Morphological filtering on graphs

Discrete Applied Mathematics, 2015

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyper edge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. Afterward, we propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of H and (ii) on the subhypergraphs of a hypergraph.

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyper edge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. Afterward, we propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of H and (ii) on the subhypergraphs of a hypergraph.

Proceedings X Brazilian Symposium on Computer Graphics and Image Processing, 1997

Mathematical Morphology is a theory that studies the decomposition of lattice operators in terms of some families of elementary lattice operators. When the lattices considered have a sup-generating family, the elementary operators can be characterized by structuring functions. The representation of structuring functions by neighborhood graphs is a powerful model for the construction of image operators. This model, that is a conceptual improvement of the one proposed by Vincent, permits a natural polymorphic extension of classical softwares for image processing by Mathematical Morphology. These systems constitute a complete framework for implementations of connected filters, that are one of the most modern and powerful approaches for image segmentation, and of operators that extract information from populations of objects in images. In this paper, besides presenting the formulation of the model, we present the polymorphic extension of a system for morphological image processing and some applications of it in image analysis.

2015 IEEE International Advance Computing Conference (IACC), 2015

Mathematical morphology (MM) helps to describe and analyze shapes using set theory. MM can be effectively applied to binary images which are treated as sets. Basic morphological operators defined can be used as an effective tool in image processing. Morphological operators are also developed based on graph and hypergraph. These operators have found better performance and applications in image processing. Bino et al. [8], [9] developed the theory of morphological operators on hypergraph. A hypergraph structure is considered and basic morphological operation erosion/dilation is defined. Several new operators opening/closing and filtering are also defined on the hypergraphs. Hypergraph based filtering have found comparatively better performance with morphological filters based on graph. In this paper we evaluate the effectiveness of hypergraph based ASF on binary images. Experimental results shows that hypergraph based ASF filters have outperformed graph based ASF.

Image Processing and Analysis with Graphs, 2017

sersc.org

The present paper focuses on a new class of mesh filter for grayscale images, called grid smoothing filter. The framework presented considers an image as a sampling grid associated to a set of gray levels. Furthermore, the sampling grid is seen as mesh composed by vertices and edges, the number of vertices being equal to the number of pixels in the image. Embedding the mesh in a 2D Euclidian space, each vertex has two spatial coordinates and one attribute, the value of the gray level. Starting from the classical formulation of Laplacian mesh filtering, a novel objective function is introduced. The minimization of the objective function leads to new spatial coordinates for the vertices in the mesh. A reconstruction mechanism is then applied to the non-uniform mesh to reconstruct a grayscale image. Whereas the Laplacian mesh filter aims at smoothing an image, the grid smoothing tends at sharpening the edges of the image. The grid smoothing framework is applied to image enhancement in this paper.

2008 19th International Conference on Pattern Recognition, 2008

In this paper, a novel approach to Mathematical Morphology operations is proposed. Morphological operators based on partial differential equations (PDEs) are extended to weighted graphs of the arbitrary topologies by considering partial difference equations. We focus on a general class of morphological filters, the levelings; and propose a novel approach of such filters. Indeed, our methodology recovers classical local PDEs-based levelings in image processing, generalizes them to nonlocal configurations and extends them to process any discrete data that can be represented by a graph. Experimental results show applications and the potential of our levelings to textured image processing, region adjacency graph based multiscale leveling and unorganized data set filtering.

IEEE Transactions on Image Processing, 2000

Mathematical morphology (MM) offers a wide range of operators to address various image processing problems. These operators can be defined in terms of algebraic (discrete) sets or as partial differential equations (PDEs). In this paper, we introduce a nonlocal PDEs-based morphological framework defined on weighted graphs. We present and analyze a set of operators that leads to a family of discretized morphological PDEs on weighted graphs. Our formulation introduces nonlocal patch-based configurations for image processing and extends PDEs-based approach to the processing of arbitrary data such as nonuniform high dimensional data. Finally, we show the potentialities of our methodology in order to process, segment and classify images and arbitrary data.

Journal of Visual Communication and Image Representation, 1992

This paper presents a systematic theory for the construction of morphological operators on graphs. Graph morphology extracts structural information from graphs using predefined test probes called structuring graphs. Structuring graphs have a simple structure and are relatively small compared to the graph that is to be transformed. A neighborhood function on the set of vertices of a graph is constructed by relating individual vertices to each other whenever they belong to a local instantiation of the structuring graph. This function is used to construct dilations and erosions. The concept of the structuring graph is also used to define openings and closings. The resulting morphological operators are invariant under symmetries of the graph. Graph morphology resembles classical morphology (which uses structuring elements to obtain translation-invariant operators) to a large extent. However, not all results from classical morphology have analogues in graph morphology because the local graph structure may be different at different vertices.

Springer eBooks, 2015

In recent years, variational methods, i.e., the formulation of problems under optimization forms, have had a great deal of success in image processing. This may be accounted for by their good performance and versatility. Conversely, mathematical morphology (MM) is a widely recognized methodology for solving a wide array of image processingrelated tasks. It thus appears useful and timely to build bridges between these two fields. In this article, we propose a variational approach to implement the four basic, structuring element-based operators of MM: dilation, erosion, opening, and closing. We rely on discrete calculus and convex analysis for our formulation. We show that we are able to propose a variety of continuously varying operators in between the dual extremes, i.e., between erosions and dilation; and perhaps more interestingly between openings and closings. This paves the way to the use of morphological operators in a number of new applications.