halanay type inequalities on time scales with applications a b s t r a c t this paper aims to introduce halanay type inequalities on time scales by means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales giving several examples we show that besides generalization and extension to q difference case our results also provide improvements for the existing theory regarding differential and difference inequalities which are the most important particular cases of dynamic inequalities on time scales

a luminance and contrast invariant edge similarity measure a novel similarity measure for edge detection that is robust to varying luminance and contrast is presented it incorporates a regularization term and employs directional fir edge filters with hyperbolic tangent profiles to ensure improved noise performance and edge localization compared to classical methods

periodic solutions for a neutral nonlinear dynamical equation on a time scale abstract let t be a periodic time scale we use a fixed point theorem due to krasnosel ski to show that the nonlinear neutral dynamic system with delay x t a t x t c t x t k q t x t x t k t t has a periodic solution we assume that k is a fixed constant if t r and is a multiple of the period of t if t r under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

floquet theory for integral and integro differential equations we study the extension of hill s method of infinite determinants to the case of integro differential equations with periodic coefficients and kernels we develop the analytical theory of such methods and we obtain certain qualitative properties of the equations that determine the boundaries between regions of dynamic stability and dynamic instability

generalization of the floquet theory abstract this paper introduces the concept of topological equivalence for the periodic systems necessary and sufficient conditions of topological equivalence are derived furthermore we generalize the floquet theory for a linear system with a periodic coefficient

second order q difference equations solvable by factorization method by solving an infinite nonlinear system of q difference equations one constructs a chain of q difference operators the eigenproblems for the chain are solved and some applications including the one related to q hahn orthogonal polynomials are discussed it is shown that in the limit q 1 the present method corresponds to the one developed by infeld and hull

the bivariate rogers szego polynomials we present an operator approach to deriving mehler s formula and the rogers formula for the bivariate rogers szego polynomials hn x y q the proof of mehler s formula can be considered as a new approach to the nonsymmetric poisson kernel formula for the continuous big q hermite polynomials hn x a q due to askey rahman and suslov mehler s formula for hn x y q involves a 32 sum and the rogers formula involves a 21 sum the proofs of these results are based on parameter augmentation with respect to the q exponential operator and the homogeneous q shift operator in two variables by extending recent results on the rogers szego polynomials hn x q due to hou lascoux and mu we obtain another rogers type formula for hn x y q finally we give a change of base formula for hn x a q which can be used to evaluate some integrals by using the askey wilson integral

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

q differential operator identities and applications abstract in this paper we construct a new q exponential operator and obtain some operator identities using these operator identities we give a formal extension of jackson s 2 2 transformation formula a formal extension of bailey s 3 3 summation and an extension of the sears terminating balanced 3 4 transformation formula are also derived by our operator method in addition we also derive several interesting a formal extensions involving multiple sum about three terms of sears 2 3 transformation formula and heine s 1 2 transformation formula

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

piecewise approximation of contours through scale space selection of dominant points this paper describes a method of approximating a shape contour with a polygon the polygon vertices are extracted from the curvature extremes through a scale space description of the contour via linear diffusion these vertices are located on the contour points where the sharper changes of the contour directions occur using a proper strategy a set of extremes that result in a given approximation level is chosen by adding new vertices the approximation level can be improved and a scalable representation of the contour is identified this method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour this polygonal approximation method is used for scalable encoding of the shape contours in this regard an encoding technique suitable for scalable polygonal approximation has been developed we show that encoding the approximated polygons result in a good relation between the distortion and the bitrate finally we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

optimal edge detection in two dimensional images this paper presents a new edge detection scheme that detects two dimensional 2 d edges by a curve segment based detection functional guided by the zero crossing contours of the laplacian of gaussian log to approach the true edge locations the detection functional is shown to be optimal in terms of signal to noise ratio snr and edge localization accuracy it also preserves the nice scaling property held uniquely by the log in scale space

on detecting dominant points abstract detecting dominant points is a crucial preprocessing step for shape recognition and point based motion estimation polygonal approximation has been a commonly used approach in detecting dominant points this paper presents two alternatives which detect stable dominant points in the first method we find a set of positive maximum and negative minimum curvature points along the gaussian smoothed boundary followed by a split and merge polygonal approximation algorithm the resulting break points vertices of the approximated polygon are the dominant points experimental results show that dominant points obtained by this method are less sensitive to the orientation of the boundary than other polygonal approximation algorithms in the sense that the number and the location of the dominant points along the contour remain relatively unchanged in the second method we smooth a boundary by a gaussian filter using various widths until the extreme curvature points remain relatively unchanged for a range of filter widths the resulting extreme curvature points which are stable to orientation and a reasonable range of scaling are the dominant points

an adaptable threshold detector a data set often comprises some data classes for example a gray scale image may consist of some objects each of which has similar pixels gray scales the threshold obtained by otsu s thresholding method otm is biased towards certain data class with larger variance or larger number of data when the variances or the numbers of data among classes are quite different in this paper adaptable threshold detector atd is proposed to improve the effectiveness of otm in determining proper thresholds by dividing class variance by class interval atd is more versatile at selecting application dependent thresholds by changing two parameter values which describe the relative importance among data size standard deviation and class interval of a class in this paper atd is applied to crop the expected objects from images to verify its effect upon thresholding experimental results demonstrate that atd is able to perform better than otm in segmenting objects from images besides excelling over the valley emphasis method vem and the minimum class variance thresholding method mcvtm atd is also suitable for separating objects from serialized video images i e computerized tomography

generalizations of certain carlitz s trilinear and srivastava agarwal type generating functions abstract in this paper a class of trilinear generating functions for generalized hahn polynomials is generalized in addition a question of carlitz is answered and some known results are deduced as special cases moreover some expansions of srivastava agarwal type generating functions for hahn polynomials are gained finally additional proofs of certain carlitz s generating functions are given

an operator approach to the al salam carlitz polynomials we present an operator approach to rogers type formulas and mehler s formula for the al salam carlitz polynomials un x y a q by using the q exponential operator we obtain a rogers type formula which leads to a linearization formula with the aid of a bivariate augmentation operator we get a simple derivation of mehler s formula due to al salam and carlitz some orthogonal q polynomials math nachr 30 47 1965 by means of the cauchy companion augmentation operator we obtain an equivalent form of mehler s formula we also give several identities on the generating functions for products of the al salam carlitz polynomials which are extensions of the formulas for the rogers szego polynomials

notes on askey roy integral and certain generating functions for q polynomials abstract in this paper generalizations of askey roy integral are given by the technique of transformation in addition a recurring formula for askey roy integral and an integral involving cauchy polynomial are deduced by askey roy integral besides a new transformation identity is gained by askey roy integral moreover a certain generating function for generalized al salam carlitz polynomials is obtained by askey roy integral at last two generalizations of u n 1 type generating function for generalized al salam carlitz polynomials are derived by askey roy integral

two operator identities and their applications to terminating basic hypergeometric series and q integrals abstract in this paper we first give two interesting operator identities and then using them and the q exponential operator technique to some terminating summation formulas of basic hypergeometric series and q integrals we obtain some q series identities and q integrals involving 3 2

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

floquet theory based on new periodicity concept for hybrid systems involving q difference equations using the new periodicity concept based on shifts we construct a unified floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous discrete or hybrid structure new periodicity concept based on shifts enables the construction of floquet theory on hybrid domains that are not necessarily additive periodic this makes periodicity and stability analysis of hybrid periodic systems possible on non additive domains in particular this new approach can be useful to know more about floquet theory for linear q difference systems defined on q z q n n z 0 where q 1 by constructing the solution of matrix exponential equation we establish a canonical floquet decomposition theorem determining the relation between floquet multipliers and floquet exponents we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts finally we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts

edge detection by maximum entropy application to omnidirectional and perspective images in the edge detection the classical operators based on the derivation are sensitive to noise which causes detection errors it is even more erroneous in the case of omnidirectional images due to geometric distortions caused by the used sensors this paper proposes a statistical method of edge detection invariant to image resolution applied to omnidirectional images without preliminary treatments it is based on the entropy measure the authors compared its behavior with existing methods on omnidirectional images and perspectives images the criteria of comparisons are the parameters of fram and deutsch for omnidirectional images the authors used two types of neighborhood fixed and adapted to the parameters of the sensor the authors compared the results of detection visually the tests are performed on grayscale images

on carlitz s trilinear generating functions abstract in this paper carlitz s trilinear generating functions for rogers szego and hahn polynomials are obtained by using carlitz s q operator in addition an equivalent proof of srivastava jain s mixed trilinear generating function is gained moreover a carlitz s trilinear generating function is deduced at last an application to the multiple carlitz s generating functions for hahn polynomials is given

some q generating functions of the carlitz and srivastava agarwal types associated with the generalized hahn polynomials and the generalized rogers szego polynomials motivated essentially by the fact that generating functions play an important role in the investigation of many potentially useful properties of the sequences which they generate various families of q generating functions of the carlitz and srivastava agarwal types have been investigated in the literature for the continuous q hermite polynomials the hahn polynomials the rogers szego polynomials and indeed also for many other q polynomial systems the main object of this paper is to apply a certain q exponential decomposition technique with a view to deriving several bilinear and trilinear q generating functions associated with the generalized hahn polynomials as a simple consequence of one of the main results derived in this paper we are led naturally to the corrected version of a bilinear q generating function which appeared in one of carlitz s papers

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

on the localization performance measure and optimal edge detection the localization performance measure of edge detectors is addressed a one dimensional formulation of the problem is considered a linear space invariant filter is used for the detection the locations of local maxima in the thresholded output of the filter are declared to be the edges the limitations of conventional performance measures are shown and a localization performance measure for edge detection is suggested this performance measure is based on the theory of zero crossings of stochastic processes it is shown that the derivative of a gaussian is the optimal edge detector for the measure

new proofs of generating functions for rogers szego polynomials abstract in this paper we show how to use the q exponential operator to prove four generating functions for rogers szego polynomials of l carlitz generating functions for certain q orthogonal polynomials collectanea math 23 1972 91 104 in addition four auxiliary ones under the replacement q q 1 are derived by this method