the set of all iterates is nowhere dense in c 0 1 0 1 we prove that if a mixing map f 0 1 0 1 belongs to the c 0 closure of the set of iterates and f 0 0 f 1 1 then f is an iterate itself together with some one dimensional techniques it implies that the set of all iterates is nowhere dense in c 0 1 0 1 giving the final answer to the question of a bruckner p humke and m laczkovich

perturbation based successive approximate topology optimization for a displacement minimization problem abstract this paper discusses a new approach for displacement based topology optimization of continuum topology optimization of continuum can be performed with the optimization of material distribution this optimization problem will include a large number of design variables and it may require a high computational cost especially for computation of gradient components with respect to an objective function and determination of an appropriate updating increment a conventional approximate optimization technique may not be effective for this problem because it is a very large scale problem from this viewpoint a new optimization technique which is based on a successive approximation algorithm is proposed the proposed method consists of two main processes 1 a linear approximation for determination of an updating direction and 2 a nonlinear approximation for determining a scale coefficient of the updating vector the first approximation is performed using the perturbation based finite element analysis and the second approximation is performed using the kriging method in order to investigate effectiveness of the proposed method it is applied to some typical examples for topology optimization problem from the numerical results effectiveness and validity of the proposed method are discussed

theory of two dimensional multirate filter banks results are presented on 2 d fir two dimensional finite impulse response filter banks for multirate applications the theory is valid for all sampling lattices conditions for alias free and perfect signal reconstruction are derived synthesis structures for paraunitary and nonparaunitary polynomial matrices are derived which yield perfect reconstruction filter banks the degrees of freedom are given for these systems linear phase conditions are posed on the polyphase form of filter banks which is used to derive a design structure for the restricted but important case of linear phase filter banks

wavelets fractals and radial basis functions wavelets and radial basis functions rbfs lead to two distinct ways of representing signals in terms of shifted basis functions rbfs unlike wavelets are nonlocal and do not involve any scaling which makes them applicable to nonuniform grids despite these fundamental differences we show that the two types of representation are closely linked together through fractals first we identify and characterize the whole class of self similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases conversely we prove that for any compactly supported scaling function spl phi x there exists a one sided central basis function spl rho sub x that spans the same multiresolution subspaces the central property is that the multiresolution bases are generated by simple translation of spl rho sub without any dilation we also present an explicit time domain representation of a scaling function as a sum of harmonic splines the leading term in the decomposition corresponds to the fractional splines a recent continuous order generalization of the polynomial splines

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2 abstract it is known that any piecewise monotonic function of nonmonotonicity height not less than 2 has no continuous iterative roots of order greater than the number of forts of the function so the following problem arises naturally does such a function have an iterative root of order n not greater than the number of forts we consider the case that the number of forts is equal to n in which there appear possibly only two types t 1 and t 2 of iterative roots i e the roots strictly increasing on the interval stretched on all forts of the given function and the roots strictly decreasing on such an interval respectively we characterize all type t 1 roots of order n and give a necessary condition for a root of order n to be of type t 2 a full description of type t 2 is still an open question

construction of generators of quasi interpolation operators of high approximation orders in spaces of polyharmonic splines the paper presents a simple procedure for the construction of quasi interpolation operators in spaces of m harmonic splines in r d which reproduce polynomials of high degree the procedure starts from a generator f 0 which is easy to derive but with corresponding quasi interpolation operator reproducing only linear polynomials and recursively defines generators f 1 f 2 f m 1 with corresponding quasi interpolation operators reproducing polynomials of degree up to 3 5 2m 1 respectively the construction of f j from f j 1 is explicit simple and independent of m the special case d 1 and the special cases d 2 m 2 3 4 are discussed in details

isotropic polyharmonic b splines scaling functions and wavelets in this paper we use polyharmonic b splines to build multidimensional wavelet bases these functions are nonseparable multidimensional basis functions that are localized versions of radial basis functions we show that rabut s elementary polyharmonic b splines do not converge to a gaussian as the order parameter increases as opposed to their separable b spline counterparts therefore we introduce a more isotropic localization operator that guarantees this convergence resulting into the isotropic polyharmonic b splines next we focus on the two dimensional quincunx subsampling scheme this configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach however up until now the design of appropriate filters for the quincunx scheme has mainly been done using the mcclellan transform in our approach we start from the scaling functions which are the polyharmonic b splines and as such explicitly known and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi orthogonal wavelet transform e g orthonormal b spline and dual the filters are automatically specified by the scaling relations satisfied by these functions we prove that the isotropic polyharmonic b spline wavelet converges to a combination of four gabor atoms which are well separated in the frequency domain we also show that these wavelets are nearly isotropic and that they behave as an iterated laplacian operator at low frequencies we describe an efficient fast fourier transform based implementation of the discrete wavelet transform based on polyharmonic b splines

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

on the construction of polyharmonic b splines in this paper we present two constructions providing polyharmonic b splines with a quasi rotational invariant behavior their properties are discussed and it is shown that in this way we improve the decay when email protected

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order

computing iterative roots of polygonal functions based on the iterative root theory for monotone functions an algorithm for computing polygonal iterative roots of increasing polygonal functions was given by j kobza in this paper we not only give an algorithm for roots of decreasing polygonal functions but also generalize kobza s results to the general n furthermore we extend our algorithms for polygonal pm functions a class of non monotonic functions

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

an orthogonal family of quincunx wavelets with continuously adjustable order we present a new family of two dimensional and three dimensional orthogonal wavelets which uses quincunx sampling the orthogonal refinement filters have a simple analytical expression in the fourier domain as a function of the order spl lambda which may be noninteger we can also prove that they yield wavelet bases of l sub 2 r sup 2 for any spl lambda 0 the wavelets are fractional in the sense that the approximation error at a given scale a decays like o a sup spl lambda they also essentially behave like fractional derivative operators to make our construction practical we propose a fast fourier transform based implementation that turns out to be surprisingly fast in fact our method is almost as efficient as the standard mallat algorithm for separable wavelets

quincunx fundamental refinable functions and quincunx biorthogonal wavelets we analyze the approximation and smoothness properties of quincunx fundamental refinable functions in particular we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in r2 their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order in addition these examples are minimally supported with symmetry for two special families of such quincunx interpolatory masks we prove that their symbols are nonnegative finally a general way of constructing quincunx biorthogonal wavelets is presented several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed

iterative solution of systems originating from integral equations and surface interpolation a simple procedure for generating diagonal dominance in certain linear systems is presented and the iterative solution of the transformed systems is discussed the method is presented for systems encountered in the numerical solution of fredholm equations of the first kind with singular kernels and for systems arising in the interpolation of surfaces by certain sets of radial functions in the second case the method provides new sets of basis functions generalizing in some sense the univariate b splines

polyharmonic splines an approximation method for noisy scattered data of extra large size the aim of this paper is to provide a fast method with a good quality of reproduction to recover functions from very large and irregularly scattered samples of noisy data which may present outliers to the given sample of size n we associate a uniform grid and around each grid point we condense the local information given by the noisy data by a suitable estimator the recovering is then performed by a stable interpolation based on isotropic polyharmonic b splines due to the good approximation rate we need only m n degrees of freedom to recover the phenomenon faithfully

non monotonic iterative roots extended from characteristic intervals it has been treated as a difficult problem to find iterative roots of non monotonic functions for some pm functions which do not increase the number of forts under iteration a method was given to obtain a non monotonic iterative root by extending a monotone iterative root from the characteristic interval in this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of pm functions

topology optimization by penalty top method in traditional structural topology optimization to the material properties of continuum finite elements of fixed form and coupling are varied to find the optimal topology that satisfies the design problem we develop an alternative fundamental formulation where the design space search is dependent on the coupling and the goal of the topology optimization by penalty top method is to determine the optimal finite element coupling constraints by this approach seemingly disparate topology design problems e g the design of structural supports topology optimization for fluid mechanics problems and topology optimization by the element connectivity parameterization ecp method can be understood as related formulations under a common topology optimization umbrella and more importantly this general framework can be applied to new design problems for example in modern multibody dynamics synthesis the geometric form of finite elements of fixed material properties and interconnectivity are varied to find the optimal topology that satisfies the mechanism design problem the a priori selection of coupling e g by revolute or translational joints severely limits the design space search the top method addresses this limitation in a novel way we develop the methodology and apply the top method to the diverse design problems discussed above

zeta functions formulas and applications abstract the existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the wodzicki residue exponentially fast convergent formulas valid in the whole of the complex plane and yielding the pole positions and residua that extend the ones by chowla and selberg for the epstein zeta function quadratic form and by barnes affine form are then given after briefly recalling the zeta function regularization procedure in quantum field theory some applications of these expressions in physics are described

iterative roots of piecewise monotonic functions with finite nonmonotonicity height abstract it is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function in this paper we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts by investigating the trajectory of possible continuous roots we give a general method to find all iterative roots of those functions with finite nonmonotonicity height

the finite ridgelet transform for image representation the ridgelet transform was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines we propose an orthonormal version of the ridgelet transform for discrete and finite size images our construction uses the finite radon transform frat as a building block to overcome the periodization effect of a finite transform we introduce a novel ordering of the frat coefficients we also analyze the frat as a frame operator and derive the exact frame bounds the resulting finite ridgelet transform frit is invertible nonredundant and computed via fast algorithms furthermore this construction leads to a family of directional and orthonormal bases for images numerical results show that the frit is more effective than the wavelet transform in approximating and denoising images with straight edges

quasi interpolation operators on hexagonal grids with high approximation orders in spaces of polyharmonic splines in this paper we discuss the problem of approximating data given on hexagonal lattices the construction of continuous representation from sampled data is an essential element of many applications as for instance image resampling numerical solution of pde boundary problems etc a useful tool is quasi interpolation that doesn t need the solution of a linear system it is then important to have quasi interpolation operators with high approximation orders and capable to provide an efficient computation of the quasi interpolant function to this end we show that the idea proposed by bozzini et al 1 for the construction of quasi interpolation operators in spaces of m harmonic splines with knots in z 2 which reproduce polynomials of high degree can be generalized to any spaces of m harmonic splines with knots on a lattice of r 2 and in particular on hexagonal grids then by a simple procedure which starts from a generator 0 with corresponding quasi interpolation operator reproducing only linear polynomials it is possible to define recursively generators 1 m 1 with corresponding quasi interpolation operators reproducing polynomials up to degree 3 5 2 m 1 we show that this new generators of quasi interpolation operators on a general lattice are positive definite functions and are scaling functions whenever 0 has those properties moreover we are able to associate with j a dyadic convergent subdivision scheme that allows a fast computation of the quasi interpolant

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

topology optimization of regions of darcy and stokes flow this paper treats the topology optimization problem of obtaining an optimal layout of regions of darcy and stokes flow where the objective is the total potential power functional representing aver

a note on iterative roots of pm functions abstract in this paper we study iterative roots of pm functions a special class of non monotone functions problem 2 in w zhang pm functions their characteristic intervals and iterative roots ann polon math lxv 1997 119 128 is solved partly and theorem 4 in that paper is generalized

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

kernel b splines on general lattices the aim of the paper is to provide a computationally effective way to construct stable bases on general non degenerate lattices in particular we define new stable bases on hexagonal lattices and we give some numerical examples which show their usefulness in applications

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

directional dyadic wavelet transforms design and algorithms we propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms dwts in two dimensions our technique relies on an extension of the work of duval destin et al 1993 where dyadic decompositions are constructed starting from the continuous wavelet transform the main advantage of this framework is that it allows for a lot of freedom in designing two dimensional 2 d dyadic wavelets we use this property to construct directional wavelets whose orientation filtering capabilities are very important in image processing we address the efficient implementation of these decompositions by constructing approximate qmfs through an l sup 2 optimization we also propose and study an efficient implementation in the fourier domain for dealing with large filters

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

image representation using 2d gabor wavelets this paper extends to two dimensions the frame criterion developed by daubechies for one dimensional wavelets and it computes the frame bounds for the particular case of 2d gabor wavelets completeness criteria for 2d gabor image representations are important because of their increasing role in many computer vision applications and also in modeling biological vision since recent neurophysiological evidence from the visual cortex of mammalian brains suggests that the filter response profiles of the main class of linearly responding cortical neurons called simple cells are best modeled as a family of self similar 2d gabor wavelets we therefore derive the conditions under which a set of continuous 2d gabor wavelets will provide a complete representation of any image and we also find self similar wavelet parametrization which allow stable reconstruction by summation as though the wavelets formed an orthonormal basis approximating a tight frame generates redundancy which allows low resolution neural responses to represent high resolution images

theoretical aspects of the internal element connectivity parameterization approach for topology optimization the internal element connectivity parameterization i ecp method is an alternative approach to overcome numerical instabilities associated with low stiffness element states in non linear problems in i ecp elements are connected by zero length links while their link stiffness values are varied therefore it is important to interpolate link stiffness properly to obtain stably converging results the main objective of this work is two fold 1 the investigation of the relationship between the link stiffness and the stiffness of a domain discretizing patch by using a discrete model and a homogenized model and 2 the suggestion of link stiffness interpolation functions the effects of link stiffness penalization on solution convergence are then tested with several numerical examples the developed homogenized i ecp model can also be used to physically interpret an intermediate design variable state

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed