Discrete mathematical morphology
Mathematical morphology provides the algebraic and algorithmic language for connected operators, shape analysis, filtering, and segmentation on discrete structures.
- lattices
- connected filters
- shape spaces
Research
Discrete structures for visual data
My research develops mathematical and algorithmic tools for images and data represented by graphs, simplicial complexes, hierarchies, trees, and topological structures. The common thread is to make discrete geometry useful: for segmentation, filtering, optimization, biomedical imaging, computer vision, deep learning, and interpretable models.
Themes
Research areas
Graphs and simplicial complexes
A central thread is the representation of images and data by graphs and simplicial complexes, including minimum spanning trees, saliency maps, component trees, the tree of shapes, and hierarchical segmentations.
- watersheds
- saliency maps
- simplicial complexes
Discrete topology and Morse theory
The work connects topological ideas with computable models on graphs, cubical grids, simplicial complexes, Morse functions, and gradient vector fields.
- Morse theory
- simplicial complexes
- persistent homology
Optimization and segmentation
Graph cuts, power watersheds, hierarchical cuts, convex optimization, and related algorithms make segmentation models precise and computationally tractable.
- power watershed
- graph cuts
- Kruskal algorithms
Biomedical and scientific imaging
Applications include PET/CT, MRI, cardiac imaging, vessel and catheter segmentation, and other visual data where topology and shape are strong priors.
- PET/CT
- MRI
- medical segmentation
Deep learning and interpretable vision
Recent work connects deep learning with graph neural networks, self-supervised learning, few-shot classification, and interpretable visual models built from hierarchies and trees.
- deep learning
- graph neural networks
- explainable AI
Selected research
Curated entry points
These threads are a selective map of the work, with representative HAL records chosen for orientation rather than exhaustiveness.
Watersheds and hierarchical segmentation
A long-running line connects watershed cuts, minimum spanning forests, saliency maps, and graph-based hierarchies. The point is not only to segment images, but to make the hierarchy itself a mathematically controlled object.
Representative works
- 2011On the equivalence between hierarchical segmentations and ultrametric watershedsJournal article · Journal of Mathematical Imaging and Vision
- 2018Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency mapsJournal article · Journal of Mathematical Imaging and Vision
- 2025Playing with Kruskal: algorithms for flat and hierarchical watershed cutsOther
Discrete calculus, power watersheds, and graph optimization
Discrete calculus gives graph-based counterparts of continuous variational tools, including combinatorial continuous max-flow. Power watersheds then connect watershed cuts with graph optimization; the SIAM gamma-convergence paper gives the proof framework, and power spectral clustering extends the same ideas to spectral clustering.
Representative works
- 2011Combinatorial Continuous Maximal FlowsJournal article · SIAM Journal on Imaging Sciences
- 2011Power Watersheds: A Unifying Graph Based Optimization FrameworkJournal article · IEEE Transactions on Pattern Analysis and Machine Intelligence
- 2017Extending the PowerWatershed framework thanks to Γ-convergenceJournal article · SIAM Journal on Imaging Sciences
- 2020Power Spectral ClusteringJournal article · Journal of Mathematical Imaging and Vision
Component trees, tree of shapes, and morphology
Connected operators and tree-based representations give image analysis a structural language: component trees, tree of shapes, shape spaces, attributes, and efficient algorithms for multiscale reasoning.
Representative works
- 2006Building the component tree in quasi-linear timeJournal article · IEEE Transactions on Image Processing
- 2013A quasi-linear algorithm to compute the tree of shapes of n-D imagesConference paper · International Symposium on Mathematical Morphology
- 2016Connected Filtering on Tree-Based Shape-SpacesJournal article · IEEE Transactions on Pattern Analysis and Machine Intelligence
Discrete topology, Morse theory, and persistence
This thread studies how topological objects can be computed on discrete data: Morse functions, gradient vector fields, well-composedness, persistent homology, and their relation to morphological dynamics.
Representative works
- 2023Discrete Morse Functions and WatershedsJournal article · Journal of Mathematical Imaging and Vision
- 2025Computing gradient vector fields with Morse sequencesConference paper · 4th International Conference on Discrete Geometry and Mathematical Morphology (DGMM 2025)
- 2022Some equivalence relation between persistent homology and morphological dynamicsJournal article · Journal of Mathematical Imaging and Vision
From Deep Learning to Self-Supervised Representations
This thread connects early deep-learning work on scene labeling with Clément Farabet, Camille Couprie, and Yann LeCun to recent self-supervised representation learning with Quentin Garrido and Yann LeCun.
Representative works
- 2013Learning Hierarchical Features for Scene LabelingJournal article · IEEE Transactions on Pattern Analysis and Machine Intelligence
- 2014Convolutional Nets and Watershed Cuts for Real-Time Semantic Labeling of RGBD VideosJournal article · Journal of Machine Learning Research
- 2023On the duality between contrastive and non-contrastive self-supervised learningConference paper · ICLR 2023 - Eleventh International Conference on Learning Representations
- 2023Self-supervised learning of Split Invariant Equivariant representationsConference paper · The Fortieth International Conference on Machine Learning
- 2023RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rankConference paper · The Fortieth International Conference on Machine Learning
Structural priors for deep learning
A complementary line brings the older mathematical objects back into learning: watersheds, component trees, and hierarchies become priors, filters, explanations, or constraints for modern models.
Representative works
- 2022Triplet-Watershed for Hyperspectral Image ClassificationJournal article · IEEE Transactions on Geoscience and Remote Sensing
- 2025Dual structure-aware image filterings for semi-supervised medical image segmentationJournal article · Medical Image Analysis
- 2026Explaning with trees: interpreting CNNs using hierarchiesJournal article · Transactions of Machine Learning Research
- 2024Unsupervised discovery of Interpretable Visual ConceptsJournal article · Information Sciences
Applications: personalized medicine and scientific imaging
This thread shows how mathematical and algorithmic tools move into real scientific data: patient-specific cardiac perfusion, vascular networks, PET image analysis, astronomical source detection, and other imaging domains where structure matters.
Representative works
- 2019Generation of patient-specific cardiac vascular networks: a hybrid image-based and synthetic geometric modelJournal article · IEEE Transactions on Biomedical Engineering
- 2021Myocardial Perfusion Simulation for Coronary Artery Disease: A Coupled Patient-Specific Multiscale ModelJournal article · Annals of Biomedical Engineering
- 2024Convex optimization for binary tree-based transport networksConference paper · International Conference on Discrete Geometry and Mathematical Morphology (DGMM)
- 2024Finite Volume Informed Graph Neural Network for Myocardial Perfusion SimulationConference paper · Proceedings of The 7nd International Conference on Medical Imaging with Deep Learning
- 2020Shaping for PET image analysisJournal article · Pattern Recognition Letters
- 2021Object Detection with Component-Graphs in Multi-band Images: Application to Source Detection in Astronomical ImagesJournal article · IEEE Access
Software and reproducible methods
The software thread makes the mathematical objects usable by others, from hierarchical graph analysis to Morse-based constructions. Higra and MorseFrames are the current public entry points.
Representative works
- 2019Higra: Hierarchical Graph AnalysisJournal article · SoftwareX
- 2024Morse framesConference paper · International Conference on Discrete Geometry and Mathematical Morphology (DGMM)
Research highlights
Three visible contributions
These highlights make three contributions visible through concrete images: personalized cardiac modeling, hierarchical graph-analysis software, and the theory of hierarchies.
Application
Personalized medicine: myocardial perfusion simulation
A patient-specific multiscale model linking coronary FFRCT, segmented and synthetic vascular networks, and myocardial microcirculation to simulate blood flow from epicardial arteries to cardiac tissue.
Software contribution
Higra: hierarchical graph analysis
A C++/Python library for efficient sparse-graph analysis, focused on constructing, processing, filtering, clustering, and evaluating hierarchical representations. Benjamin Perret is the main maintainer; I contribute to the mathematical morphology and hierarchy line that feeds the library.
Theory
Theory of hierarchies
A theoretical and algorithmic line connecting dendrograms, saliency maps, minimum spanning trees, and hierarchical watersheds, with constructive results for characterization, enumeration, transformation, and out-of-core computation. This line started within the A3SI team at LIGM, and remains a current thread in my work.
Archive highlights
Visual material from the previous site
The previous website carried a useful visual memory of papers, tutorials, books, and applications. This selection keeps that material available as a compact archive, while the full research map above remains organized around themes.
Survey / tutorial
Graph-based mathematical morphology
A trace of the survey and tutorial material that connected morphology, graphs, watersheds, and hierarchies.
PAMI papers / Colab
Shape-space filtering
The old shape-filtering visual now points to the two Xu PAMI papers and the Higra Colab scripts that make the shaping framework executable.
Evaluation paper
Cardiac and vascular imaging
Archive imagery for the medical-imaging line around coronary artery stenosis detection, quantification, and lumen segmentation.
PAMI / deep learning
Scene labeling and hierarchical features
A visual marker of the work with Clément Farabet, Camille Couprie, and Yann LeCun showing that deep, hierarchical, multiscale convolutional features could be learned directly from images for dense scene labeling, making it a key precursor to modern semantic segmentation systems.
Featured paper
Dual-constrained total variation
An old featured-paper image from the optimization and graph-based image-processing thread.
Guide-wire / stent
Interventional X-ray imaging
A visual trace of the interventional imaging work on curvilinear structures, guide-wire segmentation, and stent visualization enhancement.
Books
Long-form references
Modern Approaches to Discrete Curvaturewith Pascal Romon, Springer, 2017
Mathematical Morphology: from theory to applicationswith Hugues Talbot, ISTE-Wiley, 2010
Systèmes d'exploitation - Synthèse de cours & exercices corrigéswith Bart Lamiroy and Hugues Talbot, Pearson Education France, 2006