scaling factors of id fdtd scheme for dispersive media based on the auxiliary differential equation ade method the isotropic dispersion finite difference time domain id fdtd scheme is modified for a dispersive medium the proposed id fdtd scheme is based on the ade method in this correspondence three typical dispersive media are considered debye drude and lorentz media for which scaling factors are formulated the dispersion property is then examined and the accuracy of the proposed scheme is verified by consideration of two scattering problems

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

image segmentation by variational methods mumford and shah functional and the discrete approximations in this paper we discuss the links between mumford and shah s variational problem for signal and image segmentation based on an energy functional of a continuous grey level function and the numerical algorithms proposed to solve it these numerical approaches are based on a discrete functional we recall that in one dimension this discrete functional is asymptotically equivalent to the continuous functional this can be summarized in a gamma convergence result we show that the same result holds in dimension two provided that the continuous energy is adapted to the anisotropy of the discrete approaches we display a few experimental results in dimensions one and two

piecewise linear recursive convolution for dispersive media using fdtd electromagnetic propagation through linear dispersive media can be analyzed using the finite difference time domain fdtd method by employing the recursive convolution rc approach to evaluate the discrete time convolution of the electric field and the dielectric susceptibility function the rc approach results in a fast and computationally efficient algorithm however the accuracy achieved is not generally as good as that obtained with other methods a new piecewise linear recursive convolution plrc method is described here that has greatly improved accuracy over the original rc approach but retains its speed and efficiency advantages

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

an improved corner detection algorithm based on chain coded plane curves abstract one possible measure of corners on chain coded plane curves is the product of the length of the uniform chain sections to either side the forward and backward arms of a point times the angle of discontinuity of that particular point this paper describes a corner detection algorithm which performs close to humans expectations the improved algorithm is based on previous work done by freeman and davis the algorithm uses an approach of averaging multiple s values to avoid the selection of spurious corners the algorithm also uses a maximum cut off value a certain fraction of the entire chain to determine the length of the forward and backward arms of each point and hence eliminates the detection of spurious corners the improved algorithm was tested with results close to humans expectations the results of the improved algorithm are shown by way of examples

approximation by γ convergence of a curvature depending functional in visual reconstruction where r2 is a bounded open set the image domain h1 denotes the one dimensional hausdorff measure and g l is the input image the functional has to be minimized over all closed sets c and all u c1 c the function u represents a denoised approximation of the input image g and c represents the set of boundaries of the segmentation the mumford and shah variational model can be extended to several visual reconstruction problems see march 29 computation of depth from stereo images shah 39 computation of optical flow nesi 34 shape from shading shah 40 the existence of minimizers of gms has been proved independently by dal maso morel and solimini 21 and de giorgi carriero and leaci 23 using the compactness and lower semicontinuity theorems of ambrosio 3 mumford and shah 33 studied the properties of minimizers u c of gms assuming that c is a finite union of simple c1 1 curves meeting and meeting each other only at their endpoints they proved that the vertices of c may only be i triple points where three curves meet with equal angles ii points on the boundary of where one curve meets perpendicularly iii crack tips where a curve ends and meets nothing in computer vision the constraints imposed on the segmentations highlighted by such results constitute a drawback of the variational model in particular corners and t junctions which are relevant features for pattern recognition are distorted since the length measure is not sensitive to corners and junctions to allow for such singularities in the segmentations it is necessary to consider curvature depending energies functionals

finite difference and pseudospectral time domain methods applied to backward wave metamaterials backward wave bw materials that have simultaneously negative real parts of their electric permittivity and magnetic permeability can support waves where phase and power propagation occur in opposite directions these materials were predicted to have many unusual electromagnetic properties among them amplification of the near field of a point source which could lead to the perfect reconstruction of the source field in an image j pendry phys rev lett vol 85 pp 3966 2000 often systems containing bw materials are simulated using the finite difference time domain technique we show that this technique suffers from a numerical artifact due to its staggered grid that makes its use in simulations involving bw materials problematic the pseudospectral time domain technique on the other hand uses a collocated grid and is free of this artifact it is also shown that when modeling the dispersive bw material the linear frequency approximation method introduces error that affects the frequency of vanishing reflection while the auxiliary differential equation the z transform and the bilinear frequency approximation method produce vanishing reflection at the correct frequency the case of vanishing reflection is of particular interest for field reconstruction in imaging applications

lower semicontinuity properties of functionals with free discontinuities this paper is concerned with functionals with free discontinuities considered as depending on the set variable a splitting operation divides the sets in which the functional takes low values in the union of a noise part and a macroscopic part then we show lower semicontinuity results on the limit of the macroscopic parts by hausdorff distance along minimizing sequences

existence and convergence for quasi static evolution in brittle fracture this paper investigates the mathematical well posedness of the variational model of quasi static growth for a brittle crack proposed by francfort and marigo in 15 the starting point is a time discretized version of that evolution which results in a sequence of minimization problems of mumford and shah type functionals the natural weak setting is that of special functions of bounded variation and the main difficulty in showing existence of the time continuous quasi static growth is to pass to the limit as the time discretization step tends to 0 this is performed with the help of a jump transfer theorem which permits under weak convergence assumptions for a sequence un of sbv functions to its bv limit u to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence un in particular it is shown that the notion of minimizer of a mumford and shah type functional for its own jump set is stable under weak convergence assumptions furthermore our analysis justifies numerical methods used for computing the time continuous quasi static evolution 2003 wiley periodicals inc

monotonicity formula and regularity for general free discontinuity problems we give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality of order d 1 this result allows us to prove partial regularity results that is closure and density estimates for the jump set for a large class of free discontinuity problems involving general energies associated to the jump set as for example free boundary problems with robin conditions in particular we give a short proof to the de giorgi carriero leaci result for the mumford shah functional

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

a mumford shah level set approach for geometric image registration a new method for nonrigid registration of multimodal images is presented due to the large interdependence of segmentation and registration the approach is based on simultaneous segmentation and edge alignment the two processes are directly coupled and thus benefit from using complementary information of the entire underlying data set the approach is formulated as a bivariate variational free discontinuity problem in the mumford shah framework a geometric variable describing the contour set and a functional variable which represents the underlying deformation are simultaneously identified the contour set is represented by a level set function we derive a regularized gradient flow and describe an efficient numerical implementation using finite element discretization and multigrid techniques finally we illustrate the method in several applications such as multimodal intrapatient registration and reconstruction by registration to a reference object

on the structure of continua with finite length and goląb s semicontinuity theorem abstract the main results in this note concern the characterization of the length of continua 1 theorem 2 5 and the parametrization of continua with finite length theorem 4 4 using these results we give two independent and relatively elementary proofs of gol b s semicontinuity theorem

a multiscale algorithm for image segmentation by variational method most segmentation algorithms are composed of several procedures split and merge small region elimination boundary smoothing each depending on several parameters the introduction of an energy to minimize leads to a drastic reduction of these parameters the authors prove that the most simple segmentation tool the region merging algorithm made according to the simplest energy is enough to compute a local energy minimum belonging to a compact class and to achieve the job of most of the tools mentioned above the authors explain why merging in a variational framework leads to a fast multiscale multichannel algorithm with a pyramidal structure the obtained algorithm is o n ln n where n is the number of pixels of the picture this fast algorithm is applied to make grey level and texture segmentation and experimental results are shown

affine invariant segmentation by variational method affine invariance viewed as a simplified projective invariance is an essential requirement for analysis of natural scenes we propose an affine invariant analogue of mumford and shah energy functional to segment images and we discuss it we present a simple multiscale algorithm to minimize it based on region growing methods and we display our first numerical experiments

faber krahn inequalities for the robin laplacian a free discontinuity approach isoperimetric inequalities for the principal eigenvalues of the robin laplacian are interpreted as free discontinuity problems of unusual type we prove a full range of faber krahn inequalities in a nonlinear setting and for non smooth domains including the open case of the torsional rigidity the key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation as a byproduct we obtain the best constants for a class of poincare inequalities with trace terms in mathbb r n

on corrections to a z transform method for fdtd simulations with frequency dependent dielectric functions the impulse invariance transform method has been used to discretize frequency dependent dielectric functions for fdtd simulations this paper reviews generalizes and compares several methods for improving the accuracy of this process a slightly novel curve fitting approach is presented for the dielectric function of silver

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

multiscale corner detection by using wavelet transform a multiscale corner detection algorithm based on the wavelet transform of contour orientation is proposed it can utilize both the information of the local extrema and modulus of transform results to detect corners and arcs effectively the ramp width of contour orientation profile which can be computed using the transformed modulus of two scales reveals the difference between corner and arc and is utilized in the determination of corner points the experimental results have shown that the detector is more effective than both the single and multiple scale detectors they also demonstrate that the detector is insensitive to boundary noise in addition the proposed method is more efficient than the other multiscale corner detector because it operates on fewer number of scales which can be implemented by a fast transform algorithm

simplified excision techniques for free discontinuity problems in several variables abstract one of the methods proposed for the study of the problems with a free discontinuity set in two variables has been considered here under slightly different abstract assumptions from those in the previous papers the technical difficulties are in this new setting simplified and less background is required to the reader moreover applications to problems in several variables can be found thanks to a sobolev type theorem which ensures the holder continuity of certain functions out of a suitable neighborhood of their discontinuity set many of the results established so far for the case of two dimensions can be so extended to the general case a new property is also proved

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

on the fdtd formulations for modeling wideband lorentzian media several popular numerical approaches for modeling the dispersion characteristics of complex media are applied to simulate the wave propagation in the wideband lorentzian wbl media under the second order or fourth order accurate finite difference time domain fdtd formulation their implementing algorithms and work performance for this type of dispersive materials are studied and compared to each other the coefficients of a general second order discrete numerical model pertaining to each of these approaches are derived respectively and a unified set of fdtd updating equations for implementing them is presented in light of the analytical results calculated from the given numerical dispersion equations that determines the effective permittivity and permeability of a corresponding artificial wbl medium and the simulation results from the practical fdtd simulations for an exemplified wbl medium the modeling accuracy and stability limits of these approaches are presented by evaluating the absolute errors of both the propagation and transmission functions of a plane electromagnetic pulse wave propagating in the wbl material and transmitting through an interface between the vacuum and the wbl medium under the second order or fourth order fdtd formulation combined with each of the discussed approaches

free boundary problems and density perimeter we introduce a new concept the density perimeter which substitutes the measure of length of the perimeter of a set in free boundary problems we deduce some links between the hausdorff convergence and the char convergence for a family of domains with bounded density perimeter if this perimeter is considered as a penalty term we give existence results for the variational problem which appears in computer vision and in a bernoulli like free boundary problem we also make some considerations concerning a convergence property of the density perimeter

stable formulation of fadi fdtd method for multiterm doubly second order dispersive media this paper presents stable formulation of fundamental alternating direction implicit finite difference time domain fadi fdtd method for multiterm doubly second order dispersive media the formulation is achieved by decomposing the electric and magnetic susceptibility functions into first order poles and it is derived based on currents constitutive relations it is conveniently applicable for most commonly used second order dispersive models such as lorentz and drude models and equally applicable to first order debye model the extension for full 3 d doubly dispersive media using fadi fdtd method makes the resultant update equations much more concise and simpler than using conventional adi fdtd method the number of field variables and update coefficients are greatly reduced at the right hand sides with only first order spatial differencing and no dispersive terms and magnetic update coefficients in the implicit e update equations all these contribute to conciseness and programming simplicity as well as leading to higher efficiency due to much less memory indexing overhead and fewer arithmetic operations most importantly our present formulation is stable while the contemporary second order differential equation formulation has potential instability as to be demonstrated the stability analysis is performed using fourier method by examining the eigenvalues of the fourier amplification matrix numerically

on the accuracy and stability of several widely used fdtd approaches for modeling lorentz dielectrics a rigorous and comparative study on the approximation accuracy and stability limits of several widely used finite difference time domain fdtd approaches namely the auxiliary differential equation ade approach the bilinear transform bt approach the z transform approach zt and the piecewise linear recursive convolution plrc approach for modeling dispersive lorentz dielectrics is presented following the given updating equations between the electric flux density and the electric field intensity we find the zt approach with modified material parameters is much more accurate than the original zt approach and the other three approaches for modeling lorentz dielectrics the stability limits of the ade zt and plrc approaches in simulating lorentz dielectrics are also shown to be a bit more stringent than that of bt approach which preserves the courant stability limit as previously reported

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

frequency dependent fdtd methods using z transforms the frequency dependent finite difference time domain method fd sup 2 td method has been shown to be capable of correctly calculating electromagnetic propagation through media whose dielectric properties are frequency dependent however as researchers search for more elaborate applications the formulation of the fd sup 2 td methods becomes more complex in this work the mathematics of the fd sup 2 td method is developed using z transform theory this has the advantages of presenting a clearer formulation and allowing researchers to draw on the literature of systems analysis and signal processing disciplines

extraction of trend lines and extrema from multiscale curves abstract multiscale planar curves are curves conveying information at several levels of detail to extract polygonal approximations of such curves at different resolutions we propose a parallel method based on connecting locally computed centroids the implementation makes use of a new pyramidal data structure the chain pyramid coarser and coarser approximations are obtained through a hierarchy of increasingly coarse tessellations significant extrema of the curves are localized at the highest resolution by segmenting at a natural scale and employing the structure of the chain pyramid for down projection

a novel piecewise linear recursive convolution approach for dispersive media using the finite difference time domain method two novel methods for implementing recursively the convolution between the electric field and a time dependent electric susceptibility function in the finite difference time domain fdtd method are presented both resulting algorithms are straightforward to implement and employ an inclusive susceptibility function which holds as special cases the lorentz debye and drude media relaxations the accuracy of the new proposed algorithms is found to be systematically improved when compared to existing standard piecewise linear recursive convolution plrc approaches it is conjectured that the reason for this improvement is that the new proposed algorithms do not make any assumptions about the time variation of the polarization density in each time interval no finite difference or semi implicit schemes are used for the calculation of the polarization density the only assumption that these two new methods make is that the first time derivative of the electric field is constant within each fdtd time interval

partial shape classification using contour matching in distance transformation an algorithm is presented to recognize and locate partially distorted 2d shapes without regard to their orientation location and size the algorithm first calculates the curvature function from the digitized image of an object the points of local maxima and minima extracted from the smooth curvature are used as control points to segment the boundary and to guide the boundary matching procedure the boundary matching procedure considers two shapes at a time one shape from the template databank and the other from the object being classified the procedure tries to match the control points in the unknown shape to those of a shape from the template databank and estimates the translation rotation and scaling factors to be used to normalize the boundary of the unknown shape the chamfer 3 4 distance transformation and a partial distance measurement scheme constitute the final step in measuring the similarity between the two shapes the unknown shape is assigned to the class corresponding to the minimum distance the algorithm has been successfully tested on partial shapes using two sets of data one with sharp corners and the other with curve segments this algorithm not only is computationally simple but also works reasonably well in the presence of a moderate amount of noise

optimal and near optimal quantization of integrable functions abstract a continuous function is approximated by step functions with given number of break points the approximations are nonlinear local as well as global optimization strategies based on fixed point analysis are integrated with branch and bound techniques for multivariate lipschitz minimization branching strategies are based on the structure of attracting domains

an elasticity based covariance analysis of shapes we introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects based on the notion of nonlinear elastic deformations from one shape to another a suitable linearization of geometric shape variations is introduced once such a linearization is available a principal component analysis can be investigated this requires the definition of a covariance metric an inner product on linearized shape variations the resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation the underlying elasticity concept represents an alternative to riemannian shape statistics in this paper we compare a standard l 2 type covariance metric with a metric based on the hessian of the nonlinear elastic energy furthermore we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity for the built in pairwise elastic registration a relaxed model formulation is employed which allows for a non exact matching shape contours are approximated by single well phase fields which enables an extension of the method to a covariance analysis of image morphologies the model is implemented with multilinear finite elements embedded in a multi scale approach the characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2d and 3d