a novel eye localization method with rotation invariance this paper presents a novel learning method for precise eye localization a challenge to be solved in order to improve the performance of face processing algorithms few existing approaches can directly detect and localize eyes with arbitrary angels in predicted eye regions face images and original portraits at the same time to preserve rotation invariant property throughout the entire eye localization framework a codebook of invariant local features is proposed for the representation of eye patterns a heat map is then generated by integrating a 2 class sparse representation classifier with a pyramid like detecting and locating strategy to fulfill the task of discriminative classification and precise localization furthermore a series of prior information is adopted to improve the localization precision and accuracy experimental results on three different databases show that our method is capable of effectively locating eyes in arbitrary rotation situations 360 in plane

stable steady state pattern formation on isothermal catalyst surfaces a general class of reaction diffusion systems is considered this includes a large family of models for chemical activity on isothermal catalyst surfaces necessary and sufficient conditions are given for when a stable spatially nonuniform solution can bifurcate from a steady uniform solution as parameters in the equations are varied in particular such a bifurcation occurs for the dirichlet problem but not for the neumann problem a rather complete description is given concerning the stability properties direction of bifurcation and global continuation of the branch of bifurcating solutions

robust object detection with interleaved categorization and segmentation this paper presents a novel method for detecting and localizing objects of a visual category in cluttered real world scenes our approach considers object categorization and figure ground segmentation as two interleaved processes that closely collaborate towards a common goal as shown in our work the tight coupling between those two processes allows them to benefit from each other and improve the combined performance the core part of our approach is a highly flexible learned representation for object shape that can combine the information observed on different training examples in a probabilistic extension of the generalized hough transform the resulting approach can detect categorical objects in novel images and automatically infer a probabilistic segmentation from the recognition result this segmentation is then in turn used to again improve recognition by allowing the system to focus its efforts on object pixels and to discard misleading influences from the background moreover the information from where in the image a hypothesis draws its support is employed in an mdl based hypothesis verification stage to resolve ambiguities between overlapping hypotheses and factor out the effects of partial occlusion an extensive evaluation on several large data sets shows that the proposed system is applicable to a range of different object categories including both rigid and articulated objects in addition its flexible representation allows it to achieve competitive object detection performance already from training sets that are between one and two orders of magnitude smaller than those used in comparable systems

on the numerical approximation of the rotation number abstract the starting point of this paper is a polygonal approximation of an invariant curve of a map using this polygonal approximation an approximation for the circle map the restriction of the map to the invariant curve is obtained the rotation number of the circle map is then approximated by the rotation number of the approximated circle map the error in the obtained approximate rotation number is discussed and related to the error in the polygonal approximation of the invariant curve simple algorithms for the approximation of the rotation number are described a numerical example illustrates the theory

global bifurcations in rayleigh be nard convection experiments empirical maps and numerical bifurcation analysis abstract we use nonlinear signal processing techniques based on artificial neural networks to construct an empirical mapping from experimental rayleigh benard convection data in the quasiperiodic regime the data in the form of a one parameter sequence of poincare sections in the interior of a mode locked region resonance horn are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied rayleigh number the dynamic phenomena apparent in the data include period doublings complex intermittent behavior secondary hopf bifurcations and chaotic dynamics we use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space using numerical bifurcation techniques we locate the stable and unstable periodic solutions calculate eigenvalues approximate invariant manifolds of saddle type solutions and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

a performance evaluation of local descriptors in this paper we compare the performance of descriptors computed for local interest regions as for example extracted by the harris affine detector mikolajczyk k and schmid c 2004 many different descriptors have been proposed in the literature it is unclear which descriptors are more appropriate and how their performance depends on the interest region detector the descriptors should be distinctive and at the same time robust to changes in viewing conditions as well as to errors of the detector our evaluation uses as criterion recall with respect to precision and is carried out for different image transformations we compare shape context belongie s et al april 2002 steerable filters freeman w and adelson e setp 1991 pca sift ke y and sukthankar r 2004 differential invariants koenderink j and van doorn a 1987 spin images lazebnik s et al 2003 sift lowe d g 1999 complex filters schaffalitzky f and zisserman a 2002 moment invariants van gool l et al 1996 and cross correlation for different types of interest regions we also propose an extension of the sift descriptor and show that it outperforms the original method furthermore we observe that the ranking of the descriptors is mostly independent of the interest region detector and that the sift based descriptors perform best moments and steerable filters show the best performance among the low dimensional descriptors

weak classifier for density estimation in eye localization and tracking in this paper we propose a fast weak classifier that can detect and track eyes in video sequences the approach relies on a least squares detector based on the inner product detector ipd that can stimate a probability density distribution for a feature s location which fits naturally with a bayesian estimation cycle such as a kalman or particle filter as a least squares sliding window detector it possesses tolerance to small variations in the desired pattern while maintaining good generalization capabilities and computational efficiency we propose two approaches to integrating the ipd with a particle filter tracker we use the bioid feret lfpw and cofw public datasets as well as five manually annotated high definition video sequences to quantitatively evaluate the algorithms performance the video data set contains four subjects different types of backgrounds blurring due to fast motion and occlusions all code and data are available

period doubling and quadrupling for some second order difference schemes may exhibit stable n cycles that bifurcate into stable 2n cycles as a increases creating period doubling bifurcation sequences each beginning with a cycle of odd period these dynamics were first described by may i 21 and have since been investigated by many others see 3 61 for example a number of universal features of this bifurcation process have been observed primarily for quadratic maps 7 lo by comparison much less is known concerning the dynamics of higher order nonlinear schemes although there have been many numerical investigations of specific second order equations see i l 181 for example the full range of dynamics possible for such problems has not yet been determined in this work we investigate some general second order nonlinear difference schemes sufficient conditions are derived for the existence stability and bifurcations of periodic cycles for certain classes of such problems firstly the correspondence between periodic cycles of 1 and those of the second order problem x k l f xk l 3

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

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

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

numerical approximation of rough invariant curves of planar maps in this paper we describe a simple algorithm for the approximation of an invariant curve gamma of a planar map f we are particularly interested in the case where gamma is attracting but not smooth results for the delayed logistic map illustrate the performance of the algorithm

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

continuous measures in one dimension families of unimodal maps satisfying 1 t 1 1 1 1 witht 1 1 and t 1 1 2 t x isc2 inx2 and and symmetric inx 3 t0 0 0 t1 0 1 with frac d d lambda t 0 0 are considered the results of guckenheimer 1982 are extended to show that there is a positive measure of for whicht has a finite absolutely continuous invariant measure

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

numerical computation of invariant circles of maps abstract a method for the numerical computation of invariant circles of maps is presented along with appropriate techniques for its implementation the method involves solution of a functional equation by discretization and newton iteration the resulting algorithm is applied to a map of the cylinder and some examples of bifurcations of invariant circles are illustrated generalization of the method to the computation of invariant circles in more than two dimensions as well as sections of invariant tori is discussed

efficient subwindow search a branch and bound framework for object localization most successful object recognition systems rely on binary classification deciding only if an object is present or not but not providing information on the actual object location to estimate the object s location one can take a sliding window approach but this strongly increases the computational cost because the classifier or similarity function has to be evaluated over a large set of candidate subwindows in this paper we propose a simple yet powerful branch and bound scheme that allows efficient maximization of a large class of quality functions over all possible subimages it converges to a globally optimal solution typically in linear or even sublinear time in contrast to the quadratic scaling of exhaustive or sliding window search we show how our method is applicable to different object detection and image retrieval scenarios the achieved speedup allows the use of classifiers for localization that formerly were considered too slow for this task such as svms with a spatial pyramid kernel or nearest neighbor classifiers based on the lambda2 distance we demonstrate state of the art localization performance of the resulting systems on the uiuc cars data set the pascal voc 2006 data set and in the pascal voc 2007 competition

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

a simple proof for persistence of snap back repellers abstract in this article we show that if f has a snap back repeller then any small c 1 perturbation of f has a snap back repeller and hence has li yorke chaos and positive topological entropy by simply using the implicit function theorem we also give some examples

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

dependencies between competition and predation and their consequences for initial value sensitivity we commence with a discussion of dependencies between various processes in discrete consumer resource models two basic models are formulated these models are closely related and differ only in terms of which dependencies are assumed the models can be regarded as hybrids of the lotka volterra model the beverton holt model and the nicholson bailey model a complete stability analysis of the equilibria is given with the conclusion that the stability properties of these models are closely related to gause type predator prey systems the numerical part of our analysis brings out remarkable qualitative differences concerning the initial value sensitivity of the oscillations of the two basic models initial value sensitivity was measured by lyapunov exponents and the observed differences cannot be brought to light by a study of limiting cases or by equilibrium analysis the actual reasons behind these results remain a delicate mathematical question properties associated directly with the time series like

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3