Volume 5, Issue 1

In many domains of information processing, such as knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, spatial reasoning, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools, such as fusion and aggregation, similarity and distances, mathematical morphology. Here we propose to extend these tools by defining algebraic and topological relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators. These definitions are shown to have the desired properties and to be consistent with existing definitions on sets and fuzzy sets, while providing an additional bipolar feature. The proposed relations can be used for instance for preference modeling or spatial reasoning. They apply more generally to any type of functions taking values in a poset or a complete lattice, such as L-fuzzy sets.

We propose a novel efficient seed-based method for the multi-object segmentation of images based on graphs, named Hierarchical Layered Oriented Image Foresting Transform (HLOIFT). It uses a tree of the relations between the image objects, with each node in the tree representing an object. Each tree node may contain different individual high-level priors of its corresponding object and defines a weighted digraph, named as layer. The layer graphs are then integrated into a hierarchical graph, considering the hierarchical relations of inclusion and exclusion. A single energy optimization is performed in the hierarchical layered weighted digraph leading to globally optimal results satisfying all the high-level priors. The experimental evaluations of HLOIFT, on medical, natural, and synthetic images, indicate promising results comparable to the related baseline methods that include structural information, but with lower computational complexity. Compared to the hierarchical segmentation by the min-cut/max-flow algorithm, our approach is less restrictive, leading to globally optimal results in more general scenarios, and has a better running time.

The extension of mathematical morphology to multivariate data has been an active research topic in recent years. In this paper we propose an approach that relies on the consensus combination of several stochastic permutation orderings. The latter are obtained by searching for a smooth shortest path on a graph representing an image. This path is obtained with a randomized version nearest of neighbors heuristics on a graph. The construction of the graph is of crucial importance and can be based on both spatial and spectral information to enable the obtaining of smoother shortest paths. The starting vertex of a path being taken at random, many different permutation orderings can be obtained and we propose to build a consensus ordering from several permutation orderings. We show the interest of the approach with both quantitative and qualitative results.

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.

Representing an image through a tree structure as provided with a morphological hierarchy enables efficient image analysis and processing methods operating directly on the tree structure. Max-tree and min-tree can be built with efficient algorithms but they only focus on brighter and darker components of the image respectively. Conversely, the Tree-of-Shapes is a self-complementary image representation that provides access to all regional extrema of the image (both brighter and darker components), but its computation is more time-consuming. In this paper, we introduce a new, simple and efficient tree structure called median-tree. It relies on a median image that is straightforwardly constructed by subtracting the median pixel value from an image to decompose it into positive and negative parts. The median tree can then be obtained by applying the efficient max-tree algorithms available in the literature on this median image. We show through theoretical and experimental studies that the median-tree offers similar characteristics to the Tree-of-Shapes, but comes with a considerably lower construction complexity.

The extension of Mathematical Morphology to colour and multivariate images is challenging due to the need to define a total ordering in the colour space. No one general way of ordering multivariate data exists and, therefore, there is no single, definitive way of performing morphological operations on colour images. In this paper, we propose an extension to mathematical morphology, based on reduced ordering, specifically the morphological Hit-or-Miss Transform which is used for object detection. The reduced ordering employed transforms multivariate observations to scalar comparisons allowing for an order to be derived and for both flat and non-flat structuring elements to be used. We also compare other definitions of the Hit-or-Miss Transform and test alternative colour ordering schemes presented in the literature. Our proposed method is shown to be intuitive and outperforms other approaches to multivariate Hit-or-Miss Transforms. Furthermore, methods of setting the parameters of the proposed Hit-or-Miss Transform are introduced in order to make the transform robust to noise and partial occlusion of objects and, finally, a set of design tools are presented in order to obtain optimal values for setting these parameters accordingly.