fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

efficient parallel algorithms for the computation of two dimensional image moments abstract moments constitute an important set of parameters for image analysis the low order moments contain significant information about a simple object they have been used in finding the location and orientation of an object moment invariants have been used as features for pattern recognition to compute moments of a two dimensional image a large number of multiplications and additions are required in a direct approach multiplications which are the most time consuming operations in simple processors can be completely avoided in the proposed algorithms for low order moments in this paper parallel algorithms are proposed for efficient implementation in processor arrays the basic idea is to decompose a 2 d moment into many vertical moments and a horizontal moment and to use the data parallelism for the vertical moments and the task parallelism for the horizontal moment

high order moment computation of gray level images describes an efficient approach to calculate geometric moments of a 2 d gray level image it is shown both theoretically and experimentally that the new method compares favorably with previous techniques especially for high order moments

the revised fundamental theorem of moment invariants a revised fundamental theorem of moment invariants for pattern recognition which corrects the fundamental theorem proposed by m k hu 1962 is presented the correction affects neither similitude scale nor rotation invariants derived using the original theorem but it does affect features invariant to general linear transformations four of the latter invariants were presented originally by hu these are revised to take the correction to the fundamental theorem into account furthermore these four invariants are combined to yield three new invariants which are additionally invariant to changes in the illumination of an image

approximation by chlodowsky type jakimovski leviatan operators we introduce a generalization of the jakimovski leviatan operators constructed by a jakimovski and d leviatan 1969 in 1 and the theorems on convergence and the degree of convergence are established we also give a voronovskaya type theorem furthermore we study the convergence of these operators in a weighted space of functions on a positive semi axis and estimate the approximation by using a new type of weighted modulus of continuity introduced by a d gadjiev and a aral 2007 in 9

computing a shape s moments from its boundary abstract moments have been among the most commonly used tools in the analysis of 2d shapes important information about a shape such as its size center location and orientation are all moment based attributes there is also a number of very useful shape features such as maximum and minimum moments of inertia moment invariants shape spreadness and shape elongation which are also derived from moments traditionally moments of a shape are computed from the occupancy array representation of the shape that is all the shape s pixels are involved in the computation in this paper we propose a method which computes the same set of moments using only the corner pixels along a shape s boundary the basic approach is to construct a set of triangles using the shape s corners and the origin of the coordinate system the moments of these triangles are computed first the moments of the shape are then derived from the triangles moments we have found the suggested method to be much more efficient than the traditional occupancy array based methods

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

fast computation of moment invariants abstract the subject of moment invariants has found wide application in pattern recognition since it was proposed in the present paper we propose a fast algorithm for the calculation of moment invariants firstly we propose using green s theorem for double integral to transform moment calculation to a simple integration along the boundary which reduces the complexity from o n 2 to o n secondly the relation between the monomials for successive pixels along a curve is analysed and the pascal triangle matrix transform is proposed to calculate the monomials for a pixel from those for the preceding one an iterative algorithm is thus proposed for moment calculation which needs no multiplications and the number of additions needed is reduced to o n moreover a very simple systolic structure is proposed to implement the pascal triangle transformation and the proposed algorithm can thus be realized by simple hardware to further accelerate the calculation comparison of the computational complexity with some known methods is also given which shows that our algorithm significantly reduces the complexity

a sequence of positive linear operators associated with an approximation process considering a general class of discrete linear positive operators by using a one to one function we associate to the class mentioned above a new sequence of operators our aim is to establish the transfer of approximation properties on this construction the study is carried out in a weighted space and our results are materialized in obtaining both a convergence theorem of korovkin type and an inequality for the approximation error expressed in terms of a certain weighted modulus of smoothness two particular cases are analyzed

a general approach to moment calculation for polygons and line segments abstract the calculation of moments from the boundary of a solid digital area is a quite useful technique used in pattern recognition a straightforward way of doing the calculation that does not have any numerical instabilities is presented all the symmetries available in moment calculations are preserved and it is shown how different moment formulas are related to each other this includes relations between different moments and the relationships among different methods for calculating moments finally a direct relation between the moment expansion of a polygonal area and a stick figure made from one dimensional lines embedded in a two dimensional plane is derived

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

moment computation for objects with spline curve boundary a new approach is proposed for computation of area and geometric moments for a plane object with a spline curve boundary the explicit formulae are obtained for area and low order moment calculation the complexity of calculation depends on the moment order spline degree and the number of control points used in spline representation the formulae proposed use the advantage that the sequence of spline control points is cyclic it allowed us to reduce substantially the number of summands in them the formulae might be useful in different applications where it is necessary to perform measurements for shapes with a smooth boundary

an efficient algorithm for computation of shape moments from run length codes or chain codes abstract moments are very useful for shape analysis zero to third order moments have been used for computer vision applications such as shape recognition and orientation they can serve for composition of the well known moment invariants used as desirable features as well as for the detection of the location and the principal axis direction of a shape the shape is often represented by a binary image and its moments can be obtained by use of fast algorithms considering the shape as a discrete point array in this paper a new algorithm based on the double integral formulation is presented the shape is considered as a continuous region and the contribution of boundary points is used for fast computation of shape moments this method can be used to calculate moments from either the run length codes or the chain codes of shape

indeterminate bifurcational phenomena in hardening systems indeterminate bifurcations are emerging as an important ingredient of nonlinear dissipating dynamical systems in this paper we show that indeterminate bifurcations with an unpredictable outcome are a typical ingredient of nonlinear hardening systems such phenomena are clearly important concepts in the theory of nonlinear resonance and their discovery complements the earlier work carried out on softening systems as an illustrative example we examine the dynamics of a parametrically excited hardening system and show that when the trivial solution is located on a highly intertwined basin boundary its loss of stability gives a dynamic jump whose outcome is indeterminate in the sense that we cannot predict on to which coexisting attractor the system will settle

invariant image recognition by zernike moments the problem of rotation scale and translation invariant recognition of images is discussed a set of rotation invariant features are introduced they are the magnitudes of a set of orthogonal complex moments of the image known as zernike moments scale and translation invariance are obtained by first normalizing the image with respect to these parameters using its regular geometrical moments a systematic reconstruction based method for deciding the highest order zernike moments required in a classification problem is developed the quality of the reconstructed image is examined through its comparison to the original one the orthogonality property of the zernike moments which simplifies the process of image reconstruction make the suggest feature selection approach practical features of each order can also be weighted according to their contribution to the reconstruction process the superiority of zernike moment features over regular moments and moment invariants was experimentally verified

simple and fast computation of moments abstract in this paper we address the problem of efficient computation of moments from the boundary of a digital area boundary based computation is superior to usual region based approaches as the data dimension of boundary representations is substantially smaller than that of region representations we investigate the inter order relationship of moments one of our results is that moments of higher order can be computed from moments of lower order based on this relationship a simple iterative algorithm is proposed for the computation of moments from a polygonal approximation of the boundary in comparison with a direct computation method our algorithm is simpler to program the memory requirement is minimum simulation results show that a speed up of factor 8 can be achieved using our algorithm a special version of the algorithm can be utilized to compute moments from the run length chain code of the boundary our algorithm can be applied to compute the most popular geometric moments as well as other types of moments like legendre zernike rotational and complex moments

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

pattern recognition with moment invariants a comparative study and new results abstract moment invariants have been proposed as pattern sensitive features in classification and recognition applications in this paper the authors present a comprehensive study of the effectiveness of different moment invariants in pattern recognition applications by considering two sets of data handwritten numerals and aircrafts the authors also present a detailed study of zernike and pseudo zernike moment invariants including a new procedure for deriving the moment invariants in addition the authors introduce a new normalization scheme that reduces the large dynamic range of these invariants as well as implicit redundancies in these invariants based on a comprehensive study with both handwritten numerals and aircraft data the authors show that the new method of deriving zernike moment invariants along with the new normalization scheme yield the best overall performance even when the data are degraded by additive noise

the estimates of approximation by using a new type of weighted modulus of continuity in this paper we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of c 0 and show that it has some properties of ordinary modulus of continuity we obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity finally we give some examples

a novel feature extraction method for image recognition based on similar discriminant function sdf abstract the extraction of image features is the most fundamental and important problem in image recognition in this paper a similarity measure of matrices is first presented and a similar discriminant function sdf of images is established based on the discriminant function we further propose a novel feature extraction method for image recognition for each class of training image samples an optimal projection axis maximizing the similarity among these training image samples for the class is calculated unlike the common methods of feature extraction we extract a projective feature vector for a training image sample by projecting the image on the optimal projection axis of the class itself and a set of projective feature vectors for a testing image sample by projecting the image on all the optimal projection axes finally a hierarchical classifier in the optimal discriminant space is designed to recognize images in order to test the efficiency of our method it is used to recognize human faces and english characters experimental results have shown that our method has good recognition performance and the extracted projective feature vectors contain more recognition information than commonly used image features

three dimensional shape analysis using moments and fourier descriptors a procedure for using moment based feature vectors to identify a three dimensional object from a two dimensional image recorded at an arbitrary viewing angle and range is presented a moment form called standard moments rather than the usual moment invariants is considered a standard six airplane experiment was used to compare different techniques fourier descriptors and moment invariants were both compared to the present scheme for normalized moments various experiments were conducted using mixtures of silhouette and boundary moments and different normalization techniques standard moments gave slightly better results than fourier descriptors for this experiment both of these techniques were much better than moment invariants

indeterminate jumps to resonance from a tangled saddle node bifurcation we identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics it arises when the unstable manifold of the saddle of a saddle node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors at the saddle node fold a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump we show here that such an indeterminate tangled saddle node bifurcation is a common ingredient in the resonance of softening systems

radial and angular moment invariants for image identification radial and angular moments of images are presented and methods are shown for deriving moment functions that are invariant with respect to rotation translation reflection and size changes without the aid of the theory of algebraic invariants hu s invariants are expressed in terms of these radial and angular moments and it is claimed that this facilitates visual inspection of invariance properties

computation of volume and surface body moments abstract the paper presents explicit formulae for calculation of surface moments for polytopes in r n we describe also efficient algorithms for calculation of 3d body volume and surface moments the algorithms are based on explicit formulae for moment calculation and take advantages of shape polygonal representation they use only coordinates of the body vertices and faces orientation

the moment calculation of polyhedra abstract this paper proposes a novel approach to calculate polyhedra moments at first the gaussian theorem is used to convert a volume integral into a surface one by which we decrease the moment computational complexity from o n 3 to o n 2 then an iterative method is proposed to calculate the surface integral finally some comparison shows that the computational complexity of the new method is independent of voxel numbers of polyhedra therefore the new moment computation is more efficient

computing horizontal vertical convex shape s moments on reconfigurable meshes abstract zero to third order moments are very useful tools for analysing 2 d two dimensional shapes given an n n digital image this paper presents a novel constant time parallel algorithm for computing a horizontal vertical convex shape s moments each row column in the shape without holes on an n n reconfigurable mesh in the sense of the product of time and the number of processors used our algorithm is time and cost optimal

fast computation of 2 d image moments using biaxial transform abstract in this paper a new transform is introduced this transform is based on dividing the image into four quadrants the transform can be generated by adding or subtracting the gray levels of each two axially symmetric points in the original image the addition or subtraction is dependent on the quadrant position the newly transformed quadrants can then be used to compute the moments only one quadrant of transformed image is needed to compute the moments of specific orders this can lead to considerable savings in both processing time as well as storage other advantages of this transform such as image reconstruction and using multiple filters on the same image frame to remove noise are also explored

catastrophes with indeterminate outcome a catastrophe in a dissipative dynamical system which causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in phase space to some other attractor in systems where more than one other attractor is available the attractor chosen may depend very sensitively on how the catastrophe is realized two examples in forced oscillators of duffing type illustrate how the probabilities of different outcomes can be estimated using the phase space geometry of invariant manifolds

fast algorithm for the computation of moment invariants abstract moment invariants have been used as feature descriptors in a variety of object recognition applications when assuming a continuous image function moments calculated using a double integral formulation are invariant to variations in translation rotation and size of the object however due to the recursive nature of the calculations and the limited speed of microprocessors the moments were not computable in real time in this paper we present fast invariant moment computations using the delta method as a means of scene representation this method computes moments of contiguous images by using the contribution of each line rather than individual pixel

block decomposition and segmentation for fast hough transform evaluation the decomposition of binary images using rectangular blocks of foreground pixels as primitives is considered based on this type of decomposition a fast method for evaluating the hough transform is introduced a complexity analysis of the proposed block hough transform algorithm sets constraints on the complexity of algorithms used for block decomposition so that the total decomposition and hough transform application time is much less than the time consumed by the usual point hough transform using this analysis we propose two algorithms for the decomposition and segmentation of binary images into rectangular blocks a combination of these methods leads to significant acceleration in the identification of linear features which is demonstrated in various image processing experiments 1999 pattern recognition society published by elsevier science ltd all rights reserved

influences of harmonic coupling on bifurcations in duffing oscillator with bounded potential well abstract the dynamics of bifurcations in a dufhng oscillator with a bounded potential well is investigated special attention is paid to the prominent phenomena of repeated sets of symmetry and reverse symmetry breaking the harmonic balance method and floquet theory are employed to account for this peculiar phenomena

3 d moment forms their construction and application to object identification and positioning the 3 d moment method is applied to object identification and positioning a general theory of deriving 3 d moments invariants is proposed the notion of complex moments is introduced complex moments are defined as linear combinations of moments with complex coefficients and are collected into multiplets such that each multiplet transforms irreducibly under 3 d rotations the application of the 3 d moment method to motion estimation is also discussed using group theoretic techniques various invariant scalars are extracted from compounds of complex moments via clebsch gordon expansion twelve moment invariants consisting of the second order and third order moments are explicitly derived based on a perturbation formula it is shown that the second order moment invariants can be used to predict whether the estimation using noisy data is reliable or not the new derivation of vector forms also facilities the calculation of motion estimation in a tensor approach vectors consisting of the third order moments can be derived in a similar manner

applications of tensor theory to object recognition and orientation determination a method is developed by which images resulting from orthogonal projection of rigid planar patch objects arbitrarily oriented in three dimensional 3 d space may be used to form systems of linear equations which are solved for the affine transform relating the images the technique is applicable to complete images and to unlabeled feature sets derived from images and with small modification may be used to transform images of unknown objects such that they represent images of those objects from a known orientation for use in object identification no knowledge of point correspondence between images is required theoretical development of the method and experimental results are presented the method is shown to be computationally efficient requiring o n multiplications and additions where depending on the computation algorithm n may equal the number of object or edge picture elements

sampling and reconstruction of shapes with algebraic boundaries we present a sampling theory for a class of binary images with finite rate of innovation fri every image in our model is the restriction of mathbb 1 p leq 0 to the image plane where mathbb 1 denotes the indicator function and p is some real bivariate polynomial this particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial p we show that the image parameters i e the polynomial coefficients satisfy a set of linear annihilation equations with the coefficients being the image moments the inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels as a remedy to these problems we replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel the benefits are threefold 1 it relaxes the requirements on the sampling kernels 2 produces annihilation equations that are robust at numerical precision and 3 extends the results to images with unbounded boundaries we further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points and sequentially enforcing measurement consistency we consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images including low to moderate noise levels and a range of realistic sampling kernels

an exact method for computing the area moments of wavelet and spline curves we present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis using green s theorem we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product the multidimensional filter coefficients are precomputed exactly as the solution of a two scale relation to demonstrate the performance improvement of the new method we compare it with existing methods such as pixel based approaches and approximation of the region by a polygon we also propose an alternate scheme when the scaling function is sinc x

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

the approximate approach to chaos phenomena in oscillators having single equilibrium position abstract the study of the nature of the chaotic response of oscillators having a single equilibrium position shows a relationship between the stability limit of secondary resonances and the chaotic motion zones computer simulations reveal that continuous segments in the frequency spectrum develop around those harmonic components of pre chaotic periodic response which approach their stability limit and are due to decay the major point in the study is an analysis of the filtered response i e the component of chaotic motion which corresponds to an individual narrow band continuous frequency spectrum segment because the filtered response resembles a single harmonic function of time with randomly modulated amplitude and amplitude dependent frequency the idea of considering it to be a sort of free vibration is developed the theoretical explanation of the phenomen is found by considering the concept of the stability limit of secondary resonances in the light of the approximate theory of non linear oscillators from this point of view chaotic motion is seen as a particular type of instability of the secondary resonance periodic solution the instability which manifests itself by random like oscillations of the subultra harmonic amplitudes and precedes the strict loss of stability i e the decay of the amplitudes

on image analysis by the methods of moments various types of moments have been used to recognize image patterns in a number of applications a number of moments are evaluated and some fundamental questions are addressed such as image representation ability noise sensitivity and information redundancy moments considered include regular moments legendre moments zernike moments pseudo zernike moments rotational moments and complex moments properties of these moments are examined in detail and the interrelationships among them are discussed both theoretical and experimental results are presented

fast and exact computation of cartesian geometric moments using discrete green s theorem abstract green s theorem evaluates a double integral over a planar object by a simple integration along the boundary of the object it can be used in fast computation of cartesian geometric moments of two dimensional binary objects since the object shape is totally determined by the boundary li and shen pattern recognition 24 807 813 1991 used an approximation of green s theorem to deal with discrete objects this method is fast but not accurate in this paper we present a discrete version of green s theorem which evaluates a double sum over a two dimensional discrete object by a simple summation along the discrete boundary of the object by using the discrete green s theorem we propose a new algorithm for fast computation of the geometric moments the new algorithm is faster than the previous methods and gives exactly the same results as if a double sum were used the importance of the exact computation is discussed by examining the accuracy of hu s moment invariants and some other moment based shape features which are frequently used in various shape description and pattern recognition tasks a fast method for computing moments of regions in gray level images using the discrete green s theorem is also presented we show that the moments of this type are useful in structural texture analysis