true scale invariant random spatial networks some aspects of real world road networks seem to have an approximate scale invariance property motivating study of mathematical models of random networks whose distributions are exactly invariant under euclidean scaling this requires working in the continuum plane so making a precise definition is not trivial we introduce an axiomatization of a class of processes we call scale invariant random spatial networks whose primitives are routes between each pair of points in the plane one concrete model based on minimum time routes in a binary hierarchy of roads with different speed limits has been shown to satisfy the axioms and two other constructions based on poisson line processes and on dynamic proximity graphs are expected also to do so we initiate study of structure theory and summary statistics for general processes in the class many questions arise in this setting via analogies with diverse existing topics from geodesics in first passage percolation to transit node based route finding algorithms

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

pattern recognition by affine moment invariants abstract the paper deals with moment invariants which are invariant under general affine transformation and may be used for recognition of affine deformed objects our approach is based on the theory of algebraic invariants the invariants from second and third order moments are derived and shown to be complete the paper is a significant extension and generalization of recent works several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described

uniqueness for the signature of a path of bounded variation and the reduced path group we introduce the notions of tree like path and tree like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length we show that the equivalence classes form a group with some similarity to a free group and that in each class there is a unique path that is tree reduced the set of these paths is the reduced path group it is a continuous analogue of the group of reduced words the signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path we identify the paths with trivial signature as the tree like paths and prove that two paths are in tree like equivalence if and only if they have the same signature in this way we extend chen s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parameterisation in the general setting it is suggestive to think of this result as a non commutative analogue of the result that integrable functions on the circle are determined up to lebesgue null sets by their fourier coefficients as a second theme we give quantitative versions of chen s theorem in the case of lattice paths and paths with continuous derivative and as a corollary derive results on the triviality of exponential products in the tensor algebra

from random lines to metric spaces consider an improper poisson line process marked by positive speeds so as to satisfy a scale invariance property actually scale equivariance the line process can be characterized by its intensity measure which belongs to a one parameter family if scale and euclidean invariance are required this paper investigates a proposal by aldous namely that the line process could be used to produce a scale invariant random spatial network sirsn by means of connecting up points using paths which follow segments from the line process at the stipulated speeds it is shown that this does indeed produce a scale invariant network under suitable conditions on the parameter in fact it then produces a parameter dependent random geodesic metric for dd dimensional space d 2d 2 where geodesics are given by minimum time paths moreover in the planar case it is shown that the resulting geodesic metric space has an almost everywhere unique geodesic property that geodesics are locally of finite mean length and that if an independent poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region it is an open question whether the result is a sirsn in aldous sense so placing finite mean length in each compact region but it may be called a pre sirsn

projective and affine symmetries and equivalences of rational and polynomial surfaces abstract it is known that proper parameterizations of rational curves in reduced form are unique up to bilinear reparameterizations i e projective transformations of its parameter domain this observation has been used in a series of papers by alcazar et al to formulate algorithms for detecting euclidean equivalences and symmetries as well as similarities we generalize this approach to projective equivalences of rationally parametrized surfaces more precisely we observe that a birational base point free parameterization of a surface is unique up to projective transformations of the domain furthermore we use this insight to find all projective equivalences between two given surfaces in particular we formulate a polynomial system of equations whose solutions specify the projective equivalences i e the reparameterizations associated with them furthermore we investigate how this system simplifies for the special case of affine equivalences for polynomial surfaces and how we can use our method to detect projective symmetries of surfaces this method can be used for classifying the generic cases of quadratic surfaces

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

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

the conditions for the coincidence or overlapping of two bezier curves in a recent article chen et al 2013 derive the condition for the coincidence of two quartic bezier curves different from straight lines the condition reads that either their control polygons coincide or the curves are degenerate representations of the same quadratic curve we point out that this uniqueness of the control points for properly parameterized polynomial curves is already well known not only for quartics but for curves of arbitrary degree this is a direct consequence of a previous more general result by berry and patterson namely the uniqueness of the control points for rational bezier curves in addition two conjectures posed in chen et al 2013 also derive as straightforward corollaries finally the condition for overlapping of two properly parameterized curves is also derived unsurprisingly it amounts to the coincidence of the bezier polygons for the overlapping segments of the given curves corresponding to the overlapping portions of their domains

computing projective equivalences of special algebraic varieties abstract this paper is devoted to the investigation of selected situations when computing projective and other equivalences of algebraic varieties can be efficiently solved via finding projective equivalences of finite sets of points on the projective line in particular we design a method that finds for two algebraic varieties x y from special classes an associated set of automorphisms of the projective line the so called good candidate set consisting of suitable candidates for the subsequent construction of possible mappings x y the functionality of the designed approach is presented for computing projective equivalences of rational curves determining projective equivalences of rational ruled surfaces detecting affine transformations between planar algebraic curves and computing similarities between two implicitly given algebraic surfaces when possible symmetries of given shapes are also discussed as special cases

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

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

detecting symmetries in polynomial bezier curves in a recent article alcazar 2014 presents algorithms for detecting central and mirror symmetries in planar polynomial curves expressed with proper parameterization in the monomial i e power basis however for practical purposes in computer graphics and cagd the usual choice is the bernstein bezier representation because of its superior numerical and geometric characteristics we point out that in this form and for properly parameterized curves detecting symmetry amounts to simply checking that the bezier points exhibit pairwise symmetry this result is a direct consequence of well known properties of the bezier representation namely its symmetry affine invariance and uniqueness for proper parameterizations detecting the existence of a symmetric segment in a bezier curve also amounts to a simple task by analysing the last non vanishing derivatives finally these results carry over in a straightforward manner to symmetries in euclidean space

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

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

wiener estimates for parabolic obstacle problems on etudie la regularite par point des solutions faibles locales des problemes d obstacle paraboliques du type suivant u q e u t lu 0 dans q 0 ou est une fonction de borel arbitraire donnee dans q q o t

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

affine invariant b spline moments for curve matching the article deals with the problem of matching and recognizing planar curves that are modeled by b splines independently of possible affine transformations to which the original curve has been subjected for example rotation translation scaling orthographic and semiperspective projections and possible occlusion it presents a fast algorithm for estimating the b spline control points that is robust to nonuniform sampling noise and local deformations curve matching is achieved by using a similarity measure based on the b spline knot points introduced by cohen et al 1991 this method however can neither handle the affine transformation between the curves nor the occlusion solutions to these two problems are presented through the use of a new class of weighted b spline curve moments that are well defined for both open and closed curves the method has been applied to classifying affine transformed aircraft silhouettes and appears to perform well

involutions of polynomially parametrized surfaces we provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant as a consequence the symmetry axes symmetry planes and symmetry center of the surface if any can be determined directly from the parametrization without computing or making use of the implicit representation the algorithm is based on the fact proven in the paper that any involution of the surface comes from an involution of the parameter space r2r2 therefore by determining the latter the former can be found the algorithm has been implemented in the computer algebra system maple 18 evidence of its efficiency for moderate degrees examples and a complexity analysis are also given

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

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

wiener s criterion for the heat equation this paper establishes a necessary and sufficient condition for a boundary point of an arbitrary bounded open subset of r 1 to be regular for the heat equation our criterion is in the sense described below the exact analogue of winner s test w for boundary regularity of harmonic functions a careful statement of our result requires that we first of all introduce a great deal of terminology let 2 q r 1 n 1 denote any bounded open subset and 8 2 its topological boundary we will often write a typical point z e r 1 as z x t xer te r if ue c2 q r we may define the heat operator

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

improving the stability of algebraic curves for applications an algebraic curve is defined as the zero set of a polynomial in two variables algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics the main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data usually arguments against using algebraic curves involve references to mathematicians wilkinson and runge the first goal of this article is to understand the stability issue of algebraic curve fitting then a fitting method based on ridge regression and restricting the representation to well behaved subsets of polynomials is proposed and its properties are investigated the fitting algorithm is of sufficient stability for very fast position invariant shape recognition position estimation and shape tracking based on invariants and new representations among appropriate applications are shape based indexing into image databases

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

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

regularity of weak solutions of parabolic variational inequalities in this paper parabolic operators of the form ut div a x t u du b x t u du are considered where a and b are borel measurable and subject to linear growth conditions let f q r1 be a borel function bounded above an obstacle where q c rn 1 let u e w1 2 q be a weak solution of the variational inequality in the following sense assume that u q e and utep a dw bw o jq whenever ep e wol 2 q and ep u t q e here q e means everywhere except for a set of classical parabolic capacity it is shown that u is continuous even though the obstacle may be discontinuous a mild condition on t which can be expressed in terms of the fine topology is sufficient to ensure the continuity of u a modulus of continuity is obtained for u in terms of the data given for

estimation of planar curves surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation the author addresses the problem of parametric representation and estimation of complex planar curves in 2 d surfaces in 3 d and nonplanar space curves in 3 d curves and surfaces can be defined either parametrically or implicitly with the latter representation used here a planar curve is the set of zeros of a smooth function of two variables x y a surface is the set of zeros of a smooth function of three variables x y z and a space curve is the intersection of two surfaces which are the set of zeros of two linearly independent smooth functions of three variables x y z for example the surface of a complex object in 3 d can be represented as a subset of a single implicit surface with similar results for planar and space curves it is shown how this unified representation can be used for object recognition object position estimation and segmentation of objects into meaningful subobjects that is the detection of interest regions that are more complex than high curvature regions and hence more useful as features for object recognition

symmetries and similarities of planar algebraic curves using harmonic polynomials abstract we present novel deterministic efficient algorithms to compute the symmetries of a planar algebraic curve implicitly defined and to check whether or not two given implicit planar algebraic curves are similar i e equal up to a similarity transformation both algorithms are based on the fact well known in harmonic analysis that the laplacian commutes with orthogonal transformations and on efficient algorithms to find the symmetries similarities of a harmonic algebraic curve two given harmonic algebraic curves in fact we show that except for some special cases the problem can be reduced to the harmonic case

rational hausdorff divisors a new approach to the approximate parametrization of curves in this paper we introduce the notion of rational hausdorff divisor we analyze the dimension and irreducibility of its associated linear system of curves and we prove that all irreducible real curves belonging to the linear system are rational and are at finite hausdorff distance among them as a consequence we provide a projective linear subspace where all irreducible elements are solutions of the approximate parametrization problem for a given algebraic plane curve furthermore we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization therefore we present a generic answer to the approximate parametrization problem in addition we introduce the notion of hausdorff curve and we prove that every irreducible hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve