structuring elements are able to adapt their shapes and sizes to the image content, some enhancement properties are improved [2,3]. This intuition leads to propose area openings [4], and more generally, to introduce attributes openings [5] .…”

Morphological Bilateral Filtering

“…On the other hand, a variability on T : variable size of structuring functions according to the local intensity or contrast [55,40]. The case of a structural adaptivity based on the product space E × T has been also considered in [4], by working on nonlinear scale-space decompositions. For an overview on the state-the-art on adaptive morphology, the interested reader is invited to the paper [41].…”

An Approach to Adaptive Quadratic Structuring Functions Based on the Local Structure Tensor

“…These methods are suitable for prolonging shapes or bridging gaps, thereby emphasizing directional structures in the processed image. Some methods of this type work in multiple scales in order to adapt the size of the structuring element to the local scale of structures in the image [2,18]. Other methods address structure by considering local orientation only [17], or by combining local orientation with other factors such as distances to edges [21] or degree of anisotropy [12].…”