Flat morphological operators from non-increasing set operators, I: general theory

Flat morphology is a general method for obtaining increasing operators on grey-level or multivalued images from increasing operators on binary images (or sets). It relies on threshold stacking and superposition; equivalently, Boolean max and min operations are replaced by lattice-theoretical sup and inf operations. In this paper we consider the construction a flat operator on grey-level or colour images from an operator on binary images that is not increasing. Here grey-level and colour images are functions from a space to an interval in ℝ m or ℤ m (m ≥ 1). Two approaches are proposed. First, we can replace threshold superposition by threshold summation. Next, we can decompose the non-increasing operator on binary images into a linear combination of increasing operators, then apply this linear combination to their flat extensions. Both methods require the operator to have bounded variation, and then both give the same result, which conforms to intuition. Our approach is very general, it can be applied to linear combinations of flat operators, or to linear convolution filters. Our work is based on a mathematical theory of summation of real-valued functions of one variable ranging in a poset. In a second paper, we will study some particular properties of non-increasing flat operators.

Shape-Based Analysis on Component-Graphs for Multivalued Image Processing

Connected operators based on hierarchical image models have been increasingly considered for the design of efficient image segmentation and filtering tools in various application fields. Among hierarchical image models, component-trees represent the structure of grey-level images by considering their nested binary level-sets obtained from successive thresholds. Recently, a new notion of component-graph was introduced to extend the component-tree to any grey-level or multivalued images. The notion of shaping was also introduced as a way to improve the anti-extensive filtering by considering a two-layer component-tree for grey-level image processing. In this article, we study how component-graphs (that extend the component-tree from a spectral point of view) and shapings (that extend the component-tree from a conceptual point of view) can be associated for the effective processing of multivalued images. We provide structural and algorithmic developments. Although the contributions of this article are theoretical and methodological, we also provide two illustration examples that qualitatively emphasize the potential use and usefulness of the proposed paradigms for image analysis purposes.

Morphological sharpening of color images

This paper presents a novel approach to morphological contrast sharpening of image using the multilevel toggle operator. The concept presented here is a generalization of toggle based contrast operator for gray-level images. The multilevel toggle operator is used to enhance the contrast of multivalued images. In order to perform necessary morphological operations the modified pairwise ordering (MPO) algorithm is proposed. It gives the total order of color pixels. For comparison four other ordering methods are used. The main advantage of the proposed sharpener is its significant contrast enhancing ability when using MPO. Theoretical considerations as well as practical results are shown. Experimental results show its applicability to low-contrast color images.