Tutorial at ECAI 2020

  Mathematical morphology and artificial intelligence

Date: September 5, 2020, 9am-12:15 (half day)

Abstract: This tutorial aims at providing an overview of mathematical morphology with examples of its use in several fields of AI. This theory, in its deterministic setting, applies on any complete lattice, and the operators it provides can then be instantiated in numerous mathematical frameworks such as sets, scalar or vectorial functions, fuzzy sets, logics, graphs, hypergraphs... Examples will cover three domains of AI. In the first one, mathematical morphology operators will be expressed in some logics (propositional, modal, description logics) to answer typical questions in knowledge representation and reasoning, such as revision or fusion. In the second one, spatial reasoning will benefit from spatial relations modeled using fuzzy sets and morphological operators, with applications in image understanding. In the third one, interactions between mathematical morphology and deep learning will be detailed.

Speaker biographies:

• Isabelle Bloch is graduated from the Ecole des Mines de Paris, Paris, France, in 1986, she received the Master's degree from the University Paris 12, Paris, in 1987, the Ph.D. degree from the Ecole Nationale Supérieure des Télécommunications (Telecom Paris), Paris, in 1990, and the Habilitation degree from the University Paris 5, Paris, in 1995. She is currently a Professor with the Image Processing and Understanding Group, LTCI, Telecom Paris. Her research interests include 3D image and object understanding, computer vision, artificial intelligence, 3D and fuzzy mathematical morphology, information fusion, fuzzy set theory, structural, graph-based, and knowledge-based object recognition, spatial knowledge representation and reasoning, and medical imaging.
• Samy Blusseau received the PhD degree in 2015 in applied mathematics at CMLA, ENS Cachan. He is now following a Tenure Track program at the Centre for Mathematical Morphology (CMM) of Mines ParisTech. His current research deals with morphological signal processing on graphs, morphological neural networks, interactions between deep learning and mathematical morphology for image processing.
• Ramon Pino Pérez is a retired Professor at the Mathematical Department of Los Andes University, Mérida, Venezuela. His current interest areas are Mathematical Logic in Artificial Intelligence, specially Knowledge Representation and Knowledge Dynamics. Social Choice Theory and Decision Theory.

Outline: This tutorial will be organized as follows: an introduction where the bases of mathematical morphology will be presented, and three parts of approximately equal length for the examples.

• Introduction: We will first present mathematical morphology, restricted to its deterministic setting and to increasing operators. The underlying structure is a complete lattice, on which two basic operators are defined: dilation and erosion, forming an adjunction. Definitions and properties will be presented, as well as a few derived operators. Concrete operators depending on so-called structuring elements will also be detailed, along with simple and intuitive examples.
• Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas, morphological operators will act on formulas (and on their models). In modal logics, dilation and erosion can define modal operators. Such examples will be described in several logics. Then we will show how they can be used to define revision operators satisfying the AGM postulates, merging operators, or explanatory relations.
• Mathematical Morphology and Spatial Reasoning: In the domain of spatial reasoning, i.e. modeling spatial entities and spatial relations to reason about them, we will address the problem of model-based image understanding. Models of a scene usually involve spatial relations, to provide information on the structure of the scene and on the spatial arrangement of the objects it contains. Moreover, such relations allow disambiguating objects with similar shape and appearance, and are more robust to deformations or pathological cases. Mathematical morphology is then useful, combined with fuzzy sets, to model such spatial relations, taking into account their intrinsic vagueness (e.g. left to, close to), and to compute them efficiently. These models can then be used in spatial reasoning processes, such as graph-based methods, of constraint satisfaction problems. Examples in medical imaging and remote sensing will illustrate these methods.
• Mathematical Morphology and Deep Learning: Interactions between mathematical morphology and deep learning have been investigated since the 1980s across several aspects. Morphological neural networks were introduced as an alternative to classical architectures, yielding a new geometry in decision surfaces. Deep networks were also trained to learn morphological operators and pipelines, and morphological algorithms were used as companion tools to machine learning, for pre/post processing or even regularization purposes. These ideas have known a large resurgence in the last few years and new ones are emerging such as including morphological methods in semi-supervised learning. In the tutorial we shall give an overview of this trend and some existing tools to get hands on this topic.

• Tutorial presentation
• Code for demos