shape description and matching using integral invariants on eccentricity transformed images matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision to keep track of changes inside the breast for example it is important for a computer aided detection system to establish correspondences between regions of interest shape transformations computed both with integral invariants ii and with geodesic distance yield signatures that are invariant to isometric deformations such as bending and articulations integral invariants describe the boundaries of planar shapes however they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure conversely eccentricity transforms ecc can match shapes by signatures of geodesic distance histograms based on information from inside the shape but they ignore the boundary information we describe a method that combines the boundary signature of a shape obtained from ii and structural information from the ecc to yield results that improve on them separately

globally consistent reconstruction of ripped up documents one of the most crucial steps for automatically reconstructing ripped up documents is to find a globally consistent solution from the ambiguous candidate matches however little work has been done so far to solve this problem in a general computational framework without using application specific features in this paper we propose a global approach for reconstructing ripped up documents by first finding candidate matches from document fragments using curve matching and then disambiguating these candidates through a relaxation process to reconstruct the original document the candidate disambiguation problem is formulated in a relaxation scheme in which the definition of compatibility between neighboring matches is proposed and global consistency is defined as the global criterion initially global match confidences are assigned to each of the candidate matches after that the overall local relationships among neighboring matches are evaluated by computing their global consistency then these confidences are iteratively updated using the gradient projection method to maximize the criterion this leads to a globally consistent solution and thus provides a sound document reconstruction the overall performance of our approach in several practical experiments is illustrated the results indicate that the reconstruction of ripped up documents up to 50 pieces is possibly accomplished automatically

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

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

noise induced chaos and phase space flux abstract we study the effect of additive noise on near integrable second order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits the noise is represented by a type of shinozuka stochastic process capable of arbitrarily closely approximating gaussian noise with any specified spectrum we derive a formula for the flux factor applicable for any asymptotic mean stationary excitation this derivation shows that to first order the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit it also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux this is true for both random and nonrandom excitations and indicates that for the dynamical systems considered here these two classes of excitation play substantially equivalent roles in the promotion of chaos

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

similarity invariant signatures for partially occluded planar shapes a methodology is described for associating local invariant signature functions to smooth planar curves in order to enable their translation rotation and scale invariant recognition from arbitrarily clipped portions the suggested framework incorporates previous approaches based on locating inflections curvature extrema breakpoints and other singular points on planar object boundaries and provides a systematic way of deriving novel invariant signature functions based on curvature or cumulative turn angle of curves these new signatures allow the specification of arbitrarily dense feature points on smooth curves whose locations are invariant under similarity transformations the results are useful for detecting and recognizing partially occluded planar objects a key task in low level robot vision

geometric computation of curvature driven plane curve evolutions we present a new numerical scheme for planar curve evolution with a normal velocity equal to f kappa where kappa is the curvature and f is a nondecreasing function such that f 0 0 and either x mapsto f x 3 is lipschitz with lipschitz constant less than or equal to 1 or f x x gamma for gamma geq 1 3 the scheme is completely geometrical and avoids some drawbacks of finite difference schemes in particular no special parameterization is needed and the scheme is monotone that is if a curve initially surrounds another one then this remains true during their evolution which guarantees numerical stability we prove consistency and convergence of this scheme in a weak sense finally we display some numerical experiments on synthetic and real data

foundations of semi differential invariants this paper elaborates the theoretical foundations of a semi differential framework for invariance semi differential invariants combine coordinates and their derivatives with respect to some contour parameter at several points of the image contour thus allowing for an optimal trade off between identification of points and the calculation of derivatives a systematic way of generating complete and independent sets of such invariants is presented it is also shown that invariance under reparametrisation can be cast in the same framework the theory is illustrated by a complete analysis of 2d affine transformations in a companion paper pauwels et al 1995 these affine semi differential invariants are implemented in the computer program form flat object recognition method for the recognition of planar contours under pseudo perspective projection

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

affine geometry curve flows and invariant numerical approximations abstract a new geometric approach to the affine geometry of curves in the plane and to affine invariant curve shortening is presented we describe methods of approximating the affine curvature with discrete finite difference approximations based on a general theory of approximating differential invariants of lie group actions by joint invariants applications to computer vision are indicated

analyzing skewed symmetries symmetry is pervasive in both man made objects and nature since symmetries project to skew symmetries finding axes of skew symmetry is an important vision task this paper presents a linear time algorithm for finding the axes of skew symmetry where the degree of symmetry is known first we present a review and critique of current methods for finding the axes of skew symmetry next we decompose the problem of finding skew symmetry into the subproblems of solving for the rotational parameter of a shear symmetry and recovering the shear parameter of a reflexive symmetry using this approach the authors derive a direct non heuristic moment based technique for finding the axes of skew symmetry for skew symmetric figures with degree of symmetry less than five we obtain a closed form solution the method does not rely on continuous contours but assumes there is no occlusion and requires knowing the contour s degree of symmetry it is the first algorithm to find the axes of skew symmetry ino n time where n is the number of contour points the method is especially suited to industrial applications where the degree of symmetry is often knowna priori examples of the method are presented for both real and synthetic images and an error analysis of the method is given

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

on the geometry of transport in phase space i transport in k degree of freedom hamiltonian systems 2 k abstract over the past several years the effect of geometrical structures such as kam tori resonance zones and cantori on phase space transport in two degree of freedom hamiltonian systems has been studied in three degree of freedom systems and higher these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the hamiltonian in this paper we show that in the 2 k 1 dimensional level sets of the hamiltonian of a class of k degree of freedom hamiltonian systems the 2 k 2 dimensional stable and unstable manifolds of normally hyperbolic invariant 2 k 3 dimensional spheres form partial barriers to transport under certain conditions we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two degree of freedom hamiltonian systems considering a foliation of the level set of the hamiltonian into disjoint regions separated by such partial barriers we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles on the other hand despite the fact that these partial barriers are codimension one for k degree of freedom systems k 3 they may not intersect in such a way so as to divide the phase space

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

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

noise resistant invariants of curves projective invariants are shape descriptors that are independent of the point of view from which the shape is seen and therefore are of major importance in object recognition they make it possible to match an image of an object to one stored in a database without the need for searching for the correct viewpoint an invariant representation of a general curve is obtained the calculation is local and does not suffer from the occlusion problem of global descriptors to make the method robust differentiation techniques that give much more reliable results than previous ones are developed these differentiation methods are useful in many other applications as well

analytic integrability of a chua system we consider the system x a z a1x3 a2x2 bx y z z b1x y b2z where a and b are parameters and b1 7 10 b2 6 25 a1 44 3 and a2 41 2 we analyze the existence of local and global analytic first integrals

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

stochastic level set method for shape optimisation we present a new method for stochastic shape optimisation of engineering structures the method generalises an existing deterministic scheme in which the structure is represented and evolved by a level set method coupled with mathematical programming the stochastic element of the algorithm is built on the methods of statistical mechanics and is designed so that the system explores a boltzmann gibbs distribution of structures in non convex optimisation problems the deterministic algorithm can get trapped in local optima the stochastic generalisation enables sampling of multiple local optima which aids the search for the globally optimal structure the method is demonstrated for several simple geometrical problems and a proof of principle calculation is shown for a simple engineering structure

hyperbolicity transversality and analytic first integrals abstract let a be a normally hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold m we assume that the stable and unstable manifold of a intersect transversally in an admissible way the dynamics on a is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold respectively satisfy a non resonance condition in the case where a is a point or a torus we prove that the discrete dynamical system associated to f does not admit an analytic first integral the proof is based on a triviality lemma which is of combinatorial nature and a geometrical lemma the same techniques allow us to prove analytic non integrability of hamiltonian systems having arnold diffusion in particular using results of xia we prove analytic non integrability of the elliptic restricted three body problem as well as the planar three body problem

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

shape matching using multiscale integral invariants we present a shape descriptor based on integral kernels shape is represented in an implicit form and it is characterized by a series of isotropic kernels that provide desirable invariance properties the shape features are characterized at multiple scales which form a signature that is a compact description of shape over a range of scales the shape signature is designed to be invariant with respect to group transformations which include translation rotation scaling and reflection in addition the integral kernels that characterize local shape geometry enable the shape signature to be robust with respect to undesirable perturbations while retaining discriminative power use of our shape signature is demonstrated for shape matching based on a number of synthetic and real examples

geometric invariants and object recognition we discuss the role of the general invariance concept in object recognition and review the classical and recent literature on projective invariance invariants help solve major problems of object recognition for instance different images of the same object often differ from each other because of the different viewpoint from which they were obtained to match the two images common methods thus need to find the correct viewpoint a difficult problem that can involve search in a high dimensional space of all possible points of view and or finding point correspondences geometric invariants are shape descriptors computed from the geometry of the shape that remain unchanged under geometric transformations such as changing the viewpoint thus they can be matched without search deformations of objects are another important class of geometric changes for which invariance is useful

planar object recognition using projective shape representation we describe a model based recognition system called lewis for the identification of planar objects based on a projectively invariant representation of shape the advantages of this shape description include simple model acquisition direct from images no need for camera calibration or object pose computation and the use of index functions we describe the feature construction and recognition algorithms in detail and provide an analysis of the combinatorial advantages of using index functions index functions are used to select models from a model base and are constructed from projective invariants based on algebraic curves and a canonical projective coordinate frame examples are given of object recognition from images of real scenes with extensive object libraries successful recognition is demonstrated despite partial occlusion by unmodelled objects and realistic lighting conditions

the complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition representations are introduced for handling 2d algebraic curves implicit polynomial curves of arbitrary degree in the scope of computer vision applications these representations permit fast accurate pose independent shape recognition under euclidean transformations with a complete set of invariants and fast accurate pose estimation based on all the polynomial coefficients the latter is accomplished by a centering of a polynomial based on its coefficients followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two dimensional subspaces or identity transformations within one dimensional subspaces result from rotations in x y measured data space angles of these rotations in the two dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x y data space by recasting this approach in terms of a complex variable i e x iy z and complex polynomial coefficients further conceptual and computational simplification results application to shape based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves

detecting similarity of rational plane curves a novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity which is a central question in pattern recognition as a by product it finds all such similarities and the particular case of equal curves yields all symmetries a complete theoretical description of the method is provided and the method has been implemented and tested in the sage system for curves of moderate degrees

maurer cartan forms for fields on surfaces application to heart fiber geometry we study the space of first order models of smooth frame fields using the method of moving frames by exploiting the maurer cartan matrix of connection forms we develop geometrical embeddings for frame fields which lie on spherical ellipsoidal and generalized helicoid surfaces we design methods for optimizing connection forms in local neighborhoods and apply these to a statistical analysis of heart fiber geometry using diffusion magnetic resonance imaging this application of moving frames corroborates and extends recent characterizations of muscle fiber orientation in the heart wall but also provides for a rich geometrical interpretation in particular we can now obtain direct local measurements of the variation of the helix and transverse angles of fiber fanning and twisting and of the curvatures of the heart wall in which these fibers lie

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

integral invariants for space motion trajectory matching and recognition motion trajectories provide a key and informative clue in motion characterization of humans robots and moving objects in this paper we propose some new integral invariants for space motion trajectories which benefit effective motion trajectory matching and recognition integral invariants are defined as the line integrals of a class of kernel functions along a motion trajectory a robust estimation of the integral invariants is formulated based on the blurred segment of noisy discrete curve then a non linear distance of the integral invariants is defined to measure the similarity for trajectory matching and recognition such integral invariants in addition to being invariant to transformation groups have some desirable properties such as noise insensitivity computational locality and uniqueness of representation experimental results on trajectory matching and sign recognition show the effectiveness and robustness of the proposed integral invariants in motion trajectory matching and recognition a meaningful definition of integral invariants for space trajectories is proposed we derive a novel estimation method of integral invariants for discrete trajectories a non linear distance is defined to measure the similarity between trajectories integral invariants show robustness to noise occlusions and transformations experiments are carried out to examine their effectiveness and robustness in applications

schwarps locally projective image warps based on 2d schwarzian derivatives image warps or just warps capture the geometric deformation existing between two images of a deforming surface the current approach to enforce a warp s smoothness is to penalize its second order partial derivatives bookstein in ieee trans pattern anal mach intell 11 567 585 1989 rueckert et al in ieee trans med imaging 18 712 721 1999 because this favors locally affine warps this fails to capture the local projective component of the image deformation this may have a negative impact on applications such as image registration and deformable 3d reconstruction we propose a novel penalty designed to smooth the warp while capturing the deformation s local projective structure our penalty is based on equivalents to the schwarzian derivatives which are projective differential invariants exactly preserved by homographies we propose a methodology to derive a set of partial differential equations with only homographies as solutions we call this system the schwarzian equations and we explicitly derive them for 2d functions using differential properties of homographies we name as schwarp a warp which is estimated by penalizing the residual of schwarzian equations experimental evaluation shows that schwarps outperform existing warps in modeling and extrapolation power and lead to better results in three deformable reconstruction methods namely shape reconstruction in shape from template camera calibration in shape from template and non rigid structure from motion

the local projective shape of smooth surfaces and their outlines this article examines projectively invariant local geometric properties of smooth curves and surfaces oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines it is applied to two problems 1 the projective generalization of koenderink s famous characterization of convexities concavities and inflections of the apparent contours of solids bounded by smooth surfaces and 2 the image based construction of rim meshes which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it

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

recognition of planar shapes under affine distortion methods for the recognition ofplanar shapes from arbitrary viewpoints are described the adopted model of projection is orthographic the invariant descriptions derived for this group are one dimensional shape signatures comparable to the well known curvature as a function of arc length description of euclidean geometry since the use of such differential invariants in the affine case would lead to unacceptably high orders of derivatives affine invariant descriptions based onsemi differential invariants are proposed as an alternative a systematic discussion of different types of these invariants is given the usefulness and viability of this methodology is demonstrated on a database containing more than 40 objects

arnold diffusion and oscillatory solutions in the planar three body problem abstract in this paper the following results concerning the newtonian three body problem are obtained 1 arnold diffusion exists in the planar three body problem as conjectured by v 1 arnold 1964 dokl acad nauk sssr 156 9 2 the oscillatory solutions as well as capture and escaping solutions among other classes of chaotic solutions exist in the three body problem 3 a special and interesting phenomenon which we call the pseudo arnold diffusion arises near infinity in the three body problem 4 as a result of the existence of arnold diffusion the planar three body problem is non integrable and there are no additional real analytic integrals besides the known ones

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

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

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 unified curvature definition for regular polygonal and digital planar curves in this paper we propose a new definition of curvature called visual curvature it is based on statistics of the extreme points of the height functions computed over all directions by gradually ignoring relatively small heights a multi scale curvature is obtained the theoretical properties and the experiments presented demonstrate that multi scale visual curvature is stable even in the presence of significant noise to our best knowledge the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves moreover it yields stable curvature estimates of curves in digital images even under sever distortions we also show a relation between multi scale visual curvature and convexity of simple closed curves

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

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

efficient detection of symmetries of polynomially parametrized curves we present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations the algorithms are based on the existence of a linear relationship between two proper polynomial parametrizations of the curve which leads to a triangular polynomial system with complex unknowns that can be solved in a very fast way in particular curves parametrized by polynomials of serious degrees can be analyzed in a few seconds in our analysis we provide a good number of theoretical results on symmetries of polynomial curves algorithms for detecting rotation and mirror symmetry and closed formulas to determine the symmetry center and the symmetry axis when they exist a complexity analysis of the algorithms is also given

recognizing solid objects by alignment with an image in this paper we consider the problem of recognizing solid objects from a single two dimensional image of a three dimensional scene we develop a new method for computing a transformation from a three dimensional model coordinate frame to the two dimensional image coordinate frame using three pairs of model and image points we show that this transformation always exists for three noncollinear points and is unique up to a reflective ambiguity the solution method is closed form and only involves second order equations we have implemented a recognition system that uses this transformation method to determine possible alignments of a model with an image each of these hypothesized matches is verified by comparing the entire edge contours of the aligned object with the image edges using the entire edge contours for verification rather than a few local feature points reduces the chance of finding false matches the system has been tested on partly occluded objects in highly cluttered scenes

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

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

numerically invariant signature curves corrected versions of the numerically invariant expressions for the affine and euclidean signature of a planar curve introduced by calabi et al in int j comput vision 26 107 135 1998 are presented the new formulas are valid for fine but otherwise arbitrary partitions of the curve we also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the euclidean signature curve namely the curvature the torsion and their derivatives with respect to arc length

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

hfes a height function method with explicit input and signed output for high order estimations of curvature and unit vectors of planar curves the author proposes a height function hf method for estimating curvature and unit vectors of planar curves the new method is named hfes since its signature consists of explicit input parameters

projective evolution of plane curves we show that projectively invariant evolution operators have unavoidable singularities in particular we see that there exists no non singular projective evolution operator well defined over straight lines nor conics

affine invariant geometry for non rigid shapes shape recognition deals with the study geometric structures modern surface processing methods can cope with non rigidity by measuring the lack of isometry deal with similarity or scaling by multiplying the euclidean arc length by the gaussian curvature and manage equi affine transformations by resorting to the special affine arc length definition in classical equi affine differential geometry here we propose a computational framework that is invariant to the full affine group of transformations similarity and equi affine thus by construction it can handle non rigid shapes technically we add the similarity invariant property to an equi affine invariant one and establish an affine invariant pseudo metric as an example we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison

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

a nonlinear probabilistic method for predicting vessel capsizing in random beam seas the capsizing of vessels in random beam seas is investigated using a single degree of freedom model which is limited to the roll motion several factors including sea wave spectrum nonlinear righting moment characteristics and nonlinear damping are taken into account in the analysis a nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics specifically the melnikov function and the phase space area flux and random vibrations conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics significant wave height and characteristic wave period and the vessel parameters damping and stiffness coefficients extensive numerical simulations are carried out to demonstrate the validity of the analytical results it is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing

a wiener filter state space flux optimal control against escape from a potential well escape from a potential well often identified with system failure is reformulated for a class of two dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well a control against system failure based on minimizing the state space flux out of this safe region is considered for systems subject to weak external forcing with available control power exceeding a certain threshold the finite lag state space flux optimal control against failure is shown to involve a wiener filter a non wiener solution is proposed for available power below this threshold these results hold for wide sense stationary forcing with any statistics and any spectrum

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

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

phase space transport and control of escape from a potential well abstract a framework for controlling a nonlinear dynamical system against escape from a potential well is presented based on reducing the phase space transport across the separatrix associated with the potential well a bandlimited open loop control with finite lag is considered for systems with weak additive stationary forcing including specifically the colored gaussian case the related multiplicative closed loop control problem is shown to reduce to an open loop problem a numerical example based on the duffing oscillator is presented to illustrate the theory

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

h sub 2 control with time domain constraints theory and an application in this paper we study the problem of minimizing the h sub 2 norm of a given transfer function subject to time domain constraints on the time response of a different transfer function to a given test signal the main result of this paper shows that this problem admits a minimizing solution in r h sub 2 moreover rational solutions with performance arbitrarily close to optimal can be found by constructing families of approximating problems each one of these problems entails solving a finite dimensional quadratic programming problem whose dimension can be determined before hand these results are illustrated and experimentally validated by designing a controller for an active vision application