Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and we characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasiflat zones preservation. In practice, these results make up a toolkit for designing new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies. More precisely, we show the practical interest of the proposed framework for: i) hierarchical watershed image segmentations, ii) combinations of different hierarchical segmentations, iii) hierarchicalizations of some non-hierarchical image segmentation methods based on regional dissimilarities, and iv) hierarchical analysis of geographical data.

International audienceRigid transformations are involved in a wide range of digital image processing applications. When applied on such discrete images, rigid transformations are however usually performed in their associated continuous space, then requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as a fully discrete process. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on any subset of Z^2 of size N * N. We describe this combinatorial structure, which presents a space complexity O (N^9) and we propose an algorithm enabling to build it in linear time with respect to this space complexity. This algorithm, which handles real (i.e. non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation

Hierarchical image segmentation provides a region-oriented scale-space, 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. In addition, for image segmentation, 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 Felzenszwalb and Huttenlocher. Quantitative and qualitative assessments of the method on Berkeley image database shows efficiency, ease of use and robustness of our method.

The original version of the article was published in Mathematical Morphology - Theory and Applications 2 (2017) 55–75. Unfortunately, the original version contains a mistake: in the definition of Dif (C1, C2) in Section 3.6, max should be replaced by min. In this erratum we correct the formula defining Dif (C1, C2).

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Rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images. We suppose that a digital image A is transformed to another digital image B by a rotation. In the discrete plane, there are many angles giving the rotation from A to B, which we called admissible rotation angles from A to B. For such a set of admissible rotation angles, there exist two angles that achieve the lower and the upper bounds. To find those lower and upper bounds, we use hinge angles as used in [7]. A sequence of hinge angles is a set of particular angles determined by a digital image in the sense that any angle between two consecutive hinge angles gives the identical rotation of the digital image. We propose a method for obtaining the lower and the upper bounds of admissible rotation angles using hinge angles from a given Euclidean angle or from a pair of corresponding digital images.

Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In R n , they are both topology and geometry preserving. Unfortunately, these properties are generally lost in Z n . In particular, when applying a rigid motion on a digital object, one generally alters its structure but also the global shape of its boundary. These alterations are mainly caused by digitization during the transformation process. In this specific context, some solutions for the handling of topological issues were proposed in Z 2 . In this article, we also focus on geometric issues in Z 2 . Indeed, we propose a rigid motion scheme that preserves geometry and topology properties of the transformed digital object: a connected object will remain connected, and some geometric properties (e.g. convexity, area and perimeter) will be preserved. To reach that goal, our main contributions are twofold. First, from an algorithmic point of view, our scheme relies on (1) a polygonization of the digital object, (2) piecewise affine object of R 2 allows us to avoid the geometric alterations generally induced by standard pointwise rigid motions. However, the final digitization of the polygon back to Z 2 has to be carried out cautiously. In particular, our second, theoretical contribution is a notion of quasi-regularity that provides sufficient conditions to be fulfilled by a continuous object for guaranteeing both topology and geometry preservation during its digitization.

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