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.Comment: 19 pages, double-colum

Writing Reusable Digital Geometry Algorithms in a Generic Image Processing Framework

Digital Geometry software should reflect the generality of the underlying mathe- matics: mapping the latter to the former requires genericity. By designing generic solutions, one can effectively reuse digital geometry data structures and algorithms. We propose an image processing framework focused on the Generic Programming paradigm in which an algorithm on the paper can be turned into a single code, written once and usable with various input types. This approach enables users to design and implement new methods at a lower cost, try cross-domain experiments and help generalize resultsComment: Workshop on Applications of Discrete Geometry and Mathematical Morphology, Istanb : France (2010

Constructive links between some morphological hierarchies on edge-weighted graphs

International audienceIn edge-weighted graphs, we provide a unified presentation of a family of popular morphological hierarchies such as component trees, quasi flat zones, binary partition trees, and hierarchical watersheds. For any hierarchy of this family, we show if (and how) it can be obtained from any other element of the family. In this sense, the main contribution of this paper is the study of all constructive links between these hierarchies

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

A study of observation scales based on Felzenswalb-Huttenlocher dissimilarity measure for hierarchical segmentation

International audienceHierarchical 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. Guimarães et al. proposed a hierarchical graph based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissimilarity. This HGB method computes, for each edge of a graph, the minimum scale in a hierarchy at which two regions linked by this edge should merge according to the dissimilarity. In order to generalize this method, we first propose an algorithm to compute the intervals which contain all the observation scales at which the associated regions should merge. Then, following the current trend in mathematical morphology to study criteria which are not increasing on a hierarchy, we present various strategies to select a significant observation scale in these intervals. We use the BSDS dataset to assess our observation scale selection methods. The experiments show that some of these strategies lead to better segmentation results than the ones obtained with the original HGB method

Climbing: A Unified Approach for Global Constraints on Hierarchical Segmentation

International audienceThe paper deals with global constraints for hierarchical segmentations. The proposed framework associates, with an input image, a hierarchy of segmentations and an energy, and the subsequent optimization problem. It is the first paper that compiles the different global constraints and unifies them as Climbing energies. The transition from global optimization to local optimization is attained by the h-increasingness property, which allows to compare parent and child partition energies in hierarchies. The laws of composition of such energies are established and examples are given over the Berkeley Dataset for colour and texture segmentation

A mutual reference shape based on information theory

International audienceIn this paper, we propose to consider the estimation of a refer-ence shape from a set of different segmentation results using both active contours and information theory. The reference shape is defined as the minimum of a criterion that benefits from both the mutual information and the joint entropy of the input segmentations and called a mutual shape. This energy criterion is here justified using similarities between informa-tion theory quantities and area measures, and presented in a continuous variational framework. This framework brings out some interesting evaluation measures such as the speci-ficity and sensitivity. In order to solve this shape optimization problem, shape derivatives are computed for each term of the criterion and interpreted as an evolution equation of an active contour. Some synthetical examples allow us to cast the light on the difference between our mutual shape and an average shape. Our framework has been considered for the estimation of a mutual shape for the evaluation of cardiac segmentation methods in MRI

Axiomatizing Discrete Spatial Relations

Qualitative spatial relations are used in artificial intelligence to model commonsense notions such as regions of space overlapping, touching only at their boundaries, or being separate. In this paper we extend earlier work on qualitative relations in discrete space by pre- senting a bi-intuitionistic modal logic with universal modalities, called UBiSKt. This logic has a semantics in which formulae are interpreted as subgraphs. We show how a variety of qualitative spatial relations can be defined in UBiSKt. We make essential use of a sound and complete axiomatisation of the logic and an implementation of a tableau based theorem prover to establish novel properties of these spatial relations. We also explore the role of UBiSKt in expressing spatial relations at more than one level of detail. The features of the logic allow it to rep- resent how a subgraph at a detailed level is approximated at a coarser level

Pattern Spectra from Different Component Trees for Estimating Soil Size Distribution

We study the pattern spectra in context of soil structure analysis. Good soil structure is vital for sustainable crop growth. Accurate and fast measuring methods can contribute greatly to soil management decisions. However, the current in-field approaches contain a degree of subjectivity, while obtaining quantifiable results through laboratory techniques typically involves sieving the soil which is labour- and time-intensive. We aim to replace this physical sieving process through image analysis, and investigate the effectiveness of pattern spectra to capture the size distribution of the soil aggregates. We calculate the pattern spectra from partitioning hierarchies in addition to the traditional max-tree. The study is posed as an image retrieval problem, and confirms the ability of pattern spectra and suitability of different partitioning trees to re-identify soil samples in different arrangements and scales

Liver segmentation in MRI: a fully automatic method based on stochastic partitions

There are few fully automated methods for liver segmentation in magnetic resonance images (MRI) despite the benefits of this type of acquisition in comparison to other radiology techniques such as computed tomography (CT). Motivated by medical requirements, liver segmentation in MRI has been carried out. For this purpose, we present a new method for liver segmentation based on the watershed transform and stochastic partitions. The classical watershed over-segmentation is reduced using a marker-controlled algorithm. To improve accuracy of selected contours, the gradient of the original image is successfully enhanced by applying a new variant of stochastic watershed. Moreover, a final classifier is performed in order to obtain the final liver mask. Optimal parameters of the method are tuned using a training dataset and then they are applied to the rest of studies (17 datasets). The obtained results (a Jaccard coefficient of 0.91 +/- 0.02) in comparison to other methods demonstrate that the new variant of stochastic watershed is a robust tool for automatic segmentation of the liver in MRI. (C) 2014 Elsevier Ireland Ltd. All rights reserved.This work has been supported by the MITYC under the project NaRALap (ref. TSI-020100-2009-189), partially by the CDTI under the project ONCOTIC (IDI-20101153), by Ministerio de Educacion y Ciencia Spain, Project Game Teen (TIN2010-20187) projects Consolider-C (SEJ2006-14301/PSIC), "CIBER of Physiopathology of Obesity and Nutrition, an initiative of ISCIII" and Excellence Research Program PROMETEO (Generalitat Valenciana. Conselleria de Educacion, 2008-157). We would like to express our gratitude to the Hospital Clinica Benidorm, for providing the MR datasets and to the radiologist team of Inscanner for the manual segmentation of the MR images.López-Mir, F.; Naranjo Ornedo, V.; Angulo, J.; Alcañiz Raya, ML.; Luna, L. (2014). Liver segmentation in MRI: a fully automatic method based on stochastic partitions. Computer Methods and Programs in Biomedicine. 114(1):11-28. https://doi.org/10.1016/j.cmpb.2013.12.022S1128114