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
Topographic distance and watershed lines
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
A connected component approach to the watershed segmentation
ISMM '98 Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing
The watershed transform: definitions, algorithms and parallelization strategies
Fundamenta Informaticae - Special issue on mathematical morphology
IFT-Watershed from Gray-Scale Marker
SIBGRAPI '02 Proceedings of the 15th Brazilian Symposium on Computer Graphics and Image Processing
The Image Foresting Transform: Theory, Algorithms, and Applications
IEEE Transactions on Pattern Analysis and Machine Intelligence
Journal of Mathematical Imaging and Vision
Quasi-Linear Algorithms for the Topological Watershed
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)
Tie-Zone Watershed, Bottlenecks, and Segmentation Robustness Analysis
SIBGRAPI '05 Proceedings of the XVIII Brazilian Symposium on Computer Graphics and Image Processing
Algorithms for the topological watershed
DGCI'05 Proceedings of the 12th international conference on Discrete Geometry for Computer Imagery
Collapses and Watersheds in Pseudomanifolds
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Relationships between some watershed definitions and their tie-zone transforms
Image and Vision Computing
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There are many watershed transform algorithms in literature but most of them do not exactly correspond to their respective definition. The solution given by such algorithms depends on their implementation. Others fit with their definition which allows multiple solutions. The solution chosen by such algorithms depends on their implementation too. It is the case of the watershed by image foresting transform that consists in building a forest of trees with minimum path-costs. The recently introduced tie-zone watershed (TZW) has the advantage of not depending on arbitrary implementation choices: it provides a unique and, thereby, unbiased solution. Indeed, the TZW considers all possible solutions of the watershed transform and keeps only the common parts of them as catchment basins whereas parts that differ form a tie zone disputed by many solutions. Although the TZW insures the uniqueness of the solution, it does not prevent from possible large tie zones, sometimes unwanted in segmentation applications. This paper presents a special thinning of the tie zone that leads to a unique solution. Observing all the possible solutions of the watershed by image foresting transform, one can deduce the frequency of the labels associated with each pixel. The thinning consists in assigning the most frequent label while preserving the segmented region connectivity. We demonstrate that the label frequency can be computed both from an immersion-like recursive formula and the proposed fragmented drop paradigm. Finally, we propose an algorithm under the IFT framework that computes the TZW, the label frequency and the thinning simultaneously and without explicit calculation of all the watershed solutions.