toeplitz and hankel operators on bergman spaces in this paper we consider toeplitz and hankel operators on the bergman spaces of the unit ball and the polydisk in c n whose symbols are bounded measurable functions we giv e necessary and sufficient conditions on the symbols for these operators to be compact we study the fredholm theory of topelitz operators for which the corresponding hankel operator is compact for these toeplitz operators the essential spectrum is computed and shown to be connected we also consider symbols that extend to continuous functions on the maximal ideal space of h for these symbols we describe when the toeplitz or hankel operators are compact

curve shortening and the rendezvous problem for mobile autonomous robots if a smooth closed and embedded curve is deformed along its normal vector field at a rate proportional to its curvature it shrinks to a circular point this curve evolution is called euclidean curve shortening and the result is known as the gage hamilton grayson theorem motivated by the rendezvous problem for mobile autonomous robots we address the problem of creating a polygon shortening flow a linear scheme is proposed that exhibits several analogues to euclidean curve shortening the polygon shrinks to an elliptical point convex polygons remain convex and the perimeter of the polygon is monotonically decreasing

stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

differential morphology and image processing image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain we provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems this includes ideas on using partial differential or difference equations pdes to model distance propagation or nonlinear multiscale processes in images we briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics we also review some nonlinear pdes that model the evolution of multiscale morphological operators and use morphological derivatives among the new ideas presented we develop some general 2 d max min sum difference equations that model the space dynamics of 2 d morphological systems including the distance computations and some nonlinear signal transforms called slope transforms that can analyze these systems in a transform domain in ways conceptually similar to the application of fourier transforms to linear systems thus distance transforms are shown to be bandpass slope filters we view the analysis of the multiscale morphological pdes and of the eikonal pde solved via weighted distance transforms as a unified area in nonlinear image processing which we call differential morphology and briefly discuss its potential applications to image processing and computer vision

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

analyzing and synthesizing images by evolving curves with theosher sethian method numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations this paper reviews the relevant concepts in numerical analysis and the relation between curve evolution hamilton jacobi partial differential equations and differential conservation laws this close relation enables us to introduce finite difference approximations based on the theory of conservation laws into curve evolution it is shown how curve evolution serves as a powerful tool for image analysis and how these mathematical relations enable us to construct efficient and accurate numerical schemes some examples demonstrate the importance of the cfl condition as a necessary condition for the stability of the numerical schemes

numerical conditioning problems and solutions for nonparametric i i d statistical active contours in this paper we propose an active contour model based on nonparametric independent and identically distributed i i d statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background this is not however the first active contour model proposed to solve the segmentation problem under these same assumptions in contrast to prior active contour models based on nonparametric i i d statistics we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour instead treating the segmentation problem as a pixel wise classification problem we formulate an active contour to minimize the unbiased pixel wise average misclassification probability amp this not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour but it also solves a numerical conditioning problem that arises with such prior active contour models as a result the amp model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i i d statistical segmentation problem via probability distances to discuss this improved numerical behavior more precisely we introduce the notion of x201c conditioning ratio x201d and demonstrate that the proposed amp active contour is numerically better conditioned i e exhibits a much smaller conditioning ratio than prior probability distance based active contours

invertibility and spectral properties of dual toeplitz operators abstract in this paper we consider dual toeplitz operators with continuous symbols and bounded harmonic symbols on the orthogonal complement of the bergman space over the open unit disk we construct a dual toeplitz operator with continuous symbol such that its spectrum is disconnected on the other hand we show that the spectra of dual toeplitz operators with certain class of bounded harmonic symbols are connected which leads to some partial answers to the open question posed by stroethoff and zheng 18 in 2002 we obtain a complete characterization for the hyponormality of dual toeplitz operators with bounded harmonic symbols moreover we establish some sufficient conditions that are convenient to check for dual toeplitz operators with continuous symbols and bounded harmonic symbols to be invertible respectively

blow up rates for the general curve shortening flow abstract the blow up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow in particular this generalizes a theorem of m e gage and r s hamilton about mean curvature flow in r 2

algebraic and spectral properties of dual toeplitz operators dual toeplitz operators on the orthogonal complement of the bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement in this paper we study algebraic and spectral properties of dual toeplitz operators

geometric active deformable models in shape modeling this paper analyzes the problem of shape modeling using the principle of active geometric deformable models while the basic modeling technique already exists in the literature we highlight many of its drawbacks and discuss their source and steps to overcome them we propose a new stopping criterion to address the stopping problem we also propose to apply a level set algorithm to implement the active geometric deformable models thereby handling topology changes automatically to alleviate the numerical problems associated with the implementation of the level set algorithm we propose a new adaptive multigrid narrow band algorithm all the proposed new changes have been illustrated with experiments with synthetic images and medical images

products of hankel and toeplitz operators on the bergman space we consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products tftg are bounded on the bergman space we prove results analogous to those obtained by the second author 17 for such toeplitz products on the hardy space we furthermore obtain similar results for hankel products hfh g where f and g are square integrable on the unit disk and for the mixed haplitz products hftg and tgh f where f and g are square integrable on the unit disk and g is analytic

piecewise approximation of contours through scale space selection of dominant points this paper describes a method of approximating a shape contour with a polygon the polygon vertices are extracted from the curvature extremes through a scale space description of the contour via linear diffusion these vertices are located on the contour points where the sharper changes of the contour directions occur using a proper strategy a set of extremes that result in a given approximation level is chosen by adding new vertices the approximation level can be improved and a scalable representation of the contour is identified this method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour this polygonal approximation method is used for scalable encoding of the shape contours in this regard an encoding technique suitable for scalable polygonal approximation has been developed we show that encoding the approximated polygons result in a good relation between the distortion and the bitrate finally we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

implementing continuous scale morphology via curve evolution abstract a new approach to digital implementation of continuous scale mathematical morphology is presented the approach is based on discretization of evolution equations associated with continuous multiscale morphological operations those equations and their corresponding numerical implementation can be derived either directly from mathematical morphology definitions or from curve evolution theory the advantages of the proposed approach over the classical discrete morphology are demonstrated

removal of partial occlusion from single images this paper examines large partial occlusions in an image which occur near depth discontinuities when the foreground object is severely out of focus we model these partial occlusions using matting with the alpha value determined by the convolution of the blur kernel with a pinhole projection of the occluder the main contribution is a method for removing the image contribution of the foreground occluder in regions of partial occlusion which improves the visibility of the background scene the method consists of three steps first the region of complete occlusion is estimated using a curve evolution method second the alpha value at each pixel in the partly occluded region is estimated third the intensity contribution of the foreground occluder is removed in regions of partial occlusion experiments demonstrate the method s ability to remove the effects of partial occlusion in single images with minimal user input

convergence past singularities for a fully discrete approximation of curvature drive interfaces consider a closed surface in rn of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term this geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter eps 0 conforming piecewise linear finite elements over a quasi uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time step tau it is shown that the zero level set of the fully discrete solution converges past singularities to the true interface provided tau h 2 approx o eps 3 and no fattening occurs if the more stringent relations tau h 2 approx o eps 4 are enforced then a linear rate of convergence o eps for interfaces is derived in the vicinity of regular points namely those for which the underlying viscosity solution is nondegenerate singularities and their smearing effect are also studied the analysis is based on constructing discrete barriers via a parabolic projection lipschitz dependence of viscosity solutions with respect to perturbations of data and discrete nondegeneracy these issues are proven along with quasi optimality in two dimensions of the parabolic projection in l infty with respect to both order and regularity requirements for functions in w 2 1 p

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

a fast semi lagrangian contouring method for moving interfaces abstract general moving interface problems are solved by a new approach evaluating an explicit semi lagrangian advection formula with efficient geometric algorithms and extracting the moving interface with a fast new contouring technique the new approach decouples spatial and temporal resolutions and grid free adaptive refinement of the interface increases accuracy dramatically a modular implementation with a fast new intrinsic geometry module computes highly accurate solutions to geometric moving interface problems involving merging anisotropy and faceting with a high order embedded geometry module it solves second order problems involving curvature dynamic topology and nonlocal interactions

invariant subspaces quasi invariant subspaces and hankel operators abstract in this paper using the theory of hilbert modules we study invariant subspaces of the bergman spaces on bounded symmetric domains and quasi invariant subspaces of the segal bargmann spaces we completely characterize small hankel operators with finite rank on these spaces

a weighted curvature flow for shape deformation abstract in this paper we shall discuss a weighted curvature flow for a regular curve in the 2d euclidean space the weighted curvature flow for planar curves is a generalization of the well known curvature flow discussed by gage hamilton and grayson under a suitable weighted curvature flow convex curves will remain convex in the deformation process however the curve may not converge to a round point for general weights indeed for a nonnegative weight function u with k isolated zeros a curve will converge to a limiting k polygon the weighted curvature flow will have many useful properties which have applications to image processing we shall also present some numerical simulations to illustrate how curves deform under the weighted curvature flow with different weight functions u moreover our algorithm is very effective and stable the approximation of higher derivatives in our new algorithm only involve in the neighboring points

a multiscanning approach based on morphological filtering it is argued that the mathematical morphology method seems to be more reasonable and powerful in studying certain multiscaling vision problems than the approach that uses derivatives of gaussian shaped filters of different sizes to show the validity of this method the authors concentrated on an application that involves forming scale space image of a 2 d shape using morphological opening filtering a proof is given to show that morphological opening filtering has a property of not introducing additional zero crossings as one moves to a coarser scale this is a different result from the conclusion by a l yuille and t a poggio ibid vol pami 8 jan 1986 that the gaussian filter is the only filter with this property in addition opening filtering is computationaly simpler than the gaussian filter

on instability of yang mills connections on demontre l instabilite des connexions de yang mills pour une grande classe de variete incluant des espaces symetriques compacts

a representation theory for morphological image and signal processing a unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented the unification requires a set theoretic methodology where signals are modeled as sets systems signal transformations are viewed as set mappings and translational invariant systems are uniquely characterized by special collections of input signals this approach leads to a general representation theory in which any translation invariant increasing upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations the theory is used to analyze some special cases of image signal analysis systems such as morphological filters median and order statistic filters linear filters and shape recognition transforms although the developed theory is algebraic its prototype operations are well suited for shape analysis hence the results also apply to systems that extract information about the geometrical structure of signals

global regularizing flows with topology preservation for active contours and polygons active contour and active polygon models have been used widely for image segmentation in some applications the topology of the object s to be detected from an image is known a priori despite a complex unknown geometry and it is important that the active contour or polygon maintain the desired topology in this work we construct a novel geometric flow that can be added to image based evolutions of active contours and polygons in order to preserve the topology of the initial contour or polygon we emphasize that unlike other methods for topology preservation the proposed geometric flow continually adjusts the geometry of the original evolution in a gradual and graceful manner so as to prevent a topology change long before the curve or polygon becomes close to topology change the flow also serves as a global regularity term for the evolving contour and has smoothness properties similar to curvature flow these properties of gradually adjusting the original flow and global regularization prevent geometrical inaccuracies common with simple discrete topology preservation schemes the proposed topology preserving geometric flow is the gradient flow arising from an energy that is based on electrostatic principles the evolution of a single point on the contour depends on all other points of the contour which is different from traditional curve evolutions in the computer vision literature

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

noise removal with gauss curvature driven diffusion in this paper we propose the use of the gauss curvature in a gauss curvature driven diffusion equation for noise removal the proposed scheme uses the gauss curvature as the conductance term and controls the amount of diffusion the main advantage of the scheme is that it preserves important structures such as straight edges curvy edges ramps corners small scaled features etc

segmenting skin lesions with partial differential equations based image processing algorithms a partial differential equations pde based system for detecting the boundary of skin lesions in digital clinical skin images is presented the image is first preprocessed via contrast enhancement and anisotropic diffusion if the lesion is covered by hairs a pde based continuous morphological filter that removes them is used as an additional preprocessing step following these steps the skin lesion is segmented either by the geodesic active contours model or the geodesic edge tracing approach these techniques are based on computing again via pdes a geodesic curve in a space defined by the image content examples showing the performance of the algorithm are given

stochastic differential equations and geometric flows in previous years curve evolution applied to a single contour or to the level sets of an image via partial differential equations has emerged as an important tool in image processing and computer vision curve evolution techniques have been utilized in problems such as image smoothing segmentation and shape analysis we give a local stochastic interpretation of the basic curve smoothing equation the so called geometric heat equation and show that this evolution amounts to a tangential diffusion movement of the particles along the contour moreover assuming that a priori information about the shapes of objects in an image is known we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise we also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features in contrast to the original geometric heat flow which tends to circularize any closed curve

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

operators on the orthogonal complement of the dirichlet space ii in this paper we first prove that a dual hankel operator r on the orthogonal complement of the dirichlet space is compact for w 1 d and then that a semicommutator of two toeplitz operators on the dirichlet space or two dual toeplitz operators on the orthogonal complement of the dirichlet space in sobolev space is compact we also prove that a dual hankel operator r with w 1 d is of finite rank if and only if b is orthogonal to the dirichlet space for some finite blaschke product b and give a sufficient and necessary condition for the semicommutator of two dual toeplitz operators to be of finite rank

on the convergence of planar curves under smoothing curve smoothing has two important applications in computer vision and image processing 1 the curvature scale space css technique for shape analysis and 2 the gaussian filter for noise suppression in this paper we study how planar curves converge as they are smoothed with increasing scales first two types of convergence behavior are clarified the coined term shrinkage refers to the reduction of arc length of a smoothed planar curve which describes the convergence of the curve latitudinally and another coined term collapse refers to the movement of each point to its limiting position which describes the convergence of the curve longitudinally a systematic study on the shrinkage and collapse of three categories of curve models is then presented the corner models helps to reveal how the local structures of planar curves collapse and what the smoothed curves may converge to the sawtooth models allows us to gain insights regarding how noise is suppressed from noisy planar curves by the gaussian filter our investigation on the closed curves shows that each curve collapses to a point at its center of mass however different curves may yield different limiting shapes at the infinity scale finally based upon the derived results the performance of the css technique in corner detection and shape representation is analyzed and a fast implementation method of the gaussian filter for noise suppression is proposed

highly accurate schemes for pde based morphology with general convex structuring elements the two fundamental operations in morphological image processing are dilation and erosion these processes are defined via structuring elements it is of practical interest to consider a variety of structuring element shapes the realisation of dilation erosion for convex structuring elements by use of partial differential equations pdes allows for digital scalability and subpixel accuracy however numerical schemes suffer from blur by dissipative artifacts in our paper we present a family of so called flux corrected transport fct schemes that addresses this problem for arbitrary convex structuring elements the main characteristics of the fct schemes are i they keep edges very sharp during the morphological evolution process and ii they feature a high degree of rotational invariance we validate the fct scheme theoretically by proving consistency and stability numerical experiments with diamonds and ellipses as structuring elements show that fct schemes are superior to standard schemes in the field of pde based morphology

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

the variational problem for a certain space time functional defined on planar closed curves abstract we consider a variational problem for a certain space time functional defined on planar closed curves the functional is related to the functional appeared in bellettini and mugnai 2008 4 the variational problem is stated as follows let 0 and 1 denote planar closed curves and t be a positive constant minimize the space time functional over family of planar closed curves which change from 0 at time t 0 into 1 at time t t concerning the variational problem we prove the existence of minimizer in a radially symmetric class and determine all the minimizers for each initial final data moreover we show that there exists a unique non radially symmetric critical point in a neighborhood of a certain radially symmetric minimizer

commuting dual toeplitz operators with pluriharmonic symbols abstract in this paper we completely characterize commuting dual toeplitz operators with bounded pluriharmonic symbols on the bergman space of the unit ball we show that for and pluriharmonic s s s on l a 2 b n only in the trivial case here the trivial case is or holomorphic

visibility and its dynamics in a pde based implicit framework we investigate the problem of determining visible regions given a set of moving obstacles and a moving vantage point our approach to this problem is through an implicit framework where the obstacles are represented by a level set function the visibility problem is formally formulated as a boundary value problem bvp of a first order partial differential equation it is based on the continuation of values along the given ray field we propose a one pass multi level algorithm for the construction of the solution on a grid furthermore we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory in all of these situations topological changes such as merging and breaking occur in the regions of interest these are automatically handled by the level set framework proposed here finally we obtain additional useful information through simple operations in the level set framework

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

minimal surfaces a geometric three dimensional segmentation approach a novel geometric approach for three dimensional object segmentation is presented the scheme is based on geometric deformable surfaces moving towards the objects to be detected we show that this model is related to the computation of surfaces of minimal area local minimal surfaces the space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected the general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework the scheme is stable robust and automatically handles changes in the surface topology during the deformation results related to existence uniqueness stability and correctness of the solution to this geometric deformable model are presented as well based on an efficient numerical algorithm for surface evolution we present a number of examples of object detection in real and synthetic images

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

on the evolution of curves via a function of curvature i the classical case abstract the problem of curve evolution as a function of its local geometry arises naturally in many physical applications a special case of this problem is the curve shortening problem which has been extensively studied here we consider the general problem and prove an existence theorem for the classical solution the main theorem rests on lemmas that bound the evolution of length curvature and how far the curve can travel

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

anisotropic morphological filters with spatially variant structuring elements based on image dependent gradient fields this paper deals with the theory and applications of spatially variant discrete mathematical morphology we review and formalize the definition of spatially variant dilation erosion and opening closing for binary and gray level images using exclusively the structuring function without resorting to complement this theoretical framework allows to build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image the shape and orientation of the structuring element at each pixel are extracted from the image under study the orientation is given by means of a diffusion process of the average square gradient field which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects the proposed filters are used on binary and gray level images for enhancement of anisotropic features such as coherent flow like structures results of spatially variant erosions dilations and openings closings based filters prove the validity of this theoretical sound and novel approach