Mathematical morphology and image processing

Since its introduction in the 1960s, mathematical morphology has rapidly become , a fundamental theory of image processing and analysis. The operators it proposes provide tools for the entire image processing chain, from pre-processing (filtering, contrast enhancement) to segmentation and scene interpretation. An important feature of these operators is that they are non-linear. They enable images to be transformed, and features, objects or measurements to be extracted, through an analysis combining properties of the objects themselves (shape, size, appearance, etc.) and properties of the context (local neighborhood or relationships with other objects).

To summarize the "toolbox" of mathematical morphology, consider the following points:

• transformations are non-linear, based on operations of type sup and inf ;

• transformations are generally non-invertible, and therefore lose information; the morphologist's job is then to determine the transformations best suited to his problem, i.e. those that will "simplify" the images while retaining the relevant information;

• analytical and algebraic properties are attached to the operations, ensuring precise properties of the objects or images resulting from the transformations; these properties are used to link the transformations in order to solve a particular problem;

• Algorithms are also associated with the transformations, enabling them to be applied efficiently.

In what follows, we'll briefly review the history of mathematical morphology, then introduce the four basic operations (dilation, erosion, opening, closing) for binary and grayscale images. Some immediate applications of these operations will be illustrated. We will then return to the mathematical foundations underlying these definitions, and in particular to the algebraic framework of complete lattices, which is...