In this article, we introduce the notion of safe predefined-time stability and address an optimal safe predefined-time stabilization problem. In particular, safe predefined-time stability characterizes parameter-dependent nonlinear dynamical systems whose trajectories starting in a given set of admissible states remain in the set of admissible states for all time and converge to an equilibrium point in a predefined time. Furthermore, we provide a Lyapunov theorem establishing sufficient conditions for safe predefined-time stability. We address the optimal safe predefined-time stabilization problem by synthesizing feedback controllers that guarantee closed-loop system safe predefined-time stability while optimizing a given performance measure. Specifically, safe predefined-time stability of the closed-loop system is guaranteed via a Lyapunov function satisfying a differential inequality while simultaneously serving as a solution to the steady-state Hamilton-Jacobi-Bellman (HJB) equation ensuring optimality. Given that the HJB equation is generally difficult to solve, we develop a physics-informed machine learning-based algorithm for learning the safely predefined-time stabilizing solution to the steady-state HJB equation. Several simulation results are provided to demonstrate the efficacy of the proposed approach.

In this paper, we consider multiple Unmanned Aerial Vehicles (UAV) serving as flying Base Stations (BS) of a wireless network and the problem of jointly optimizing their trajectory with respect to a running cost. This cost accounts for the consumed energy related to the vehicle velocity and for the amount of data traffic collected or served by the UAVs. The data traffic is supposed to be spatially distributed around a hotspot and is equivalent to a potential in Physics. Using the principles of Lagrangian Mechanics, we derive a scalable algorithm able to optimize the trajectory of thousands of drones in milliseconds on a off-the-shelf laptop. Our model allows to control the distance between the UAVs to avoid collisions by using a coupling between the drone trajectories.

We consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data.

This paper provides new theoretical connections between multi-time Hamilton--Jacobi partial differential equations and variational image decomposition models in imaging sciences. We show that the minimal values of these optimization problems are governed by multi-time Hamilton--Jacobi partial differential equations. The minimizers of these optimization problems can be represented using the momentum in the corresponding Hamilton--Jacobi partial differential equation. Moreover, variational behaviors of both the minimizers and the momentum are investigated as the regularization parameters approach zero. In addition, we provide a new perspective from convex analysis to prove the uniqueness of convex solutions to Hamilton--Jacobi equations. Finally, we consider image decomposition models that do not have unique minimizers, and we propose a regularization approach to perform the analysis using multi-time Hamilton--Jacobi partial differential equations.

We consider standard finite-dimensional variational models used in signal/image processing that consist of minimizing an energy involving a data fidelity term and a regularization term. We propose new contributions from a theoretical perspective which give a precise description on how the solutions of these optimization problems depend on the amount of smoothing effects and the data itself. The dependence of the minimal values of the energy is shown to be ruled by Hamilton--Jacobi equations, while the minimizers $\mathbf{u}(\mathbf{x},t)$ for the observed images $\mathbf{x}$ and smoothing parameters $t$ are given by $\mathbf{u}(\mathbf{x},t)\;=\;\mathbf{x}-t\,\nabla\,H(\nabla_{\mathbf{x}}E(\mathbf{x},t)),$ where $E(\mathbf{x},t)$ is the minimal value of the energy and $H$ is a Hamiltonian related to the data fidelity term. Various vanishing smoothing parameter results are derived illustrating the role played by the prior in such limits.

We consider the Chan and Vese multiphase segmentation based on the partition of an image minimizing an energy involving a region-based data fidelity term and a regularization term. The common implementation of the continuous optimization of this segmentation framework, with multiple level set functions, raises some numerical issues which lead to poor performance of the method when handling more than two phases. We propose a general formulation of the multiphase model, and a permutation method, incorporated in the level-set based implementation of the multi-phase approach to handle in an original way the so-called hidden phase problem.

In this letter, we propose a Markovian approach for interferometric synthetic aperture radar (InSAR) phase reconstruction. Recently, Markovian models based on multichannel InSAR likelihood statistics and total variation prior have been proposed to reconstruct the noisy and wrapped phase. Efficient discrete optimization algorithms based on the graph-cut technique are used to efficiently minimize the energy. Our contribution consists in extending these works to cope with continuous label sets providing more precise and accurate reconstructed profiles. The proposed approach also provides a good way to estimate local hyperparameters to adjust the prior model and preserve well discontinuities in profiles. This task is useful when working with real InSAR data where the quantization of the continuous label set leads to a loss of some physical information. The proposed method is compared to other Markovian approaches with discrete multilabel optimization algorithms. Experiments show better quality results both on simulated and real InSAR data.

We compare in this study two image restoration approaches for the pre-processing of printed documents:namely the Non-local Means filter and a total variation minimization approach. We apply these two approaches to printed document sets from various periods,and we evaluate their effectiveness through character recognition performance using an open source OCR. Our results show that for each document set, one or both pre-processing methods improve character recog-nition accuracy over recognition without preprocessing. Higher accuracies are obtained with Non-local Means when characters have a low level of degradation since they can be restored by similar neighboring parts of non-degraded characters. The Total Variation approach is more effective when characters are highly degraded and can only be restored through modeling instead of using neighboring data.

This paper addresses the problem of change detection in multispectral satellite images. We introduce a new information theoretic-based metric between the images associated with a Markov Random Field spatial regularization. The proposed metric is parametrized through copulas and implemented over component trees representation of the images. Such topographic map based metric associated to an Ising model exhibits interesting results for both abrupt and slow changes while being robust to global illumination and contrast changes. Experiments are conducted on SPOT images of the amazonian basin for landcover monitoring.

This paper considers two approaches to perform image restoration while preserving the contrast. The first one is the Total Variation-based Bregman iterations while the second consists in the minimization of an energy that involves robust edge preserving regularization. We show that these two approaches can be derived form a common framework. This allows us to deduce new properties and to extend and generalize these two previous approaches.

This paper copes with the approximate minimization of Markovian energy with pairwise interactions. We extend previous approaches that rely on graph-cuts and move making techniques. For this purpose, a new move is introduced that permits us to perform better approximate optimizations. Some experiments show that very good local minima are obtained while keeping the memory usage low.

In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409–422, 2006 ) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. 26–33, 2003 ). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Almgren et al. (SIAM Journal on Control and Optimization 31(2):387–438, 1993 ) and Luckhaus and Sturzenhecker (Calculus of Variations and Partial Differential Equations 3(2):253–271, 1995 ), and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term.

Interferometric synthetic aperture radar (SAR) images suffer from a strong noise, and their regularization is often a prerequisite for successful use of their information. Independently of the unwrapping problem, interferometric phase denoising is a difficult task due to shadows and discontinuities. In this paper, we propose to jointly filter phase and amplitude data in a Markovian framework. The regularization term is expressed by the minimization of the total variation and may combine different information (phase, amplitude, optical data). First, a fast and approximate optimization algorithm for vectorial data is briefly presented. Then, two applications are described. The first one is a direct application of this algorithm for 3-D reconstruction in urban areas with very high resolution images. The second one is an adaptation of this framework to the fusion of SAR and optical data. Results on aerial SAR images are presented.

Synthetic aperture radar (SAR) images, like other coherent imaging modalities, suffer from speckle noise. The presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms. Numerous approaches have been proposed to filter speckle noise. Markov random field (MRF) modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image. In this context, total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges. Speckle noise follows heavy-tailed distributions, and the MRF formulation leads to a minimization problem involving nonconvex log-likelihood terms. Such a minimization can be performed efficiently by computing minimum cuts on weighted graphs. Due to memory constraints, exact minimization, although theoretically possible, is not achievable on large images required by remote sensing applications. The computational burden of the state-of-the-art algorithm for approximate minimization (namely the alpha -expansion) is too heavy specially when considering joint regularization of several images. We show that a satisfying solution can be reached, in few iterations, by performing a graph-cut-based combinatorial exploration of large trial moves. This algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area SAR images.

This paper investigates the use of a popular regularization model, the total variation minimization (TV), to filter SAR interferometric images (amplitude and phase data). This model is extensively used for its property of preserving edges and is therefore well adapted for urban areas. Using a TV model adapted to multi-dimensional data, we propose to do a joint filtering of phase and amplitude images. Due to the many local minima, the minimization of such a model is hard to perform. A new fast approximate discrete algorithm is presented. The filtering is applied in the framework of 3D reconstruction. Results on real images are presented.

This paper investigates the joint use of interferometric SAR and optical data for 3D reconstruction. A framework for phase filtering constrained by the discontinuities of the optical image is presented. First, both the amplitude and the interferometric phase are projected in a 3D coordinate system. The problem is then expressed as the regularization of the amplitude and phase images with the introduction as prior knowledge of the edges detected on the optical image. We define the regularized elevation in the framework of Markov random fields (MRF) and derive a smoothness prior that both preserves sharp boundaries (based on total variation minimization) and is driven by the structures present in the optical image. We apply a recent graph-cut based algorithm to perform fast regularization of the elevation field. First results on a real pair of optical and InSAR images are presented.

We propose simple and extremely efficient methods for solving the basis pursuit problem $\min\{\|u\|_1 : Au = f, u\in\mathbb{R}^n\},$ which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem $\min_{u\in\mathbb{R}^n} \mu\|u\|_1+\frac{1}{2}\|Au-f^k\|_2^2$ for given matrix A and vector $f^k$. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving A and $A^\top$ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.

We present an identification and authentication system based on hand modality which is part of a reference system for all modalities developed within the biosecure consortium. It relies on simple geometric features extracted from hand boundary. The different steps of this system are detailed, namely: pre-processing, feature extraction and hand matching. This system has been tested on the biosecure hand database which consists of 4500 hand images of 750 individuals. Results are detailed with respect to different enrolment conditions such as population size, enrolment size, and image resolution.

The problem of person recognition and verification based on their hand images has been addressed. The system is based on the images of the right hands of the subjects, captured by a flatbed scanner in an unconstrained pose at 45 dpi. In a pre-processing stage of the algorithm, the silhouettes of hand images are registered to a fixed pose, which involves both rotation and translation of the hand and, separately, of the individual fingers. Two feature sets have been comparatively assessed, Hausdorff distance of the hand contours and independent component features of the hand silhouette images. Both the classification and the verification performances are found to be very satisfactory as it was shown that, at least for groups of about five hundred subjects, hand-based recognition is a viable secure access control scheme.

We present an original method to segment color images using a classification in the 3-D color space. In the case of ordinary images, clusters that appear in 3-D histograms usually do not fit a well-known statistical model. For that reason, we propose a classifier that relies on mathematical morphology, and more precisely on the watershed algorithm. We show on various images that the expected color clusters are correctly identified by our method. Last, to segment color images into coherent regions, we perform a Markovian labeling that takes advantage of the morphological classification results.