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

boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

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

adapting product kernels to curves in the plane we study the l p boundedness properties of convolution operators whose convolution kernels are obtained by adapting product kernels to curves in the plane

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

multi parameter singular radon transforms ii the lp theory the purpose of this paper is to study the l2 boundedness of operators of the form f x f t x k t dt where t x is a c function defined on a neighborhood of the origin in t x n n satisfying 0 x x is a c cut off function supported on a small neighborhood of 0 n and k is a multi parameter singular kernel supported on a small neighborhood of 0 n the goal is given an appropriate class of kernels k to give conditions on such that every operator of the above form is bounded on l2 the case when k is a calderon zygmund kernel was studied by christ nagel stein and wainger we generalize their conditions to the case when k has a multi parameter structure for example when k is given by a product kernel even when k is a calderon zygmund kernel our methods yield some new results this is the first paper in a three part series the later two of which are joint with e m stein the second paper deals with the related question of lp boundedness while the third paper deals with the special case when is real analytic

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

spectral compact difference hybrid computation of passive scalar in isotropic turbulence an accurate and efficient hybrid numerical method is developed for direct numerical simulation of passive scalar in homogeneous turbulence for the schmidt number 1 and 50 the hybrid method uses the standard fourier spectral method for the incompressible navier stokes equation and the combined compact difference scheme for the passive scalar transport equation accuracy of the method is carefully examined by comparing with the full spectral method regarding the spectra probability density function field structure of the passive scalar and is found to be very satisfactory the computational time for the hybrid method is decreased by 26 for the schmidt number 1 when compared to the full spectral method and by 77 for the schmidt number 50 when the number of grid points for the velocity field is reduced under the scale separation

analysis of second order differential operators on heisenberg groups ii given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

an algebra containing the two sided convolution operators abstract we present an intrinsically defined algebra of operators containing the right and left invariant calderon zygmund operators on a stratified group the operators in our algebra are pseudolocal and bounded on l p 1 p this algebra provides an example of an algebra of singular integrals that falls outside of the classical calderon zygmund theory

boundedness of multi parameter fourier multiplier operators on triebel lizorkin and besov lipschitz spaces abstract the main purpose of this paper is three fold first we prove that under the limited smoothness conditions multi parameter fourier multiplier operators are bounded on multi parameter triebel lizorkin and besov lipschitz spaces by the littlewood paley decomposition and the strong maximal operator second we offer a different and more direct method to deal with the boundedness instead of transforming fourier multiplier operators into multi parameter calderon zygmund operators third we also prove the boundedness of multi parameter fourier multiplier operators on weighted multi parameter triebel lizorkin and besov lipschitz spaces when the fourier multiplier is only assumed with limited smoothness

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

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

lp harmonic analysis and radon transforms on the heisenberg group in this paper harmonic analysis in the heisenberg group heisn cn r with group law z t z t z z t t 12im z z is taken to mean the joint spectral theory of the operators l heisenberg laplacian and t t the spectral decomposition of e l2 is given explicitly as e 0 e d where e 1 k 0 1 2 and e z t n 2 n 1 n 2k n 1 exp ie tn 2k exp z 24 n 2k lkn 1 z 22 n 2k where lkn 1 denotes the laguerre polynomial the eigenvalues are it e e n 2k e and l k e k e this decomposition is essentially well known but has not previously been described in this light it is also the decomposition into irreducibles of the representation of the heisenberg motion group semi direct product of heisn and u n on l2 heisn the main result is a summability in lp of the decomposition for e lp 1 p limr 1 rk 0 ed for p 1 only a weaker substitute is available and for p the decomposition is false there is also a plancherel formula for l2 2 2 k e n 2k 0 k e z 0 2dzd a number of explicit examples of the decomposition are computed including part of the schrodinger propagator eisl the heisenberg radon transform is defined by r z t n z t w 0 dw this operator was studied by geller and stein it is shown to be bounded from lp to lq if and only if p 2n 2 2n 1 and q 2n 2 some generalizations of these results to the free step two nilpotent lie groups are given but the results are not as complete

singular integrals with flag kernels and analysis on quadratic cr manifolds abstract we study a class of operators on nilpotent lie groups g given by convolution with flag kernels these are special kinds of product type distributions whose singularities are supported on an increasing subspace 0 v1 vk g we show that product kernels can be written as finite sums of flag kernels that flag kernels can be characterized in terms of their fourier transforms and that flag kernels have good regularity restriction and composition properties we then apply this theory to the study of the b complex on certain quadratic cr submanifolds of c n we obtain lp regularity for certain derivatives of the relative fundamental solution of b and for the corresponding szego projections onto the null space of b by showing that the distribution kernels of these operators are finite sums of flag kernels

h p boundedness of calderon zygmund operators on product spaces in this paper we prove the product h p boundedness of calderon zygmund operators which were considered by fefferman and stein the methods used in this paper are new even for the classical h p boundedness of calderon zygmund operators namely using some subtle estimates together with the h p l p boundedness of product vector valued calderon zygmund operators

on local isotropy of passive scalars in turbulent shear flows an assessment of local isotropy and universality in high reynolds number turbulent flows is presented the emphasis is on the behaviour of passive scalar fields advected by turbulence but a brief review of the relevant facts is given for the turbulent motion itself experiments suggest that local isotropy is not a natural concept for scalars in shear flows except perhaps at such extreme reynolds numbers that are of no practical relevance on earth yet some type of scaling exists even at moderate reynolds numbers the relation between these two observations is a theme of this paper

boundedness of singular integrals on the flag hardy spaces on heisenberg group abstract we prove that the classical one parameter convolution singular integrals on the heisenberg group are bounded on multiparameter flag hardy spaces which satisfy the intermediate dilation between the one parameter anisotropic dilation and the product dilation on c n r implicitly

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

exponential self similar mixing by incompressible flows we study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions the scalar solves the continuity equation with a divergence free velocity field which satisfies a bound in the sobolev space ws p where s 0 and 1 p the mixing properties are given in terms of a characteristic length scale called the mixing scale we consider two notions of mixing scale one functional expressed in terms of the homogeneous sobolev norm h 1 the other geometric related to rearrangements of sets we study rates of decay in time of both scales under self similar mixing for the case s 1 and 1 p including the case of lipschitz continuous velocities and the case of physical interest of enstrophy constrained flows we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound established in previous works on the time decay of both scales we also present several consequences for the geometry of regular lagrangian flows associated to sobolev velocity fields

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

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

analysis of the kohn laplacian on quadratic cr manifolds abstract we study the kohn laplacian b q acting on 0 q forms on quadratic cr manifolds we characterize the operators b q that are locally solvable and hypoelliptic respectively in terms of the signatures of the scalar components of the levi form

scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence abstract statistics of a passive scalar flux in a uniform mean scalar gradient convected by homogeneous isotropic steady turbulence are numerically studied by using very high resolution direct numerical simulation it is found that the nusselt number increases in proportion to the peclet number and that the one point probability density function of the scalar flux is negatively skewed and exponential and is insensitive to the variation of the peclet number the scalar field is studied by visualization and the ramp cliff structure and the mesa canyon structure are observed along the directions parallel and perpendicular to the mean scalar gradient respectively the probability density function of the scalar flux is theoretically computed and found to be in good agreement with the numerical results a lagrangian statistical theory for the scalar flux is developed which predicts that the scalar transfer flux is given by the time integral of the lagrangian velocity autocorrelation and increases in proportion to the peclet number which is consistent with the results of the direct numerical simulation a physical explanation of the asymmetry of the scalar flux pdf is explored

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

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings

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