generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

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

integrability and the painleve property for low dimensional systems we examine a system described by two first order nonlinear differential equations from the point of view of integrability the singularity analysis in the complex time plane is used to investigate the painleve property which according to the ablowitz ramani segur conjecture is a prerequisite for integrability for infinite dimensional systems we show that for such low dimensional systems the painleve analysis is still a most useful guide but integrable cases also exist which do not possess the painleve property

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

normal form solutions of dynamical systems in the basin of attraction of their fixed points abstract the normal form theory of poincare siegel and arnol d is applied to an analytically solvable lotka volterra system in the plane and a periodically forced dissipative duffing s equation with chaotic orbits in its 3 dimensional phase space for the planar model we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t plane despite such limitations however we show in the case of a periodically driven system that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the poincare map whose boundary is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby

an extension of lomonosov s techniques to non compact operators the existence of invariant subspaces for bounded linear operators acting on an infinite dimensional hilbert space appears to be one of the most difficult questions in the theory of linear transformations the question is known as the invariant subspace problem very few affirmative answers are known regarding this problem one of the most prominent ones is the theorem on the existence of hyper invariant subspaces for compact operators due to v i lomonosov

lazy random walks for superpixel segmentation we present a novel image superpixel segmentation approach using the proposed lazy random walk lrw algorithm in this paper our method begins with initializing the seed positions and runs the lrw algorithm on the input image to obtain the probabilities of each pixel then the boundaries of initial superpixels are obtained according to the probabilities and the commute time the initial superpixels are iteratively optimized by the new energy function which is defined on the commute time and the texture measurement our lrw algorithm with self loops has the merits of segmenting the weak boundaries and complicated texture regions very well by the new global probability maps and the commute time strategy the performance of superpixel is improved by relocating the center positions of superpixels and dividing the large superpixels into small ones with the proposed optimization algorithm the experimental results have demonstrated that our method achieves better performance than previous superpixel approaches

on the integrability of systems of nonlinear ordinary differential equations with superposition principles a new class of solvable nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations ode s describing each member of this class possess nonlinear superposition principles these systems of ode s are generally not derived from a hamiltonian and are classified by associated pairs of lie algebras of vector fields in this paper all such systems of n 3 ode s are integrated in a unified way by finding explicit integrals for them and relating them all to a pivotal member of their class the projective riccati equations moreover by perturbing two parametrically driven projective riccati equations thus making them nonsolvable in the above sense evidence is discovered of chaotic behavior on the poincare surface of section in the form of sensitive dependence on initial conditions near a boundary separating bounded from unbounded motion

explicit integrability for hamiltonian systems and the painleve conjecture we analyze a class of hamiltonian systems in two dimensions for which we proved that a second constant of motion exists it is shown that using the two first integrals the equations of motion can be written in a form which allows their integration by quadratures an analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex time plane it is shown explicitly that the systems belonging to this class possess the weak painleve property i e their solutions in complex time can present singularities of a specific algebraic type

completely indecomposable operators and a uniqueness theorem of cartwright levinson type abstract a bounded linear operator t on a complex hilbert space will be called completely indecomposable if its spectrum is not a singleton and is included in the spectrum of the restrictions of t and t to any of their nonzero invariant subspaces two classes of completely indecomposable operators are constructed the first consists of essentially selfadjoint operators with spectrum 2 2 and the second of bilateral weighted shifts whose spectrum is the unit circle we do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace we also establish a new uniqueness theorem of cartwright levinson type which is the main ingredient in our proofs of complete indecomposability

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

singularity clustering in the duffing oscillator an asymptotic and numerical study is made of the singularity structure in the complex t plane of the duffing oscillator the presence of logarithmic terms in the local psi series expansion of the form t4 ln t leads to a multisheeted singularity structure of great complexity this structure is built recursively from an elemental pattern which takes the form of four armed stars of singularities this construction is deduced analytically from the properties of the mapping z t4 ln t and is confirmed quite accurately numerically a systematic resummation of the psi series in terms of lame functions is developed this series exhibits the same analytic structure at all orders and provides a semi local analytical representation of the solution which is apparently valid even in the chaotic regime

on the application of normal forms near attracting fixed points of dynamical systems abstract it is shown on an integrable example in the plane that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem however as is demonstrated on a nonintegrable example with 3 dimensional phase space the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point whose invariant manifolds embrace the attracting fixed point forming a complicated basin boundary thus in such problems normal forms are shown to be useful in practice as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t

on the singularity analysis of intersecting separatrices in near integrable dynamical systems abstract it has been recently proved by s l ziglin that transversal intersections of separatrices invariant manifolds in near integrable hamiltonian systems of two degrees of freedom imply the existence of multi valued solutions with infinitely many riemann sheets ziglin s theorem is illustrated here on a simple example and then extended and applied to non hamiltonian analytic flows x f x eg x t with x x y and g x t g x t 2 which for e 0 possess the painleve property on the other hand the theoretically expected logarithmic singularities for e 0 are obtained explicitly in solutions near the intersecting separatrices thus we conjecture that dynamical systems with the painleve property can have no separatrix intersections and hence no strange attractors etc these singularities are then numerically located and found to form certain very interesting chimney patterns in the complex t plane on which they accumulate densely the upper part of the chimneys away from the re t axis is asymptotically quite insensitive to changes in parameter values or initial conditions the singularity pattern itself however becomes denser and each chimney is seen to gradually collapse towards the re t axis as the amplitude of the driving term increases and the motion becomes more chaotic

a probabilistic associative model for segmenting weakly supervised images weakly supervised image segmentation is an important yet challenging task in image processing and pattern recognition fields it is defined as in the training stage semantic labels are only at the image level without regard to their specific object scene location within the image given a test image the goal is to predict the semantics of every pixel superpixel in this paper we propose a new weakly supervised image segmentation model focusing on learning the semantic associations between superpixel sets graphlets in this paper in particular we first extract graphlets from each image where a graphlet is a small sized graph measures the potential of multiple spatially neighboring superpixels i e the probability of these superpixels sharing a common semantic label such as the sky or the sea to compare different sized graphlets and to incorporate image level labels a manifold embedding algorithm is designed to transform all graphlets into equal length feature vectors finally we present a hierarchical bayesian network to capture the semantic associations between postembedding graphlets based on which the semantics of each superpixel is inferred accordingly experimental results demonstrate that 1 our approach performs competitively compared with the state of the art approaches on three public data sets and 2 considerable performance enhancement is achieved when using our approach on segmentation based photo cropping and image categorization

an efficient mrf embedded level set method for image segmentation this paper presents a fast and robust level set method for image segmentation to enhance the robustness against noise we embed a markov random field mrf energy function to the conventional level set energy function this mrf energy function builds the correlation of a pixel with its neighbors and encourages them to fall into the same region to obtain a fast implementation of the mrf embedded level set model we explore algebraic multigrid amg and sparse field method sfm to increase the time step and decrease the computation domain respectively both amg and sfm can be conducted in a parallel fashion which facilitates the processing of our method for big image databases by comparing the proposed fast and robust level set method with the standard level set method and its popular variants on noisy synthetic images synthetic aperture radar sar images medical images and natural images we comprehensively demonstrate the new method is robust against various kinds of noises in particular the new level set method can segment an image of size 500 500 within 3 s on matlab r2010b installed in a computer with 3 30 ghz cpu and 4 gb memory

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

application of normal forms to the lorenz model in the subcritical region abstract it is shown that within a certain parameter interval below the hopf bifurcation the lorenz equations can be decoucpled in the normal form representation we normalize only up to monomials of finite order and combine the unnormalized part of the flow in a remainder the formalism avoids small denominators and is well defined if the inverse of a finite polynomial map exists numerical results are presented for the basin of attraction in some neighborhood of the stable fixed points and for the hopf limit cycle

the recognition of the point symbols in the scanned topographic maps it is difficult to separate the point symbols from the scanned topographic maps accurately which brings challenges for the recognition of the point symbols in this paper based on the framework of generalized hough transform ght we propose a new algorithm which is named shear line segment ght sls ght to recognize the point symbols directly in the scanned topographic maps sls ght combines the line segment ght ls ght and the shear transformation on the one hand ls ght is proposed to represent the features of the point symbols more completely its r table has double level indices the first one is the color information of the point symbols and the other is the slope of the line segment connected a pair of the skeleton points on the other hand the shear transformation is introduced to increase the directional features of the point symbols it can make up for the directional limitation of ls ght indirectly in this way the point symbols are detected in a series of the sheared maps by ls ght and the final optimal coordinates of the setpoints are gotten from a series of the recognition results sls ght detects the point symbols directly in the scanned topographic maps totally different from the traditional pattern of extraction before recognition moreover several experiments demonstrate that the proposed method allows improved recognition in complex scenes than the existing methods

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

the discrete shearlet transform a new directional transform and compactly supported shearlet frames it is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying image is essential in many applications including image processing in order to achieve this several directional image representation schemes have been proposed in this paper we develop the discrete shearlet transform dst which provides efficient multiscale directional representation and show that the implementation of the transform is built in the discrete framework based on a multiresolution analysis mra we assess the performance of the dst in image denoising and approximation applications in image approximations our approximation scheme using the dst outperforms the discrete wavelet transform dwt while the computational cost of our scheme is comparable to the dwt also in image denoising the dst compares favorably with other existing transforms in the literature

minimal normal forms in harmonic oscillations with small nonlinear perturbations the freedom of choice of the zeroth order approximation in the perturbative expansion of solutions to oscillatory problems with small nonlinearities is exploited within the framework of the method of normal forms a priori the normal form may contain an infinite number of resonant terms it is shown that in a large class of problems of interest the normal form can be simplified so that the number of such terms is small or to have other desired properties conservative as well as dissipative systems are discussed

asymptotic analysis of time singularities for a class of time dependent hamiltonians we analyse the local and global structure of time singularities for a class of quasi integrable hamiltonian systems in the arnold liouville sense we show that there is good agreement between the numerically observed local behaviour of the solutions and the perturbative scheme we produce using asymptotic approximations of the solution around the singularities we also prove the convergence of the psi series associated to the movable singularities of the systems considered we also propose a simple model in order to analyse the global structure of the singularities in the directions of exponential growth of the potential in time

the existence of translation invariant subspaces of symmetric self adjoint sequence spaces on z we prove that if x is a reflexive translation invariant banach space of complex sequences on z that contains all finitely supported sequences in which the coordinate functionals are continuous and for every sequence c n in the space the sequences c n and c n are also in the space then x has a nontrivial translation invariant subspace this provides in particular a positive solution to the translation invariant subspace problem for weighted lp spaces on z with even weights for 1 p the proof is based on an intermediate result that asserts that if a is an operator on a reflexive real banach space of dimension greater than one and there exist non zero vectors u in the space and v in the dual space such that anu v n 0 is a moment sequence of a finite positive borel measure on a bounded interval on the real line then a has a nontrivial invariant subspace

globally optimal estimation of nonrigid image distortion image alignment in the presence of non rigid distortions is a challenging task typically this involves estimating the parameters of a dense deformation field that warps a distorted image back to its undistorted template generative approaches based on parameter optimization such as lucas kanade can get trapped within local minima on the other hand discriminative approaches like nearest neighbor require a large number of training samples that grows exponentially with respect to the dimension of the parameter space and polynomially with the desired accuracy 1 in this work we develop a novel data driven iterative algorithm that combines the best of both generative and discriminative approaches for this we introduce the notion of a pull back operation that enables us to predict the parameters of the test image using training samples that are not in its neighborhood not close in the parameter space we prove that our algorithm converges to the global optimum using a significantly lower number of training samples that grows only logarithmically with the desired accuracy we analyze the behavior of our algorithm extensively using synthetic data and demonstrate successful results on experiments with complex deformations due to water and clothing