A discrete expression of Canny's criteria for step edge detectorperformances evaluation

Optimal filters for edge detection are usually developed in the continuous domain and then transposed by sampling to the discrete domain. Simpler filters are directly defined in the discrete domain. We define criteria to compare filter performances in the discrete domain. Canny has defined (1983, 1986) three criteria to derive the equation of an optimal filter for step edge detection: good detection, good localization, and low-responses multiplicity. These criteria seem to be good candidates for filter comparison. Unfortunately, they have been developed in the continuous domain, and their analytical expressions cannot be used in the discrete domain. We establish three criteria with the same meaning as Canny's. Some comparisons with experimental results confirm the validity of our approach. This study highlights the existence of two classes of derivative operators that are distinguished by whether or not the impulse response of the filter in continuous space domain is continuous on its center. These classes exhibit very different properties for the second and third criteria. We extend the use of the first and third criteria to the smoothing filters. We also define an optimal continuous filter according to the continuous third criterion and an optimal discrete filter according to the discrete third criterion. We compare the performances of the sampled version of the continuous filter to those of the optimal discrete filter     

A nonfragile controller for suppressing exogenous disturbances

We suggest an approach to the problem of building a nonfragile controller, i.e., one that would allow for variation in its parameters, in order to suppress bounded exogenous disturbances in a linear dynamic system. We consider both continuous and discrete version of the problem and also its robust version. As an example, we study the control problem for a double oscillator.     

Insights on Aliasing Driven Instabilities for Advection Equations with Application to Gauss–Lobatto Discontinuous Galerkin Methods

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional \(L^2\) norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of the split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term properties (e.g. DG with Gauss–Lobatto points). Numerical experiments are included to illustrate the theoretical findings.     

Decidability results in First–Order Hybrid Petri Nets

In this paper we tackle the decidabilityof marking reachability for a hybrid formalism based on Petrinets. The model we consider is the untimed version of First–OrderHybrid Petri Nets: it combines a discrete Petri net and a continuousPetri net, the latter being a fluid version of a usual discretePetri net. It is suggested that the decidability results shouldbe pursued exploiting a hierarchy of models as it has been donein the framework of Hybrid Automata. In this paper we definethe class of Single–Rate Hybrid Petri Nets: the continuousdynamics of these nets is such that the vector of the markingderivatives of the continuous places is constant but for a scalarfactor. This class of nets can be seen as the counterpart oftimed automata with skewed clocks. We prove that the reachabilityproblem for this class can be reduced to the reachability problemof an equivalent discrete net and thus it is decidable.     

A quasi-Newton method for unconstrained non-smooth problems

We present a method, based on a variational problem, for solving a non-smooth unconstrained optimization problem. We assume that the objective function is a Lipschitz continuous and a regular function. In this case the function of our variational problem is semismooth and a quasi-Newton method may be used to solve the variational problem. A convergence theorem for our algorithm and its discrete version is also proved. Preliminary computational results show that the method performs quite well and can compete with other methods.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Heuristic methods for the optimal statistic median problem

We consider the problem of finding the convex combination of vectors for which the median is maximum. The objective function of this problem is non-concave and non-differentiable, with many local optima that can trap any subgradient algorithm. So we analyzed and tested some heuristic procedures to find optimal or near-optimal solutions. First, we introduced a variant of Random Restart, in which starting solutions are the vertices of the simplex, and we proved that small size problems are solved by this procedure. Then, we analyzed two versions of Variable Neighborhood Search that are used to explore the whole space of the feasible solutions, and we show that the continuous version is more powerful than the discrete version.     

Discrete cuckoo search algorithm for the travelling salesman problem

In this paper, we present an improved and discrete version of the Cuckoo Search (CS) algorithm to solve the famous traveling salesman problem (TSP), an NP-hard combinatorial optimisation problem. CS is a metaheuristic search algorithm which was recently developed by Xin-She Yang and Suash Deb in 2009, inspired by the breeding behaviour of cuckoos. This new algorithm has proved to be very effective in solving continuous optimisation problems. We now extend and improve CS by reconstructing its population and introducing a new category of cuckoos so that it can solve combinatorial problems as well as continuous problems. The performance of the proposed discrete cuckoo search (DCS) is tested against a set of benchmarks of symmetric TSP from the well-known TSPLIB library. The results of the tests show that DCS is superior to some other metaheuristics.