Simulation tool for morphological analysis

International audienceTo understand the mechanisms underlying the morphogenesis of multicellular organisms we study the dynamic system of cells (cell multiplication, cell migration, apoptosis); local interactions between cells for understanding the convergence of the system to a stable form that is constantly renewed.and the controls established by the nature of the growth of the organism, and its convergence to a stable form. We must be able to formalize it in a proper metric space a metaphor of cell dynamics to nd conditions (decisions, states) in which operational constraints (such as those induced by the tissue or the use of resources) are always satis ed and therefore in which the system is viable and maintain its shape while renewing. The aim of this paper is to explain the mathematical foundations of this work and describe a simulation tool to study the morphogenesis of a virtual organism and to describe a simulation tool to study the morphogenesis of a virtual multicellular organism. We formalize mathematically a model of cell dynamic on the principles of morphological analysis. Morphological analysis and viability theory are the mathematical foundations that motivate this work and this tool will test whether a system generated by morphological equations can maintain its shape and remains "viable" in a given environment

Shape evolutions under state constraints: A viability theorem

AbstractThe aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space).In this morphological framework, the evolution of compact subsets of RN is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis.Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order)—correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space.We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in RN, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set M⊂RN

Control problems for nonlocal set evolutions

In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space. Their family, however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance, it is a metric space. Here Aubin's framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space of nonempty compact subsets of the Euclidean space supplied with Pompeiu-Hausdorff distance. Now various control problems are formulated for compact sets depending on time: open-loop, relaxed and closed-loop control problems – each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions

Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions

Evolution equations in ostensible metric spaces : Definitions and existence.

The primary aim is to unify the definition of solution for completely different types of evolutions. Such a common approach is to lay the foundations for solving systems like, for example, a semilinear evolution equation (of parabolic type) in combination with a first order geometric evolution. In regard to geometric evolutions, this concept is to fulfill 3 conditions : First, consider nonempty compact subsets K(t) of R^N without a priori restrictions on the regularity of the boundary. Second, the evolution of K(t) might depend on nonlocal properties of the set K(t) and its normal cones. Last, but not least, no inclusion principle. The approach here is based on generalizing the mutational equations of Aubin for metric spaces in two respects : Replacing the metric by a countable family of (possibly nonsymmetric) distances (called ostensible metrics) and extending the basic idea of distributions

A viability theorem for morphological inclusions

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of the Euclidean space is described by means of flows along bounded Lipschitz vector fields (similarly to the velocity method alias speed method in shape analysis). Now for each compact subset, more than just one vector field is admitted - correspondingly to the set-valued map of a differential inclusion in finite dimensions. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in finite dimensions, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally an application to shape optimization under state constraints is sketched

Critical Transitions In a Model of a Genetic Regulatory System

We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).Comment: 19 pages, 8 figure

Evolution of Coalitions Governed by Mutational Equations

In cooperative game theory as well as in some domains of economic regulation by shortages (queues or unemployment), one is confronted to the problem of evolution of coalitions of players or economic agents. Since coalitions are subsets and cannot be represented by vectors -- except if we embed subsets in the family of fuzzy sets, which are functions -- the need to adapt the theoy of differential equations and dynamical systems to govern the evolution of coalitions or subsets instead of vectors did emerge. Evolution of subsets (regarded as shapes or images) was also motivated by evolution equations of "tubes" in "visual servoing" on one hand, mathematical morphology on the other. The "power spaces" in which coalitions, images, shapes, etc. evolve are metric spaces without a linear structure. However, one can extend the differential calculus to a mutational calculus for maps from one metric space to another, as we shall explain in this paper. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by "transitions" with which we can also define differential quotients of a map. Their various limits are called "mutations" of a map. Many results of differential calculus do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended to mutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations

Understanding Physiological and Degenerative Natural Vision Mechanisms to Define Contrast and Contour Operators

BACKGROUND:Dynamical systems like neural networks based on lateral inhibition have a large field of applications in image processing, robotics and morphogenesis modeling. In this paper, we will propose some examples of dynamical flows used in image contrasting and contouring. METHODOLOGY:First we present the physiological basis of the retina function by showing the role of the lateral inhibition in the optical illusions and pathologic processes generation. Then, based on these biological considerations about the real vision mechanisms, we study an enhancement method for contrasting medical images, using either a discrete neural network approach, or its continuous version, i.e. a non-isotropic diffusion reaction partial differential system. Following this, we introduce other continuous operators based on similar biomimetic approaches: a chemotactic contrasting method, a viability contouring algorithm and an attentional focus operator. Then, we introduce the new notion of mixed potential Hamiltonian flows; we compare it with the watershed method and we use it for contouring. CONCLUSIONS:We conclude by showing the utility of these biomimetic methods with some examples of application in medical imaging and computed assisted surgery

Objective Definition of Rosette Shape Variation Using a Combined Computer Vision and Data Mining Approach

Computer-vision based measurements of phenotypic variation have implications for crop improvement and food security because they are intrinsically objective. It should be possible therefore to use such approaches to select robust genotypes. However, plants are morphologically complex and identification of meaningful traits from automatically acquired image data is not straightforward. Bespoke algorithms can be designed to capture and/or quantitate specific features but this approach is inflexible and is not generally applicable to a wide range of traits. In this paper, we have used industry-standard computer vision techniques to extract a wide range of features from images of genetically diverse Arabidopsis rosettes growing under non-stimulated conditions, and then used statistical analysis to identify those features that provide good discrimination between ecotypes. This analysis indicates that almost all the observed shape variation can be described by 5 principal components. We describe an easily implemented pipeline including image segmentation, feature extraction and statistical analysis. This pipeline provides a cost-effective and inherently scalable method to parameterise and analyse variation in rosette shape. The acquisition of images does not require any specialised equipment and the computer routines for image processing and data analysis have been implemented using open source software. Source code for data analysis is written using the R package. The equations to calculate image descriptors have been also provided