shape elongation from optimal encasing rectangles let s be a shape with a polygonal boundary we show that the boundary of the maximally elongated rectangle r s which encases the shape s contains at least one edge of the convex hull of s such a nice property enables a computationally efficient construction of r s in addition we define the elongation of a given shape s as the ratio of the length of r s determined by the longer edge of r s and the width of r s determined by the shorter edge of r s and show that a so defined shape elongation measure has several desirable properties several examples are given in order to illustrate the behavior of the new elongation measure as a by product of the method developed here we obtain a new method for the computation of the shape orientation where the orientation of a given shape s is defined by the direction of the longer edge of r s

curvature weighted gradient based shape orientation determining the orientation of a shape is a common task in many image processing applications it is usually part of the image preprocessing stages and further processing may rely on an adequate method to determine the orientation there are several methods for computing the orientation of a shape each of them with its own strengths and weaknesses a method which performs outstandingly for one application may have a poor performance for a different application in this paper we present a new method for computing shape orientation based on the projection of the tangent vectors of a shape onto a line and weighting them using a function of the curvature some of the results from zunic 2008 14 are particular cases of the results presented here

symmetry detection by generalized complex gc moments a close form solution this paper presents a unified method for detecting both reflection symmetry and rotation symmetry of 2d images based on an identical set of features i e the first three nonzero generalized complex gc moments this method is theoretically guaranteed to detect all the axes of symmetries of every 2d image if more nonzero gc moments are used in the feature set furthermore we establish the relationship between reflectional symmetry and rotational symmetry in an image which can be used to check the correctness of symmetry detection this method has been demonstrated experimentally using more than 200 images

the effects of model selection on confidence intervals for the size of a closed population one encounters in the literature estimates of some rates of genetic and congenital disorders based on log linear methods to model possible interactions among sources often the analyst chooses the simplest model consistent with the data for estimation of the size of a closed population and calculates confidence intervals on the assumption that this simple model is correct however despite an apparent excellent fit of the data to such a model we note here that the resulting confidence intervals may well be misleading in that they can fail to provide an adequate coverage probability we illustrate this with a simulation for a hypothetical population based on data reported in the literature from three sources the simulated nominal 95 per cent confidence intervals contained the modelled population size only 30 per cent of the time only if external considerations justify the assumption of plausible interactions of sources would use of the simpler model s interval be justified

boundary based shape orientation the computation of a shape s orientation is a common task in the area of computer vision and image processing being used for example to define a local frame of reference and is helpful for recognition and registration robot manipulation etc it is usually an initial step or a part of data preprocessing in many image processing and computer vision tasks thus it is important to have a good solution for shape orientation because an unsuitable solution could lead to a big cumulative error at the end of the computing process there are several approaches to the problem most of them could be understood as the area based ones or at least they do not take into account all the boundary points if a shape orientation measure is based on its encasing rectangle only the convex hull points count for example thus the demand for a pure boundary based method where the orientation of the shape is dependent on the boundary points seems to be very reasonable such a method is presented in this paper we are initially focused on the shapes having polygonal boundaries we define the orientation of a polygonal shape by the line that maximises the total sum of squared lengths of all the boundary edge projections onto this line the advantages and limitations of the new method are analysed next we suggested how the method can be adapted in order to be applicable to a wider class than the initial method is finally we introduced another modification of the method in such a way that the modified method can be applied to shapes with arbitrary boundaries several illustrative experiments are provided

universal principal axes an easy to construct tool useful in defining shape orientations for almost every kind of shape abstract a convenient tool for defining shape orientations called universal principal axes is introduced these axes are formed of half lines starting from the shape centroid with their directional angles expressed as functions of the polar angle of the first non zero complex number encountered in the sequence x iy 1 d a with 1 2 3 4 provided that the integration domain is taken to be the given two dimensional 2d shape and the origin is assumed to be the shape centroid universal principal axes are shown to be independent of the translation scaling and rotation of the coordinate system used and they are therefore qualified in defining shape orientations a major benefit of using the proposed universal principal axes is that they exist for almost every kind of shape another benefit is that there is no need to judge in advance whether a given shape is mirror symmetric rotationally symmetric irregular etc moreover in the case of rotationally symmetric shapes the universal principal axes make the preprocessing procedure for detecting the number of folds contained in the given shape unnecessary although such procedure is essential for many existing tools designed to define orientations of rotationally symmetric shapes defining shape orientations by universal principal axes is therefore quite convenient an algorithm describing the construction of universal principal axes and several examples showing the detected universal principal axes for some shapes of distinct types are given the relationship between universal and conventional principal axes is also discussed

generalized affine invariant image normalization we provide a generalized image normalization technique which basically solves all problems in image normalization the orientation of any image can be uniquely defined by at most three non zero generalized complex moments the correctness of our method is demonstrated theoretically as well as in practice by applying them to a number of degenerate images which have failed other previously reported techniques for image normalization

asymptotic behavior of solutions to nonlinear volterra integral equations in section 2 we establish the existence of solutions to 1 1 and to related equations theorems 2 1 and 2 3 in contrast with previous papers e g 7 we do not assume that a is m accretive instead we require a to satisfy a weaker cone condition certain other facts are also included in this section throughout this paper we also assume that b is completely positive 6 in the third section we present two mean ergodic theorems for the solutions of 1 1 in hilbert space the first theorem theorem 3 1 deals with weak almost convergence and the second theorem 3 4 with strong almost convergence these results improve upon those of 1 and include the known results for nonlinear semigroups 191 it is important to establish almost convergence because it then follows that the solution is summed by every strongly regular kernel corollaries 3 3 and 3 5 a weak convergence criterion corollary 3 2 is also obtained if the banach space x is reflexive and strictly convex and 0 e r a then

the family of universal axes the family of universal axes ua is introduced in this paper this family has infinitely many members each of which generates a method that is universal in the sense that each generated method can define shape orientation for almost any kind of shape and once the numerical computing system has been set up the users of this orientation detection system need not supply the system the information about whether a given shape is mirror symmetric rotationally symmetric irregular etc some mathematical properties of the proposed ua are investigated and numerical techniques for implementing ua are discussed the method of universal principal axes upa introduced recently is just one of the infinitely many members of the proposed family of ua the upa method yields at least two orientations for each shape and thus must be combined with another non universal single orientation method in order to reduce the time needed to match irregular shapes most members of the proposed ua family do not need to be supplemented by a single orientation method in this way

normalization of rotationally symmetric shapes for pattern recognition abstract image normalization is useful for image recognition by the matching method recently a normalization algorithm was developed to normalize images under the distortion of translation rotation scaling and skew however the method fails when the image is rotationally symmetric a method is introduced called the modified fourier descriptor to normalize rotationally symmetric shapes this method is simple and fast and can be combined with the preceding normalization algorithm to make it more general and realizable

on the asymptotic behavior of nonlinear semigroups and the range of accretive operators ii abstract it is known that certain problems in partial differential equations may be interpreted as initial value problems for ordinary differential equations in banach spaces when such an evolution equation is governed by an accretive operator then its solutions give rise to a nonlinear contraction semigroup in this paper we study certain aspects of the asymptotic behavior of nonlinear semigroups and or resolvents of accretive operators we also derive new results on their behavior at the origin it turns out that the behavior of a nonlinear semigroup resembles that of the resolvent of its generator both at infinity and at the origin author

image normalization by complex moments the role of moments in image normalization and invariant pattern recognition is addressed the classical idea of the principal axes is analyzed and extended to a more general definition the relationship between moment based normalization moment invariants and circular harmonics is established invariance properties of moments as opposed to their recognition properties are identified using a new class of normalization procedures the application of moment based normalization in pattern recognition is demonstrated by experiment

frequency domain classification of cyclic and dihedral symmetries of finite 2 d patterns this paper presents a very simple and effective algorithm for classifying cyclic and dihedral symmetries from images of finite two dimensional patterns it consists in an extension of a frequency domain algorithm proposed by the author for estimating two dimensional rotation of rigid objects within this framework the estimation of rotation is conveniently turned into the problem of detecting two orthogonal lines within the locus of the zero crossings of a certain function represented in orthogonal cartesian coordinates the main contribution of this paper is the use and interpretation of the ambiguities that arise in such a locus in the form of additional pairs of orthogonal lines when applied to symmetric patterns and the development of other algorithmic steps which allow a simple and fast discrimination between cyclic and dihedral symmetries unlike any other available method this algorithm does not require any conversion from cartesian to polar image representations besides classification reliability and consistency observed through several dozen testing experiments the nice features of this new method also include robustness to noise and ease of implementation with fast fourier transform algorithms which makes it amenable to almost real time pattern classifications applications several examples representative of the performance of the algorithm through extensive testing are reported and discussed in the paper

recognitive aspects of moment invariants moment invariants are evaluated as a feature space for pattern recognition in terms of discrimination power and noise tolerance the notion of complex moments is introduced as a simple and straightforward way to derive moment invariants through this relation properties of complex moments are used to characterize moment invariants aspects of information loss suppression and redundancy encountered in moment invariants are investigated and significant results are derived the behavior of moment invariants in the presence of additive noise is also described

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

asymptotic behavior of solutions of nonlinear volterra equations 0 recently baillon and clement 3 established mean ergodic theorems for solutions of nonlinear volterra equations in hilbert space in the present paper we study the asymptotic behavior of solutions of 1 1 by using nonlinear ergodic theorems the first mean ergodic theorem was given by baillon l and it was improved by several authors 3 8 141 163

orientation and anisotropy of multi component shapes from boundary information we define a method for computing the orientation of compound shapes based on boundary information the orientation of a given compound shape s is taken as the direction a that maximises the integral of the squared length of projections of all the straight line segments whose end points belong to particular boundaries of components of s to a line that has the slope a just as the concept of orientation can be extended from single component shapes to multiple components elongation can also be applied to multiple components and we will see that it effectively produces a measure of anisotropy since it is maximised when all components are aligned in the same direction the presented method enables a closed formula for an easy computation of both orientation and anisotropy

automatic symmetry determination and normalization for rotationally symmetric 2d shapes and 3d solid objects abstract determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision application however the usual normalization methods are inapplicable when the shape is n fold rotationally symmetric n 3 in this paper an efficient and reliable new method using modified fourier transform is introduced to normalize such a 2d rotationally symmetric shape furthermore we extend this new method to the normalization of a n fold axial rotationally symmetric 3d object there are two obvious advantages when using this new method 1 it prevents the high computation load and the difficulty in normalizing the shape s orientation which occurred in the old methods using full matching or correlation functions and 2 the weakness introduced by the old methods using data reduction from 2d data to 1d data is remedied experiments will show the validity of our method

revisiting complex moments for 2 d shape representation and image normalization when comparing 2 d shapes a key issue is their normalization translation and scale are easily taken care of by removing the mean and normalizing the energy however defining and computing the orientation of a 2 d shape is not so simple in fact although for elongated shapes the principal axis can be used to define one of two possible orientations there is no such tool for general shapes as we show in the paper previous approaches fail to compute the orientation of even noiseless observations of simple shapes we address this problem and show how to uniquely define the orientation of an arbitrary 2 d shape in terms of what we call its principal moments we start by showing that a small subset of these moments suffices to describe the underlying 2 d shape i e that they form a compact representation which is particularly relevant when dealing with large databases then we propose a new method to efficiently compute the shape orientation principal moment analysis finally we discuss how this method can further be applied to normalize gray level images besides the theoretical proof of correctness we describe experiments demonstrating robustness to noise and illustrating with real images

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

a unifying approach to moment based shape orientation and symmetry classification in this paper the problem of moment based shape orientation and symmetry classification is jointly considered a generalization and modification of current state of the art geometric moment based functions is introduced the properties of these functions are investigated thoroughly using fourier series analysis and several observations and closed form solutions are derived we demonstrate the connection between the results presented in this work and symmetry detection principles suggested from previous complex moment based formulations the proposed analysis offers a unifying framework for shape orientation symmetry detection in the context of symmetry classification and matching the second part of this work presents a frequency domain method aiming at computing a robust moment based feature set based on a true polar fourier representation of image complex gradients and a novel periodicity detection scheme using subspace analysis the proposed approach removes the requirement for accurate shape centroid estimation which is the main limitation of moment based methods operating in the image spatial domain the proposed framework demonstrated improved performance compared to state of the art methods

similarity and symmetry measures for convex shapes using minkowski addition this paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on minkowski addition and the brunn minkowski inequality this means in particular that these measures are region based in contrast to most of the literature where one considers contour based measures all measures considered in this paper are invariant under translations furthermore they can be chosen to be invariant under rotations multiplications reflections or the class of affine transformations it is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle spl theta is a piecewise concave function of spl theta this and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures various results obtained in this paper are illustrated by experimental data although the paper deals exclusively with the two dimensional case many of the theoretical results carry over almost directly to higher dimensional spaces

the asymptotics of nonexpansive iterations abstract let d be a closed subset of a banach space x and t d d a nonexpansive mapping conditions are given on the space x for t to satisfy the following property of ergodic type t n x n converges either weakly or strongly to a vector v rather unexpectedly d is not assumed to be convex nor is i t assumed to satisfy any range condition in addition it is shown that v is the unique point of least norm in the closure of r i t if and only if i t satisfies a certain range condition at infinity several interesting applications to accretive operator and nonlinear semigroup theory are also included

maximum likelihood applied to a capture recapture model maximum likelihood is applied to a capture recapture model in order to estimate the population size n the likelihoods to which this model gives rise can be highly skew it is shown that the use of n 3 gives an approximately normal likelihood this results in a function that has an approximately standard normal distribution and is linear in n 3 thence complete sets of approximate confidence intervals can be obtained merely by the use of a pocket calculator

an alternative approach to computing shape orientation with an application to compound shapes we consider the method that computes the shape orientation as the direction that maximises the integral of the length of projections taken to the power of 2n of all the straight line segments whose end points belong to the shape to a line that has the slope we show that for n 1 such a definition of shape orientation is consistent with the shape orientation defined by the axis of the least second moment of inertia for n 1 this is not the case and consequently our new method can produce different results as an additional benefit our approach leads to a new method for computation of the orientation of compound objects

detection of generalized principal axes is rotationally symmetric shapes abstract automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper the inapplicability of a well known equation which is used to compute the direction of the principal axis of a given shape using moment functions to a class of so called degenerate shapes is first pointed out rotationally symmetric shapes often encountered in real applications are shown to belong to this class this problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions analytic equations for computing the direction of high order principal axes are derived they include the well known equation for computing the direction of the second order principal axis as a special case some experimental results are included finally to show the effectiveness of the derived analytic equations

a moment based approach for deskewing rotationally symmetric shapes the covariance matrix of a pattern is composed by its second order central moments for a rotationally symmetric shape its covariance matrix is a scalar identity matrix in this work we apply this property to restore the skewed shape of rotational symmetry the relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived by computing the covariance matrix of the skewed shape and letting the covariance matrix of the original shape be a scalar identity matrix the skew transformation matrix can be solved then the rotationally symmetric shape can be recovered by multiplying the inverse transformation matrix with the skewed shape the method does not rely on continuous contours and is robust to noise because only the second order moments of the input shape are required experimental results are also presented

goodness of fit based confidence intervals for estimates of the size of a closed population methods for estimating the size of a closed population often consist of fitting some model e g a log linear model to data with a missing cell corresponding to the members of the population missed by all reporting sources although the use of the asymptotic standard error is the usual method for forming confidence intervals for the population total the sample sizes are not always large enough to produce valid confidence intervals we propose a method for forming confidence intervals based upon changes in a goodness of fit statistic associated with changes in trial values of the population total

affine permutation invariance of 2 d shapes shapes provide a rich set of clues on the identity and topological properties of an object in many imaging environments however the same object appears to have different shapes due to distortions such as translation rotation reflection scaling or skewing further the order by which the object s feature points are scanned changes i e the order of the pixels may be permuted relating two dimensional shapes of the same object distorted by different affine and permutation transformations is a challenge we introduce a shape invariant that we refer to as the intrinsic shape of an object and describe an algorithm blaiser to recover it the intrinsic shape is invariant to affine permutation distortions it is a uniquely defined representative of the equivalence class of all affine permutation distortions of the same object blaiser computes the intrinsic shape from any arbitrarily affine permutation distorted image of the object without prior knowledge regarding the distortions or the undistorted shape of the object the critical step of blaiser is the determination of the shape orientation and we provide a detailed discussion on this topic the operations of blaiser are based on low order moments of the input shape and thus robust to error and noise examples illustrate the performance of the algorithm

on the detection of the axes of symmetry of symmetric and almost symmetric planar images a simple algorithm is presented for finding all the axes of symmetry of symmetric and almost symmetric planar images having nonuniform gray level intensity images the technique which can also be used in conjunction with planar curves is based on the identification of the centroids of the given image and other related sets of points followed by a maximization of a specially defined coefficient of symmetry owing to the nature of the required procedures which are strictly based on the computation of mean values the method is not particularly affected by the presence of noise and is especially suited for application in conjunction with digitized figures

notes on shape orientation where the standard method does not work in this paper we consider some questions related to the orientation of shapes with particular attention to the situation where the standard method does not work there are irregular and non symmetric shapes whose orientation cannot be computed in a standard way but in the literature the most studied situations are those where the shape under consideration has more than two axes of symmetry or where it is an n fold rotationally symmetric shape with n 2 the basic reference for our work is w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 we give a very simple proof of the main result from w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 and suggest a modification of the proposal on how the principal axes of rotationally symmetric shapes should be computed we show some desirable property in defining the orientation of such shapes if the modified approach is applied also we give some comments on the problems that arise when computing shape elongation

asymptotic properties of some non autonomous systems in banach spaces abstract let x be a reflexive banach space we introduce the notion of weakly almost nonexpansive sequences x n n 0 in x and study their asymptotic behavior by showing that the nonempty weak limit set of the sequence x n n n 1 always lies on a convex subset of a sphere centered at the origin of radius d lim n x n n subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi autonomous dissipative system d u d t a u f where a is an accretive possibly multivalued operator in x x and f f l p 0 x for some f x and 1 p these results extend recent results of j s jung and j s park j s jung j s park asymptotic behavior of nonexpansive sequences and mean points proc amer math soc 124 1996 475 480 and j s jung j s park and e h park j s jung j s park e h park asymptotic behaviour of generalized almost nonexpansive sequences and applications proc nonlinear funct anal 1 1996 65 79 as well as our results cited below containing previous results by several authors

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

automatic rotational symmetry determination for shape analysis abstract the determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications in this paper a simple but effective algorithm to analyse the rotational symmetry of a given closed curve shape s is proposed a circle c with the centroid of s as the circle center and the average radius of s as the circle radius is superimposed on s resulting in the intersection of c and s at a set of points by theoretical analysis the relationship between the order n s of the rotational symmetry of s and the number of intersection points between c and s is established all the possible values for n s are determined and finally n s is determined by evaluating the similarity between s and its rotated versions in the proposed algorithm only simple pixel operations and second order moment function computation are involved several problems caused by the use of discrete image coordinates are analysed and solved for correct decision making in the algorithm good experimental results are also included

irregular shape symmetry analysis theory and application to quantitative galaxy classification this paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the amount of symmetry of arbitrary shapes the definitions of both bilateral symmetricity and rotational symmetricity give new insight into analyzing the geometrical property of a shape and enable characterizing arbitrary shapes in a new way we developed a set of criteria for quantitative galaxy classification using our proposed irregular shape symmetry measures our study has demonstrated the effectiveness of the proposed method for the characterization of the shape of the celestial bodies the concepts described in the paper are applicable to many fields such as mathematics artificial intelligence digital image processing robotics biomedicine etc

bayesian methods for estimation of the size of a closed population summary a bayesian methodology for estimating the size of a closed population from multiple incomplete administrative lists is proposed the approach allows for a variety of dependence structures between the lists can make use of covariates and explicitly accounts for model uncertainty interval estimates from this approach are compared to frequentist and previously published bayesian approaches several examples are considered

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h