Christophe Fiorio

Two linear time Union-Find strategies for image processing

Abstract We consider Union-Find as an appropriate data structure to obtain two linear time algorithms for the segmentation of images. The linearity is obtained by restricting the order in which Unions are performed. For one algorithm the complexity bound is proven by amortizing the Find operations. For the other we use periodic updates to keep the relevant part of our Union-Find-tree of constant height. Both algorithms are generalized and lead to new linear strategies for Union-Find that are neither covered by the algorithm of Gabow and Tarjan (1984) nor by the one of Dillencourt et al. (1992).

Topological model for two-dimensional image representation: definition and optimal extraction algorithm

We define the two-dimensional topological map, a model which represents both topological and geometrical information of a two-dimensional labeled image. Since this model is minimal, complete, and unique, we can use it to define efficient image processing algorithms. The topological map is the last level of a map hierarchy. Each level represents the region boundaries of the image and is defined from the previous one in the hierarchy, thus giving a simple constructive definition. This model is similar to two existing structures but the main innovation of our approach is the progressive definition based on the successive map levels. These different maps can easily be extended in order to define the topological map in any dimension. Furthermore we provide an optimal extraction algorithm which extracts the different maps of the hierarchy in a single image scan. This algorithm is based on local configurations called precodes. Due to our constructive definition, different configurations are factorized which simplifies the implementation.

Border Map: A Topological Representation for nD Image Analysis

This article presents an algorithm computing a border map of an image that generalizes to the n dimension graph structures used in image analysis. Such a map represents simple and multiple adjacencies, inclusion of regions, as well as the frontier type between two adjacent regions. An algorithm computing a border map, linear to the number of elements of an image, is defined in 2D, then generalized in 3D and in nD.

Discrete circles: an arithmetical approach with non-constant thickness

In the present paper, we introduce an arithmetical definition of discrete circles with a non-constant thickness and we exhibit different classes of them depending on the arithmetical discrete lines. On the one hand, it results in the characterization of regular discrete circles with integer parameters as well as J. Bresenhams circles. As far as we know, it is the first arithmetical definition of the latter one. On the other hand, we introduce new discrete circles, actually the thinnest ones for the usual discrete connectedness relations.

Arithmetic discrete hyperspheres and separatingness

In the framework of the arithmetic discrete geometry, a discrete object is provided with its own analytical definition corresponding to a discretization scheme It can thus be considered as the equivalent, in a discrete space, of a Euclidean object Linear objects, namely lines and hyperplanes, have been widely studied under this assumption and are now deeply understood This is not the case for discrete circles and hyperspheres for which no satisfactory definition exists In the present paper, we try to fill this gap Our main results are a general definition of discrete hyperspheres and the characterization of the k-minimal ones thanks to an arithmetic definition based on a non-constant thickness function To reach such topological properties, we link adjacency and separatingness with norms.

Curvature estimation for discrete curves based on auto-adaptive masks of convolution

We propose a method that we call auto-adaptive convolution which extends the classical notion of convolution in pictures analysis to function analysis on a discrete set. We define an averaging kernel which takes into account the local geometry of a discrete shape and adapts itself to the curvature. Its defining property is to be local and to follow a normal law on discrete lines of any slope. We used it together with classical differentiation masks to estimate first and second derivatives and give a curvature estimator of discrete functions.

Generalized functionality for arithmetic discrete planes

The discrete plane β (a,b,c,μ,ω) is the set of integer points (x,y,z)∈ℤ satisfying 0 ≤ ax+by+cz + μ < ω. In the case ω=max(|a|,|b|,|c|),the discrete plane is said naive and is well-known to be functional on one of the coordinate planes, that is, for any point of P of this coordinate plane, there exists a unique point in the discrete plane obtained by adding to P a third coordinate. Naive planes have been widely studied, see for instance [Rev91, DRR94, DR95, AAS97, VC97, Col02, BB02].

Sorted region merging to maximize test reliability

We discuss an algorithmic approach to region merging which is built on a recent statistical work on the way to decide merging while keeping the complexity optimal. In that latter work, a concentration-based statistical test is proposed, having the particularity to reduce the error occurring when rejecting the merging of two observed regions coming from the same true region. We propose a preliminary ordered-based algorithmic procedure to cope with the errors occurring when merging two different regions in the first approach, thereby leading to a fast algorithm tailor-made for the reduction of both kinds of error. Experimentations proposed on images used without any preprocessing shed light on the quality of the segmentations obtained.

Adaptive discrete Laplace operator

Diffusion processes capture information about the geometry of an object such as its curvature, symmetries and particular points. The evolution of the diffusion is governed by the LAPLACE-BELTRAMI operator which presides to the diffusion on the manifold. In this paper, we define a new discrete adaptive Laplacian for digital objects, generalizing the operator defined on meshes. We study its eigenvalues and eigenvectors recovering interesting geometrical informations. We discuss its convergence towards the usual Laplacian operator especially on lattice of diamonds. We extend this definition to 3D shapes. Finally we use this Laplacian in classical but adaptive denoising of pictures preserving zones of interest like thin structures.

Image segmentation using a generic, fast and non-parametric approach

We investigate image segmentation by region merging. Given any similarity measure between regions, satisfying some weak constraints, we give a general predicate for answering if two regions are to be merged or not during the segmentation process. Our predicate is generic and has six properties. The first one is its independence with respect to the similarity measure, that leads to a user-independent and adaptative predicate. Second, it is non-parametric, and does not rely on any assumption concerning the image. Third, due to its weak constraints, knowledge may be included in the predicate to fit better to the users behaviour. Fourth, provided the similarity is well chosen by the user, we are able to upperbound one type of error made during the image segmentation. Fifth, it does not rely on a particular segmentation algorithm and can be used with almost all region merging algorithms in various application domains. Sixth, it is calculated quickly, and can lead with appropriated algorithms to very efficient segmentation.