new exact solutions for polynomial oscillators in large dimensions a new type of exact solvability is reported the schrodinger equation is considered in a very large spatial dimension d 1 and its central polynomial potential is allowed to depend on many 2q coupling constants in a search for its bound states possessing an exact and elementary wavefunction proportional to a harmonic oscillator like polynomial of a freely varying i e not just small degree n the solvability conditions are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of d q and n assuming that d is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all n and up to q 5 at least

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

minimax entropy principle and its application to texture modeling this article proposes a general theory and methodology called the minimax entropy principle for building statistical models for images or signals in a variety of applications this principle consists of two parts the first is the maximum entropy principle for feature binding or fusion for a given set of observed feature statistics a distribution can be built to bind these feature statistics together by maximizing the entropy over all distributions that reproduce them the second part is the minimum entropy principle for feature selection among all plausible sets of feature statistics we choose the set whose maximum entropy distribution has the minimum entropy computational and inferential issues in both parts are addressed in particular a feature pursuit procedure is proposed for approximately selecting the optimal set of features the minimax entropy principle is then corrected by considering the sample variation in the observed feature statistics and an information criterion for feature pursuit is derived the minimax entropy principle is applied to texture modeling where a novel markov random field mrf model called frame filter random field and minimax entropy is derived and encouraging results are obtained in experiments on a variety of texture images the relationship between our theory and the mechanisms of neural computation is also discussed

scaling generator coordinate method and the eigensolutions of the schrodinger equation an extremely simple approximate variational method for the study of the eigensolutions of the one dimensional schrodinger equation is introduced based upon the idea of employing scaled eigenstates of the harmonic oscillator as generating functions for the generator coordinate method the approach is successfully applied to a variety of situations

frames a maximum entropy statistical estimate of the inverse problem a maximum entropy statistical treatment of an inverse problem concerning frame theory is presented the problem arises from the fact that a frame is an overcomplete set of vectors that defines a mapping with no unique inverse although any vector in the concomitant space can be expressed as a linear combination of frame elements the coefficients of the expansion are not unique frame theory guarantees the existence of a set of coefficients which is optimal in a minimum norm sense we show here that these coefficients are also optimal from a maximum entropy viewpoint

the anharmonic oscillator problem a new algebraic solution the second order one dimensional or radial differential schrodinger equation with the potential v r mu r2 nu r4 may be re interpreted as a difference equation of the fourth order indeed the hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n n 0 1 thus the authors construct its four independent general solutions by purely algebraic means via expansions in powers of n 1 14 next preserving the analogy between difference and differential equations the physical wavefunctions n mod psi and their energies are determined by the matching of the jost and regular solutions finally using the simplest matching conditions at the origin the resulting numerical algorithm is demonstrated to be both stable and quickly convergent

a class of exact solutions for anharmonically coupled oscillators the author studies the solutions of the two dimensional schrodinger equation for anharmonically coupled oscillators the author obtains a finite set of exact solutions of the form psi x y polynomial x y exp polynomial x y provided that certain constraints on the potential parameters are satisfied

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

extended continued fractions and energies of the anharmonic oscillators we describe the analytic solution to the schrodinger eigenvalue problem for the class of the central potentials v r za r where a 2 1 4 amax 0 z is an arbitrary finite set of the integer or rational exponents 2 1 0 the formalism is based on an analysis of the asymptotic behavior of the explicit regular solution r and issues in the formulation of the secular equation 1 l1 e 0 which determines the binding energies the main result is the rigorous construction of l1 e as a generalized extended and convergent continued fraction the method cannot be applied to the a i 1 0 cases this disproves the closely related hill determinant approach as conjectured recently by singh et al for the simplest potentials with z 2 2 4 6 and z 2 1 1 2

analytic treatment of the αx2 βx4 oscillator we carry out an investigation of the schrodinger equation with the x2 x4 interaction for 0 re x and exp i in the sectors 0 arg x 3 6 0 and 6 arg x 3 6 0 a subdominant solution is constructed essentially in terms of the laplace transform of a function f s 0 s which is expressible as a converging power series in s the solution y x thus obtained is compared and contrasted with the one valid in the usual case where re x 0 is included and it is argued that y x may not be accessible to perturbative approaches

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

embedding gestalt laws in markov random fields the goal of this paper is to study a mathematical framework of 2d object shape modeling and learning for middle level vision problems such as image segmentation and perceptual organization for this purpose we pursue generic shape models which characterize the most common features of 2d object shapes in this paper shape models are learned from observed natural shapes based on a minimax entropy learning theory the learned shape models are gibbs distributions defined on markov random fields mrfs the neighborhood structures of these mrfs correspond to gestalt laws colinearity cocircularity proximity parallelism and symmetry thus both contour based and region based features are accounted for stochastic markov chain monte carlo mcmc algorithms are proposed for learning and model verification furthermore this paper provides a quantitative measure for the so called nonaccidental statistics and thus justifies some empirical observations of gestalt psychology by information theory our experiments also demonstrate that global shape properties can arise from interactions of local features

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

perceptual scale space and its applications when an image is viewed at varying resolutions it is known to create discrete perceptual jumps or transitions amid the continuous intensity changes in this paper we study a perceptual scale space theory which differs from the traditional image scale space theory in two aspects i in representation the perceptual scale space adopts a full generative model from a gaussian pyramid it computes a sketch pyramid where each layer is a primal sketch representation guo et al in comput vis image underst 106 1 5 19 2007 an attribute graph whose elements are image primitives for the image structures each primal sketch graph generates the image in the gaussian pyramid and the changes between the primal sketch graphs in adjacent layers are represented by a set of basic and composite graph operators to account for the perceptual transitions ii in computation the sketch pyramid and graph operators are inferred as hidden variables from the images through bayesian inference by stochastic algorithm in contrast to the deterministic transforms or feature extraction such as computing zero crossings extremal points and inflection points in the image scale space studying the perceptual transitions under the bayesian framework makes it convenient to use the statistical modeling and learning tools for a modeling the gestalt properties of the sketch graph such as continuity and parallelism etc b learning the most frequent graph operators i e perceptual transitions in image scaling and c learning the prior probabilities of the graph operators conditioning on their local neighboring sketch graph structures in experiments we learn the parameters and decision thresholds through human experiments and we show that the sketch pyramid is a more parsimonious representation than a multi resolution gaussian wavelet pyramid we also demonstrate an application on adaptive image display showing a large image in a small screen say pda through a selective tour of its image pyramid in this application the sketch pyramid provides a means for calculating information gain in zooming in different areas of an image by counting a number of operators expanding the primal sketches such that the maximum information is displayed in a given number of frames

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

modeling visual patterns by integrating descriptive and generative methods this paper presents a class of statistical models that integrate two statistical modeling paradigms in the literature i descriptive methods such as markov random fields and minimax entropy learning zhu s c wu y n and mumford d 1997 neural computation 9 8 and ii generative methods such as principal component analysis independent component analysis bell a j and sejnowski t j 1997 vision research 37 3327 3338 transformed component analysis frey b and jojic n 1999 iccv wavelet coding mallat s and zhang z 1993 ieee trans on signal processing 41 3397 3415 chen s donoho d and saunders m a 1999 journal on scientific computing 20 1 33 61 and sparse coding olshausen b a and field d j 1996 nature 381 607 609 lewicki m s and olshausen b a 1999 josa a 16 7 1587 1601 in this paper we demonstrate the integrated framework by constructing a class of hierarchical models for texton patterns the term texton was coined by psychologist julesz in the early 80s at the bottom level of the model we assume that an observed texture image is generated by multiple hidden texton maps and textons on each map are translated scaled stretched and oriented versions of a window function like mini templates or wavelet bases the texton maps generate the observed image by occlusion or linear superposition this bottom level of the model is generative in nature at the top level of the model the spatial arrangements of the textons in the texton maps are characterized by minimax entropy principle which leads to embellished versions of gibbs point process models stoyan d kendall w s and mecke j 1985 stochastic geometry and its applications the top level of the model is descriptive in nature we demonstrate the integrated model by a set of experiments

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

the generalized continued fractions and potentials of the lennard jones type for a broad class of the strongly singular potentials v r which are defined as superpositions of separate power law components the general solution of the corresponding schrodinger differential equation is constructed as an analog of mathieu functions the analogy is supported by the use of the generalized continued fractions the questions of convergence are analyzed in detail

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

intrackability characterizing video statistics and pursuing video representations videos of natural environments contain a wide variety of motion patterns of varying complexities which are represented by many different models in the vision literature in many situations a tracking algorithm is formulated as maximizing a posterior probability in this paper we propose to measure the video complexity by the entropy of the posterior probability called the intrackability to characterize the video statistics and pursue optimal video representations based on the definition of intrackability our study is aimed at three objectives firstly we characterize video clips of natural scenes by intrackability we calculate the intrackabilities of image points to measure the local inferential uncertainty and collect the histogram of the intrackabilities over the video in space and time as the global video statistics we find that a pca scatter plot based on the first two principle components of intrackability histograms can reflect the major variations i e image scaling and object density in natural video clips secondly we show that different video representations including deformable contours tracking kernels with various appearance features dense motion fields and dynamic texture models are connected by the change of intrackability and thus develop a simple criterion for model transition and for pursuing the optimal video representation thirdly we derive the connections between the intrackability measure and other criteria in the literature such as the shi tomasi texturedness measure conditional number and harris stephens r score and compare with the shi tomasi measure in tracking experiments

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases

on the inverse windowed fourier transform the inversion problem concerning the windowed fourier transform is considered it is shown that out of the infinite solutions that the problem admits the windowed fourier transform is the optimal solution according to a maximum entropy selection criterion