There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

From Discourse to Logic

Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models-the ability to automatically handle topology changes-turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved. We present a new class of geometric deformable models designed using a novel topology-preserving level set method, which achieves topology preservation by applying the simple point concept from digital topology. These new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces. Moreover, since the topology-preserving constraint is enforced efficiently through local computations, the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models. Several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm.

/pdf/a-topology-preserving-level-set-method-for-geometric-4fk3qlpafn.pdf

A topology preserving level set method for geometric deformable models

Curve-skeletons are thinned 1D representations of 3D objects useful for many visualization tasks including virtual navigation, reduced-model formulation, visualization improvement, animation, etc. There are many algorithms in the literature describing extraction methodologies for different applications; however, it is unclear how general and robust they are. In this paper, we provide an overview of many curve-skeleton applications and compile a set of desired properties of such representations. We also give a taxonomy of methods and analyze the advantages and drawbacks of each class of algorithms.

/pdf/curve-skeleton-properties-applications-and-algorithms-8n6p5tz4yb.pdf

Curve-Skeleton Properties, Applications, and Algorithms

On the dynamics

We study the watersheds in edge-weighted graphs. We define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface. We first establish the consistency of these watersheds: they can be equivalently defined by their "catchment basinsrdquo (through a steepest descent property) or by the "dividing linesrdquo separating these catchment basins (through the drop of water principle). Then, we prove, through an equivalence theorem, their optimality in terms of minimum spanning forests. Afterward, we introduce a linear-time algorithm to compute them. To the best of our knowledge, similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm, both in theory and in practice. Finally, the defined concepts are illustrated in image segmentation, leading to the conclusion that the proposed approach improves, on the tested images, the quality of watershed-based segmentations.

/pdf/watershed-cuts-minimum-spanning-forests-and-the-drop-of-3uvf1qi1ht.pdf

Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle

Simple points, topological numbers and geodesic neighborhoods in cubic grids

A Boolean characterization of three-dimensional simple points

An approach to the segmentation of a discrete 3-D object into a structure of characteristic topological primitives with attached qualitative features is proposed. This structure can be seen as a qualitative description of the object. The approach concentrates on topological properties of discrete surfaces. These surfaces may correspond to the external surface of the objects, or to the skeleton surface. The labeling algorithm is based on very local computations, allowing massively parallel and real-time computations. >

/pdf/topological-segmentation-of-discrete-surfaces-2gu6k8ltt9.pdf

Gilles Bertrand

Papers

On the dynamics

Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle

Simple points, topological numbers and geodesic neighborhoods in cubic grids

A Boolean characterization of three-dimensional simple points

Topological segmentation of discrete surfaces