boson realization of sp 4 r ii the generating kernel formulation in a previous paper of this series the matrix elements were discussed with respect to boson states of an operator k2 required for the boson realization of the sp 4 r lie algebra in the present paper it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp 4 r coherent states the results have relevance for the determination of the matrix elements of the generators of the sp 4 r lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group and are in principle generalizable to symplectic algebras of higher dimensions

differential morphology and image processing image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain we provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems this includes ideas on using partial differential or difference equations pdes to model distance propagation or nonlinear multiscale processes in images we briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics we also review some nonlinear pdes that model the evolution of multiscale morphological operators and use morphological derivatives among the new ideas presented we develop some general 2 d max min sum difference equations that model the space dynamics of 2 d morphological systems including the distance computations and some nonlinear signal transforms called slope transforms that can analyze these systems in a transform domain in ways conceptually similar to the application of fourier transforms to linear systems thus distance transforms are shown to be bandpass slope filters we view the analysis of the multiscale morphological pdes and of the eikonal pde solved via weighted distance transforms as a unified area in nonlinear image processing which we call differential morphology and briefly discuss its potential applications to image processing and computer vision

symplectic groups symplectic spreads codes and unimodular lattices abstract it is known that the symplectic group sp 2 n p has two complex conjugate irreducible representations of degree p n 1 2 realized over q p provided that p 3 mod 4 in the paper we give an explicit construction of an odd unimodular sp 2 n p 2 invariant lattice p n in dimension p n 1 for any p n 3 mod 4 such a lattice has been constructed by r bacher and b b venkov in the case p n 27 a second main result says that these lattices are essentially unique we show that for n 3 the minimum of p n is at least p 1 2 and at most p n 1 2 the interrelation between these lattices symplectic spreads of f p 2 n and self dual codes over f p is also investigated in particular using new results of u dempwolff and l bader w m kantor and g lunardon we come to three extremal self dual ternary codes of length 28

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

quasiprimitive linear groups with quadratic elements abstract a semisimple complex linear transformation with exactly two distinct eigenvalues is called a quadratic element in this paper the finite irreducible linear groups generated by quadratic elements of order 3 for which the multiplicities of the eigenvalues are distinct are determined this is used to obtain bounds on the degree of an irreducible primitive linear group containing such elements there are no irreducible primitive linear groups containing quadratic elements of order 4 with eigenvalues 1 i with distinct multiplicities where i is a primitive fourth root of 1

the remarkable properties of the associated romanovski functions for and with n m q 0 a non negative integer we introduce a differential equation for the associated romanovski functions r q n m x so that its form remains invariant under the transformation n 2q n 1 introducing the rodrigues formula for both regions and we obtain simultaneously and separately realization of the laddering equations with respect to the labels n and m by means of two pairs of ladder operators also the square integrability and orthogonality of the functions are considered in cases 0 and 0 with respect to the weight functions and 1 x2 1 q respectively exponential generating functions for the infinite sequences with the same n in both regions and as well as for the infinite ones with given even and odd values of n m q are separately calculated as an application to supersymmetric quantum mechanics we show that there exist four different methods for generating the spectrum of the noncentral hyperbolic scarf potential

reconstructions from zero crossings in scale space in computer vision the one parameter family of images obtained from the laplacian of a gaussian filtered version of the image parameterized by the width of the gaussian has proved to be a useful data structure for the extraction of feature data in particular the zero crossings of this so called scale space data are associated with edges and have been proposed by d marr 1982 and others as the basis of a representation of the image data the question arises as to whether the representation is complete and stable the authors survey some of the studies and results related to these questions as well as several studies that attempt reconstructions based on this or related representations they formulate a novel method for reconstruction from zero crossings in scale space that is based on minimizing equation error and they present results showing that the reconstruction is possible but can be unstable they further show that the method applies when gradient data along the zero crossings are included in the representation and they demonstrate empirically that the reconstruction is then stable

sl 2 modules by sl 2 coherent states irreducible sp 4 module with highest weight labeled by the azimuthal and magnetic quantum numbers l and m is split into the direct sums of the irreducible su 2 and su 1 1 submodules in four different ways finite integer unitary irreducible subspaces corresponding to the orbital angular momentum algebra su 2 infinite positive discrete series of su 1 1 with an arbitrary half integer bargmann index and the positive and negative discrete series of su 1 1 with both the bargmann indices 1 4 and 3 4 even and odd coherent states for the positive su 1 1 submodules with the bargmann indices 1 4 and 3 4 are constructed and it is shown that they enjoy the property of completeness by two appropriate positive definite measures we show that the even and odd coherent states themselves form the positive discrete series of su 1 1 with the bargmann indices 1 4 and 3 4 respectively for these even and odd coherent states we consider the uncertainty relations for the x and y components of the angular m

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

extending romanovski polynomials in quantum mechanics some extensions of the third class romanovski polynomials also called romanovski pseudo jacobi polynomials which appear in bound state wavefunctions of rationally extended scarf ii and rosen morse i potentials are considered for the former potentials the generalized polynomials satisfy a finite orthogonality relation while for the latter an infinite set of relations among polynomials with degree dependent parameters is obtained both types of relations are counterparts of those known for conventional polynomials in the absence of any direct information on the zeros of the romanovski polynomials present in denominators the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second order differential equations of schrodinger type it is also shown that on going from scarf i to scarf ii or from rosen morse ii to rosen morse i potentials the variety of rational extensions is narrowed down from types i ii and iii to type iii only

evolution equations for continuous scale morphological filtering multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges the authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous space multiscale morphological erosions dilations openings and closings these equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals the nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators generally taking the form of a nonlinear function of certain partial differential operators

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

spatially variant morphological restoration and skeleton representation the theory of spatially variant sv mathematical morphology is used to extend and analyze two important image processing applications morphological image restoration and skeleton representation of binary images for morphological image restoration we propose the sv alternating sequential filters and sv median filters we establish the relation of sv median filters to the basic sv morphological operators i e sv erosions and sv dilations for skeleton representation we present a general framework for the sv morphological skeleton representation of binary images we study the properties of the sv morphological skeleton representation and derive conditions for its invertibility we also develop an algorithm for the implementation of the sv morphological skeleton representation of binary images the latter algorithm is based on the optimal construction of the sv structuring element mapping designed to minimize the cardinality of the sv morphological skeleton representation experimental results show the dramatic improvement in the performance of the sv morphological restoration and sv morphological skeleton representation algorithms in comparison to their translation invariant counterparts

asymptotically optimal algorithms for running max and min filters on random inputs given a d dimensional array of size n d and an integer p the running max or min filter is the set of maximum or minimum elements within a d dimensional sliding window of edge length p inside the array this problem is useful in many signal processing applications such as pattern analysis adaptive signal processing and morphological analysis the current best algorithm for computing the one dimensional 1 d max or min filter due to the work of h yuan and m j atallah running max min filters using 1 o 1 comparisons per sample ieee trans pattern anal mach intell vol 33 no 12 pp 2544 2548 dec 2011 uses 1 o 1 comparisons per sample in the worst case as a direct consequence the d dimensional max or min filter max and min filters respectively can be computed in d o 1 2d o 1 respectively comparisons per sample in this paper we first present an algorithm for computing d dimensional max and min filters simultaneously on i i d inputs that uses 1 5 o 1 expected comparisons per sample this is the first algorithm on i i d inputs that gets rid of the dependence on d in the dominating term with respect to n and p of the expected number of comparisons needed it is also asymptotically optimal when d is a fixed constant as n rightarrow infty and p rightarrow infty we also consider the dynamic version of the problem of d dimensional max and min filters simultaneously on i i d inputs where we want to maintain the filters after changes in the input array we design a linear sized data structure that stores precomputed information for efficient update using o p d 1 log 2 p expected comparisons per update

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

nonliner inversion of seismic reflection data in a laterally invariant medium interpretation of seismic waveforms can be expressed as an optimization problem based on a non linear least squares criterion to find the model which best explains the data an initial model is corrected iteratively using a gradient method conjugate gradient at each iteration computation of the direction of the model perturbation requires the forward propagation of the actual sources and the reverse time propagation of the residuals misfit between the data and the synthetics the two wave fields thus obtained are then correlated an extra forward propagation is required to compute the amplitude of the perturbation along the conjugate gradient direction the number of propagations to be simulated numerically in each iteration equals three times the number of shots since a 2 d finite difference code is employed to solve forward and backward propagation problems the method is general and can handle arbitrary 2 d source receiver configurations and lateral heterogeneities using conventional velocity

time domain multiscale full waveform inversion using the rapid expansion method and efficient step length estimation full waveform inversion fwi is rapidly becoming a standard tool for high resolution velocity estimation however the application of this method is usually limited to low frequencies due to the high computational cost of the wavefield propagation and the inversion scheme to mitigate this problem we have developed a rapid expansion method rem for numerical wavefield extrapolation inside the fwi workflow this method approximated the partial time derivative of the wave equation using chebyshev polynomials instead of the conventional finite difference fd approximation this allowed the rem to accurately propagate wavefields with bigger time steps thus improving the computational efficiency of fwi we have compared high frequency fwi results using rem and the traditional fd approximation of the time derivative to illustrate the ability of rem to remain stable and free of numerical dispersion noise even with large grid and time samplings in addition we studied a step length estimation scheme with the objective of avoiding an extra forward wavefield propagation during the line search at each fwi iteration in this scheme we estimated the step length value based on prior iterations and validated this value using the wolfe conditions if the value was accepted then the forward modeling was availed in the next iteration and no extra propagation was necessary we tried seven different step length estimation methods from which we highlighted the adaptive barzilai borwein method when combined with the steepest descent inversion scheme and the unitary step length for the l bfgs algorithm through synthetic numerical results we showed how this scheme could achieve convergence while keeping the number of extra forward modelings way below the number of fwi iterations

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

object based image analysis using multiscale connectivity this paper introduces a novel approach for image analysis based on the notion of multiscale connectivity we use the proposed approach to design several novel tools for object based image representation and analysis which exploit the connectivity structure of images in a multiscale fashion more specifically we propose a nonlinear pyramidal image representation scheme which decomposes an image at different scales by means of multiscale grain filters these filters gradually remove connected components from an image that fail to satisfy a given criterion we also use the concept of multiscale connectivity to design a hierarchical data partitioning tool we employ this tool to construct another image representation scheme based on the concept of component trees which organizes partitions of an image in a hierarchical multiscale fashion in addition we propose a geometrically oriented hierarchical clustering algorithm which generalizes the classical single linkage algorithm finally we propose two object based multiscale image summaries reminiscent of the well known morphological pattern spectrum which can be useful in image analysis and image understanding applications

a reducible weil representation of sp 4 realized by differential operators in the space of smooth functions on h2 s1 this work presents a novel way to obtain the associated romanovski functions rn m x with n m in the three separate regions in terms of n and m we obtain the raising and lowering relations with respect to the both indices simultaneously in the three regions then a reducible weil representation of the real lie algebra sp 4 is realized in the space of complex valued smooth functions on h2 s1 by differential forms for the cartan weyl basis its invariant subspace is the second rare instance of the highest weight irreducible representation of sp 4 all whose weight spaces are one dimensional

some characterizations of the weil representations of the symplectic and unitary groups abstract let g be a finite symplectic or unitary group we characterize the weil representations of g via their restriction to a standard subgroup then we complete the determination of complex representations of g with specific minimal polynomials of certain elements by showing that they coincide with the weil representations

branching rules for restriction of the weil representations of sp n r to its maximal parabolic subgroup cm n the symplectic group sp n r is the group of linear canonical transformations of a real 2n dimensional phase space and cm n sp n r is a maximal parabolic subgroup the symplectic groups are the fundamental dynamical groups of classical and quantal hamiltonian mechanics in particular sp 3 r is the dynamical group of the spherical harmonic oscillator and its weil harmonic series representations are important for the microscopic shell model description of the collective motions of many particle systems the subgroup cm 3 sp 3 r also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus thus the sp 3 r cm 3 branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model some new developments are made in the vector coherent state theory of induced representations

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

multiscale seismic waveform inversion iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models e g the marmousi model the primary difficulty being the presence of numerous local minima in the objective function the presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum the multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale at long scales there are fewer local minima and those that remain are further apart from each other thus at long scales iterative methods can get closer to the neighborhood of the global minimum we apply the multigrid method to a subsampled low frequency version of the marmousi data set although issues of source estimation source bandwidth and noise are not treated results show that iterative inversion methods perform much better when employed with a decomposition by scale furthermore the method greatly reduces the computational burden of the inversion that will be of importance for 3 d extensions to the method

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

matrix representation of the generators of symplectic algebras i the case of sp 4 r because of their many physical applications there has been considerable interest lately in the basis for irreducible representations irreps of symplectic groups and in the matrix elements of the generators of these groups with respect to this basis in the present paper the authors carry out this programme for the irreps in the positive discrete series of sp 4 r by building up this basis from powers of the raising generators applied to the lowest weight state by using the dyson boson realisation of the generators their matrix representation can be obtained by direct differentiation while the overlap of the states of the basis can be determined with the help of coherent states the extension of the analysis to symplectic groups of arbitrary dimension is straightforward and will be implemented explicitly for sp 6 r in a later paper

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

a multiscanning approach based on morphological filtering it is argued that the mathematical morphology method seems to be more reasonable and powerful in studying certain multiscaling vision problems than the approach that uses derivatives of gaussian shaped filters of different sizes to show the validity of this method the authors concentrated on an application that involves forming scale space image of a 2 d shape using morphological opening filtering a proof is given to show that morphological opening filtering has a property of not introducing additional zero crossings as one moves to a coarser scale this is a different result from the conclusion by a l yuille and t a poggio ibid vol pami 8 jan 1986 that the gaussian filter is the only filter with this property in addition opening filtering is computationaly simpler than the gaussian filter

laplace fourier domain elastic full waveform inversion using time domain modeling to obtain subsurface information from onshore seismic exploration data using full waveform inversion fwi based on the acoustic wave equation elastic waves such as ground roll and mode converted waves should be suppressed through heavy preprocessing however such preprocessing deforms not only the elastic waves but also the acoustic waves moreover it is not easy to separate body waves from surface waves in seismic traces for these reasons we need to generate both types of waves in the modeling step to obtain seismic waves that are similar to real seismic waves therefore elastic fwi using the elastic wave equation is necessary to achieve a more accurate fwi in addition elastic fwi can provide better geologic information than acoustic fwi because it inverts the p wavevelocity s wave velocity and density laplace fourier domain elastic fwi is an effective method because it inverts these multiple parameters and can be applied to real seismic data that lack low frequency components however the conventional laplace fourier domain fwi requires a matrix solver with a huge memory cost to perform the modeling in the laplace fourierdomain and memory usage becomes more intensive in the elastic case in the present study we combined time domain wave propagation modeling and laplace fourier domain elastic fwi to invert multiple parameters with less memory cost by using time domain modeling which does not require a matrix solver we obtainedthe forwardand adjoint wavefields withless memory cost the residuals between the recorded and modeled data the virtual sources the hessian matrices and the gradient directions were calculated in the laplace fourier domain to validate the proposed algorithm we performed numerical tests with model 94 synthetic data and benjamin creek real seismic data

quasi newton full waveform inversion with a projected hessian matrix abstractwe present a method in realistic size full waveform inversion fwi to explicitly construct a projected hessian matrix and its inverse matrix which we subsequently used to solve fwi with a quasi newton method newton s method is practically unfeasible in solving realistic size fwi problems because of the prohibitive cost computing time and memory consumption of calculating the hessian matrix and the inverse hessian therefore the gauss newton method and various quasi newton methods are proposed to approximate a hessian matrix particularly current quasi newton fwi qnfwi commonly uses the limited memory bfgs l bfgs method which however only implicitly approximates an inverse hessian we repose fwi as a sparse optimization problem in a sparse model space which contains substantially fewer model parameters that are constrained by structures of the model with respect to fewer parameters in the sparse model we can avoid the limited memory approximation and are able to explicitly comp

symplectic scheme and the poynting vector in reverse time migration abstractwe developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method rem this solution can be used for seismic modeling and reverse time migration rtm in seismic modeling and rtm spatial derivatives are usually calculated by finite differences fds or by the fourier method and the time evolution is normally obtained by a second order fd approach if the spatial derivatives are computed by higher order fd schemes then the time step needs to be small enough to avoid numerical dispersion therefore increasing the computational time however by using rem with the fourier method for the spatial derivatives we can apply the proposed method to propagate the wavefield for larger time steps moreover if the appropriate number of expansion terms is chosen this method is unconditionally stable and propagates seismic waves free of numerical dispersion the use of a symplectic numerical scheme provides the solution of the wave equation an

weil representations via abstract data and heisenberg groups a comparison abstract let b be a ring not necessarily commutative having an involution and let u 2 m b be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to when b is finite we construct a weil representation of u 2 m b via heisenberg groups and find its explicit matrix form on the bruhat elements as a consequence we derive information on generalized gauss sums on the other hand there is an axiomatic method to define a weil representation of u 2 m b and we compare the two weil representations thus obtained under fairly general hypotheses when b is local not necessarily finite we compute the index of the subgroup of u 2 m b generated by its bruhat elements besides the independent interest this subgroup and index are involved in the foregoing comparison of weil representations

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

vector coherent state representations of so 5 in an so 3 basis vector coherent state theory is used to give an algorithm for constructing the so 5 irreps of type n 0 and n n in the physical so 3 basis the construction is remarkably simple and permits the matrices of the so 5 operators for irreps of low dimension and or small multiplicities to be computed by hand the construction is applied to the 2 0 and 2 2 irreps to illustrate the process

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

the weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring we find all irreducible constituents of the weil representation of a unitary group um a of rank m associated to a ramified quadratic extension a of a finite commutative local and principal ring r of odd characteristic we show that this weil representation is multiplicity free with monomial ir reducible constituents we also find the number of these constituents and describe them in terms of clifford theory with respect to a congruence sub group we find all character degrees in the special case when r is a field

comparison of multifrequency selection strategies for simultaneous source full waveform inversion abstractthe computational cost of full waveform inversion fwi is a major obstacle to estimating the velocity model for large scale problems one way to reduce the overall cost of waveform inversion is by adopting a simultaneous source strategy in other words multiple sources are simultaneously fired to simulate supershot gathers and thereby reduce the number of seismic modeling simulations that are performed during the inversion however the use of simultaneous sources introduces crosstalk artifacts that arise from the interference among the sources that constitute a supershot we analyzed the influence of different simultaneous multifrequency selection strategies on crosstalk artifacts our analysis focused on a frequency domain fwi algorithm that is implemented with simultaneous sources that are randomly encoded with random time shifts in the multiscale conventional fwi strategies a finite set of discrete frequencies was selected and the inversion was carried out sequentially from low to high fr

even unimodular lattices associated with the weil representation of the finite symplectic group soit q une puissance d un nombre premier impair p et soit g le groupe symplectique sp 2n q quand n est pair et q p est un nombre premier avec p 3 mod 4 il existe un treillis integral rationnel g invariant l dans m qui supporte une forme unimodulaire definie positive symetrique paire m est un qg module et q le corps des rationnels

using the rapid expansion method for accurate time stepping in modeling and reverse time migration reverse time migration is based on seismic forward modeling algorithms where spatial derivatives usually are calculated by finite differences or by the fourier method time integration in general is done by finite difference time stepping of low orders if the spatial derivatives are calculated by high order methods and time stepping is based on low order methods there is an imbalance that might require that the time step size needs to be very small to avoid numerical dispersion as a result computing times increase using the rapid expansion method rem avoids numerical dispersion if the number of expansion terms is chosen properly comparisons with analytical solutions show that the rem is preferable especially at larger propagation times for reverse time migration the rem needs to be applied in a time stepping manner this is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time this is not appropriate for reverse time migration where the sources have different time histories in reverse time migration it might be desirable to use the poynting vector information to estimate opening angles to improve the quality of the image in the solution of the wave equation this requires that one calculates not only the pressure wavefield but also its time derivative the rapid expansion method can be extended easily to provide this time derivative with negligible extra cost

measuring the fractal dimension of signals morphological covers and iterative optimization fractals can model many classes of time series data the fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales the covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal s graph in this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods it is theoretically estab lished that for the fractal dimension computation covering one dimensional signals with planar sets is equivalent to mor phologically transforming the signal by one dimensional func tions which reduces the computational complexity from quad ratic in the signal s length to linear then a morphological covering algorithm is developed and applied to discrete time signals synthesized from weierstrass functions fractal inter polation functions and fractional brownian motion further for deterministic parametric fractals depending on a single pa rameter related to their dimension we develop an optimization method that starts from an initial estimate and iteratively con verges to the true fractal dimension by searching in the param eter space and minimizing a distance between the original sig nal and all such signals from the same class experimental results are also provided to demonstrate the good performance of the developed methods

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

the digital morphological sampling theorem morphological sampling reduces processing time and cost and yet produces results sufficiently close to the result of full processing a morphological sampling theorem is described which states 1 how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling 2 to what precision an appropriate morphologically filtered image can be reconstructed after sampling and 3 the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element the digital sampling theorem is developed first for the case of binary morphology and then it is extended to gray scale morphology through the use of the umbra homomorphism theorems

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

strong consistency of the lad l sub 1 estimator of parameters of stationary autoregressive processes with zero mean strong consistency almost sure convergence to the true parameters of the lad least absolute deviations ar autoregressive parameter estimator has been proven by s gross and w l steiger 1979 under the condition that i i d noise driving a stationary autoregressive process has zero median this work extends their proof to include the case when the driving noise has zero mean thus when the noise pdf probability density function is asymmetric with distinct mean and median the lad estimator will be strongly consistent with the pdf centered with either mean or median at the origin the results of this work extend computer simulations which further indicate that under these conditions the lad estimator is ms consistent mean squared convergence to the true parameters the importance of these results in lad signal processing applications is discussed

boson realisation of symplectic algebras the boson realisation of the two dimensional symplectic lie algebra sp 2 for a given irreducible representation irrep of the sp 2 group has been known for a long time more recently the boson realisation of the 2d dimensional symplectic lie algebras sp 2d have been derived for particular irreps of the sp 2d group the author outlines the corresponding result for an arbitrary irrep of sp 2d

partial difference operators on weighted graphs for image processing on surfaces and point clouds partial difference equations pdes and variational methods for image processing on euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems with the recent advent of many 3d sensors there is a growing interest in transposing and solving pdes on surfaces and point clouds in this paper we propose a simple method to solve such pdes using the framework of pdes on graphs this latter approach enables us to transcribe for surfaces and point clouds many models and algorithms designed for image processing to illustrate our proposal three problems are considered 1 p laplacian restoration and inpainting 2 pdes mathematical morphology and 3 active contours segmentation

hall higman type theorems for semisimple elements of finite classical groups minimum polynomials of semisimple elements of prime power order p a of finite classical groups in nontrivial irreducible cross characteristic representations are studied in particular an analogue of the hall higman theorem is established which shows that the degree of such a polynomial is at least p a 1 p 1 with a few explicit exceptions

hypergeometric type operators and their supersymmetric partners the generalization of the factorization method performed by mielnik j math phys 25 3387 1984 opened new ways to generate exactly solvable potentials in quantum mechanics we present an application of mielnik s method to hypergeometric type operators it is based on some solvable riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

gradient watersheds in morphological scale space after introducing several results relating to the modification of the homotopy of gradient functions based on extrema in the base image and building on earlier results in morphological scale space we introduce a scale space monotonicity theorem for regions of an image defined by watersheds of a gradient function modified to retain only the local minima or maxima of its smoothed parent image we then illustrate the theorem with an example of the scale space extraction of texture features from the nuclei of cervical cells

optimal morphological pattern restoration from noisy binary images a theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented the problem is formulated in a general form and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory consideration is given to the set difference distance function as a measure of comparison between images this function is used to introduce the mean difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering it is proved that the class of alternating sequential filters is a set of parametric smoothing morphological filters that best preserve the crucial structure of input images in the least mean difference sense

an efficient multiscale method for time domain waveform tomography this efficient multiscale method for time domain waveform tomography incorporates filters that are more efficient than hamming window filters a strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain a staggered grid explicit finite difference method with fourth order accuracy in space and second order accuracy in time is used for forward modeling and the adjoint calculation the adjoint method is utilized in inverting for an efficient computation of the gradient directions in the multiscale approach multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography the method is applied successfully to 1d and 2d heterogeneous models it can accurately recover low and high wavenumber components of the velocity models the inversion result for the 2d model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional single scale method

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks

a general methodology for the determination of 2d bodies elastic deformation invariants application to the automatic identification of parasites a novel methodology is introduced here that exploits 2d images of arbitrary elastic body deformation instances so as to quantify mechanoelastic characteristics that are deformation invariant determination of such characteristics allows for developing methods offering an image of the undeformed body general assumptions about the mechanoelastic properties of the bodies are stated which lead to two different approaches for obtaining bodies deformation invariants one was developed to spot a deformed body s neutral line and its cross sections while the other solves deformation pdes by performing a set of equivalent image operations on the deformed body images both of these processes may furnish a body undeformed version from its deformed image this was confirmed by obtaining the undeformed shape of deformed parasites cells protozoa fibers and human lips in addition the method has been applied to the important problem of parasite automatic classification from their microscopic images to achieve this we first apply the previous method to straighten the highly deformed parasites and then apply a dedicated curve classification method to the straightened parasite contours it is demonstrated that essentially different deformations of the same parasite give rise to practically the same undeformed shape thus confirming the consistency of the introduced methodology finally the developed pattern recognition method classifies the unwrapped parasites into six families with an accuracy rate of 97 6 percent