Initially introduced and developed by Georges Matheron and Jean Serra in the 1960s, mathematical morphology provides a powerful framework for analyzing and processing images based on their geometric structure and shape. Using fundamental concepts from set theory, lattice theory, and topology, it offers efficient solutions for common Image Processing and Pattern Recognition tasks through non-linear filtering techniques.

Its key strengths lie in producing interpretable results through geometrically meaningful operations, being invariant to contrast changes, requiring no training data, and working effectively across various types of images - from medical scans to satellite imagery. While rooted in classical mathematics, mathematical morphology continues to evolve and finds numerous applications in contemporary challenges such as noise reduction, edge detection, and object recognition. These solutions can also be seamlessly integrated with modern machine learning approaches when needed.

Our contributions in mathematical morphology

The Image Processing and Pattern Recognition group has established itself as a key contributor to both theoretical foundations and practical applications of mathematical morphology. Our research spans several complementary axes:

• Hierarchical Image Representations We specialize in developing efficient algorithms for computing tree-based image representations (such as the min/max-tree, the tree of shape, the α-tree and the binary partition tree structures). Our innovations include extending these algorithms to handle multivariate images [3][5], as well as implementing efficient, and recently GPU-accelerated versions [4][6].
• Multivariate Image Processing While mathematical morphology is well-established for univariate (grayscale) images, its extension to multivariate (such as color, multispectral, or multi-sensor) images presents significant theoretical and practical challenges. Our team works on developing robust and theoretically sound approaches to extend classical morphological operations to these complex types of images [8][9].
• Discrete topology After having generalized digital well-composedness to n-dimensional spaces, our team has established strong relations between its various forms and developed algorithms to achieve digital well-composedness through interpolation and topological repair [1]. Recently, we introduced a powerful mathematical tool that extends the concept of discrete surfaces, free from topological issues, applicable in both discrete topology and discrete geometry [2].
• Connecting mathematical morpholgy with (deep) neural network models We develop theoretical frameworks for learning morphological operations and structuring elements through neural networks [7], advancing towards automated morphological processing pipelines, as well as learning robust hierarchical representations directly from given input images.

Related Projects

Related Publications