Jacques-Olivier Lachaud

Deformable meshes with automated topology changes for coarse-to-fine three-dimensional surface extraction

This work presents a generic deformable model for extracting objects from volumetric data with a coarse-to-fine approach. This model is based on a dynamic triangulated surface which alters its geometry according to internal and external constraints to perform shape recovery. A new framework for topology changes is proposed to extract complex objects: within this framework, the model dynamically adapts its topology to the geometry of its vertices according to simple distance constraints. In order to speed up the process, an algorithm of pyramid construction with any reduction factor transforms the image into a set of images with progressively higher resolutions. This organization into a hierarchy, combined with a model which can adapt its sampling to the resolution of the workspace, enables a fast estimation of the shapes included in the image. After that, the model searches for finer and finer details while relying successively on the different levels of the pyramid.

Continuous Analogs of Digital Boundaries: A Topological Approach to Iso-Surfaces

The definition and extraction of objects and their boundaries within an image are essential in many imaging applications. Classically, two approaches are followed. The first considers the image as a sample of a continuous scalar field: boundaries are implicit surfaces in this field; they are often called iso-surfaces. The second considers the image as a digital space with adjacency relations and classifies elements of this space as inside or outside: boundaries are pairs composed of one inside element and one outside element; they are called digital boundaries. In this paper, we show that these two approaches are closely related. This statement holds for arbitrary dimensions. To do so, we propose a local method to construct a continuous analog of a digital boundary. The continuous analog is designed to satisfy properties in the Euclidean space that are similar to the properties of its counterpart in the digital space (e.g., connectedness, closeness, separation). It appears that this continuous analog is indeed a piecewise linear approximation of an iso-(hyper)surface (i.e., a triangulated iso-surface in the three-dimensional case). Furthermore, we derive significant digital boundary properties from its continuous analog using the Jordan–Brouwer separation theorem: new Jordan pairs, new adjacencies between boundary elements, new Jordan triples. We conclude this paper by illustrating the 3D case more precisely. In particular, we show that a digital boundary can be transformed directly into a triangulated iso-surface. The implementation of this transformation and its efficiency are discussed with a comparison with the classical marching-cubes algorithm.

Fast, accurate and convergent tangent estimation on digital contours

This paper presents a new tangent estimator to digitized curves based on digital line recognition. It outperforms existing ones on important criteria while keeping the same computation time: accuracy on smooth or polygonal shapes, isotropy, preservation of inflexion points and convexity, asymptotic behaviour. Its asymptotic convergence (sometimes called multigrid convergence) is proved in the case of convex shapes with C^3 boundary.

Analysis and comparative evaluation of discrete tangent estimators

This paper presents a comparative evaluation of tangent estimators based on digital line recognition on digital curves. The comparison is carried out with a comprehensive set of criteria: accuracy on smooth or polygonal shapes, behaviour on convex/concave parts, computation time, isotropy, asymptotic convergence. We further propose a new estimator mixing the qualities of existing ones and outperforming them on most mentioned points.

Delaunay conforming iso-surface, skeleton extraction and noise removal

Iso-surfaces are routinely used for the visualization of volumetric structures. Further processing (such as quantitative analysis, morphometric measurements, shape description) requires volume representations. The skeleton representation matches these requirements by providing a concise description of the object. This paper has two parts. First, we exhibit an algorithm which locally builds an iso-surface with two significant properties: it is a 2-manifold and the surface is a subcomplex of the Delaunay tetrahedrization of its vertices. Secondly, because of the latter property, the skeleton can in turn be computed from the dual of the Delaunay tetrahedrization of the iso-surface vertices. The skeleton representation, although informative, is very sensitive to noise. This is why we associate a graph to each skeleton for two purposes: (i) the amount of noise can be identified and quantified on the graph and (ii) the selection of the graph subpart that does not correspond to noise induces a filtering on the skeleton. Finally, we show some results on synthetic and medical images. An application, measuring the thickness of objects (heart ventricles, bone samples) is also presented.

Curvature estimation along noisy digital contours by approximate global optimization

In this paper, we introduce a new curvature estimator along digital contours, which we called global min-curvature (GMC) estimator. As opposed to previous curvature estimators, it considers all the possible shapes that are digitized as this contour, and selects the most probable one with a global optimization approach. The GMC estimator exploits the geometric properties of digital contours by using local bounds on tangent directions defined by the maximal digital straight segments. The estimator is then adapted to noisy contours by replacing maximal segments with maximal blurred digital straight segments. Experiments on perfect and damaged digital contours are performed and in both cases, comparisons with other existing methods are presented.

Convex Digital Polygons, Maximal Digital Straight Segments and Convergence of Discrete Geometric Estimators

Discrete geometric estimators approach geometric quantities on digitized shapes without any knowledge of the continuous shape. A classical yet difficult problem is to show that an estimator asymptotically converges toward the true geometric quantity as the resolution increases. For estimators of local geometric quantities based on Digital Straight Segment (DSS) recognition this problem is closely linked to the asymptotic growth of maximal DSS for which we show bounds both about their number and sizes on Convex Digital Polygons. These results not only give better insights about digitized curves but indicate that curvature estimators based on local DSS recognition are not likely to converge. We indeed invalidate a conjecture which was essential in the only known convergence theorem of a discrete curvature estimator. The proof involves results from arithmetic properties of digital lines, digital convexity, combinatorics and continued fractions.

Lyndon + Christoffel = digitally convex

Discrete geometry redefines notions borrowed from Euclidean geometry creating a need for new algorithmical tools. The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis. To the many different approaches of digital convexity, we propose the combinatorics on words point of view, unnoticed until recently in the pattern recognition community. In this paper, we provide first a fast optimal algorithm checking digital convexity of polyominoes coded by their contour word. The result is based on linear time algorithms for both computing the Lyndon factorization of the contour word and the recognition of Christoffel factors that are approximations of digital lines. By avoiding arithmetical computations the algorithm is much simpler to implement and much faster in practice. We also consider the convex hull computation and relate previous work in combinatorics on words with the classical Melkman algorithm.

Deformable model with adaptive mesh and automated topology changes

Due to their general and robust formulation deformable models offer a very appealing approach to 3D image segmentation. However there is a trade-off between model genericity, model accuracy and computational efficiency. In general, fully generic models require a uniform sampling of either the space or their mesh. The segmentation accuracy is thus a global parameter. Recovering small image features results in heavy computational costs whereas generally only restricted parts of images require a high segmentation accuracy. We present a highly deformable model that both handles fully automated topology changes and adapts its resolution locally according to the geometry of image features. The main idea is to replace the Euclidean metric with a Riemannian metric that expands interesting parts of the image. Then, a regular sampling is maintained with this new metric. This allows to automatically handle topology changes while increasing the model resolution locally according to the geometry of image components. By this way high quality segmentation is achieved with reduced computational costs.

Deformable model with a complexity independent from image resolution

We present a parametric deformable model which recovers image components with a complexity independent from the resolution of input images. The proposed model also automatically changes its topology and remains fully compatible with the general framework of deformable models. More precisely, the image space is equipped with a metric that expands salient image details according to their strength and their curvature. During the whole evolution of the model, the sampling of the contour is kept regular with respect to this metric. By this way, the vertex density is reduced along most parts of the curve while a high quality of shape representation is preserved. The complexity of the deformable model is thus improved and is no longer influenced by feature-preserving changes in the resolution of input images. Building the metric requires a prior estimation of contour curvature. It is obtained using a robust estimator which investigates the local variations in the orientation of image gradient. Experimental results on both computer generated and biomedical images are presented to illustrate the advantages of our approach.