Mohamed Tajine

Recognizing arithmetic straight lines and planes

The problem of recognizing a straight line in the discrete plane ℤ2 (resp. a plane in ℤ3) is to find an algorithm deciding wether a given set of points in ℤ2 (resp. ℤ3) belongs to a line (resp. a plane). In this paper the lines and planes are arithmetic, as defined by Reveilles [Rev91], and the problem is translated, for any width that is a linear function of the coefficients of the normal to the searched line or plane, into the problem of solving a set of linear inequalities. This new problem is solved by using the Fouriers elimination algorithm. If there is a solution, the family of solutions is given by the algorithm as a conjunction of linear inequalities. This method of recognition is well suited to computer imagery, because any traversal algorithm of the given set is possible, and also because any incomplete segment of line or plane can be recognized.

Discretization in Hausdorff Space

In this paper, a new approach to the discretization of n-dimensional Euclidean figures is studied: the discretization of a compact Euclidean set K is a discrete set S whose Hausdorff distance to K is minimal; in particular such a discretization depends on the choice of a metric in the Euclidean space, for example the Euclidean or a chamfer distance. We call such a set S a Hausdorff discretizing set of K. The set of Hausdorff discretizing sets of K is nonvoid, finite, and closed under union; we consider thus in particular the greatest one among such sets, which we call the maximal Hausdorff discretization of K. We give a mathematical description of Hausdorff discretizing sets: it is related to the discretization by dilation considered by Heijmans and Toet and the cover discretization studied by Andrès. We have a bound on the Hausdorff distance between a compact set and its maximal Hausdorff discretization, and the latter converges (for the Hausdorff metric) to the compact set when the spacing of the discrete grid tends to zero. Such a convergence result holds also for the discretization by dilation when the structuring element satisfies the covering assumption. Our approach is here the most general possible. In a next paper we will consider the case where the underlying metric on points satisfies some general constraints in relation to the cells associated to the discrete points, and we will then see that these constraints guarantee that the usual supercover and cover discretizations give indeed Hausdorff discretizing sets.

The recursive deterministic perceptron neural network

We introduce a feedforward multilayer neural network which is a generalization of the single layer perceptron topology (SLPT), called recursive deterministic perceptron (RDP). This new model is capable of solving any two-class classification problem, as opposed to the single layer perceptron which can only solve classification problems dealing with linearly separable sets (two subsets X and Y of R(d) are said to be linearly separable if there exists a hyperplane such that the elements of X and Y lie on the two opposite sides of R(d) delimited by this hyperplane). We propose several growing methods for constructing a RDP. These growing methods build a RDP by successively adding intermediate neurons (IN) to the topology (an IN corresponds to a SLPT). Thus, as a result, we obtain a multilayer perceptron topology, which together with the weights, are determined automatically by the constructing algorithms. Each IN augments the affine dimension of the set of input vectors. This augmentation is done by adding the output of each of these INs, as a new component, to every input vector. The construction of a new IN is made by selecting a subset from the set of augmented input vectors which is LS from the rest of this set. This process ends with LS classes in almost n-1 steps where n is the number of input vectors. For this construction, if we assume that the selected LS subsets are of maximum cardinality, the problem is proven to be NP-complete. We also introduce a generalization of the RDP model for classification of m classes (m>2) allowing to always separate m classes. This generalization is based on a new notion of linear separability for m classes, and it follows naturally from the RDP. This new model can be used to compute functions with a finite domain, and thus, to approximate continuous functions. We have also compared - over several classification problems - the percentage of test data correctly classified, or the topology of the 2 and m classes RDPs with that of the backpropagation (BP), cascade correlation (CC), and two other growing methods.

Tree matching applied to vascular system

In this paper, we propose an original tree matching algorithm for intra-patient hepatic vascular system registration. The vascular systems are segmented from CT-Scan images acquired at different time, and then modeled as trees. The goal of this algorithm is to find common bifurcations (nodes) and vessels (edges) in both trees. Starting from the tree root, edges and nodes are iteratively matched. The algorithm works on a set of matching hypotheses which is updated to keep best matches. It is robust against topological modification, as the segmentation process can fail to detect some branches. Finally, this algorithm is validated on the Visible Human with synthetic deformations thanks to the simulator prototype developed at the INRIA which provides realistic deformations for liver and its vascular network.

Liver registration for the follow-up of hepatic tumors

In this paper we propose a new two step method to register the liver from two acquisitions. This registration helps experts to make an intra-patient follow-up for hepatic tumors. Firstly, an original and efficient tree matching is applied on different segmentations of the vascular system of a single patient. These vascular systems are segmented from CT-scan images acquired (every six months) during disease treatement, and then modeled as trees. Our method matches common bifurcations and vessels. Secondly, an estimation of liver deformation is computed from the results of the first step. This approach is validated on a large synthetic database containing cases with various deformation and segmentation problems. In each case, after the registration process, the liver recovery is very accurate (around 95%) and the mean localization error for 3D landmarks in liver is small (around 4 mm).

New methods for testing linear separability

This paper introduces latest advances in the subject of linear separability. New methods for testing linear separability are introduced. This is a very important area of work which can help simplify the topology of a neural network by using a single layer perceptron when the problem at hand is linearly separable. The research presented in this paper has allowed researchers to enhance the performance of the RDP neural network. It appears in one of the leading journals of Neural Networks.

On Local Definitions of Length of Digital Curves

In this paper we investigate the ‘local’ definitions of length of digital curves in the digital space r ℤ2 where r is the resolution of the discrete space. We prove that if μ r is any local definition of the length of digital curves in r ℤ2, then for almost all segments S of ℝ2, the measure μ r (S r ) does not converge to the length of S when the resolution r converges to 0, where S r is the Bresenham discretization of the segment S in r ℤ2. Moreover, the average errors of classical local definitions are estimated, and we define a new one which minimizes this error.

Design of robust vascular tree matching: validation on liver

In this paper, we propose an original and efficient tree matching algorithm for intra-patient hepatic vascular system registration. Vascular systems are segmented from CT-scan images acquired at different times, and then modeled as trees. The goal of this algorithm is to find common bifurcations (nodes) and vessels (edges) in both trees. Starting from the tree root, edges and nodes are iteratively matched. The algorithm works on a set of match solutions that are updated to keep the best matches thanks to a quality criterion. It is robust against topological modifications due to segmentation failures and against strong deformations. Finally, this algorithm is validated on a large synthetic database containing cases with various deformation and segmentation problems.

Growing methods for constructing recursive deterministic perceptron neural networks and knowledge extraction

Abstract The Recursive Deterministic Perceptron (RDP) feedforward multilayer neural network is a generalization of the single layer perceptron topology (SLPT). This new model is capable of solving any two-class classification problem, as opposed to the single layer perceptron which can only solve classification problems dealing with linearly separable (LS) sets (two subsets X and Y of R d are said to be linearly separable if there exists a hyperplane such that the elements of X and Y lie on the two opposite sides of R d delimited by this hyperplane). For all classification problems, the construction of an RDP is done automatically and thus, the convergence to a solution is always guaranteed. We propose three growing methods for constructing an RDP neural network. These methods perform, respectively, batch, incremental, and modular learning. We also show how the knowledge embedded in an RDP neural network model can always be expressed, transparently, as a finite union of open polytopes. The combination of the decision region of RDP models, by using boolean operations, is also discussed.

Hausdorff Discretization and Its Comparison to Other Discretization Schemes

We study the problem of discretization in a Hausdorff space followed in [WTR 98]. We recall the definitions and properties of the Hausdorff discretization of a compact set. We also study the relationship between the covering discretizations and the Hausdorff discretization. For a cellular metric every covering discretization minimizes the Hausdorff distance, and conversely, if the supercover discretization minimizes the Hausdorff distance then the metric is cellular. The supercover discretization is the Hausdorff discretization i? the metric is proportional to d1. We compare also the Hausdorff discretization and the Bresenham discretization [Bres 65]. Actually, the Bresenham discretization of a segment of IR2 is not always a good discretization relatively to a Hausdorff metric.