projection operators and moment invariants to image blurring in this paper we introduce a new theory of blur invariants blur invariants are image features which preserve their values if the image is convolved by a point spread function psf of a certain class we present the invariants to convolution with an arbitrary n fold symmetric psf both in fourier and image domain we introduce a notion of a primordial image as a canonical form of all blur equivalent images it is defined in spectral domain by means of projection operators we prove that the moments of the primordial image are invariant to blur and we derive recursive formulae for their direct computation without actually constructing the primordial image we further prove they form a complete set of invariants and show how to extent their invariance also to translation rotation and scaling we illustrate by simulated and real data experiments their invariance and recognition power potential applications of this method are wherever one wants to recognize objects on blurred images

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

phragmen lindelof principles on algebraic varieties from several results in recent years starting with h6rmander s characterization of the constant coefficient partial differential equations p d u f that have a real analytic solution u for every real analytic function f it has become clear that certain properties of the partial differential operator p d are equivalent to estimates of phragmen lindelof type for plurisubharmonic functions on the algebraic variety

the geometry of analytic varieties satisfying the local phragmen lindelof condition and a geometric characterization of the partial differential operators that are surjective on ℝ the local phragmen lindelof condition for analytic subvarieties of c n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators as hormander has shown here necessary geometric conditions for this phragmen lindelof condition are derived they are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in c 3 the latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on r 4

convex univalent functions and continuous linear right inverses abstract for surjective infinite order differential operators on the space of holomorphic functions on a convex domain g other than the entire complex plane we characterize when they admit a continuous linear right inverse the characterization is given by a smoothness condition on g which is phrased in terms of the boundary behavior of the riemann mapping of the unit disc onto g

on the dependence of analytic solutions of partial differential equations on the right hand side given a nonzero polynomial p z a m a0z0 on cn martineau proved in the 1960s that for each convex domain g of cn the partial differential operator p d f eai m aa0f c acting on the frechet space a g of all analytic functions on g is surective in the present paper it is investigated whether solutions f of the equation p d f g can be chosen as f r g with a continuous linear operator r a g a g for bounded g we give a necessary and sufficient condition for the existence of such an r

a no reference objective image sharpness metric based on the notion of just noticeable blur jnb this work presents a perceptual based no reference objective image sharpness blurriness metric by integrating the concept of just noticeable blur into a probability summation model unlike existing objective no reference image sharpness blurriness metrics the proposed metric is able to predict the relative amount of blurriness in images with different content results are provided to illustrate the performance of the proposed perceptual based sharpness metric these results show that the proposed sharpness metric correlates well with the perceived sharpness being able to predict with high accuracy the relative amount of blurriness in images with different content

isoparametric foliations a problem of eells lemaire and conjectures of leung in this paper two sequences of minimal isoparametric hypersurfaces are constructed via representations of clifford algebras based on these we give estimates on eigenvalues of the laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres this improves results of z z tang and w j yan isoparametric foliation and yau conjecture on the first eigenvalue j differential geom 94 2013 521 540 z z tang y q xie and w j yan isoparametric foliation and yau conjecture on the first eigenvalue ii j funct anal 266 2014 6174 6199 eells and lemaire selected topics in harmonic maps c b m s regional conference series in mathematics 50 american mathematical society providence ri 1983 posed a problem to characterize the compact riemannian manifold m for which there is an eigenmap from m to sn as another application of our constructions the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres most importantly by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres we provide infinitely many counterexamples to two conjectures of leung minimal submanifolds in a sphere ii bull london math soc 23 1991 387 390 posed in 1991 on minimal submanifolds in unit spheres note that these conjectures of leung have been proved in the case that the normal connection is flat t hasanis and t vlachos ricci curvatures and minimal submanifolds pacific j math 197 2001 13 24

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

spectral symmetry analysis we present a spectral approach for detecting and analyzing rotational and reflectional symmetries in n dimensions our main contribution is the derivation of a symmetry detection and analysis scheme for sets of points irn and its extension to image analysis by way of local features each object is represented by a set of points s irn where the symmetry is manifested by the multiple self alignments of s the alignment problem is formulated as a quadratic binary optimization problem with an efficient solution via spectral relaxation for symmetric objects this results in a multiplicity of eigenvalues whose corresponding eigenvectors allow the detection and analysis of both types of symmetry we improve the scheme s robustness by incorporating geometrical constraints into the spectral analysis our approach is experimentally verified by applying it to 2d and 3d synthetic objects as well as real images

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

exterior monge ampere solutions abstract we discuss the siciak zaharjuta extremal function of a real convex body in c n a solution of the homogeneous complex monge ampere equation on the exterior of the convex body we determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic we study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity the extremal curves are all complex quadratic curves and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria as a byproduct we encounter a new invariant of an exterior domain the robin indicatrix which is in some cases the dual of the kobayashi indicatrix for a bounded domain finally we construct extremal curves for two non convex bodies in r 2

siciak zahariuta extremal functions analytic discs and polynomial hulls we prove two disc formulas for the siciak zahariuta extremal function of an arbitrary open subset of complex affine space we use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs similar results in previous work of ours required the subsets to be connected

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

finding axes of symmetry from potential fields this paper addresses the problem of detecting axes of bilateral symmetry in images in order to achieve robustness to variation in illumination only edge gradient information is used to overcome the problem of edge breaks a potential field is developed from the edge map which spreads the information in the image plane pairs of points in the image plane are made to vote for their axes of symmetry with some confidence values to make the method robust to overlapping objects only local features in the form of taylor coefficients are used for quantifying symmetry we define an axis of symmetry histogram which is used to accumulate the weighted votes for all possible axes of symmetry to reduce the computational complexity of voting a hashing scheme is proposed wherein pairs of points whose potential fields are too asymmetric are pruned by not being counted for the vote experimental results indicate that the proposed method is fairly robust to edge breaks and is able to detect symmetries even when only 0 05 of the possible pairs are used for voting

water reflection recognition based on motion blur invariant moments in curvelet space water reflection a typical imperfect reflection symmetry problem plays an important role in image content analysis existing techniques of symmetry recognition however cannot recognize water reflection images correctly because of the complex and various distortions caused by the water wave hence we propose a novel water reflection recognition technique to solve the problem first we construct a novel feature space composed of motion blur invariant moments in low frequency curvelet space and of curvelet coefficients in high frequency curvelet space second we propose an efficient algorithm including two sub algorithms low frequency reflection cost minimization and high frequency curvelet coefficients discrimination to classify water reflection images and to determine the reflection axis through experimenting on authentic images in a series of tasks the proposed techniques prove effective and reliable in classifying water reflection images and detecting the reflection axis as well as in retrieving images with water reflection