non linear observer based on the euler discretization in this paper we present an estimator based on the euler discretization first we show that the proposed estimator is an exponential observer in the case when the error between trajectories of continuous time and its associated euler discretized system can be neglected otherwise we show that the error of estimation is exponentially attracted by some neighbourhood of the origin which depends only on the sampling time finally numerical simulations using a bioreactor model are given in order to highlight the performance of the proposed estimator

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

distributed set membership observers for interconnected multi rate systems abstract this paper presents a new distributed observer for interconnected multi rate systems the developed observer belongs to the family of set membership estimators and the use of zonotopes is proposed to mathematically describe the sets a choice motivated by the available mathematical background in operations such as intersections combinations and linear manipulations the main features of the proposed distributed observer are a the actual state of any subsystem always belongs to the computed sets b the volume of these sets is minimized in real time c under equivalent assumptions the performance of the observer approaches to that of an analogous distributed kalman filter

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

rapid and accurate computation of the distance function using grids we present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids the algorithms are then generalized to compute the distance to piecewise linear objects by incorporating the geometry of huygens principle in the reverse order with the classical viscosity solution theory for the eikonal equation u 1 the algorithms become almost purely algebraic and yield very accurate approximations the generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries without the construction of the voronoi diagrams this method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods

constrained zonotopes this article introduces a new class of sets called constrained zonotopes that can be used to enclose sets of interest for estimation and control the numerical representation of these sets is sufficient to describe arbitrary convex polytopes when the complexity of the representation is not limited at the same time this representation permits the computation of exact projections intersections and minkowski sums using very simple identities efficient and accurate methods for computing an enclosure of one constrained zonotope by another of lower complexity are provided the advantages and disadvantages of these sets are discussed in comparison to ellipsoids parallelotopes zonotopes and convex polytopes in halfspace and vertex representations moreover extensive numerical comparisons demonstrate significant advantages over other classes of sets in the context of set based state estimation and fault detection

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

convergence analysis of the extended kalman filter used as an observer for nonlinear deterministic discrete time systems in this paper convergence analysis of the extended kalman filter ekf when used as an observer for nonlinear deterministic discrete time systems is presented based on a new formulation of the first order linearization technique sufficient conditions to ensure local asymptotic convergence are established furthermore it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified ekf significantly the efficiency of this approach compared to the classical version of the ekf is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete time systems

sonar based robot navigation using non linear robust discrete time observers in this paper we address the sonar based nagivation of mobile robots the extended kalman filtering ekf technique is considered but from a deterministic not a stochastic point of view for this problem we present results on the robustness of the non linear discrete time observation scheme this work is strongly based on our previous paper on continuous time ekf here we provide the discrete time counterpart of these results the original feature of our approach is that the region of convergence question is posed in its complete non linear framework that is considering the dynamics not only of the estimation error but also of the covariance matrix p in this way the approach followed makes the treatment less conservative and improves the convergence analysis in discrete time new problems and difficulties appeared for proving convergence the jacobians of f and h a k and c k respectively are evaluated at different trajectories and the exponential weighting factor has a multiplicative effec

fast computation of weighted distance functions and geodesics on implicit hyper surfaces 730 an algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper surfaces is introduced in this paper the basic idea is to approximate the intrinsic weighted distance by the euclidean weighted distance computed in a band surrounding the implicit hyper surface in the embedding space thereby performing all the computations in a cartesian grid with classical and efficient numerics based on work on geodesics on riemannian manifolds with boundaries we bound the error between the two distance functions we show that this error is of the same order as the theoretical numerical error in computationally optimal hamilton jacobi based algorithms for computing distance functions in cartesian grids therefore we can use these algorithms modified to deal with spaces with boundaries and obtain also for the case of intrinsic distance functions on implicit hyper surfaces a computationally efficient technique the approach can be extended to solve a more general class of hamilton jacobi equations defined on the implicit surface following the same idea of approximating their solutions by the solutions in the embedding euclidean space the framework here introduced thereby allows for the computations to be performed on a cartesian grid with computationally optimal algorithms in spite of the fact that the distance and hamilton jacobi equations are intrinsic to the implicit hyper surface c 2001 academic press

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

interval observers for time varying discrete time systems this techical note deals with interval state observer design for time varying discrete time systems the problem of a similarity transformation computation which connects a time varying matrix and its nonnegative representation is studied three solutions are proposed for a generic time varying system a system with positive state and for a particular class of periodical systems numerical simulations are provided to demonstrate advantages of the developed techniques

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

diffusion tensors for processing sheared and rotated rectangles image restoration and simplification methods that respect important features such as edges play a fundamental role in digital image processing however known edge preserving methods like common nonlinear diffusion methods tend to round vertices for large diffusion times in this paper we adapt the diffusion tensor for anisotropic diffusion to avoid this effects in images containing rotated and sheared rectangles respectively in this context we propose a new method for estimating rotation angles and shear parameters based on the so called structure tensor further we show how the knowledge of appropriate diffusion tensors can be used in variational models numerical examples including orientation estimation denoising and segmentation demonstrate the good performance of our methods

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

nonsmooth neural network for convex time dependent constraint satisfaction problems this paper introduces a nonsmooth ns neural network that is able to operate in a time dependent td context and is potentially useful for solving some classes of ns td problems the proposed network is named nonsmooth time dependent network ntn and is an extension to a td setting of a previous ns neural network for programming problems suppose c t t ge 0 is a nonempty td convex feasibility set defined by td inequality constraints the constraints are in general ns nondifferentiable functions of the state variables and time ntn is described by the subdifferential with respect to the state variables of an ns td barrier function and a vector field corresponding to the unconstrained dynamics this paper shows that for suitable values of the penalty parameter the ntn dynamics displays two main phases in the first phase any solution of ntn not starting in c 0 at t 0 is able to reach the moving set c cdot in finite time t h whereas in the second phase the solution tracks the moving set i e it stays within c t for all subsequent times t ge t h ntn is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times this new and peculiar dynamics displayed by ntn is potentially useful for addressing some significant td signal processing tasks as an illustration this paper discusses a number of examples where ntn is applied to the solution of ns td convex feasibility problems

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

interval state estimation for uncertain polytopic systems the aim of this article is the state estimation of uncertain polytopic dynamic systems the parametric uncertainties affecting the system are time varying unknown and bounded with known bounds the objective is to determine the state estimates consisting in the smallest interval containing the real state value caused by the parametric uncertainties this set will be characterized by the lower and upper bounds of the state trajectory given the uncertainty bounds the set can be computed by a direct simulation of the system but a more accurate estimation is obtained with a luenberger type observer fed with the system measurements the proposed observer is designed to minimize the interval width of the estimates the observer gains are obtained by solving an optimization problem under lmi constraints the efficiency of the proposed approach is illustrated by numerical examples

on set valued kalman filtering and its application to event based state estimation motivated by challenges in state estimation with event based measurement updates the properties of the exact and approximate set valued kalman filters with multiple sensor measurements for linear time invariant systems are investigated in this work first we show that the exact and the proposed approximate set valued filters are independent of the fusion sequence at each time instant second the boundedness of the size of the set of estimation means is proved for the exact set valued filter for the approximate set valued filter if the closed loop matrix is contractive then the set of estimation means has a bounded size asymptotically otherwise a nonsingular linear transform is constructed such that the size of the set of estimation means for the transformed states is asymptotically bounded third the effect of set valued measurements on the size of the set of estimation means is analyzed and conditions for performance improvement in terms of smaller size of the set of estimation means are proposed finally the results are applied to event based estimation which allow the event triggering conditions to be designed by considering requirements on performance and communication rates the efficiency of the proposed results are illustrated by simulation examples and comparison with the approximate event based mmse estimator and the kalman filter with intermittent observations

brief a set theoretic approach for time to contact estimation in dynamic vision this paper deals with the estimation of the time to contact in dynamic vision it is well known that differential invariants of the image velocity field can be used to characterize the shape changes of objects in the scene due to relative motion between the observer and the scene under the hypothesis of constant velocity along the optical axis the time to contact turns out to be a function of the area enclosed by the object contour and its time derivative in the paper a novel approach based on set membership estimation theory is proposed to estimate the variables involved in the computation of the time to contact both errors in the motion model and image measurement noise are described as unknown but bounded disturbances without requiring any statistical assumption the proposed technique allows for the computation of guaranteed bounds on the time to contact estimates in finite time a crucial issue in all problems where a robust evaluation of the time to contact is in order

the generalized moment based filter can we solve the filtering problem from the only knowledge of few moments of the noise terms in this technical note by exploiting set of distributions based filtering we solve this problem without introducing additional assumptions on the distributions of the noises e g gaussianity or on the final form of the estimator e g linear estimator given the moments e g mean and variance of random variable x it is possible to define the set of all distributions that are compatible with the moments information this set can be equivalently characterized by its extreme distributions a family of mixtures of dirac s deltas the lower and upper expectation of any function g of x are obtained in correspondence of these extremes and can be computed by solving a linear programming problem the filtering problem can then be solved by running iteratively this linear programming problem in this technical note we discuss theoretical properties of this filter we show the connection with set membership estimation and its practical applications

set membership filtering with incomplete observations abstract this study addresses the set membership estimation problem for a class of discrete time varying systems with incomplete observations a set membership filter is developed and a recursive algorithm is proposed to calculate the state estimate ellipsoid which contains the true value to solve the problem that the conventional set membership filter cannot guarantee the stability when applied to discrete time varying systems with incomplete observations a quantitative analysis method about incomplete observations is developed and a tight bound of the estimation error is found based on interval analysis and some bounded noise assumptions in terms of bounded assumptions the relationship between the bound of estimated error and the data dropout rate is obtained if the data dropout rate is less than a maximal value the set membership filter is asymptotically stable the proposed filter is applied to a two state example to demonstrate the effectiveness of theoretical results

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

a robust variational approach for simultaneous smoothing and estimation of dti abstract estimating diffusion tensors is an essential step in many applications such as diffusion tensor image dti registration segmentation and fiber tractography most of the methods proposed in the literature for this task are not simultaneously statistically robust and feature preserving techniques in this paper we propose a novel and robust variational framework for simultaneous smoothing and estimation of diffusion tensors from diffusion mri our variational principle makes use of a recently introduced total kullback leibler tkl divergence for dti regularization tkl is a statistically robust dissimilarity measure for diffusion tensors and regularization by using tkl ensures the symmetric positive definiteness of tensors automatically further the regularization is weighted by a non local factor adapted from the conventional non local means filters finally for the data fidelity we use the nonlinear least squares term derived from the stejskal tanner model we present experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

brief on robustness and precision of mobile robots missions this paper proposes solutions for the robust achievement of displacement missions by a wheeled mobile robot the considered disturbances are generated by the use of inaccurate configuration estimates as feedback information those errors are characterized in a bounded error context we introduce a trajectory tracking controller whose stability is analysed by a lyapunov function the time derivative of which is negative definite inside a large domain around the zero error this property makes it possible to define a bounded attractive domain around the nominal trajectory quantifying thus the guaranteed precision of the mission the method is extended to a parking non stationary controller and illustrated by a real experiment

brief paper a set membership state estimation algorithm based on dc programming this paper presents a new approach to guaranteed state estimation for nonlinear discrete time systems with a bounded description of noise and parameters the sets of states that are consistent with the evolution of the system the measured outputs and bounded noise and parameters are represented by zonotopes dc programming and intersection operations are used to obtain a tight bound an example is given to illustrate the proposed algorithm

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

analysis of consensus protocols with bounded measurement errors this paper analyzes two classes of consensus algorithms in the presence of bounded measurement errors the considered protocols adopt an updating rule based either on constant or vanishing weights under the assumption of bounded error the consensus problem is cast in a set membership framework and the agreement of the team is studied by analyzing the evolution of the feasible state set bounds on the asymptotic difference between the states of the agents are explicitly derived in terms of the bounds on the measurement noise and the values of the weight matrix

cyclic schemes for pde based image analysis we investigate a class of efficient numerical algorithms for many partial differential equations pdes in image analysis they are applicable to parabolic or elliptic pdes that have bounded coefficients and lead to space discretisations with symmetric matrices our schemes are easy to implement and well suited for parallel implementations on gpus since they are based on the explicit diffusion scheme in the parabolic case and the jacobi method in the elliptic case by supplementing these methods with cyclically varying time step sizes or relaxation parameters we achieve efficiency gains of several orders of magnitude we call the resulting algorithms fast explicit diffusion fed and fast jacobi fj methods to achieve a good compromise between efficiency and accuracy we show that one should use parameter cycles that result from factorisations of box filters for these cycles we establish stability results in the euclidean norm our schemes perform favourably in a number of applications including isotropic nonlinear diffusion filters with widely varying diffusivities as well as anisotropic diffusion methods for image filtering inpainting and regularisation in computer vision moreover they are equally suited for higher dimensional problems as well as higher order pdes and they can also be interpreted as efficient first order methods for smooth optimisation problems

tight error bounds for projection algorithms in conditional set membership estimation abstract in set membership estimation conditional problems arise when the estimate must belong to a given set of assigned structure conditional projection algorithms provide estimates that are suboptimal in terms of the worst case estimation error in order to precisely evaluate the suboptimality level of these estimators tight upper bounds on the estimation errors must be computed as a function of the conditional radius of information which represents the minimum achievable error in this paper tight bounds are derived for l and l 1 estimation errors in a general setting which allows to consider any compact set of feasible problem elements and linearly parameterized estimates

estimating shortest paths and minimal distances on digitized three dimensional surfaces abstract the estimation of minimal distances and shortest paths between points on a continuous three dimensional 3d surface given in digitized form is considered the suggested approach combines recently developed length estimators for digitized 3d curves with well known algorithms for computing shortest paths in sparse graphs to estimate the minimal distances and the corresponding shortest paths in a systematic computationally efficient way in comparison to differential geometry techniques for solving the continuous geodesic problem and to computational geometry algorithms for solving the discrete geodesic probelm the suggested hybrid continuous discrete approach is well suited to applications in which continuous surfaces are given in digitized form computation time must be relatively short and an approximate solution is acceptable in particular the suggested method can be used to efficiently obtain a good initial approximation as an input to other algorithms thus improving their speed of convergence and reducing their tendency to converge to insignificant local minima some confusion has recently arisen with respect to the relation between the design of length estimators and the optimization of local distances for distance transformations since 3d length estimation is a central concept in the suggested approach an effort is made to resolve discrepancies and to highlight the similarities and differences between these problems it is shown that while in two dimensions the design of length estimators and the design of local weights for distance transformations are rather similar in three dimensions digital geometry dictates that these problems are different the contexts in which each of the two approaches should be preferred are identified

brief paper an ekf based nonlinear observer with a prescribed degree of stability the topic of this article is a nonlinear observer based on a slight modification of the extended kalman filter the purpose of this modification is twofold first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities using the second method of lyapunov it is proved that the proposed observer is an exponential observer to examine the practical usefulness the proposed observer is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines

brief paper recursive state bounding by parallelotopes in this paper the problem of recursively estimating the state uncertainty set of a discrete time linear dynamical system is addressed a novel approach based on minimum volume bounding parallelotopes is introduced and an algorithm of polynomial complexity is derived simulation results and performance comparisons with ellipsoidal recursive state bounding algorithms are also given

orientation matching minimization for image denoising and inpainting in this paper we propose an orientation matching functional minimization for image denoising and image inpainting following the two step tv stokes algorithm rahman et al in scale space and variational methods in computer vision pp 473 482 springer heidelberg 2007 tai et al in image processing based on partial differential equations pp 3 22 springer heidelberg 2006 bertalmio et al in proc conf comp vision pattern rec pp 355 362 2001 a regularized tangential vector field with zero divergence condition is first obtained then a novel approach to reconstruct the image is proposed instead of finding an image that fits the regularized normal direction from the first step we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction this functional yields a new nonlinear partial differential equation pde for reconstructing denoised and inpainted images the equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field providing reconstructed images which have sharp edges and smooth regions the additive operator splitting aos scheme is used for discretizing euler lagrange equations we present the results of various numerical experiments that illustrate the improvements obtained with the new functional

new riemannian techniques for directional and tensorial image data this paper develops new geometrical filtering and edge detection algorithms for processing non euclidean image data we view image data as residing on a riemannian manifold and we work with a representation based on the exponential map for this manifold together with the riemannian weighted mean of image data we show how the weighted mean can be efficiently computed using newton s method which converges faster than the gradient descent method described elsewhere in the literature based on geodesic distances and the exponential map we extend the classical median filter and the perona malik anisotropic diffusion technique to smooth non euclidean image data we then propose an anisotropic gaussian kernel for image filtering and we also show how both the median filter and the anisotropic gaussian filter can be combined to develop a new edge preserving filter which is effective at removing both gaussian noise and impulse noise by using the intrinsic metric of the feature manifold we also generalise di zenzo s structure tensor to non euclidean images for edge detection we demonstrate the applications of our riemannian filtering and edge detection algorithms both on directional and tensor valued images

multiresolution bilateral filtering for image denoising the bilateral filter is a nonlinear filter that does spatial averaging without smoothing edges it has shown to be an effective image denoising technique an important issue with the application of the bilateral filter is the selection of the filter parameters which affect the results significantly there are two main contributions of this paper the first contribution is an empirical study of the optimal bilateral filter parameter selection in image denoising applications the second contribution is an extension of the bilateral filter multiresolution bilateral filter where bilateral filtering is applied to the approximation low frequency subbands of a signal decomposed using a wavelet filter bank the multiresolution bilateral filter is combined with wavelet thresholding to form a new image denoising framework which turns out to be very effective in eliminating noise in real noisy images experimental results with both simulated and real data are provided

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

brief guaranteed state estimation by zonotopes this paper presents a new approach to guaranteed state estimation for non linear discrete time systems with a bounded description of noise and parameters the main result is an algorithm to compute a set that contains the states consistent with the measured output and the given noise and parameters this set is represented by a zonotope the size of the zonotope is minimized each sample time by an analytic expression or by solving a convex optimization problem interval arithmetic is used to calculate a guaranteed trajectory of the process state two examples have been provided to clarify the algorithm

a short time beltrami kernel for smoothing images and manifolds we introduce a short time kernel for the beltrami image enhancing flow the flow is implemented by convolving the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a gaussian kernel the kernel is appropriate for smoothing regular flat 2 d images for smoothing images painted on manifolds and for simultaneously smoothing images and the manifolds they are painted on the kernel combines the geometry of the image and that of the manifold into one metric tensor thus enabling a natural unified approach for the manipulation of both additionally the derivation of the kernel gives a better geometrical understanding of the beltrami flow and shows that the bilateral filter is a euclidean approximation of it on a practical level the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the beltrami flow in addition the kernel works with equal ease on regular 2 d images and on images painted on parametric or triangulated manifolds we demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds

high order two dimensional nonoscillatory methods for solving hamilton jacobi scalar equations for the computation of nonlinear solutions of hamilton jacobi scalar equations in two space dimensions we develop high order accurate numerical schemes that can be applied to complicated geometries previously the recently developed essentially nonoscillatory eno technology has been applied in simple domains like squares or rectangles using dimension by dimension algorithms on arbitrary two dimensional closed or multiply connected domains first order monotone methods were used in this paper we propose two different techniques to construct high order accurate methods using the eno philosophy namely any arbitrary domain is triangulated by finite elements into which two dimensional eno polynomials are constructed these polynomials are then differentiated to compute a high order accurate numerical solution these new techniques are shown to be very useful in the computation of numerical solutions of various applications without significantly increasing cpu running times as compared to dimension by dimension algorithms furthermore these methods are stable and no spurious oscillations are detected near singular points or curves

construction of interval observers for continuous time systems with discrete measurements we consider continuous time systems with input output and additive disturbances in the particular case where the measurements are only available at discrete instants and have disturbances to solve a state estimation problem we construct continuous discrete interval observers that are asymptotically stable in the absence of disturbances these interval observers are composed of two copies of the studied system and of a framer accompanied with appropriate outputs which give componentwise upper and lower bounds for the solutions of the studied system

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

edge preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods this paper introduces a novel image dependent filtering approach derived from concepts known in mathematical morphology and aiming at edge preserving smoothing of natural images like other adaptive methods it assumes that the neighbourhood of a pixel contains the essential information required for the estimation of local features in the image the proposed strategy essentially consists in a weighted averaging combining both spatial and tonal information for that purpose a twofold similarity measure is calculated from local geodesic time functions this way no prior operator definition is required since a weighting neighbourhood and a weighting kernel are determined automatically from the unfiltered input data for each pixel location by designing relevant geodesic masks two adaptive filtering algorithms are derived that are particularly efficient at smoothing heterogeneous areas while preserving relevant structures in greyscale and multichannel images

zonotopes and kalman observers state bounding observation based on zonotopes is the subject of this paper dealing with zonotopes is motivated by set operations resulting in simple matrix calculations with regard to the often huge number of facets and vertices of the equivalent polytopes discrete time ltv lpv systems with state and measurement uncertainties are considered based on a new zonotope size criterion called f w radius and by merging optimal and robust observer gain designs a zonotopic kalman filter zkf is proposed with a proof of robust convergence the notion of covariation is introduced and results in an explicit bridge between the zonotopic set membership and the stochastic paradigms for kalman filtering no intersection is used and the influence of the reduction operator limiting to a tunable maximum the size of the matrices involved in the zonotopic set computations is fully taken into account in the lmi based robust stability analysis a numerical example illustrates the effectiveness of the proposed zkf

new developments in state estimation for nonlinear systems based on an interpolation formula accurate state estimators for nonlinear systems can be derived the estimators do not require derivative information which makes them simple to implement

interval state estimation for a class of nonlinear systems the goal of this technical note is to design interval observers for a class of nonlinear continuous time systems the first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time invariant changes of coordinates even if the system is not cooperative this result is extended to a class of nonlinear systems using partial exact linearisations the proposed observers guarantee to enclose the set of system states that is consistent with the model the disturbances and the measurement noise moreover it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity uncorrelation or type of distribution the proposed observer is illustrated through numerical simulations

brief applying the ekf to stochastic differential equations with level effects a transformation is introduced to effectively remove level effects i e the state dependency of the diffusion function in a restricted class of multivariate stochastic differential equations such that the general continuous discrete time nonlinear filtering problem may be solved using new or existing implementations of the extended kalman filter ekf an implementation of a quasi maximum likelihood qml method for direct estimation of embedded parameters in nonlinear multivariate stochastic differential equations using discrete time input output data encumbered with additive measurement noise is discussed and its properties are compared with those provided by another software package

processing textured surfaces via anisotropic geometric diffusion a multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy textured parametric surfaces the aim is to smooth noisy triangulated surfaces and accompanying noisy textures as they are delivered by new scanning technology while enhancing geometric and texture features for an initial textured surface a fairing method is described which simultaneously processes the texture and the surface considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges the method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture here the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients a spatial finite element discretization on arbitrary unstructured triangular grids and a semi implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm a normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution different applications underline the efficiency and flexibility of the presented surface processing tool

scale space analysis and active contours for omnidirectional images a new generation of optical devices that generate images covering a larger part of the field of view than conventional cameras namely catadioptric cameras is slowly emerging these omnidirectional images will most probably deeply impact computer vision in the forthcoming years provided that the necessary algorithmic background stands strong in this paper we propose a general framework that helps define various computer vision primitives we show that geometry which plays a central role in the formation of omnidirectional images must be carefully taken into account while performing such simple tasks as smoothing or edge detection partial differential equations pdes offer a very versatile tool that is well suited to cope with geometrical constraints we derive new energy functionals and pdes for segmenting images obtained from catadioptric cameras and show that they can be implemented robustly using classical finite difference schemes various experimental results illustrate the potential of these new methods on both synthetic and natural images

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

global minimum for active contour models a minimal path approach a new boundary detection approach for shape modeling is presented it detects the global minimum of an active contour model s energy between two end points initialization is made easier and the curve is not trapped at a local minimum by spurious edges we modify the snake energy by including the internal regularization term in the external potential term our method is based on finding a path of minimal length in a riemannian metric we then make use of a new efficient numerical method to find this shortest path it is shown that the proposed energy though based only on a potential integrated along the curve imposes a regularization effect like snakes we explore the relation between the maximum curvature along the resulting contour and the potential generated from the image the method is capable to close contours given only one point on the objects boundary by using a topology based saddle search routine we show examples of our method applied to real aerial and medical images

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation