modified multiscale cross sample entropy for complex time series we introduce the cmcse and further propose the rcmcse method mcse cmcse and rcmcse methods are employed to artificial and financial time series rcmcse reduces standard deviation and the probability of inducing undefined entropy rcmcse can provide better robustness and more accurate entropies rcmcse is more applicable for the study between us and chinese stock markets in this paper we introduce the composite multiscale cross sample entropy cmcse which may induce undefined entropies and then further propose the refined composite multiscale cross sample entropy rcmcse which modifies cmcse first we apply multiscale cross sample entropy mcse cmcse and rcmcse methods to three types of artificial time series in order to test the validity and accuracy of these methods results show that rcmcse reduces not only standard deviation but also the probability of inducing undefined entropy effectively which can provide better robustness and more accurate entropies then these three methods are employed to investigate financial time series including us and chinese stock indices for the study between stock indices in the same region some conclusions which are consistent with previous study are drawn by the rcmcse results meanwhile it can be found that undefined entropies are induced and the numbers of inducing undefined entropy by three methods for investigation between three us stock indices and two chinese mainland stock indices are given compared with mcse and cmcse rcmcse method is capable of reducing the number of undefined entropy and providing more accurate entropies moreover the differences on inducing undefined entropy between results for us stock indices two chinese mainland stock indices and results for us stock indices hsi demonstrate a much closer relation between us stock markets and hsi than between us stock markets and two chinese mainland stock markets hence it can be concluded that rcmcse is more applicable for the study between us and chinese stock markets

recovering surface shape and orientation from texture texture provides an important source of information about the three dimensional structure of visible surfaces particularly for stationary monocular views to recover 3d structure the distorting effects of projection must be distinguished from properties of the texture on which the distortion acts this requires that assumptions must be made about the texture yet the unpredictability of natural textures precludes the use of highly restrictive assumptions the recovery method reported in this paper exploits the minimal assumption that textures do not mimic projective effects this assumption determines the strategy of attributing as much as possible of the variation observed in the image to projection equivalently the interpretation is chosen for which the texture prior to projection is made as uniform as possible this strategy was implemented using statistical methods first for the restricted case of planar surfaces and then by extension for curved surfaces the technique was applied successfully to natural images

shape from texture estimation isotropy and moments a theory is proposed for the interpretation of 3d textures with oriented elements it builds on two previous theories a statistical one due to witkin and kanatani s buffon transform we show that the statistical estimation of orientation of a textured plane is one of a general family of optimal backprojection problems which also includes maximal compactness general uniqueness and convergence results are obtained for an efficient algorithm employing second moment feedback this framework greatly enhances the usefulness of surface orientation estimation for texture in two ways first an error distribution is derived for the estimator and this is crucial for integration of shape information second hypotheses about intrinsic texture statistics can be verified by a 2 test which if failed warns that the orientation estimator is not to be believed finally it is argued that these results suggest a general moment tensor approach to analysis of 3d texture

complexity analysis of time series based on generalized fractional order cumulative residual distribution entropy abstract based on the theory of cumulative residual entropy and distribution entropy this paper proposes a new model cumulative residual distribution entropy crde which is more suitable for the complexity and risk analysis of time series the new model makes full use of the known information including not only the information of probability but also the information about the value of random variables additionally the crde considers the potential inherent information of vector to vector distance in the state space by combining theoretical analysis with empirical research this paper verifies that the new model has more advantages in measuring extreme events and small probability events with higher consistency stability and practicability in this paper the cumulative residual distribution entropy model is extended to the fractional order the generalized fractional cumulative residual distribution entropy gcrde can better capture the tiny evolution of time series which is more advantageous for studying the dynamic characteristics of complex systems the new model makes up for some shortcomings of traditional models and it can play a guiding role in the study of complex systems in the real world

line drawing interpretation bilateral symmetry results previously derived by the author are used to investigate the implication of bilateral symmetry in line drawings it is shown that the line drawing of an orthographically protected surface of revolution exhibits bilateral symmetry about the projection of its axis of revolution irrespective of the viewing direction barring one exception a bilaterally symmetric line drawing is necessarily the orthographic projection of a local surface of revolution whenever its symmetry axis continues to be the projection of a fixed line in space under perturbation of viewpoint the axis of revolution is the invariant preimage of the symmetry axis various line drawing causes are detailed which facilitate the deduction of invariant preimages of symmetry axes and consequently of local surfaces of revolution

shape from regular patterns niis paper describes an aigoritlitr1 to obtaitl local surface orierltatiorr frofti the apparetlt surface pattertl distortion in an image we propose a spherical projection to model perspctiue imaging a trtappitlg is debtled based of1 the measurement of the local distortions of a repeated known rexture pattern due io the image projection 77tis mapping maps an apparent shape on the image sphere lo a locus of possible surface orienrations on the gaussian sphere an ireratiw constraint propagation algorithm with the orientations at occluding boundaries reduces possible surface orientations to a unique orientation 7lis algorithm can recover local surface orientation as well as interpolate surjace orientations where no information is auailable xis algorithm is applied to a real image ro demonstrate its perfontlance

weighted multiscale renyi permutation entropy of nonlinear time series abstract in this paper based on renyi permutation entropy rpe which has been recently suggested as a relative measure of complexity in nonlinear systems we propose multiscale renyi permutation entropy mrpe and weighted multiscale renyi permutation entropy wmrpe to quantify the complexity of nonlinear time series over multiple time scales first we apply mpre and wmpre to the synthetic data and make a comparison of modified methods and rpe meanwhile the influence of the change of parameters is discussed besides we interpret the necessity of considering not only multiscale but also weight by taking the amplitude into account then mrpe and wmrpe methods are employed to the closing prices of financial stock markets from different areas by observing the curves of wmrpe and analyzing the common statistics stock markets are divided into 4 groups 1 dji s p500 and hsi 2 nasdaq and ftse100 3 dax40 and cac40 and 4 shangzheng and shencheng results show that the standard deviations of weighted methods are smaller showing wmrpe is able to ensure the results more robust besides wmpre can provide abundant dynamical properties of complex systems and demonstrate the intrinsic mechanism

refined composite multiscale weighted permutation entropy of financial time series abstract for quantifying the complexity of nonlinear systems multiscale weighted permutation entropy mwpe has recently been proposed mwpe has incorporated amplitude information and been applied to account for the multiple inherent dynamics of time series however mwpe may be unreliable because its estimated values show large fluctuation for slight variation of the data locations and a significant distinction only for the different length of time series therefore we propose the refined composite multiscale weighted permutation entropy rcmwpe by comparing the rcmwpe results with other methods results on both synthetic data and financial time series rcmwpe method shows not only the advantages inherited from mwpe but also lower sensitivity to the data locations more stable and much less dependent on the length of time series moreover we present and discuss the results of rcmwpe method on the daily price return series from asian and european stock markets there are significant differences between asian markets and european markets and the entropy values of hang seng index hsi are close to but higher than those of european markets the reliability of the proposed rcmwpe method has been supported by simulations on generated and real data it could be applied to a variety of fields to quantify the complexity of the systems over multiple scales more accurately

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

binary indices of time series complexity measures and entropy plane abstract complexity of time series is an important feature of dynamical systems such as financial systems in order to bring out the complete non linear behavior of financial time series non linear tools of complexity measurement like entropy measures are indispensable sample entropy sampen and distribution entropy disten are popular methods of assessing the complexity in various fields however both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets therefore we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane in order to further improve the accuracy of the research results we take the embedding dimension m as the variable expand the entropy point into the entropy curve for further research furthermore considering the shortcomings of sample entropy and distribution entropy in practical application we generalize them by i replacing sample entropy and distribution entropy with multi scale sample entropy and multi scale distribution entropy ii generalizing shannon entropy as tsallis entropy also scale factor and entropic index q are taken respectively as variables to get the entropy curves by using the artificial data we confirm the rationality of using entropy points and entropy curves to study the complexity of time series finally we apply this method to measure the complexity of real world financial time series the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences

intrinsic chaos and external noise in population dynamics abstract we address the problem of the relative importance of the intrinsic chaos and the external noise in determining the complexity of population dynamics we use a recently proposed method for studying the complexity of nonlinear random dynamical systems the new measure of complexity is defined in terms of the average number of bits per time unit necessary to specify the sequence generated by the system this measure coincides with the rate of divergence of nearby trajectories under two different realizations of the noise in particular we show that the complexity of a nonlinear time series model constructed from sheep populations comes completely from the environmental variations however in other situations intrinsic chaos can be the crucial factor this method can be applied to many other systems in biology and physics

chaotic and stochastic functions abstract we study chaotic functions that are exact solutions to nonlinear maps a generalization of these functions which cannot be expressed as a recursive procedure anymore can produce truly random sequences even if the initial conditions are known exactly the next values are in principle unpredictable we present a mathematical formulation of an elementary stochastic process several applications are discussed

blind source separation using temporal predictability a measure of temporal predictability is defined and used to separate linear mixtures of signals given any set of statistically independent source signals it is conjectured here that a linear mixture of those signals has the following property the temporal predictability of any signal mixture is less than or equal to that of any of its component source signals it is shown that this property can be used to recover source signals from a set of linear mixtures of those signals by finding an un mixing matrix that maximizes a measure of temporal predictability for each recovered signal this matrix is obtained as the solution to a generalized eigenvalue problem such problems have scaling characteristics of o n3 where n is the number of signal mixtures in contrast to independent component analysis the temporal predictability method requires minimal assumptions regarding the probability density functions of source signals it is demonstrated that the method can separate signal mixtures in which each mixture is a linear combination of source signals with supergaussian subgaussian and gaussian probability density functions and on mixtures of voices and music

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

on three dimensional surface reconstruction methods a survey is presented of some of the surface reconstruction methods that can be found in the literature the focus is on a small recent and important subset of the published reconstruction techniques the techniques are classified based on the surface representation used implicit versus explicit functions a study is made of the important aspects of the surface reconstruction techniques one aspect is the viewpoint invariance of the methods this is an important property if object recognition is the ultimate objective the robustness of the various methods is examined it is determined whether the parameter estimates are biased and the sensitivity to obscuration is addressed the latter two aspects are particularly important for fitting functions in the implicit form a detailed description is given of a parametric reconstruction method for three dimensional object surfaces that involves numeric grid generation techniques and variational principle formulations this technique is invariant to rigid motion in dimensional space

on the geometric interpretation of image contours abstract in this paper we suggest a computational model for the 3d interpretation of a 2d view based on contour classification and contour interpretation we concentrate on those contours arising from discontinuities in surface orientation we combine a generic surface description well suited for visual tasks with a model of the image formation process in order to derive image contour configurations that are likely to be interpreted in terms of surface contours next we describe a computer algorithm which attempts to interpret image contours on the following grounds first an image analysis process produces a description in terms of contours and relationships between them second among these contours we select those which form a desired configuration third the selected contours are combined with constraints available with the image formation process in order to be interpreted in terms of discontinuities in surface orientation as a consequence there is a dramatic reduction in the number of possible orientations of the associated scene surfaces

a recipe for randomness despite many diverse theories that address closely related themes e g probability theory algorithmic complexity cryptoanalysis and pseudorandom number generation a near void remains in constructive methods certified to yield the desired random output herein we provide explicit techniques to produce broad sets of both highly irregular finite and normal infinite sequences based on constructions and properties derived from approximate entropy apen a computable formulation of sequential irregularity furthermore for infinite sequences we considerably refine normality by providing methods for constructing diverse classes of normal numbers classified by the extent to which initial segments deviate from maximal irregularity

direct estimation of shape from texture witkin 1981 has proposed a maximum likelihood ml estimator of surface orientation based on the observed directional bias of projected texture elements however a drawback of this procedure is that the estimate is only defined indirectly in terms of a set of nonlinear equations an alternative method is proposed which allows an estimate of the surface orientation to be computed directly in a single step from certain simple statistics of the image data we also show that this direct estimate allows witkin s ml estimate to be computed to within 0 05 degrees in only two or three iterative steps the performance of the new estimator is demonstrated experimentally and compared to that of the ml estimator using both synthetic data and real gray level images

surface parametrization and curvature measurement of arbitrary 3 d objects five practical methods curvature sampling of arbitrary fully described 3 d objects e g tomographic medical images is difficult because of surface patch parameterization problems five practical solutions are presented and characterized the sander zucker approach two novel methods based on direct surface mapping a piecewise linear manifold technique and a turtle geometry method one of the new methods called the cross patch cp method is shown to be very fast robust in the presence of noise and is always based on a proper surface parameterization provided the perturbations of the surface over the patch neighborhood are isotropically distributed

recovering three dimensional shape from a single image of curved objects an algorithm to recover three dimensional shape i e surface orientation and relative depth from a single segmented image is presented it is assumed that the scene is composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or textures it is also assumed that the reflectance map r n is known for the canonical case of lambertian surfaces illuminated by a point light source this implies knowing the light source direction a variational formulation of line drawing and shading constraints in a common framework is developed the global constraints are partitioned into constraint sets corresponding to the faces edges and vertices in the scene for a face the constraints are given by horn s image irradiance equation a variational formulation of the constraints at an edge both from the known direction of the image curve corresponding to the edge and shading is developed at a vertex the constraints are modeled by a system of nonlinear equations an algorithm is presented to solve this system of constraints

an extremum principle for shape from contour an extremum principle is developed that determines three dimensional surface orientation from a two dimensional contour the principle maximizes the ratio of the area to the square of the perimeter a measure of the compactness or symmetry of the three dimensional surface the principle interprets regular figures correctly and it interprets skew symmetries as oriented real symmetries the maximum likelihood method approximates the principle on irregular figures but we show that it consistently overestimates the slant of an ellipse

maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations in this paper we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations we consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid the grid points on the subdomain boundaries need not match the grid points from adjacent subdomains to obtain a global finite difference discretization of the elliptic problem we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain by interpolating the discrete solution on adjacent subdomains if the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains we describe also how the resulting linear system can be solved iteratively by a parallel schwarz alternating method or a schwarz preconditioned krylov subspace iterative method several numerical results are included to support the theory

determining the complexity of fh ss sequence by approximate entropy high complexity of frequency hopping fh spreadspectrum ss sequence is of great importance to high security multiple access communication systems for it makes fh ss sequence difficult to be analyzed with the growing development in the design of fh ss sequence in much wider fields the wellknown complexity measuresiquestthe linear complexity lc the linear complexity profile lcp and the k error linear complexity k error lc iquestare widely used but not sufficient to evaluate the complexities of the sequences available such as the cryptographical sequence and the chaotic sequence families in this paper a new complexity metric to evaluate the unpredictability of fh ss sequence based on the approximate entropy apen is proposed in the view of the maximal randomness of the sequences with arbitrary length and the theoretical bounds of the apen are derived from a probabilistic point of view simulations and analysis results show that the proposed apen works effectively to discern the changing complexities of the fh ss sequences with small number of samples which provide superior performance over its candidates

recovery of 3 d objects with multiple curved surfaces from 2 d contours abstract inference of 3 d shape from 2 d contours in a single image is an important problem in machine vision often techniques to solve this problem examine each surface in the scene separately whereas our perception of their shapes clearly depends on the interplay between them as well in this paper we describe a technique that attempts to recover the shapes of all the surfaces of an object simultaneously though it is limited to objects made of zero gaussian curvature surfaces our technique is based on an analysis of three kinds of symmetries defined in the paper and the constraints that derive from them and from other boundaries this technique uses some of the constraints developed in an earlier paper that was limited to examining a zero gaussian curvature surface cut by parallel planes this restriction has been removed here and the constraints have been reformulated to allow integration of constraints from all the neighboring surfaces results on some complex examples are shown

interpreting line drawings as three dimensional surfaces understanding how line drawings convey tri dimensionality is of fundamental importance in explaining surface perception when photometry is either uninformative or too complex to model analytically we put forward here a computational model for interpreting line drawings as three dimensional surfaces based on constraints on local surface orientation along extremal and discontinuity boundaries specific techniques are described for two key processes recovering the three dimensional conformation of a space curve e g a surface boundary from its two dimensional projection in an image and interpolating smooth surfaces from orientation constraints along extremal boundaries the relevance of the model to a general theory of low level vision is discussed

randomness and degrees of irregularity the fundamental question are sequential data random arises in myriad contexts often with severe data length constraints furthermore there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness e g to evaluate the efficacy of therapy in medicine we address both these issues from a computable framework via a quantification of regularity apen approximate entropy defining maximal randomness for sequences of arbitrary length indicating the applicability to sequences as short as n 5 points an infinite sequence formulation of randomness is introduced that retains the operational and computable features of the finite case in the infinite sequence setting we indicate how the foundational definition of independence in probability theory and the definition of normality in number theory reduce to limit theorems without rates of convergence from which we utilize apen to address rates of convergence of a deficit from maximal randomness refining the aforementioned concepts in a computationally essential manner representative applications among many are indicated to assess i random number generation output ii well shuffled arrangements and iii the quality of bootstrap replicates

fundamental limitations on projective invariants of planar curves in this paper some fundamental limitations of projective invariants of non algebraic planar curves are discussed it is shown that all curves within a large class can be mapped arbitrarily close to a circle by projective transformations it is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves thus a continuous projective invariant on closed curves is constant this also limits the possibility of finding so called projective normalisation schemes for closed planar curves

stochastic model for heart rate fluctuations a normal human heart rate shows complex fluctuations in time which is natural because the heart rate is controlled by a large number of different feedback control loops these unpredictable fluctuations have been shown to display fractal dynamics long term correlations and 1 f noise these characterizations are statistical and they have been widely studied and used but much less is known about the detailed time evolution dynamics of the heart rate control mechanism here we show that a simple one dimensional langevin type stochastic difference equation can accurately model the heart rate fluctuations in a time scale from minutes to hours the model consists of a deterministic nonlinear part and a stochastic part typical to gaussian noise and both parts can be directly determined from the measured heart rate data studies of 27 healthy subjects reveal that in most cases the deterministic part has a form typically seen in bistable systems there are two stable fixed points and one unstable one

from image quadrilaterals to symmetrical trapezia abstract this paper concerns the 3 d interpretation of image quadrilaterals as the perspective projections of bilateral symmetric trapezia simple algorithms are described for reconstructing a symmetric trapezium from a given image quadrilateral and for determining the 3 d motion of a symmetric trapezium from two matched image quadrilaterals several results are established and or further verified including the necker reversal ambiguity in the interpretation of image quadrilaterals as the perspective projection of bilateral symmetric trapezia experiments with both synthetic and real images are included to demonstrate the validity of the theoretic results

polyhedron recognition using three view analysis abstract using three view analysis this paper proposes a new approach for single polyhedron recognition both the 3d surface data and 2d image projection patterns are used to find the exact corresponding model of an unknown input object the recognition process first applies a hypothesis generation verification process to match the 3d partial object of the unknown input which is reconstructed by applying the three view matching strategy chiou et al proc ics 88 pp 1516 1521 1988 and the polyhedron reconstruction process hung et al pattern recognition 22 231 246 1989 against the data based models for selecting the possible corresponding models candidates it then backprojects each candidate onto the three image planes and matches the outer boundaries of the three backprojections by a simple method for judging the exact corresponding model measurement error is also analyzed theoretically to give a bound of the threshold value so that it can be automatically chosen in each decision making process the approach is implemented on a pc level vision system experimental results show that the system is feasible and reliable even in a noisy environment