A graph-based mathematical morphology reader

This survey paper aims at providing a literary anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research.

Dimensional operators for mathematical morphology on simplicial complexes

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at url{https://code.google.com/p/math-morpho-simplicial-complexes.}

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing.

An efficient hierarchical graph based image segmentation

Hierarchical image segmentation provides region-oriented scalespace, i.e., a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Most image segmentation algorithms, such as region merging algorithms, rely on a criterion for merging that does not lead to a hierarchy, and for which the tuning of the parameters can be difficult. In this work, we propose a hierarchical graph based image segmentation relying on a criterion popularized by Felzenzwalb and Huttenlocher. We illustrate with both real and synthetic images, showing efficiency, ease of use, and robustness of our method.

Scene Parsing with Multiscale Feature Learning, Purity Trees, and Optimal Covers

Scene parsing, or semantic segmentation, consists in labeling each pixel in an image with the category of the object it belongs to. It is a challenging task that involves the simultaneous detection, segmentation and recognition of all the objects in the image. The scene parsing method proposed here starts by computing a tree of segments from a graph of pixel dissimilarities. Simultaneously, a set of dense feature vectors is computed which encodes regions of multiple sizes centered on each pixel. The feature extractor is a multiscale convolutional network trained from raw pixels. The feature vectors associated with the segments covered by each node in the tree are aggregated and fed to a classifier which produces an estimate of the distribution of object categories contained in the segment. A subset of tree nodes that cover the image are then selected so as to maximize the average purity of the class distributions, hence maximizing the overall likelihood that each segment will contain a single object. The convolutional network feature extractor is trained end-to-end from raw pixels, alleviating the need for engineered features. After training, the system is parameter free. The system yields record accuracies on the Stanford Background Dataset (8 classes), the Sift Flow Dataset (33 classes) and the Barcelona Dataset (170 classes) while being an order of magnitude faster than competing approaches, producing a 320 times 240 image labeling in less than 1 second.

On the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.