Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
Data structures and network algorithms
Data structures and network algorithms
Digital topology: introduction and survey
Computer Vision, Graphics, and Image Processing
Introduction to algorithms
Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations
IEEE Transactions on Pattern Analysis and Machine Intelligence
Watershed of a continuous function
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
Attribute openings, thinnings, and granulometries
Computer Vision and Image Understanding
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Connected filtering and segmentation using component trees
Computer Vision and Image Understanding
The watershed transform: definitions, algorithms and parallelization strategies
Fundamenta Informaticae - Special issue on mathematical morphology
A Comparison of Algorithms for Connected Set Openings and Closings
IEEE Transactions on Pattern Analysis and Machine Intelligence
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
Tree Representation and Implicit Tree Matching for a Coarse to Fine Image Matching Algorithm
MICCAI '99 Proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention
A Theoretical Tour of Connectivity in Image Processing and Analysis
Journal of Mathematical Imaging and Vision
Journal of Mathematical Imaging and Vision
Watersheds, mosaics, and the emergence paradigm
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Antiextensive connected operators for image and sequence processing
IEEE Transactions on Image Processing
Journal of Mathematical Imaging and Vision
Watersheds, mosaics, and the emergence paradigm
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Image and Vision Computing
Uniquely-Determined Thinning of the Tie-Zone Watershed Based on Label Frequency
Journal of Mathematical Imaging and Vision
Fusion Graphs: Merging Properties and Watersheds
Journal of Mathematical Imaging and Vision
Weighted fusion graphs: Merging properties and watersheds
Discrete Applied Mathematics
Parallel Algorithm for Concurrent Computation of Connected Component Tree
ACIVS '08 Proceedings of the 10th International Conference on Advanced Concepts for Intelligent Vision Systems
ISMM '09 Proceedings of the 9th International Symposium on Mathematical Morphology and Its Application to Signal and Image Processing
Watersheds, mosaics, and the emergence paradigm
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Automated, accurate and fast segmentation of 4D cardiac MR images
FIMH'07 Proceedings of the 4th international conference on Functional imaging and modeling of the heart
Novel classification and segmentation techniques with application to remotely sensed images
Transactions on rough sets VII
Segmentation of 4D cardiac MRI: Automated method based on spatio-temporal watershed cuts
Image and Vision Computing
On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds
Journal of Mathematical Imaging and Vision
Towards a parallel topological watershed: first results
ISMM'11 Proceedings of the 10th international conference on Mathematical morphology and its applications to image and signal processing
Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts
ISMM'11 Proceedings of the 10th international conference on Mathematical morphology and its applications to image and signal processing
Fusion graphs, region merging and watersheds
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
Algorithms for the topological watershed
DGCI'05 Proceedings of the 12th international conference on Discrete Geometry for Computer Imagery
Grayscale watersheds on perfect fusion graphs
IWCIA'06 Proceedings of the 11th international conference on Combinatorial Image Analysis
Automatic photoreceptor detection in in-vivo adaptive optics retinal images: statistical validation
ICIAR'12 Proceedings of the 9th international conference on Image Analysis and Recognition - Volume Part II
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The watershed transformation is an efficient tool for segmenting grayscale images. An original approach to the watershed (Bertrand, Journal of Mathematical Imaging and Vision, Vol. 22, Nos. 2/3, pp. 217--230, 2005.; Couprie and Bertrand, Proc. SPIE Vision Geometry VI, Vol. 3168, pp. 136--146, 1997.) consists in modifying the original image by lowering some points while preserving some topological properties, namely, the connectivity of each lower cross-section. Such a transformation (and its result) is called a W-thinning, a topological watershed being an "ultimate" W-thinning. In this paper, we study algorithms to compute topological watersheds. We propose and prove a characterization of the points that can be lowered during a W-thinning, which may be checked locally and efficiently implemented thanks to a data structure called component tree. We introduce the notion of M-watershed of an image F, which is a W-thinning of F in which the minima cannot be extended anymore without changing the connectivity of the lower cross-sections. The set of points in an M-watershed of F which do not belong to any regional minimum corresponds to a binary watershed of F. We propose quasi-linear algorithms for computing M-watersheds and topological watersheds. These algorithms are proved to give correct results with respect to the definitions, and their time complexity is analyzed.