sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

regular multivariate sampling and approximation in lp shift invariant spaces abstract in this paper some results about regular multivariate generalized sampling in the l p setting 1 p are proven thus stable multivariate regular sampling formulas are derived for l p shift invariant spaces v p i e the l p closure of the linear span of the shifts of a finite set of generators these sampling formulas include regular samples of the function its derivatives and or some filtered versions of the function itself taken at a lattice of r d approximation schemes using these generalized sampling formulas are also included

frames arising from irreducible solvable actions i abstract in this work we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable lie groups in contrast to other well known techniques such as the coorbit theory the generalized coorbit theory and other discretization schemes we make no assumption on the integrability or square integrability of the representations of interest moreover our scheme produces explicit constructions of frames with precise frame bounds as an illustration of the scope of our results we highlight that a large class of representations which naturally occur in wavelet theory and time frequency analysis is handled by our scheme for example the affine group the generalized heisenberg groups the shearlet groups solvable extensions of vector groups and various solvable extensions of non commutative nilpotent lie groups are a few examples of groups whose irreducible representations are handled by our method the class of representations studied in this work is described as follows let g be a simply connected connected completely solvable lie group with lie algebra g p m next let be an infinite dimensional unitary irreducible representation of g obtained by inducing a character from a closed normal subgroup p exp p of g additionally we assume that g p m m exp m is a closed subgroup of g d m is a fixed haar measure on the solvable lie group m and there exists a linear functional p such that the representation ind p g is realized as acting in l 2 m d m making no assumption on the integrability of we describe explicitly a discrete subset of g and a vector f l 2 m d m such that f is a tight frame for l 2 m d m we also construct compactly supported smooth functions s and discrete subsets g such that s is a frame for l 2 m d m

fast surface interpolation using hierarchical basis functions an alternative to multigrid relaxation that is much easier to implement and more generally applicable is presented conjugate gradient descent is used in conjunction with a hierarchical multiresolution set of basis functions the resultant algorithm uses a pyramid to smooth the residual vector before the direction is computed simulation results showing the speed of convergence and its dependence on the choice of interpolator the number of smoothing levels and other factors are presented the relationship of this approach to other multiresolution relaxation and representation schemes is also discussed

noise reduction in surface reconstruction from a given gradient field we present a gradient space technique for noise reduction in surfaces reconstructed from a noisy gradient field we first analyze the error sources in the recovered gradient field of a surface using a three image photometric stereo method based on this analysis we propose an additive noise model to describe the errors in the surface gradient estimates we then use a vector space formulation and construct a multiscale orthonormal expansion for gradient fields using the sparse representation properties of this expansion we develop techniques for reducing the gradient field noise by coefficient selection with thresholding the simulation results indicate that the proposed technique provides significant improvement on the noise levels of both the estimated gradient fields and the reconstructed surfaces under heavy noise levels furthermore the experiments using noisy photometric stereo image triplets of real range data suggest that the additive model remains viable after the nonlinear photometric stereo operation to provide accurate noise removal

stochastic flows on the boundaries of sl n r we study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in sl n r or gl n r these compact spaces are called boundaries of sl n r which include so n the flag manifold the spheresn 1 and the grassmannians the one point motions of these flows are brownian motions for almost every we determine the set of stable points this is a random open set whose complement has zero lebesgue measure the distance between any two points in the same component of this set tends to zero exponentially fast under the flow the lyapunov exponents at stable points are computed explicitly we apply our results to a stochastic flow onsn 2 generated by a stochastic differential equation which exhibits some nice symmetry

average sampling and reconstruction of reproducing kernel signals in mixed lebesgue spaces abstract in this paper we mainly study the average sampling and reconstruction of signals in a reproducing kernel subspace of the mixed lebesgue space l p q r m n first the sampling stability for two kinds of average sampling functionals is considered then the corresponding iterative approximation projection algorithms with exponential convergence are established finally the asymptotic pointwise error estimates are given for reconstructing a signal from its average samples corrupted by additive random noise

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

the computation of visible surface representations a computational theory of visible surface representations is developed the visible surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of 1 integrating multiscale constraints on surface depth and orientation from multiple visual sources 2 interpolating dense piecewise smooth surfaces from these constraints 3 detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation and 4 structuring large scale distributed surface representations to achieve computational efficiency visible surface reconstruction is an inverse problem a well posed variational formulation results from the use of a controlled continuity surface model discontinuity detection amounts to the identification of this generic model s distributed parameters from the data finite element shape primitives yield a local discretization of the variational principle the result is an efficient algorithm for visible surface reconstruction

stability of localized operators let lp 1 p be the space of all p summable sequences and ca be the convolution operator associated with a summable sequence a it is known that the lp stability of the convolution operator ca for different 1 p are equivalent to each other i e if ca has lp stability for some 1 p then ca has lq stability for all 1 q in the study of spline approximation wavelet analysis time frequency analysis and sampling there are many localized operators of non convolution type whose stability is one of the basic assumptions in this paper we consider the stability of those localized operators including infinite matrices in the sjostrand class synthesis operators with generating functions enveloped by shifts of a function in the wiener amalgam space and integral operators with kernels having certain regularity and decay at infinity we show that the lp stability or lp stability of those three classes of localized operators are equivalent to each other and we also prove that the left inverse of those localized operators are well localized

surface from gradients without discrete integrability enforcement a gaussian kernel approach representative surface reconstruction algorithms taking a gradient field as input enforce the integrability constraint in a discrete manner while enforcing integrability allows the subsequent integration to produce surface heights existing algorithms have one or more of the following disadvantages they can only handle dense per pixel gradient fields smooth out sharp features in a partially integrable field or produce severe surface distortion in the results in this paper we present a method which does not enforce discrete integrability and reconstructs a 3d continuous surface from a gradient or a height field or a combination of both which can be dense or sparse the key to our approach is the use of kernel basis functions which transfer the continuous surface reconstruction problem into high dimensional space where a closed form solution exists by using the gaussian kernel we can derive a straightforward implementation which is able to produce results better than traditional techniques in general an important advantage of our kernel based method is that the method does not suffer discretization and finite approximation both of which lead to surface distortion which is typical of fourier or wavelet bases widely adopted by previous representative approaches we perform comparisons with classical and recent methods on benchmark as well as challenging data sets to demonstrate that our method produces accurate surface reconstruction that preserves salient and sharp features the source code and executable of the system are available for downloading

generalized coorbit space theory and inhomogeneous function spaces of besov lizorkin triebel type coorbit space theory is an abstract approach to function spaces and their atomic decom positions the original theory developed by feichtinger and grochenig in the late 1980ies heavily uses integrable representations of locally compact groups their theory covers in particular homogeneous besov lizorkin triebel spaces modulation spaces bergman spaces and the recent shearlet spaces however inhomogeneous besov lizorkin triebel spaces cannot be covered by their group theoretical approach later it was recognized by fornasier and the first named author 24 that one may replace coherent states related to the group representation by more general abstract continuous frames in the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces a unified approach towards atomic decompositions and banach frames with new results for general coorbit spaces is presented in the second part we apply the abstract setting to a specific framework and study coorbits of what we call peetre spaces they allow to recover inhomogeneous besov lizorkin triebel spaces of var ious types of interest as coorbits we obtain several old and new wavelet characterizations based on explicit smoothness decay and vanishing moment assumptions of the respective wavelet as main examples we obtain results for weighted spaces muckenhoupt doubling general 2 microlocal spaces besov lizorkin triebel morrey spaces spaces of dominating mixed smoothness and even mixtures of the mentioned ones due to the generality of our approach there are many more examples of interest where the abstract coorbit space theory is applicable

sampling and exact reconstruction of bandlimited signals with additive shot noise in this correspondence we consider sampling continuous time periodic bandlimited signals which contain additive shot noise the classical sampling scheme does not perfectly recover these particular nonbandlimited signals but only reconstructs a lowpass filtered approximation by modeling the shot noise as a stream of dirac pulses we first show that the sum of a bandlimited signal with a stream of dirac pulses falls into the class of signals that contain a finite rate of innovation that is a finite number of degrees of freedom second by taking into account the degrees of freedom of the bandlimited signal in the sampling and reconstruction scheme developed previously for streams of dirac pulses we derive a sampling and perfect reconstruction scheme for the bandlimited signal with additive shot noise

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

banach spaces of distributions having two module structures abstract after a discussion of a space of test functions and the corresponding space of distributions a family of banach spaces b b in standard situation is described these are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution the main results concern relations between the different spaces associated to b established by means of well known methods from the theory of banach modules among them b0 and b the closure of the test functions in b and the weak relative completion of b respectively the latter is shown to be always a dual banach space the main diagram given in theorem 4 7 gives full information concerning inclusions between these spaces showing also a complete symmetry a great number of corresponding formulas is established how they can be applied is indicated by selected examples in particular by certain segal algebras and the ap algebras of herz various further applications are to be given elsewhere

wiener s lemma for infinite matrices in this paper we introduce a class of infinite matrices related to the beurling algebra of periodic functions and we show that it is an inverse closed subalgebra of mathcal b ell q w the algebra of all bounded linear operators on the weighted sequence space ell q w for any 1 q and any discrete muckenhoupt aq weight w

fast facial shape recovery from a single image with general unknown lighting by using tensor representation in this paper we propose a fast 3 d facial shape recovery algorithm from a single image with general unknown lighting in order to derive the algorithm we formulate a nonlinear least square problem with two parameter vectors which are related to personal identity and light conditions we then combine the spherical harmonics for the surface normals of a human face with tensor algebra and show that in a certain condition the dimensionality of the least square problem can be further reduced to one tenth of the regular subspace based model by using tensor decomposition n mode svd which greatly speeds up the computations in order to enhance the shape recovery performance we have incorporated prior information in updating the parameters in the experiment the proposed algorithm takes less than 0 4s to reconstruct a face and shows a significant performance improvement over other reported schemes

sampling theorems on bounded domains this paper concerns with iterative schemes for the perfect reconstruction from nonuniform sampling of functions belonging to multiresolution spaces on bounded manifolds since the iterations converge uniformly we can produce the corresponding iterative integration schemes that allow to recover the integral of functions belonging to multiresolution spaces from nonuniform sampling we present also an error analysis and in particular we estimate the l 2 error we produce in recovering smooth functions in h s but not necessarily in any multiresolution space the error analysis extends to integrals our results hold regardless of the dimension of the domain and for a variety of multiresolution spaces constructed from certain refinable bases formed by the so called gp functions

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

direct analytical methods for solving poisson equations in computer vision problems direct analytical methods are discussed for solving poisson equations of the general form delta u f on a rectangular domain some embedding techniques that may be useful when boundary conditions obtained from stereo and occluding boundary are defined on arbitrary contours are described the suggested algorithms are computationally efficient owing to the use of fast orthogonal transforms applications to shape from shading lightness and optical flow problems are also discussed a proof for the existence and convergence of the flow estimates is given experiments using synthetic images indicate that results comparable to those using multigrid can be obtained in a very small number of iterations

an almost periodic noncommutative wiener s lemma we develop a theory of almost periodic elements in banach algebras and present an abstract version of a noncommutative wiener s lemma the theory can be used for example to derive some of the recently obtained results in time frequency analysis such as the spectral properties of the finite linear combinations of time frequency shifts

asymptotic behaviour of stochastic flows of diffeomorphisms two case studies the two stochastic flows studied are i the canonical stochastic flow on the orthonormal frame bundle of hyperbolic space which gives stochastic parallel translation along brownian paths in hyperbolic space and ii a stochastic flow on the spheres n 1 arising from its embedding as the unit sphere in n both flows are controlled by the same stochastic differential equation in a finite dimensional lie group in each case the lyapunov exponents are computed and a complete description is given of the local and global stability of the flow

sampling and quasi optimal approximation for signals in a reproducing kernel space of homogeneous type abstract this paper considers the reconstruction of signals in a reproducing kernel space of homogeneous type from finite samples first a pre reconstruction operator based on finite samples and probability measure is proposed and its bounded property is studied secondly the stability and an iterative algorithm with exponential convergence are established for sampling and recovering signals in a subspace of homogeneous reproducing kernel space then we show that the proposed algorithm also provides a quasi optimal approximation to signals in a reproducing kernel space of homogeneous type finally some numerical simulations are given to reconstruct signals on an interval

characterization of coorbit spaces with phase space covers abstract we show that coorbit spaces can be characterized in terms of arbitrary phase space covers which are families of phase space multipliers associated with partitions of unity this generalizes previously known results for time frequency analysis to include time scale decompositions as a by product we extend the existing results for time frequency analysis to an irregular setting

error analysis of frame reconstruction from noisy samples this paper addresses the problem of reconstructing a continuous function defined on rd from a countable collection of samples corrupted by noise the additive noise is assumed to be i i d with mean zero and variance sigma2 we sample the continuous function on the uniform lattice 1 m zd and show for large enough m that the variance of the error between the frame reconstruction fepsiv m from noisy samples of f and the function f satisfy var fepsiv m x f x ap sigma2 md cx where cx is the best constant for every x isin rd we also prove a similar result in the case that our data are weighted average samples of corrupted by additive noise

slanted matrices banach frames and sampling in this paper we present a rare combination of ab stract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sam pling problems we show that for a large class of slanted matrices boundedness below of the corresponding operator in p for some p implies boundedness below in p for all p we use the established result to enrich our understanding of banach frames and obtain new results for irregular sampling problems we also present a version of a non commutative wiener s lemma for slanted matri ces

iterative actions of normal operators abstract let a be a normal operator in a hilbert space h and let g h be a countable set of vectors we investigate the relations between a g and l that make the system of iterations a n g g g 0 n l g complete bessel a basis or a frame for h the problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis including frame theory and spectral theory it also has relations to topics in applied harmonic analysis including wavelet theory and time frequency analysis

maximum likelihood estimation of depth maps using photometric stereo photometric stereo and depth map estimation provide a way to construct a depth map from images of an object under one viewpoint but with varying illumination directions while estimating surface normals using the lambertian model of reflectance is well established depth map estimation is an ongoing field of research and dealing with image noise is an active topic using the zero mean gaussian model of image noise this paper introduces a method for maximum likelihood depth map estimation that accounts for the propagation of noise through all steps of the estimation process solving for maximum likelihood depth map estimates involves an independent sequence of nonlinear regression estimates one for each pixel followed by a single large and sparse linear regression estimate the linear system employs anisotropic weights which arise naturally and differ in value to related work the new depth map estimation method remains efficient and fast making it practical for realistic image sizes experiments using synthetic images demonstrate the method s ability to robustly estimate depth maps under the noise model practical benefits of the method on challenging imaging scenarios are illustrated by experiments using the extended yale face database b and an extensive data set of 500 reflected light microscopy image sequences

symmetry seeking models and 3d object reconstruction we propose models of 3d shape which may be viewed as deformable bodies composed of simulated elastic material in contrast to traditional purely geometric models of shape deformable models are active their shapes change in response to externally applied forces we develop a deformable model for 3d shape which has a preference for axial symmetry symmetry is represented even though the model does not belong to a parametric shape family such as generalized cylinders rather a symmetry seeking property is designed into internal forces that constrain the deformations of the model we develop a framework for 3d object reconstruction based on symmetry seeking models instances of these models are formed from monocular image data through the action of external forces derived from the data the forces proposed in this paper deform the model in space so that the shape of its projection into the image plane is consistent with the 2d silhouette of an object of interest the effectiveness of our approach is demonstrated using natural images

a method for enforcing integrability in shape from shading algorithms an approach for enforcing integrability a particular implementation of the approach an example of its application to extending an existing shape from shading algorithm and experimental results showing the improvement that results from enforcing integrability are presented a possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes the integrability projection constraint was applied to extending an iterative shape from shading algorithm of m j brooks and b k p horn 1985 experimental results show that the extended algorithm converges faster and with less error than the original version good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces

pseudodifferential operators on locally compact abelian groups and sjoestrand s symbol class we investigate pseudodifferential operators on arbitrary locally compact abelian groups as symbol classes for the kohn nirenberg calculus we introduce a version of sjoestrand s class pseudodifferential operators with such symbols form a banach algebra that is closed under inversion since hard analysis techniques are not available on locally compact abelian groups a new time frequency approach is used with the emphasis on modulation spaces gabor frames and banach algebras of matrices sjoestrand s original results are thus understood as a phenomenon of abstract harmonic analysis rather than hard analysis and are proved in their natural context and generality

depth analogy data driven approach for single image depth estimation using gradient samples inferring scene depth from a single monocular image is a highly ill posed problem in computer vision this paper presents a new gradient domain approach called depth analogy that makes use of analogy as a means for synthesizing a target depth field when a collection of rgb d image pairs is given as training data specifically the proposed method employs a non parametric learning process that creates an analogous depth field by sampling reliable depth gradients using visual correspondence established on training image pairs unlike existing data driven approaches that directly select depth values from training data our framework transfers depth gradients as reconstruction cues which are then integrated by the poisson reconstruction the performance of most conventional approaches relies heavily on the training rgb d data used in the process and such a dependency severely degenerates the quality of reconstructed depth maps when the desired depth distribution of an input image is quite different from that of the training data e g outdoor versus indoor scenes our key observation is that using depth gradients in the reconstruction is less sensitive to scene characteristics providing better cues for depth recovery thus our gradient domain approach can support a great variety of training range datasets that involve substantial appearance and geometric variations the experimental results demonstrate that our depth gradient domain approach outperforms existing data driven approaches directly working on depth domain even when only uncorrelated training datasets are available

nonideal sampling and interpolation from noisy observations in shift invariant spaces digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous time signal from a sequence of corrupted discrete time samples in this paper a general formulation of this problem is developed that treats the interpolation problem from ideal noisy samples and the deconvolution problem in which the signal is filtered prior to sampling in a unified way the signal reconstruction is performed in a shift invariant subspace spanned by the integer shifts of a generating function where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter several alternative approaches to designing the correction filter are suggested which differ in their assumptions on the signal and noise the classical deconvolution solutions least squares tikhonov and wiener are adapted to our particular situation and new methods that are optimal in a minimax sense are also proposed the solutions often have a similar structure and can be computed simply and efficiently by digital filtering some concrete examples of reconstruction filters are presented as well as simple guidelines for selecting the free parameters e g regularization of the various algorithms

memory estimation of inverse operators we use methods of harmonic analysis and group rep resentation theory to estimate memory decay of the inverse oper ators in banach spaces the memory of the operators is defined using the notion of the beurling spectrum we obtain a general continuous non commutative version of the celebrated wiener s tauberian lemma with estimates of the fourier coefficients of inverse operators in particular we generalize various estimates of the elements of the inverse matrices the results are illus trated with a variety of examples including integral and integro differential operators

perturbed sampling formulas and local reconstruction in shift invariant spaces abstract let v be a closed subspace of l 2 r generated from the integer shifts of a continuous function with a certain decay at infinity and a non vanishing property for the function n z n e i n on in this paper we determine a positive number so that the set n n n z is a set of stable sampling for the space v for any selection of the elements n within the ranges we demonstrate the resulting sampling formula called perturbation formula for functions f v and also we establish a finite reconstruction formula approximating f on bounded intervals we compute the corresponding error and we provide estimates for the jitter error as well

shannon sampling and function reconstruction from point values i first met rene at the well known 1956 meeting on topology in mexico city he then came to the university of chicago where i was starting my job as instructor for the fall of 1956 he suzanne clara and i became good friends and saw much of each other for many decades especially at ihes in paris thom s encouragement and support were important for me especially in my first years after my ph d i studied his work in cobordism singularities of maps and transversality gaining many insights i also enjoyed listening to his provocations for example his disparaging remarks on complex analysis 19th century mathematics and bourbaki there was also a stormy side in our relationship neither of us could hide the pain that our public conflicts over catastrophe theory caused rene thom was a great mathematician leaving his impact on a wide part of mathematics i will always treasure my memories of him steve smale

polynomial control on stability inversion and powers of matrices on simple graphs abstract spatially distributed networks of large size arise in a variety of science and engineering problems such as wireless sensor networks and smart power grids most of their features can be described by properties of their state space matrices whose entries have indices in the vertex set of a graph in this paper we introduce novel algebras of beurling type that contain matrices on a connected simple graph having polynomial off diagonal decay and we show that they are banach subalgebras of b l p 1 p the space of all bounded operators on the space l p of all p summable sequences the l p stability of state space matrices is an essential hypothesis for the robustness of spatially distributed networks in this paper we establish the equivalence among l p stabilities of matrices in beurling algebras for different exponents 1 p with quantitative analysis for the lower stability bounds admission of norm control inversion plays a crucial role in some engineering practice in this paper we prove that matrices in beurling subalgebras of b l 2 have norm controlled inversion and we find a norm controlled polynomial with close to optimal degree polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks in this paper we apply our results on norm controlled inversion to obtain a polynomial estimate to powers of matrices in beurling algebras the polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary markov chain on a spatially distributed network

coherent state lattices and square integrability of representations in this paper we discretize the continuous theory of coherent states on a general semidirect product group g v x s where v is a vector space and s subset of gl v is a semisimple connected lie group we show that it is always possible to construct a discrete frame associated with a unitary irreducible representation of g which is square integrable modulo an affine admissible section we also prove the converse result that the existence of a discrete frame associated with a unitary representation u of g indexed by a homogeneous space gamma implies the square integrability of u on gamma

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

distributed multi parametric quadratic programming one of the fundamental problems in the area of large scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints such problems can motivate the development of fast and well conditioned distributed algorithms in this paper we study spatial locality features of large scale multi parametric quadratic programming mpqp problems with linear inequality constraints our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints we propose a new approach for analysis of large scale mpqp problems by blending tools from duality theory with operator theory the class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints we show that the optimal solution of a convex mpqp is piecewise affine represented as convolution sums more importantly we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem

deformation of gabor systems abstract we introduce a new notion for the deformation of gabor systems such deformations are in general nonlinear and in particular include the standard jitter error and linear deformations of phase space with this new notion we prove a strong deformation result for gabor frames and gabor riesz sequences that covers the known perturbation and deformation results our proof of the deformation theorem requires a new characterization of gabor frames and gabor riesz sequences it is in the style of beurling s characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of gabor frames without inequalities from lattices to non uniform sets

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

time sampling and reconstruction in weighted reproducing kernel subspaces abstract this paper mainly studies the time sampling and reconstruction of signals in weighted reproducing kernel subspaces of l p firstly a class of examples for weighted reproducing kernel subspaces are given then using samples produced by a crossing time encoding machine or an integrate and fire time encoding machine satisfying some density constraints iterative approximation algorithms are established for perfect recovery of signals in weighted reproducing kernel subspaces finally an integrate and fire sampling scheme is discussed and an approximate reconstruction procedure is proposed for signals in weighted reproducing kernel subspaces moreover we give detailed analysis for the approximate reconstruction method in the setting of shift invariant subspaces

sampling and reconstruction of signals with finite rate of innovation in the presence of noise recently it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals namely certain signals of finite rate of innovation a common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples in order to prove sampling theorems vetterli et al considered the case of deterministic noiseless signals and developed algebraic methods that lead to perfect reconstruction however when noise is present many of those schemes can become ill conditioned in this paper we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction we analyze in detail a signal made up of a stream of diracs and develop algorithmic tools that will be used as a basis in all constructions while some of the techniques have been already encountered in the spectral estimation framework we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components for classes of periodic signals such as piecewise polynomials and nonuniform splines we propose novel algebraic approaches that solve the sampling problem in the laplace domain after appropriate windowing building on the results for periodic signals we extend our analysis to finite length signals and develop schemes based on a gaussian kernel which avoid the problem of ill conditioning by proper weighting of the data matrix our methods use structured linear systems and robust algorithmic solutions which we show through simulation results

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

symmetry and inverse closedness of matrix algebras and functional calculus for infinite matrices we investigate the symbolic calculus for a large class of matrix algebras that are defined by the off diagonal decay of infinite matrices applications are given to the symmetry of some highly non commutative banach algebras to the analysis of twisted convolution and to the theory of localized frames

reconstructing surfaces by volumetric regularization using radial basis functions we present a new method of surface reconstruction that generates smooth and seamless models from sparse noisy nonuniform and low resolution range data data acquisition techniques from computer vision such as stereo range images and space carving produce 3d point sets that are imprecise and nonuniform when compared to laser or optical range scanners traditional reconstruction algorithms designed for dense and precise data do not produce smooth reconstructions when applied to vision based data sets our method constructs a 3d implicit surface formulated as a sum of weighted radial basis functions we achieve three primary advantages over existing algorithms 1 the implicit functions we construct estimate the surface well in regions where there is little data 2 the reconstructed surface is insensitive to noise in data acquisition because we can allow the surface to approximate rather than exactly interpolate the data and 3 the reconstructed surface is locally detailed yet globally smooth because we use radial basis functions that achieve multiple orders of smoothness

normal estimation of a transparent object using a video reconstructing transparent objects is a challenging problem while producing reasonable results for quite complex objects existing approaches require custom calibration or somewhat expensive labor to achieve high precision when an overall shape preserving salient and fine details is sufficient we show in this paper a significant step toward solving the problem when the object s silhouette is available and simple user interaction is allowed by using a video of a transparent object shot under varying illumination specifically we estimate the normal map of the exterior surface of a given solid transparent object from which the surface depth can be integrated our technical contribution lies in relating this normal estimation problem to one of graph cut segmentation unlike conventional formulations however our graph is dual layered since we can see a transparent object s foreground as well as the background behind it quantitative and qualitative evaluation are performed to verify the efficacy of this practical solution

the noncommutative wiener lemma linear independence and spectral properties of the algebra of time frequency shift operators in this paper we analyze the banach algebra of time frequency shifts with absolutely summable coecients we prove a noncommutative version of the wiener lemma we also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators as a corollary we obtain a special case of the heil ramanathan topiwala conjecture regarding linear independence of nitely many time frequency shifts of one l 2 function we also estimate the coecient decay of the inverse of nite linear combinations of time frequency shifts

generalized metaplectic operators and the schrodinger equation with a potential in the sjostrand class it is well known that the matrix of a metaplectic operator with respect to phase space shifts is concentrated along the graph of a linear symplectic map we show that the algebra generated by metaplectic operators and by pseudodifferential operators in a sjostrand class enjoys the same decay properties we study the behavior of these generalized metaplectic operators and represent them by fourier integral operators our main result shows that the one parameter group generated by a hamiltonian operator with a potential in the sjostrand class consists of generalized metaplectic operators as a consequence the schrodinger equation preserves the phase space concentration as measured by modulation space norms

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

geometric dynamics of a harmonic oscillator arbitrary minimal uncertainty states and the smallest step 3 nilpotent lie group the paper presents a new method of geometric solution of a schrodinger equation by constructing an equivalent first order partial differential equation with a bigger number of variables the equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform the method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three lie group the group also known under many names e g quartic group we obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector in contrast it is shown that the well known fock segal bargmann transform and the heisenberg group require a specific fiducial vector to produce a geometric solution a technical aspect considered in this paper is that a certain modification of a coherent state transform is required although the irreducible representation of the group is square integrable modulo a subgroup the obtained dynamic is transverse to the homogeneous space

weighted sampling and reconstruction in weighted reproducing kernel spaces in this paper we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator we show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively separated set with sufficiently small gap we also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively separated set with sufficiently small gap

left inverses of matrices with polynomial decay abstract it is known that the algebra of schur operators on l 2 namely operators bounded on both l 1 and l is not inverse closed when l 2 l 2 x where x is a metric space one can consider elements of the schur algebra with certain decay at infinity for instance if x has the doubling property then q sun has proved that the weighted schur algebra a x for a strictly polynomial weight is inverse closed in this paper we prove a sharp result on left invertibility of the these operators namely if an operator a a x satisfies a f p f p for some 1 p then it admits a left inverse in a x the main difficulty here is to obtain the above inequality in l 2 the author was both motivated and inspired by a previous work of aldroubi baskarov and krishtal 2008 1 where similar results were obtained through different methods for x z d under additional conditions on the decay