A numerical approach to compute the topology of the Apparent Contour of a smooth mapping from R2 to R2

A rigorous algorithm for computing the topology of the Apparent Contour of a generic smooth map is designed and studied in this paper. The source set is assumed to be a simply connected compact subset of the plane and the target space is the plane. Whitney proved that, generically, critical points of a smooth map are folds or cusps (Whitney, 1955). The Apparent Contour is the set of critical values, that is, the image of the critical points. Generically speaking, the Apparent Contour does not have triple points and double points are normal crossings (i.e. crossing without tangency). Each of those particular cases, cusp and normal crossing, is described in order to be rigorously handled by an interval analysis based scheme. The first step of the presented method provides an enclosure of those particular points. The second part of the designed method is a computation of a graph which is homeomorphic to the Apparent Contour. Edges of this graph are computed by testing connectivity of those particular points in the source space. This paper also defines a concept called portrait. Relations between this notion and the more classical notion of Apparent Contour are discussed

Guaranteeing the homotopy type of a set defined by non-linear inequalities

This paper provides an effective method to create an abstract simplicial complex homotopy equivalent to a given set S described by non-linear inequalities (polynomial or not). To our knowledge, no other numerical algorithm is able to deal with this type of problem. The proposed approach divides S into subsets that have been proven to be contractible using interval arithmetic. The method is close to Čech cohomology and uses the nerve theorem. Some examples illustrate the principle of the approach. This algorithm has been implemented

Cell cycle genes regulate vestigial and scalloped to ensure normal proliferation in the wing disc of Drosophila melanogaster.

In Drosophila, the Vestigial-Scalloped (VG-SD) dimeric transcription factor is required for wing cell identity and proliferation. Previous results have shown that VG-SD controls expression of the cell cycle positive regulator dE2F1 during wing development. Since wing disc growth is a homeostatic process, we investigated the possibility that genes involved in cell cycle progression regulate vg and sd expression in feedback loops. We focused our experiments on two major regulators of cell cycle progression: dE2F1 and the antagonist dacapo (dap). Our results reinforce the idea that VG/SD stoichiometry is critical for correct development and that an excess in SD over VG disrupts wing growth. We reveal that transcriptional activity of VG-SD and the VG/SD ratio are both modulated upon down-expression of cell cycle genes. We also detected a dap-induced sd upregulation that disrupts wing growth. Moreover, we observed a rescue of a vg hypomorphic mutant phenotype by dE2F1 that is concomitant with vg and sd induction. This regulation of the VG-SD activity by dE2F1 is dependent on the vg genetic background. Our results support the hypothesis that cell cycle genes fine-tune wing growth and cell proliferation, in part, through control of the VG/SD stoichiometry and activity. This points to a homeostatic feedback regulation between proliferation regulators and the VG-SD wing selector

On Sufficient Conditions of the Injectivity: Development of a Numerical Test Algorithm via Interval Analysis

This paper presents a new numerical algorithm based on interval analysis able to verify that a continuously differentiable function is injective. The efficiency of the method is demonstrated by illustrative examples. These examples have been treated by a C++ solver which is made available

Approach for sequential image interpretation using a priori binary perceptual topological and photometric knowledge and k-means-based segmentation

International audienc

A graph based image interpretation method using a priori qualitative inclusion and photometric relationships

This paper presents a method for recovering and identifying image regions from an initial oversegmentation using qualitative knowledge of its content. Compared to recent works favoring spatial information and quantitative techniques, our approach focuses on simple a priori qualitative inclusion and photometric relationships such as "region A is included in region B", "the intensity of region A is lower than the one of region B" or "regions A and B depict similar intensities" (photometric uncertainty). The proposed method is based on a two steps" inexact graph matching approach. The first step searches for the best subgraph isomorphism candidate between expected regions and a subset of regions resulting from the initial oversegmentation. Then, remaining segmented regions are progressively merged with appropriate already matched regions, while preserving the coherence with a priori declared relationships. Strengths and weaknesses of the method are studied on various images (grayscale and color), with various intial oversegmentation algorithms (k-means, meanshift, quickshift). Results show the potential of the method to recover, in a reasonable runtime, expected regions, a priori described in a qualitative manner. For further evaluation and comparison purposes, a Python opensource package implementing the method is provided, together with the specifically built experimental database