improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

multiple correction and continued fraction approximation abstract the main aim of this paper is to further develop a multiple correction method formulated in a previous work 6 as applications we find a kind of hybrid type finite continued fraction approximations in two cases of landau constants and lebesgue constants in addition we refine the previous results of lu 29 and xu and you 48 for the euler mascheroni constant

the integration of image segmentation maps using region and edge information we present an algorithm that integrates multiple region segmentation maps and edge maps it operates independently of image sources and specific region segmentation or edge detection techniques user specified weights and the arbitrary mixing of region edge maps are allowed the integration algorithm enables multiple edge detection region segmentation modules to work in parallel as front ends the solution procedure consists of three steps a maximum likelihood estimator provides initial solutions to the positions of edge pixels from various inputs an iterative procedure using only local information without edge tracing then minimizes the contour curvature finally regions are merged to guarantee that each region is large and compact the channel resolution width controls the spatial scope of the initial estimation and contour smoothing to facilitate multiscale processing experimental results are demonstrated using data from different types of sensors and processing techniques the results show an improvement over individual inputs and a strong resemblance to human generated segmentation

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

a class of integral approximations for the factorial function we introduce stirling s formula in a more general class of approximation formulas to extend the integral representation of liu z liu a new version of the stirling formula tamsui oxf j math sci 23 4 2007 389 392 finally an accurate approximation for the factorial function is established

classifier design with incomplete knowledge abstract pattern classification is a well studied problem in which the identity of an unknown pattern is determined to be one of m classes spanning the pattern space a new criterion function the inck incomplete knowledge criterion function is proposed which approximates the error probability when the m classes do not span the entire pattern space this criterion is based on the probabilistic measures obtained from a modified version of dubuisson and masson s statistical decision rule with reject the error probability or conversely the probability of correct classification can be determined without having complete knowledge about the class distributions the attractiveness of this criterion is that it is highly correlated with the statistical decision rule with reject which can be used by the classifier thus the criterion provides a good indication of the classifier performance that can be expected with the reduced feature space a classifier design is presented which incorporates the new criterion function as well it is described how the rejection classes of the statistical decision rule with reject can be used in a multiresolution classifier design to determine the class identity of patterns which cannot be determined at a given resolution experimental evidence is provided which demonstrates the advantages of the inck criterion function used in conjunction with a multiresolution classifier design

geodesic active regions and level set methods for supervised texture segmentation this paper presents a novel variational framework to deal with frame partition problems in computer vision this framework exploits boundary and region based segmentation modules under a curve based optimization objective function the task of supervised texture segmentation is considered to demonstrate the potentials of the proposed framework the textured feature space is generated by filtering the given textured images using isotropic and anisotropic filters and analyzing their responses as multi component conditional probability density functions the texture segmentation is obtained by unifying region and boundary based information as an improved geodesic active contour model the defined objective function is minimized using a gradient descent method where a level set approach is used to implement the obtained pde according to this pde the curve propagation towards the final solution is guided by boundary and region based segmentation forces and is constrained by a regularity force the level set implementation is performed using a fast front propagation algorithm where topological changes are naturally handled the performance of our method is demonstrated on a variety of synthetic and real textured frames

spectral concentrations and resonances of a second order block operator matrix and an associated λ rational sturm liouville problem this paper studies the resonances and points of spectral concentration of the block operator matrix egin pmatrix frac d 2 d x 2 q sqrt tw sqrt tw u end pmatrix in the space l 2 0 1 oplus l 2 0 1 in particular we study the dynamics of the resonance eigenvalue t showing that an embedded eigenvalue can evolve into a resonance and that eigenvalues which are absorbed by the essential spectrum give rise to resonance points a connection is also established between resonances and points of spectral concentration finally some numerical examples are given which show that each of the above theoretical possibilities can be realized

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

game theoretic integration for image segmentation robust segmentation of structures from an image is essential for a variety of image analysis problems however the conventional methods of region based segmentation and gradient based boundary finding are often frustrated by poor image quality here we propose a method to integrate the two approaches using game theory in an effort to form a unified approach that is robust to noise and poor initialization this combines the perceptual notions of complete boundary information using edge data and shape priors with gray level homogeneity using two computational modules the novelty of the method is that this is a bidirectional framework whereby both computational modules improve their results through mutual information sharing a number of experiments were performed both on synthetic datasets and datasets of real images to evaluate the new approach and it is shown that the integrated method typically performs better than conventional gradient based boundary finding

mac magnetostatic active contour model we propose an active contour model using an external force field that is based on magnetostatics and hypothesized magnetic interactions between the active contour and object boundaries the major contribution of the method is that the interaction of its forces can greatly improve the active contour in capturing complex geometries and dealing with difficult initializations weak edges and broken boundaries the proposed method is shown to achieve significant improvements when compared against six well known and state of the art shape recovery methods including the geodesic snake the generalized version of gradient vector flow gvf snake the combined geodesic and gvf snake and the charged particle model

continued fraction estimates for the psi function we present continued fraction estimates for the psi function as a consequence we educe inequality for the euler mascheroni constant which improves a result of detemple

image segmentation by unifying region and boundary information a two stage method of image segmentation based on gray level cooccurrence matrices is described an analysis of the distributions within a cooccurrence matrix defines an initial pixel classification into both region and interior or boundary designations local consistency of pixel classification is then implemented by minimizing the entropy of local information where region information is expressed via conditional probabilities estimated from the cooccurrence matrices and boundary information via conditional probabilities which are determined a priori the method robustly segments an image into homogeneous areas and generates an edge map the technique extends easily to general edge operators an example is given for the canny operator applications to synthetic and forward looking infrared flir images are given

on the location of resonances of exponentially decaying sturm liouville potentials the paper is concerned with analytic continuation into the unphysical sheet for sturm liouville potentials whose decay at infinity is no faster than o e ax a resonance free region is found and the existence of a natural boundary is discussed an example related to the bessel equation is examined in some detail

efficient calculation of spectral density functions for specific classes of singular sturm liouville problems new families of approximations to sturm liouville spectral density functions are derived for cases where the potential function has one of several specific forms this particular form dictates the type of expansion functions used in the approximation error bounds for the residuals are established for each case in the case of power potentials the approximate solutions of an associated terminal value problem at are shown to be asymptotic power series expansions of the exact solution numerical algorithms have been implemented and several examples are given demonstrating the utility of the approach

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

limits and inequalities associated with the euler mascheroni constant i we present several limits associated with the euler mascheroni constant ii let c 0 577215 be the euler mascheroni constant and let t n k 1 n1k lnn 12 124n and p n k 1 n22k 1 ln 4n we determine the best possible constants a b a and b such that the inequalities148 n a 3 c t n 148 n b 3and124 n a 2 1

numerical results on the transcendence of constants involving pi e and euler s constant the existence of simple polynomial equations integer relations for the constants e pi e pi log pi gamma euler s constant e exp gamma gamma e gamma pi and log gamma is investigated by means of numerical computations the recursive form of the ferguson fourcade algorithm ferguson and fourcade 1979 ferguson 1986 and 1987 is implemented on the cray 2 supercomputer at nasa ames applying multiprecision techniques similar to those described by bailey 1988 except that ffts are used instead of dual prime modulus transforms for multiplication it is shown that none of the constants has an integer relation of degree eight or less with coefficients of euclidean norm 10 to the 9th or less

relaxation evaluation and applications probabilistic relaxation labeling processes are iterative parallel schemes that use contextual information to reduce local ambiguities the behavior of these processes can be described by examining the rates of change and entropies of the probability vectors at each iteration examples are given comparing three relaxation processes as applied to several basic image analysis tasks

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

new sequence converging towards the euler mascheroni constant we propose new sequences containing a modified logarithmic term which converge to the euler mascheroni constant faster than sequences known from the literature

best estimates of the generalized stirling formula the aim of this paper is to introduce an approximations family of the factorial function that contains stirling s formula burnside s formula and gosper s formula the parameters which provide the best approximations are indicated finally numerical computations are made to show the superiority of our formulas over other known formulas

optimal rate of convergence for sequences of a prescribed form abstract we provide a highly convergent version of the generalized euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form the new approach extends and unifies many previous efforts in this direction in addition we answer an open problem on the asymptotic expansion of the harmonic number sequence

some properties of the gamma and psi functions with applications in this paper some monotoneity and concavity properties of the gamma beta and psi functions are obtained from which several asymptotically sharp inequalities follow applying these properties the authors improve some well known results for the volume n of the unit ball b n r n the surface area n 1 of the unit sphere s n 1 and some related constants

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

bscan image segmentation by thresholding using cooccurrence matrix analysis abstract in this paper we present innovative research for the processing of an uktrasonic image obtained in non destructive testing ndt we also present an algorithm for automatic thresholding segmentation the image analysis is based on the calculation of cooccurrence matrices the method we used is the spatial grey level dependence method sgldm it shows the distribution of the intensities of pairs of pixels separated by a vector d we show that it is necessary to build optimized matrices in order to improve the existing techniques while adapting to the specifics of the ultrasonic image obtained in ndt we solve for the vector d through an automatic analysis of the matrix behaviour for several vectors based on the characterization curves the developed method allows us to present an efficient thresholding algorithm that is tested on real images

bounds on scattering poles in one dimension for the class of super exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated in the case of nonnegative potentials explicit estimates for the width of a pole free strip are obtained

analytic continuation and resonance free regions for sturm liouville potentials with power decay the analytic continuation of the jost solution into a sector of the unphysical sheet is established for analytic potentials with power decay the method also yields resonance free regions which in the case of super exponential decay are confirmed to be the best possible of their kind by the known asymptotic distribution of resonances

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

active contouring based on gradient vector interaction and constrained level set diffusion this paper presents an extension of our recently introduced mac model to deal with the initialization dependency problem that commonly appears in edge based approaches its dynamic force field unique bidirectionality and constrained diffusion based level set evolution provide great freedom in contour initialization and show significant improvements in initialization independency compared to other edge based techniques it can handle more sophisticated topological changes than splitting and merging it provides new potentials for edge based active contour methods particularly when detecting and localizing objects with unknown location geometry and topology

the hurwitz theorem for bessel functions and antibound states in spectral theory an 1889 result of hurwitz is that the bessel function j v c v 0 c 0 has no zeros for 2 n v n 1 with integer n a new application of this result is given which describes the behaviour of jost functions and antibound states for one dimensional schrodinger equations with exponentially decaying potential a new hurwitz type theorem with similar applications is developed for hypergeometric functions extensions are obtained where the power series are terminated after 2 n 1 terms and larger zero free intervals 2 n v are found

a solution to an open problem on the euler mascheroni constant the classical le verrier algorithm for the calculation of the characteristic polynomial of a matrix has led us to a solution of an open problem concerning the euler mascheroni constant c more exactly we prove that there exists uniquely a sequence of rational numbers n n 1 such that for any integer q 1 the sequence k 1n1k logn log1 1n qn q is an n q 1 order approximation of c

estimating the digamma and trigamma functions by completely monotonicity arguments abstract we establish new sharp bounds for the digamma and trigamma functions using the complete monotonicity of some class of functions

improved asymptotic formulas for the gamma function the aim of this paper is to improve the ramanujan formula for approximation the gamma function a fast asymptotic series is constructed

resonances and analytic continuation for exponentially decaying sturm liouville potentials abstract the sturm liouville equation on the half line 0 is considered with a potential which decays exponentially at infinity it is known that green s function has an analytic continuation into part of the unphysical sheet im here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay the methods are constructive and they also provide a new and rigorous way of locating resonances