zero crossings of a wavelet transform the completeness stability and application to pattern recognition of a multiscale representation based on zero crossings is discussed an alternative projection algorithm is described that reconstructs a signal from a zero crossing representation which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero crossings the reconstruction algorithm has a fast convergence and each iteration requires o n log sup 2 n computation for a signal of n samples the zero crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems as an example the implementation and results of a coarse to fine stereo matching algorithm are described

affine invariant pattern recognition using multiscale autoconvolution this paper presents a new affine invariant image transform called multiscale autoconvolution msa the proposed transform is based on a probabilistic interpretation of the image function the method is directly applicable to isolated objects and does not require extraction of boundaries or interest points and the computational load is significantly reduced using the fast fourier transform the transform values can be used as descriptors for affine invariant pattern classification and in this article we illustrate their performance in various object classification tasks as shown by a comparison with other affine invariant techniques the new method appears to be suitable for problems where image distortions can be approximated with affine transformations

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

chaotic phase similarities and recurrences in a damped driven duffing oscillator we report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped driven duffing oscillator as the amplitude of the drive increases families of period adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension controlling the alternation of chaos and order in parameter space of certain flows

what is the effective impact of the explosive orbital growth in discrete time one dimensional polynomial dynamical systems abstract we study the distribution of periodic orbits in one dimensional two parameter maps specifically we report an exact expression to quantify the growth of the number of periodic orbits for discrete time dynamical systems governed by polynomial equations of motion of arbitrary degree in addition we compute high resolution phase diagrams for quartic and for both normal forms of cubic dynamics and show that their stability phases emerge all distributed in a similar way preserving a characteristic invariant ordering such coincidences are remarkable since our exact expression shows the total number of orbits of these systems to differ dramatically by more than several millions even for quite low periods all this seems to indicate that surprisingly the total number and the distribution of stable phases is not significantly affected by the specific nature of the nonlinearity present in the equations of motion

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

a linear signal decomposition approach to affine invariant contour identification means for the identification of objects from contours despite affine transform induced distortions using a linear signal space decomposition are described this technique also yields robust estimates of the 3 d rotations of a near planar object the ability to determine object identity and orientation from a single model representation without iteration or combinatorial search proceeds from the use of affine invariant differential measures derived via lie group theory the technique is extremely robust owing to the error rejection properties of signal space projections results illustrating the resilience of the solutions in the presence of severe non affine distortion and pixelization are given

geometric and illumination invariants for object recognition we propose invariant formulations that can potentially be combined into a single system in particular we describe a framework for computing invariant features which are insensitive to rigid motion affine transform changes of parameterization and scene illumination perspective transform and view point change this is unlike most current research on image invariants which concentrates on either geometric or illumination invariants exclusively the formulations are widely applicable to many popular basis representations such as wavelets short time fourier analysis and splines exploiting formulations that examine information about shape and color at different resolution levels the new approach is neither strictly global nor local it enables a quasi localized hierarchical shape analysis which is rarely found in other known invariant techniques such as global invariants furthermore it does not require estimating high order derivatives in computing invariants unlike local invariants whence is more robust we provide results of numerous experiments on both synthetic and real data to demonstrate the validity and flexibility of the proposed framework

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

dissecting shrimps results for some one dimensional physical models this paper describes how certain shrimp like clusters of stability organize themselves in the parameter space of dynamical systems clusters are composed of an infinite affine similar repetition of a basic elementary cell containing two primary noble points a head and a tail defining an axis of approximate symmetry knowledge of the axis and the skewness of the k periodic main cell of a k 2 cluster is enough to define the orientation of the whole cluster in space peculiarly simple directions along which shrimp like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head in addition we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example that shrimps are diffeomorphic copies of shrimps

geometric invariants and object recognition we discuss the role of the general invariance concept in object recognition and review the classical and recent literature on projective invariance invariants help solve major problems of object recognition for instance different images of the same object often differ from each other because of the different viewpoint from which they were obtained to match the two images common methods thus need to find the correct viewpoint a difficult problem that can involve search in a high dimensional space of all possible points of view and or finding point correspondences geometric invariants are shape descriptors computed from the geometry of the shape that remain unchanged under geometric transformations such as changing the viewpoint thus they can be matched without search deformations of objects are another important class of geometric changes for which invariance is useful

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

a dyadic wavelet affine invariant function for 2d shape recognition dyadic wavelet transform has been used to derive an affine invariant function first an invariant function using two dyadic levels is derived then this invariant function is used to derive another invariant function using six dyadic levels we introduce the wavelet based conic equation the invariant function is based on analyzing the object boundary using the dyadic wavelet transform experimental results on both synthetic and real data are used to demonstrate the discriminating power of the proposed invariant function it has also been compared with some traditional methods the stability of the proposed invariant function is examined in addition the stability under large perspective transformation is tested

invariant descriptors for 3d object recognition and pose invariant descriptors are shape descriptors that are unaffected by object pose by perspective projection or by the intrinsic parameters of the camera these descriptors can be constructed using the methods of invariant theory which are briefly surveyed a range of applications of invariant descriptors in 3d model based vision is demonstrated first a model based vision system that recognizes curved plane objects irrespective of their pose is demonstrated curves are not reduced to polyhedral approximations but are handled as objects in their own right models are generated directly from image data once objects have been recognized their pose can be computed invariant descriptors for 3d objects with plane faces are described all these ideas are demonstrated using images of real scenes the stability of a range of invariant descriptors to measurement error is treated in detail

compound windows of the henon map abstract for the two parameter second order henon map the shapes and locations of the periodic windows continua of parameter values for which solutions x 0 x 1 can be stably periodic embedded in larger regions where chaotic solutions or solutions of other periods prevail are found by a random searching procedure and displayed graphically many windows have a typical shape consisting of a central body from which four narrow antennae extend such windows to be called compound windows are often arranged in bands to be called window streets that are made up largely of small detected but poorly resolved compound windows for each fundamental subwindow the portion of a window where a fundamental period prevails a stability measure u is introduced where the solution is stable u 1 curves of constant u are found by numerical integration along one line in parameter space the henon map reduces to the one parameter first order logistic map and two antennae from each compound window intersect this line the curves where u 1 and u 1 that bound either antenna are close together within these intersections but as either curve with u 1 leaves the line it diverges from the curve where u 1 crosses the other curve where u 1 and nears the other curve where u 1 forming another antenna the region bounded by the numerically determined curves coincides with the subwindow as found by random searching a fourth degree equation for an idealized curve of constant u is established points in parameter space producing periodic solutions where x 0 x m 0 for given values of m are found to lie on cantor sets of curves that closely fit the window streets points producing solutions where x 0 x m 0 and satisfying a third condition approximating the condition that x n be bounded as n lie on curves to be called street curves of order m that approximate individual members of the cantor set and individual window streets compound windows of period m m tend to occur near the intersections of street curves of orders m and m some exceptions to what appear to be fairly general results are noted the exceptions render it difficult to establish general theorems

from mandelbrot like sets to arnold tongues a transition from mandelbrot like sets to arnold tongues is characterized via a coupling of two non identical quadratic maps proposed by us a two dimensional parameter space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event the location of the parameter sets where naimark sacker bifurcations occur which is exactly the place where arnold tongues of arbitrary periods are born was computed analytically

global structure of periodicity hubs in lyapunov phase diagrams of dissipative flows infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow namely a r ossler s oscillator we show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal like sheaves of shilnikov homoclinic bifurcations of a common saddle focus equilibrium the specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor

wavelet based affine invariant representation a tool for recognizing planar objects in 3d space a technique is developed to construct a representation of planar objects undergoing a general affine transformation the representation can be used to describe planar or nearly planar objects in a three dimensional space observed by a camera under arbitrary orientations the technique is based upon object contours parameterized by an affine invariant parameter and the dyadic wavelet transform the role of the wavelet transform is the extraction of multiresolution affine invariant features from the affine invariant contour representation a dissimilarity function is also developed and used to distinguish among different object representations this function makes use of the extrema on the representations thus making its computation very efficient a study of the effect of using different wavelet functions and their order or vanishing moments is also carried out experimental results show that the performance of the proposed representation is better than that of other existing methods particularly when objects are heavily corrupted with noise

comments on vibration analysis of arbitrary shaped membranes using non dimensional dynamic influence function in reference 1 the authors presented the so called non dimensional dynamic in uence function method for membrane vibration a regular formulation and singularity free method were obtained also a symmetry and meshless formulation can be achieved the auxiliary system is a complementary solution instead of fundamental function it is source free in the in uence function many successful examples of the dirichlet types were demonstrated it seems that this method is very attractive however this method can be treated as one kind of the tre tz method 2 4 based on the dual formulation developed by chen and hong 5 6 the in uence function is nothing but the imaginary part of the fundamental solution s x ih 1 0 kr 7 the method by kang et al 1 can be treated as a special case of the imaginary part dual bem also the real part dual bem was developed and many references can be referred 8 9 mrm formulation can also be viewed as a real part formulation and its occurrence of spurious eigenvalues have been found in references 10 15 it is well known that the real part imaginary part formulations and multiple reciprocity method all result in spurious eigensolutions particularly the imaginary part formulation also results in an ill posed problem since the condition number for the in uence matrix is very large many approaches have been employed to lter out the spurious solutions and extract the true solution for example residue method 11 12 singular value decomposition svd technique 9 13 15 generalized singular value decomposition gsvd technique 16 and domain partition technique 17 it is expected that the kang s method has the problems of spurious solutions and ill conditioned behavior since it is an imaginary part formulation however no such information was addressed some points will be discussed as follows 1 spurious eigensolution it is interesting to nd that all the examples in reference 1 are of the dirichlet type according to the theoretical derivation the neumann problem has the problems of spurious solutions using the imaginary part formulation we will prove that in the following as mentioned earlier spurious eigenvalues occur in the real part of mrm formulation 9 11 12 also the imaginary part bem results in spurious solutions 7 here we will derive the true and spurious solutions in the discrete system analytically for a circular domain by using the non dimensional in uence function method 1 and the imaginary part dual bem 7 in a uni ed way the degenerate kernels and circulants are employed to study the discrete system in an exact form the relation between the non dimensional in uence function method and the imaginary part dual bem is also addressed and is summarized in table 1 the symbols in table 1 follow the dual model of chen and hong 6

an exploration of the henon quadratic map abstract the two dimensional mapping introduced by henon in 1976 which yields a strange attractor for the parameter values a 1 4 b 0 3 has been investigated over the extended region 2 a 6 1 b 1 analytical solutions have been obtained for the onset of real stable cycles of periods 1 through 6 the associated bifurcations to stable cycles of twice the period 2 to 12 have also been determined all the resulting boundary curves are displayed in the a b parameter plane finally an interesting three dimensional generalization of the henon map is announced

application of affine invariant fourier descriptors to recognition of 3 d objects the method of fourier descriptors is extended to produce a set of normalized coefficients which are invariant under any affine transformation translation rotation scaling and shearing the method is based on a parameterized boundary description which is transformed to the fourier domain and normalized there to eliminate dependencies on the affine transformation and on the starting point invariance to affine transforms allows considerable robustness when applied to images of objects which rotate in all three dimensions as is demonstrated by processing silhouettes of aircraft maneuvering in three space

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed