Michel Couprie

Building the Component Tree in Quasi-Linear Time

The level sets of a map are the sets of points with level above a given threshold. The connected components of the level sets, thanks to the inclusion relation, can be organized in a tree structure, that is called the component tree. This tree, under several variations, has been used in numerous applications. Various algorithms have been proposed in the literature for computing the component tree. The fastest ones (considering the worst-case complexity) have been proven to run in O(nln(n)). In this paper, we propose a simple to implement quasi-linear algorithm for computing the component tree on symmetric graphs, based on Tarjans union-find procedure. We also propose an algorithm that computes the n most significant lobes of a map

Watershed Cuts: Thinnings, Shortest Path Forests, and Topological Watersheds

We recently introduced watershed cuts, a notion of watershed in edge-weighted graphs. In this paper, our main contribution is a thinning paradigm from which we derive three algorithmic watershed cut strategies: The first one is well suited to parallel implementations, the second one leads to a flexible linear-time sequential implementation, whereas the third one links the watershed cuts and the popular flooding algorithms. We state that watershed cuts preserve a notion of contrast, called connection value, on which several morphological region merging methods are (implicitly) based. We also establish the links and differences between watershed cuts, minimum spanning forests, shortest path forests, and topological watersheds. Finally, we present illustrations of the proposed framework to the segmentation of artwork surfaces and diffusion tensor images.

Quasi-Linear Algorithms for the Topological Watershed

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.

Discrete bisector function and Euclidean skeleton in 2D and 3D

We propose a new definition and an algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient.

Segmentation of 4D cardiac MRI: Automated method based on spatio-temporal watershed cuts

Based on discrete mathematical morphology, we introduce in this paper a new watershed framework which allows for segmenting spatio-temporal images, that we apply to medical image analysis. Specifically, we propose a new automated and fast procedure to segment the left ventricular myocardium in 4D (3D+t) cine-MRI sequences. Both quantitative and qualitative evaluation are provided. Thanks to the comparison with manual segmentation performed by two cardiologists, we demonstrate the accuracy of the proposed method. The relevance of the ejection fraction and myocardium mass measured from segmentations is also assessed. Furthermore, we show that the proposed 4D procedure allows to keep the temporal coherency between the successive 3D segmentations obtained along the time sequence. In an effort to promote open science, both the data and the software are available on-line.

Video segmentation based on 2D image analysis

The video segmentation problem consists in the identification of the boundary between consecutive shots. The common approach to solve this problem is based on dissimilarity measures between frames. In this work, the video segmentation problem is transformed into a problem of pattern detection, where each video event is transformed into a different pattern on a 2D image, called visual rhythm, obtained by a specific transformation. In our analysis we use topological and morphological tools to detect cuts. Also, we use discrete line analysis and max tree analysis to detect fade transitions and flashes, respectively. We present a comparative analysis of our method for cut detection with respect to some other methods, which shows the better results of our method.

Prototyping DBS3, a shared-memory parallel database system

DBS3 is a database system with extended relational capabilities designed for a shared-memory multiprocessor. This paper presents the design choices, architecture and performance evaluation of the current DBS3 prototype. The major contributions of DBS3 are: a parallel dataflow execution model based on data declustering, the compile-time optimization of both independent and pipelined parallelism, and the exploitation of shared-memory for efficient concurrent execution. The current DBS3 prototype runs on a shared-memory, commercially available multiprocessor. The initial performance experiments for single-user queries are promising and show excellent response times and scalability.<<ETX>>

Some links between extremum spanning forests, watersheds and min-cuts

Minimum cuts, extremum spanning forests and watersheds have been used as the basis for powerful image segmentation procedures. In this paper, we present some results about the links which exist between these different approaches. Especially, we show that extremum spanning forests are particular cases of watersheds from arbitrary markers and that min-cuts coincide with extremum spanning forests for some particular weight functions.

Watersheds, mosaics, and the emergence paradigm

In this paper, we investigate the links between the flooding paradigm and the topological watershed. Guided by the analysis of a classical flooding algorithm, we present several notions that lead us to a better understanding of the watershed: minima extension, mosaic, pass value and separation. We first make a detailed examination of the effectiveness of the divide set produced by watershed algorithms. We introduce the mosaic to retrieve the altitude of points along the divide set. A desirable property is that, when two minima are separated by a crest in the original image, they are still separated by a crest of the same altitude in the mosaic. Our main result states that this is the case if and only if the mosaic is obtained through a topological thinning. We investigate the possibility for a flooding to produce a topological watershed, and conclude that this is not feasible. This leads us to reverse the flooding paradigm, and to propose a notion of emergence. An emergence process is a transformation based on a topological criterion, in which points are processed in decreasing altitude order while preserving the number of connected components of lower cross-sections. Our main result states that any emergence watershed is a topological watershed, and more remarkably, that any topological watershed of a given image can be obtained as an emergence watershed of the image.

Watershed Algorithms and Contrast Preservation

This paper is devoted to the study of watershed algorithms behavior. Through the introduction of a concept of pass value, we show that most classical watershed algorithms do not allow the retrieval of some important topological features of the image (in particular, saddle points are not correctly computed). An important consequence of this result is that it is not possible to compute sound measures such as depth, area or volume of basins using most classical watershed algorithms. Only one watershed principle, called topological watershed, produces correct watershed contours.