zero crossings of a wavelet transform the completeness stability and application to pattern recognition of a multiscale representation based on zero crossings is discussed an alternative projection algorithm is described that reconstructs a signal from a zero crossing representation which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero crossings the reconstruction algorithm has a fast convergence and each iteration requires o n log sup 2 n computation for a signal of n samples the zero crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems as an example the implementation and results of a coarse to fine stereo matching algorithm are described

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

unicity in piecewise polynomial l 1 approximation via an algorithm our main result shows that certain generalized convex functions on a real interval possess a unique best l 1 approximation from the family of piecewise polynomial functions of fixed degree with varying knots this result was anticipated by kioustelidis in 11 however the proof given there is nonconstructive and uses topological degree as the primary tool in a fashion similar to the proof the comparable result for the l 2 case in 5 by contrast the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors the proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness this algorithm was suggested by kioustelidis 10 in addition an asymptotic uniqueness result and a nonuniqueness result are indicated which analogize known results in the l 2 case

multiscale modeling and estimation of poisson processes with application to photon limited imaging many important problems in engineering and science are well modeled by poisson processes in many applications it is of great interest to accurately estimate the intensities underlying observed poisson data in particular this work is motivated by photon limited imaging problems this paper studies a new bayesian approach to poisson intensity estimation based on the haar wavelet transform it is shown that the haar transform provides a very natural and powerful framework for this problem using this framework a novel multiscale bayesian prior to model intensity functions is devised the new prior leads to a simple bayesian intensity estimation procedure furthermore we characterize the correlation behavior of the new prior and show that it has 1 f spectral characteristics the new framework is applied to photon limited image estimation and its potential to improve nuclear medicine imaging is examined

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

feature adaptive motion tracking of ultrasound image sequences using a deformable mesh by exploiting the correlation of ultrasound speckle patterns that result from scattering by underlying tissue elements two dimensional tissue motion theoretically can be recovered by tracking the apparent movement of the associated speckle patterns speckle tracking however is an ill posed inverse problem because of temporal decorrelation of the speckle patterns and the inherent low signal to noise ratio of medical ultrasonic images this paper investigates the use of an adaptive deformable mesh for nonrigid tissue motion recovery from ultrasound images the nodes connecting the mesh elements are allocated adaptively to stable speckle patterns that are less susceptible to temporal decorrelation the authors use the approach of finite clement analysis in manipulating the irregular mesh elements a novel deformable block matching algorithm making use of a lagrange element for higher order description of local motion is proposed to estimate a nonrigid motion vector at each node in order to ensure that the motion estimates are admissible to a physically plausible solution the nodal displacements are regularized by minimizing the strain energy associated with the mesh deformations experiments based on ultrasound images of a tissue mimicking phantom and a muscle undergoing contraction and on computer simulations have shown that the proposed algorithm can successfully track nonrigid displacement fields

algorithms for optimal discontinuous piecewise linear and constant l 2 fits to continuous functions with adjustable nodes in one and two dimensions in this paper a direct variational approach with nonstandard variations is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant l2 fits to a continuous function of one or two variables with adjustable nodes in the one variable case the algorithm is fast and robust the mesh cannot tangle and the resulting fits are continuous a e in the two variable case on an adjustable triangular grid the algorithm is less robust but gives good results for particular functions possessing a single steep feature the extension to higher dimensions is straightforward

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

f structures and integral points on semiabelian varieties over finite fields motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished generator of the ring

maximum likelihood approach to strain imaging using ultrasound a maximum likelihood ml strategy for strain estimation is presented as a framework for designing and evaluating bioelasticity imaging systems concepts from continuum mechanics signal analysis and acoustic scattering are combined to develop a mathematical model of the ultrasonic waveforms used to form strain images the model includes three dimensional 3 d object motion described by affine transformations rayleigh scattering from random media and 3 d system response functions the likelihood function for these waveforms is derived to express the fisher information matrix and variance bounds for displacement and strain estimation the ml estimator is a generalized cross correlator for pre and post compression echo waveforms that is realized by waveform warping and filtering prior to cross correlation and peak detection experiments involving soft tissuelike media show the ml estimator approaches the cramer rao error bound for small scaling deformations at 5 mhz and 1 2 compression the predicted

the isotrivial case in the mordell lang theorem we determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety g defined over a finite field with a closed subvariety x c g we also study a related question in the context of a power of the additive group scheme

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

motion estimation with quadtree splines this paper presents a motion estimation algorithm based on a new multiresolution representation the quadtree spline this representation describes the motion field as a collection of smoothly connected patches of varying size where the patch size is automatically adapted to the complexity of the underlying motion the topology of the patches is determined by a quadtree data structure and both split and merge techniques are developed for estimating this spatial subdivision the quadtree spline is implemented using another novel representation the adaptive hierarchical basis spline and combines the advantages of adaptively sized correlation windows with the speedups obtained with hierarchical basis preconditioners results are presented on some standard motion sequences

a skolem mahler lech theorem in positive characteristic and finite automata lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions this result is known as the skolem mahler lech theorem lech gave a counterexample to a similar statement in positive characteristic we will present some more pathological examples we will state and prove a correct analog of the skolem mahler lech theorem in positive characteristic the zeroes of a recurrence sequence in positive characteristic can be described using finite automata

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

fast intersections on nested tetrahedrons fint an algorithm for adaptive finite element based distributed parameter estimation a variety of biomedical imaging techniques such as optical and fluorescence tomography electrical impedance tomography and ultrasound imaging can be cast as inverse problems wherein image reconstruction involves the estimation of spatially distributed parameter s of the pde system describing the physics of the imaging process finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries a dual adaptive mesh based approach wherein one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter can result in high resolution image reconstruction at minimum computation effort if both the meshes are allowed to adapt independently till date no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes herein we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8 similar subtetrahedron subdivision scheme the algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons the procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection the computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single and a dual mesh and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements

enhanced frequency analysis using wavelets an enhanced frequency analysis method is introduced using a phase shift wavelet representation to analyse individual frequency components and to reveal features of a given signal the frequency representation provides details at each time location without the loss caused by local orthogonality between the wavelet basis functions and the given signal which occurs when a normal wavelet decomposition is employed movement of the local orthogonality locations corresponding to the phase shift of the wavelet basis functions are studied and it is shown that these movements are important and can be used to reveal features of the signal in the frequency domain such as continuities smoothness and trends several examples are presented to illustrate the method and an application to a permanent magnet synchronous drive system is provided to demonstrate the approach

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

moving finite element least squares and finite volume approximations of steady and time dependent pdes in multidimensions abstract we review recent advances in galerkin and least squares methods for approximating the solutions of first and second order pdes with moving nodes in multidimensions these methods use unstructured meshes and minimise the norm of the residual of the pde over both solutions and nodal positions in a unified manner both finite element and finite volume schemes are considered as are transient and steady problems for first order scalar time dependent pdes in any number of dimensions residual minimisation always results in the methods moving the nodes with the often inconvenient approximate characteristic speeds for second order equations however the moving finite element mfe method moves the nodes usefully towards high curvature regions in the steady limit for pdes derived from a variational principle the mfe method generates a locally optimal mesh and solution this also applies to least squares minimisation the corresponding moving finite volume mfv method based on the l2 norm does not have this property however although there does exist a finite volume method which gives an optimal mesh both for variational principles and least squares

anisotropic motion estimation on edge preserving riesz wavelets for robust video mosaicing image mosaicing is a technique widely used for extending the field of view of industrial medical outdoor or indoor scenes however image registration can be very challenging e g due to large texture variability illumination changes image blur and camera perspective changes in this paper a total variational optical flow approach is investigated to estimate dense point correspondences between image pairs an edge preserving riesz wavelet scale space combined with a novel tv regularizer is proposed for preserving motion discontinuities along the edges of weak textures and for handling strong in plane rotations present in image sequences an anisotropic weighted median filtering is implemented for minimizing outliers quantitative evaluation of the method on the middlebury image database and simulated sequences with known ground truth demonstrates high accuracy of the proposed method in comparison with other state of the art methods including a robust graph cut method and a patch matching approach qualitative results on video sequences of difficult real scenes demonstrate the robustness of the proposed method highlightsmosaicing of images with strong texture and illumination variability fast robust and accurate optical flow computation tv l1 method on a second order riesz wavelet basis for preserving texture discontinuities novel anisotropic tv regularizer accurate results for both the middlebury data set and various complicated scenes

mixed moving finite element method abstract in this paper we present a mixed moving finite element method mmfem in this method nodes are divided into several classes and move differently we concentrate on burgers equation with a linear basis function and carry out the element and global analysis the computation shows the improvement in contrast to moving finite element method

fast parametric elastic image registration we present an algorithm for fast elastic multidimensional intensity based image registration with a parametric model of the deformation it is fully automatic in its default mode of operation in the case of hard real world problems it is capable of accepting expert hints in the form of soft landmark constraints much fewer landmarks are needed and the results are far superior compared to pure landmark registration particular attention has been paid to the factors influencing the speed of this algorithm the b spline deformation model is shown to be computationally more efficient than other alternatives the algorithm has been successfully used for several two dimensional 2 d and three dimensional 3 d registration tasks in the medical domain involving mri spect ct and ultrasound image modalities we also present experiments in a controlled environment permitting an exact evaluation of the registration accuracy test deformations are generated automatically using a random hierarchical fractional wavelet based generator

mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere we describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere then we use this as a test case to compute optimal meshes in presence of sharp boundary layers for the well posedness of the continuous minimizing problem we introduce a lower semicontinuous extension of the energy with respect to weak convergence in bv and we prove that the extended minimization problem has a unique singular solution we then show how a moving finite element method in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization mimics this geometric point of view finally we present numerical computations with boundary layers of zero thickness and we give numerical evidence of the convergence of the method this last aspect is proved in another paper

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

improved transient signal detection using a wavepacket based detector with an extended translation invariant wavelet transform this paper presents the theory of m band extended translation invariant eti wavelet transforms the eti generalizes the translation invariant wavelet transform of weiss 1933 it is shown that iteration of the eti in a tree structure provides a signal decomposition into an orthonormal wavepacket basis other properties such as translation invariance and invertibility of the transform are proven the theory is then applied to transient signal detection through development of a family of translation invariant wavepacket based detectors this family of detectors provides improved the performance over previously defined wavepacket based detectors a performance analysis is conducted roc curves generated by monte carlo simulation are presented indicating the detector performance the detector performance is demonstrated to be independent of the signal translation

an optimal triangulation for second order elliptic problems abstract let be a polygonal domain in r n h an associated triangulation and uh the finite element solution of a well posed second order elliptic problem on h let m mi p qi 1 be the set of nodes which defines the vertices of the triangulation h for each i m i x i l 1 l n in rn the object of this paper is to provide a computational tool to approximate the best set of positions m of the nodes and hence the best triangulation gt h which minimizes the solution error in the natural norm associated with the problem the main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates xil 1 l n of each of the variable nodes mi i 1 p

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

the mordell lang conjecture for function fields we give a proof of the geometric mordell lang conjecture in any characteristic our method involves a model theoretic analysis of the kernel of manin s homomorphism and of a certain analog in characteristic p department of mathematics massachusetts institute of technology 2 277 cambridge massachusetts 02139 current address department of mathematics hebrew university jerusalem israel e mail address ehud math mit edu

the closed cone of a rational series is rational polyhedral abstract for a multivariate power series f let cone f denote the cone generated by the exponents of the monomials with nonzero coefficients assume that f is an expansion of a rational function p q with gcd p q 1 then we prove that the closure cone f is equal to cone p cone q as applications we show the irrationality of euler chow series of certain algebraic varieties

weak series bv convergence of a moving finite element method for singular axisymmetric harmonic maps we prove the convergence of a moving finite element method for the minimization of a relaxed dirichlet energy among axisymmetric maps from the disc into the sphere the optimal mesh and the discrete minimizer exist for a discrete relaxed energy similarly to the true solution the discrete minimizer has a boundary layer because of the consistency error introduced by the discretization of the energy this discrete minimizer is nonconforming we show that it converges to the solution of the continuous problem in an appropriate bv space the proof is done for three different piecewise linear finite element discretizations

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

theoretical limitations of the elastic wave equation inversion for tissue elastography this article examines the theoretical limitations of the local inversion techniques for the measurement of the tissue elasticity most of these techniques are based on the estimation of the phase speed or the algebraic inversion of a one dimensional wave equation to analyze these techniques the wave equation in an elastic continuum is revisited it is proven that in an infinite medium harmonic shear waves can travel at any phase speed greater than the classically known shear wave speed by demonstrating this for a special case with cylindrical symmetry hence in addition to the mechanical properties of the tissue the phase speed depends on the geometry of the wave as well the elastic waves in an infinite cylindrical rod are studied it is proven that multiple phase speeds can coexist for a harmonic wave at a single frequency this shows that the phase speed depends not only on the mechanical properties of the tissue but also on its shape the final conclusion is that the only way to avoid theore

on vanishing coefficients of algebraic power series over fields of positive characteristic let k be a field of characteristic p 0 and let f t1 td be a power series in d variables with coefficients in k that is algebraic over the field of multivariate rational functions k t1 td we prove a generalization of both derksen s recent analogue of the skolem mahler lech theorem in positive characteristic and a classical theorem of christol by showing that the set of indices n1 nd d for which the coefficient of t 1 n 1 cdots t d n d in f t1 td is zero is a p automatic set applying this result to multivariate rational functions leads to interesting effective results concerning some diophantine equations related to s unit equations and more generally to the mordell lang theorem over fields of positive characteristic

linear equations over multiplicative groups recurrences and mixing i let u1 umu1 um be linear recurrences with values in a field kk of positive characteristic pp we show that the set of integer vectors k1 km k1 km such that u1 k1 um km 0u1 k1 um km 0 is pp normal in a natural sense generalizing that of the first author who proved the result for m 1m 1 furthermore the set is effectively computable if kk is we illustrate this with an example for m 4m 4 we also show that the corresponding set for zero characteristic is not decidable for m 557844m 557844 thus verifying a conjecture of cerlienco mignotte and piras

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

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

analysis of structural similarity in mammograms for detection of bilateral asymmetry we hypothesize that quantification of structural similarity or dissimilarity between paired mammographic regions can be effective in detecting asymmetric signs of breast cancer bilateral masking procedures are applied for this purpose by using automatically detected anatomical landmarks changes in structural information of the extracted regions are investigated using spherical semivariogram descriptors and correlation based structural similarity indices in the spatial and complex wavelet domains the spatial distribution of grayscale values as well as of the magnitude and phase responses of multidirectional gabor filters are used to represent the structure of mammographic density and of the directional components of breast tissue patterns respectively a total of 188 mammograms from the ddsm and mini mias databases consisting of 47 asymmetric cases and 47 normal cases were analyzed for the combined dataset of mammograms areas under the receiver operating characteristic curves of 0 83 0 77 and 0 87 were obtained respectively with linear discriminant analysis the bayesian classifier and an artificial neural network with radial basis functions using the features selected by stepwise logistic regression and leave one patient out cross validation two view analysis provided accuracy up to 0 94 with sensitivity and specificity of 1 and 0 88 respectively

a new moving mesh algorithm for the finite element solution of variational problems the paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite dimensional spaces the optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation be minimized such a criterion has a relatively long history in the finite element literature the chief merit of the procedure presented in this paper is that each node of the mesh and the corresponding nodal value of the discrete approximation are updated by solving sequentially local minimization problems with very few degrees of freedom it is shown that this procedure reduces the energy functional monotonically without the need to solve the global discrete problem at intermediate stages applications to partial differential equations are considered numerical results in two dimensions are obtained by incorporating the algorithm into bank s well known pltmg elliptic solver

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

spatio temporal nonrigid registration for ultrasound cardiac motion estimation we propose a new spatio temporal elastic registration algorithm for motion reconstruction from a series of images the specific application is to estimate displacement fields from two dimensional ultrasound sequences of the heart the basic idea is to find a spatio temporal deformation field that effectively compensates for the motion by minimizing a difference with respect to a reference frame the key feature of our method is the use of a semi local spatio temporal parametric model for the deformation using splines and the reformulation of the registration task as a global optimization problem the scale of the spline model controls the smoothness of the displacement field our algorithm uses a multiresolution optimization strategy to obtain a higher speed and robustness we evaluated the accuracy of our algorithm using a synthetic sequence generated with an ultrasound simulation package together with a realistic cardiac motion model we compared our new global multiframe approach with a previous method based on pairwise registration of consecutive frames to demonstrate the benefits of introducing temporal consistency finally we applied the algorithm to the regional analysis of the left ventricle displacement and strain parameters were evaluated showing significant differences between the normal and pathological segments thereby illustrating the clinical applicability of our method

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

restoration of the velocity field of the heart from two dimensional echocardiograms a method to quantify the motion of the heart from digitized sequences of two dimensional echocardiograms 2 d echos was recently proposed this method computes on every point of the 2 d echoes the 2 d apparent velocity vector or optical flow which characterizes its interframe motion however further analysis is required to determine what part of this motion is due to translation rotation contraction and deformation of the myocardium a method to locally obtain this information is presented the proposed method assumes that the interframe velocity field u xy v x y can be locally described by linear equations in the form u x y a ax by v x y b cx dy the additional constraint was introduced in the computation of the local velocity field by the method of projections onto convex sets since this constraint is only valid locally the myocardium must be first divided into sectors and the velocity fields computed independently for each sector

an interacting multiple model probabilistic data association filter for cavity boundary extraction from ultrasound images this paper presents a novel segmentation technique for extracting cavity contours from ultrasound images the problem is first discretized by projecting equispaced radii from an arbitrary seed point inside the cavity toward its boundary the distance of the cavity boundary from the seed point is modeled by the trajectory of a moving object the motion of this moving object is assumed to be governed by a finite set of dynamical models subject to uncertainty candidate edge points obtained along each radius include the measurement of the object position and some false returns the modeling approach enables us to use the interacting multiple model estimator along with a probabilistic data association filter for contour extraction the convergence rate of the method is very fast because it does not employ any numerical optimization the robustness and accuracy of the method are demonstrated by segmenting contours from a series of ultrasound images the results are validated through comparison with manual segmentations performed by an expert an application of the method in segmenting bone contours from computed tomography images is also presented

une demonstration simple du theoreme de skolem mahler lech abstract a simple proof of the skolem mahler lech theorem is given which does not make the use of the p adic number theory it mainly relies on the equivalence between rational and recognizable series in particular this equivalence allows to deduce easily the general case of a commutative field of characteristic zero from the rational case

analysis of an algorithm for generating locally optimal meshes for l 2 approximation by discontinuous piecewise polynomials this paper discusses the problem of constructing a locally optimal mesh for the best l 2 approximation of a given function by discontinuous piecewise polynomials in the one dimensional case it is shown that under certain assumptions on the approximated function baines algorithm m j baines math comp 62 1994 pp 645 669 for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh the rate of convergence is investigated a two dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self adjusting numerical results in one and two dimensions are presented

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

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

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

motion estimation based on mutual information and adaptive multi scale thresholding this paper proposes a new method of calculating a matching metric for motion estimation the proposed method splits the information in the source images into multiple scale and orientation subbands reduces the subband values to a binary representation via an adaptive thresholding algorithm and uses mutual information to model the similarity of corresponding square windows in each image a moving window strategy is applied to recover a dense estimated motion field whose properties are explored the proposed matching metric is a sum of mutual information scores across space scale and orientation this facilitates the exploitation of information diversity in the source images experimental comparisons are performed amongst several related approaches revealing that the proposed matching metric is better able to exploit information diversity generating more accurate motion fields

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

a differential model of the complex cell the receptive fields of simple cells in the visual cortex can be understood as linear filters these filters can be modeled by gabor functions or gaussian derivatives gabor functions can also be combined in an energy model of the complex cell response this letter proposes an alternative model of the complex cell based on gaussian derivatives it is most important to account for the insensitivity of the complex response to small shifts of the image the new model uses a linear combination of the first few derivative filters at a single position to approximate the first derivative filter at a series of adjacent positions the maximum response over all positions gives a signal that is insensitive to small shifts of the image this model unlike previous approaches is based on the scale space theory of visual processing in particular the complex cell is built from filters that respond to the 2d differential structure of the image the computational aspects of the new model are studied in one and two dimensions using the steerability of the gaussian derivatives the response of the model to basic images such as edges and gratings is derived formally the response to natural images is also evaluated using statistical measures of shift insensitivity the neural implementation and predictions of the model are discussed

on bounds of shift variance in two channel multirate filter banks critically sampled multirate fir filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage we assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged we express these changes by a so called commutator we then derive a sharp upper bound for shift variance via the operator norm of the commutator which is independent of the input signal its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator leading to a matrix norm which depends on frequency this bound can be regarded as a worst case instance holding for all input signals for two channel fir filter banks with perfect reconstruction pr we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used moreover the framework allows to identify the signals for which the upper bound is almost reached as so called near maximizers of the frequency dependent matrix norm for unitary pr filter banks these near maximizers are shown to be narrow band signals to complement this worst case bound we derive an additional bound on shift variance for input signals with given amplitude spectra where we use wide band model spectra instead of narrow band signals like the operator norm this additional bound is based on the above frequency dependent matrix norm we provide results for various critically sampled two channel filter banks such as quadrature mirror filters pr conjugated quadrature filters wavelets and biorthogonal filters banks

information theoretic active polygons for unsupervised texture segmentation curve evolution models used in image segmentation and based on image region information usually utilize simple statistics such as means and variances hence can not account for higher order nature of the textural characteristics of image regions in addition the object delineation by active contour methods results in a contour representation which still requires a substantial amount of data to be stored for subsequent multimedia applications such as visual information retrieval from databases polygonal approximations of the extracted continuous curves are required to reduce the amount of data since polygons are powerful approximators of shapes for use in later recognition stages such as shape matching and coding the key contribution of this paper is the development of a new active contour model which nicely ties the desirable polygonal representation of an object directly to the image segmentation process this model can robustly capture texture boundaries by way of higher order statistics of the data and using an information theoretic measure and with its nature of the ordinary differential equations this new variational texture segmentation model is unsupervised since no prior knowledge on the textural properties of image regions is used another contribution in this sequel is a new polygon regularizer algorithm which uses electrostatics principles this is a global regularizer and is more consistent than a local polygon regularization in preserving local features such as corners

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