moment conditions and support theorems for radon transforms on affine grassmann manifolds abstract let g p n and g q n be the affine grassmann manifolds of p and q planes in r n respectively and let r p q be the radon transform from smooth functions on g p n to smooth functions on g q n arising from the inclusion incidence relation when p q and dim g p n dim g p n we present a range characterization theorem for r p q via moment conditions we then use this range result to prove a support theorem for r p q this complements a previous range characterization theorem for r p q via differential equations when dim g p n dim g p n we also present a support theorem in this latter case

the invertibility of rotation invariant radon transforms let r denote the radon transform on rn that integrates a function over hyperplanes in given smooth positive measures depending on the hyperplane we characterize the measures for which r is rotation invariant we prove rotation invariant transforms are all one to one and hence invertible on the domain of square integrable functions of compact support l02 rn we prove the hole theorem if f l02rn and r f 0 for hyperplanes not intersecting a ball centered at the origin then f is zero outside of that ball using the theory of fourier integral operators we extend these results to the domain of distributions of compact support on rn our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated radon transform as well as invertibility for other radon transforms

learning overcomplete representations in an overcomplete basis the number of basis vectors is greater than the dimensionality of the input and the representation of an input is not a unique combination of basis vectors overcomplete representations have been advocated because they have greater robustness in the presence of noise can be sparser and can have greater flexibility in matching structure in the data overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex previous work has focused on finding the best representation of a signal using a fixed overcomplete basis or dictionary we present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data we show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency this can be viewed as a generalization of the technique of independent component analysis and provides a method for bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures

use of radon s projection theory in electromagnetic inverse scattering in a wide variety of electromagnetic profile reconstruction or shape imaging techniques a need often arises to deduce the three two or one dimensional distribution of different physical quantities from their projections e g in radio astronomy structural biology roentgenology geophysics and also in electromagnetic imaging investigation of such problems in various specialized areas resulted in the establishment of the new interdisciplinary subject known as reconstruction from projections the underlying theory which was first rigorously formulated by radon will become of increasing importance to radar target mapping wave imaging and related electromagnetic inverse problems therefore a tutorial exposition is timely and well suited for this issue major emphasis will be placed on showing how ludwig s theorems on support determinacy and self consistency can be used favorably to analyze the data limited reconstruction cases the derived theorems will be applied to the specific case of physical optics far field inverse scattering clearly proving that the bojarski and the kennaugh identities constitute a fourier radon transform pair

the three dimensional steady state thermo elastodynamic problem of moving sources over a half space abstract a procedure based on the radon transform and elements of distribution theory is developed to obtain fundamental thermoelastic three dimensional 3d solutions for thermal and or mechanical point sources moving steadily over the surface of a half space a concentrated heat flux is taken as the thermal source whereas the mechanical source consists of normal and tangential concentrated loads it is assumed that the sources move with a constant velocity along a fixed direction the solutions obtained are exact within the bounds of biot s coupled thermo elastodynamic theory and results for surface displacements are obtained over the entire speed range i e for sub rayleigh super rayleigh subsonic transonic and supersonic source speeds this problem has relevance to situations in contact mechanics tribology and dynamic fracture and is especially related to the well known heat checking problem thermo mechanical cracking in an unflawed half space material from high speed asperity excitations our solution technique fully exploits as auxiliary solutions the ones for the corresponding plane strain and anti plane shear problems by reducing the original 3d problem to two separate 2d problems these problems are uncoupled from each other with the first problem being thermoelastic and the second one pure elastic in particular the auxiliary plane strain problem is completely analogous to the original problem not only with regard to the field equations but also with regard to the boundary conditions this makes the technique employed here more advantageous than other techniques which require the prior determination of a fictitious auxiliary plane strain problem through solving an integral equation

optimal threshold selection for segmentation of dense homogeneous objects in tomographic reconstructions in this paper we present a novel approach to segment dense homogeneous objects in a tomographic reconstruction or tomogram a popular method to extract such objects from a tomogram is global thresholding in which the threshold value is determined from the image histogram however accurate threshold selection is not straightforward since due to noise or artefacts in the reconstruction the histogram does not always contain a clear separate peak for the dense object we propose a new threshold estimation approach segmentation inconsistency minimization that exploits the available projection data to determine the optimal global threshold the proposed algorithm was tested on simulation data and on experimental ct data the results show that this method results in more accurate segmentations compared to alternative threshold selection methods

reconstruction of a three dimensional object from a limited range of views abstract we define the radon transform r and the back projection r adjoint of r on the space of c rapidly decreasing functions j and on the space of distributions of compact support e we show that r r is a convolution operator the result is true in the case of full views as well as in the case of a limited range of views an application to the medical image reconstruction is given

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

cancellation and direct summands in dimension 1 abstract let be a module finite algebra over a commutative noetherian ring of krull dimension 1 we extend roiter s direct summand theorem to arbitrary finitely generated modules obtaining a sharpened form of serre s direct summand theorem in this setting we also extend drozd s cancellation theorem to arbitrary finitely generated modules obtaining a sharpened form of bass s cancellation theorem in this setting a corollary is that over commutative reduced noetherian rings of dimension 1 direct sum cancellation holds in every genus of finitely generated modules this becomes false if the ring has nilpotent elements another corollary is that if direct sum cancellation holds in the genera of modules m and n then it holds in the genus of m n this seems to be new even for the modules that occur in integral representation theory the main thrust of this paper is to close the gap between integral representation theory and the rest of module theory by eliminating hypotheses concerning the existence of maximal orders of finite normalization in the commutative case and allowing our rings to have nilpotent ideals

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

linearizing non linear inverse problems and an application to inverse backscattering we propose an abstract approach to prove local uniqueness and conditional holder stability to non linear inverse problems by linearization the main condition is that in addition to the injectivity of the linearization a we need a stability estimate for a as well that condition is satisfied in particular if a a is an elliptic pseudo differential operator we apply this scheme to show uniqueness and holder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed

the non homogeneous biharmonic plate equation fundamental solutions abstract real materials and structural components are often non homogeneous either by design or because of the physical composition and imperfections in the underlying material thus analytical solutions for non homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications given the prevalence of these materials in fields as diverse as aerospace construction electronics etc more precisely those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties such as optimal arrangement of the material minimum weight etc to this end the displacement fundamental solution or green s function corresponding to a point force for the non homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the radon transformation these functions besides being useful in their own right can also be used within the context of integral equation formulations for the solution of boundary value problems finally a series of numerical examples that deal with the non homogeneous plate on elastic foundation problem serve to illustrate the present method

inversion of the radon transform on the laguerre hypergroup by using generalized wavelets abstract we consider the radon transform r 0 on the laguerre hypergroup k 0 r we characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that r is a bijection from this space onto itself we establish an inversion formula and a plancherel theorem for the operator r finally by using the continuous wavelet transform on the laguerre hypergroup k we deduce another expression for the inverse r 1 of the operator r

discriminative feature representation to improve projection data inconsistency for low dose ct imaging in low dose computed tomography ldct imaging the data inconsistency of measured noisy projections can significantly deteriorate reconstruction images to deal with this problem we propose here a new sinogram restoration approach the sinogram discriminative feature representation s dfr method different from other sinogram restoration methods the proposed method works through a 3 d representation based feature decomposition of the projected attenuation component and the noise component using a well designed composite dictionary containing atoms with discriminative features this method can be easily implemented with good robustness in parameter setting its comparison to other competing methods through experiments on simulated and real data demonstrated that the s dfr method offers a sound alternative in ldct

multidimensional born velocity inversion single wideband point source the inverse scattering problem for an acoustic medium is considered within the approximate direct inversion framework as opposed to iterative methods the direct inversion approach gives an estimate of the medium velocities by operating on the observed scattered field without repeated solutions of the forward problem previous solutions to the multidimensional born velocity inversion problem require either coincident source and receiver arrays or plane wave sources in both cases an array of point sources is required to collect the proper data for inversion in this paper the solution to the single point source problem is derived it is shown that by extrapolating and imaging the observed scattered field appropriately the projections of the velocity function at all angles can be obtained the velocities are then reconstructed by the inverse radon transform method of tomography

on the range of the attenuated radon transform in strictly convex sets we present new necessary and sufficient conditions for a function on s 1 to be in the range of the attenuated radon transform of a sufficiently smooth function support in the convex set r 2 the approach is based on an explicit hilbert transform associated with traces of the boundary of a analytic functions in the sense of bukhgeim in this paper we are concerned with the range characterization of the attenuated radon transform of function of compact support in the plane necessary and sufficient constraints on range of the non att enuated clas sical radon transform in the euclidean space have been known since the works in 6 7 and 10 these constraints known as the cavalieri or the moment conditions are in terms of the angular variable for function in the schwartz class they are essentially unique due to a paley wiener type the orem moreover the helgason support theorem extends the conditions to smooth functions of compact support 8 however in the case of functions of compact support it is possible to obtain essentially dif ferent range con ditions since more than one operator can annihilate functions of compact support in the range of the radon transform the results here constitute one such example inversion methods of the attenuated radon transform in the plane ap peared first in 1 and 14 and various developments can be found in 13 3 5 2 the interest in range conditions stems out fro m their appli cations to data enhancement in medical imaging methods such as single photon or positron emission computed tomography 12 for the eu clidean attenuated radon transform some range characterization based on the inversion procedure in 14 can be found in 15 these constraints are also in terms of the angular variable different from the existing results above our new characterization is in terms of a hilbert transform associated with the a analytic maps a la

hough transform from the radon transform an appropriate special case of a transform developed by j radon in 1917 is shown to have the major properties of the hough transform which is useful for finding line segments in digital pictures such an observation may be useful in further efforts to generalize the hough transform techniques for applying the radon transform to lines and pixels are developed through examples and the appropriate generalization to arbitrary curves is discussed

statistical projection completion in x ray ct using consistency conditions projection data incompleteness arises in many situations relevant to x ray computed tomography ct imaging we propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the helgason ludwig hl consistency conditions to accommodate projection data incompleteness image reconstruction is performed by the filtered backprojection fbp in a second step in our problem formulation the objective function consists of the log likelihood of the x ray ct data and a penalty term the hl condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast fourier transform fft and inverse fft we derive an iterative algorithm that increases the objective function monotonically the proposed algorithm is applied to both computer simulated data and real patient data we study different factors in the problem formulation that affect the properties of the final fbp reconstructed images including the data truncation level the amount of prior knowledge on the object support as well as different approximations of the statistical distribution of the available projection data we also compare its performance with an analytical truncation artifacts reduction method the proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field of view and to a certain extent recovers the truncated peripheral region of the object the proposed method may also be applied in areas such as limited angle tomography metal artifacts reduction and sparse sampling imaging

null spaces of radon transforms we obtain new descriptions of the null spaces of sev eral projectively equivalent transforms in integral geometry the paper deals with the hyperplane radon transform the totally ge odesic transforms on the sphere and the hyperbolic space the spherical slice transform and the cormack quinto spherical mean transform for spheres through the origin the consideration ex tends to the corresponding dual transforms and the relevant exte rior interior modifications the method relies on new results for the gegenbauer chebyshev integrals which generalize abel type fractional integrals on the positive half line

lens rigidity for manifolds with hyperbolic trapped set for a riemannian manifold m g with strictly convex boundary m the lens data consists in the set of lengths of geodesics with endpoints on m together with their endpoints x x m m and tangent exit vectors v v t x m t x m we show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points a class which contains all manifolds with negative curvature and strictly convex boundary including those with non trivial topology and trapped geodesics for the same class of manifolds in dimension 2 we prove that the set of endpoints and exit vectors of geodesics ie the scattering data determines the riemann surface up to conformal diffeomorphism

stability estimates for the calderon problem with partial data this is a follow up of a previous article where we proved local stability estimates for a potential in a schr odinger equation on an open bounded set in dimension n 3 from the dirichlet to neumann map with partial data the region under control was the penumbra delimited by a source of light outside of the convex hull of the open set these local estimates provided stability of log log type corresponding to the uniqueness results in calder on s inverse problem with partial data proved by kenig sj ostrand and uhlmann in this article we prove the corresponding global estimates in all dimensions higher than three the estimates are based on the construction of solutions of the schr odinger equation by complex geometrical optics developed in the anisotropic setting by dos santos ferreira kenig salo and uhlmann to solve the calder on problem in certain admissible geometries

analysis and synthesis of multicomponent signals using positive time frequency distributions a new approach to the analysis and reconstruction of multicomponent nonstationary signals from their time frequency distribution tfd is presented specifically we consider a tfd based on the recently introduced minimum cross entropy principle mce this positive tfd is cross terms free and hence has an advantage over the family of bilinear distributions based on the mce tfd a new algorithm for reconstructing the phase and amplitude parameters of each component of the signal is developed to evaluate the accuracy of the algorithm monte carlo simulations are presented and compared with the corresponding cramer rao bound it is shown that the new algorithm is superior to presently available methods in both efficiency and performance it is concluded that together with the mce tfd representation the proposed approach provides a powerful tool for analysis of nonstationary multicomponent signals embedded in additive gaussian noise

stable isomorphism of modules over one dimensional rings abstract two r modules m and n are said to be stably isomorphic provided m r n n r n for some n 1 in this paper r is always a commutative reduced one dimensional noetherian ring with finite normalization r and m and n are torsionfree all modules are assumed to be finitely generated there is a natural action of r c the group of units of r c where c is the conductor of r in r on torsionfree r modules and the main theorem of this paper is that the orbits of the induced action of r c are exactly the stable isomorphism classes as applications we show that stably isomorphic modules are actually isomorphic if r is a domain finitely generated as an r algebra with at most one singular real maximal ideal we give several examples of modules that are stably isomorphic but not isomorphic illustrating that these hypotheses cannot be significantly weakened we are grateful to l s levy and the referee for several suggestions that have improved the exposition of this work

unification of the inverse radon transform in odd and even dimensions it is well known that the inverse radon transform exists in two forms for odd and even dimension functions respectively this note shows that these formulas become zero when the dimension requirement is violated which results in a simple unified formula for the inverse radon transform of arbitrary dimensions

the spherical curl transform of a linear force free magnetic field the description of linear force free magnetic fields in terms of the moses eigenfunctions of the curl operator begun previously is here completed by the derivation of a general expression for the field s spherical curl transform this enables the transform space representation of a given field to be determined and compared with that of other fields assisting the analysis and classification of this type of magnetic field as well as providing a basis for generalization the result obtained gives the spherical curl transform as a weighted projection of the vector radon transform of the field on the appropriate curl eigenvector the process is exemplified by the determination of the transforms of three fields the simplest force free magnetic field and the lundquist and classical spheromak fields the latter two are both of interest as models of the magnetic fields of solar magnetic clouds while the classical spheromak field is relevant to the design of nuclear fusion reactors as well the use of the transform in generalizing the lundquist field is briefly discussed as before all results apply equally well to the description of the trkalian subset of beltrami fields in fluid dynamics

radon transformation on reductive symmetric spaces support theorems abstract we introduce a class of radon transforms for reductive symmetric spaces including the horospherical transforms and derive support theorems for these transforms a reductive symmetric space is a homogeneous space g h for a reductive lie group g of the harish chandra class where h is an open subgroup of the fixed point subgroup for an involution on g let p be a parabolic subgroup such that p is opposite to p and let n p be the unipotent radical of p for a compactly supported smooth function on g h we define r p g to be the integral of n p n g n h over n p the radon transform r p thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions for these transforms we derive support theorems in which the support of is partially characterized in terms of the support of r p the proof is based on the relation between the radon transform and the fourier transform on g h and a paley wiener shift type argument our results generalize the support theorem of helgason for the radon transform on a riemannian symmetric space

blind source separation based on time frequency signal representations blind source separation consists of recovering a set of signals of which only instantaneous linear mixtures are observed thus far this problem has been solved using statistical information available on the source signals this paper introduces a new blind source separation approach exploiting the difference in the time frequency t f signatures of the sources to be separated the approach is based on the diagonalization of a combined set of spatial t f distributions in contrast to existing techniques the proposed approach allows the separation of gaussian sources with identical spectral shape but with different t f localization properties the effects of spreading the noise power while localizing the source energy in the t f domain amounts to increasing the robustness of the proposed approach with respect to noise and hence improved performance asymptotic performance analysis and numerical simulations are provided

time frequency approach to underdetermined blind source separation this paper presents a new time frequency tf underdetermined blind source separation approach based on wigner ville distribution wvd and khatri rao product to separate n non stationary sources from m m n mixtures first an improved method is proposed for estimating the mixing matrix where the negative value of the auto wvd of the sources is fully considered then after extracting all the auto term tf points the auto wvd value of the sources at every auto term tf point can be found out exactly with the proposed approach no matter how many active sources there are as long as n 2m 1 further discussion about the extraction of auto term tf points is made and finally the numerical simulation results are presented to show the superiority of the proposed algorithm by comparing it with the existing ones

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

performance analysis of minimum spl lscr sub 1 norm solutions for underdetermined source separation results of the analysis of the performance of minimum spl lscr sub 1 norm solutions in underdetermined blind source separation that is separation of n sources from m n linearly mixed observations are presented in this paper the minimum spl lscr sub 1 norm solutions are known to be justified as maximum a posteriori probability map solutions under a laplacian prior previous works have not given much attention to the performance of minimum spl lscr sub 1 norm solutions despite the need to know about its properties in order to investigate its practical effectiveness we first derive a probability density of minimum spl lscr sub 1 norm solutions and some properties we then show that the minimum spl lscr sub 1 norm solutions work best in a case in which the number of simultaneous nonzero source time samples is less than the number of sensors at each time point or in a case in which the source signals have a highly peaked distribution we also show that when neither of these conditions is satisfied the performance of minimum spl lscr sub 1 norm solutions is almost the same as that of linear solutions obtained by the moore penrose inverse our results show when the minimum spl lscr sub 1 norm solutions are reliable

on the x ray transform of planar symmetric 2 tensors abstract in this paper we study the attenuated x ray transform of 2 tensors supported in convex bounded subsets with sufficiently smooth boundary in the euclidean plane we characterize its range and reconstruct all possible 2 tensors yielding identical x ray data the characterization is in terms of a hilbert transform associated with a analytic maps in the sense of bukhgeim

exact reconstruction filters to invert radon transforms with finite elements abstract the new reconstruction filters introduced in this paper make possible various schemes to invert the radon transform r in r n and the general integral transform over k dimensional planes in r m potential applications include nuclear magnetic resonance and projection pursuit techniques in statistics for the exploration of large data sets the first filter presented here solves a mathematical problem in computed tomography posed by z h cho and restated by l a shepp and j b kruskal the construction of a transfer function whose dual radon transform r reproduces exactly cho s cylindrical approximate dirac measure in r 2 this function then provides the basis for a finite element galerkin rayleigh ritz method to invert the radon transform with least squares estimates of accuracy the generalization of to r n exists but requires singular distributions a second reconstruction filter real analytic avoids this difficulty and its dual radon transform r equals a gaussian approximate dirac measure in r n the smooth function unifies the dirac and fourier approaches with a common convolution kernel and allows a rigorous control of pointwise accuracy and random noise with both a priori and a posteriori estimates of error in both the quadratic and uniform norms finally the same numerical methods extend to invert the k dimensional integral transform in r m

partial radon transforms this article formally defines partial radon transforms for functions of more than two dimensions it shows that a generalized projection slice theorem exists which connects planar and hyperplanar projections of a function to its fourier transform in addition a general theoretical framework is provided for carrying out n dimensional backprojection reconstruction in a multistage fashion through the use of the partial radon transform

propagation estimates for dispersive wave equations application to the stratified wave equation the plane stratified wave equation t2 h 0 with h c y 2 z2 is studied where z x y x rk y r1 and c y c 0 as y solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x directions only i e trapped modes this leads to a consideration of the pseudo differential wave equation t2 x 0 such that the dispersion relation 2 is analytic and satisfies c1 2 c2 for c 0 uniform propagation estimates like x t e utp 0 dkx c 1 t e 0 dkx are obtained where ut is the evolution group p are projection operators onto the hilbert space of initial conditions h and e is the local energy density in special cases scattering of trapped modes off a local perturbation satisfies the causality estimate p jsp k c for each 1 2 here p j p k are remote outgoing detector incoming transmitter projections for the jth kth trapped mode also r is compact so the projections lo

path integral for a hyperbolic system of the first order on considere l approche integrale de parcours pour un systeme hyperbolique n n d ordre 1 pour chaque t x 0 t xr d fixe on construit une mesure d espace de chemin denombrablement additive a valeur matricielle n n v t x 0 sur l espace de banach c 0 t r d des chemins continus x 0 t r d

the boundary rigidity problem in the presence of a magnetic field abstract for a compact riemannian manifold with boundary endowed with a magnetic potential we consider the problem of restoring the metric g and the magnetic potential from the values of the mane action potential between boundary points and the associated linearized problem we study simple magnetic systems in this case knowledge of the mane action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction this problem can only be solved up to an isometry and a gauge transformation of for the linearized problem we show injectivity up to the natural obstruction under explicit bounds on the curvature and on we also show injectivity and stability for g and in a generic class g including real analytic ones for the nonlinear problem we show rigidity for real analytic simple g rigidity for metrics in a given conformal class and locally near any g g we also show that simple magnetic systems on two dimensional manifolds are always rigid

null spaces and ranges for the classical and spherical radon transforms abstract let r be the classical radon transform that integrates a function over hyperplanes in r n and let sm be the transform that integrates a function over spheres containing the origin in r n we prove continuity results for both transforms and explicitly give the null space of r for a class of square integrable functions on the exterior of a ball in r n as well as the null space of sm for square integrable functions on a ball we show sm l 2 r n l 2 r n is one one and we characterize the range of sm on classes of smooth functions and square integrable functions by certain moment conditions if g x is a schwartz function on r n that is zero to infinite order at x 0 we prove moment conditions sufficient for g to be in the range of sm c r n we apply our results on sm to existence and uniqueness theorems for solutions to a characteristic initial value problem for the darboux partial differential equation

bandlimited and haar filtered back projection reconstructions a new way to discretize the filtered back projection fbp algorithm is presented the function basis is the haar system 2d product of rectangular windows this scheme allows one to derive the optimal shape of the apodisation window which is angle varying and the oversampling ratio between the pixel and the projection cell size the discrete equivalent filter is also derived the comparison of standard radial band limited and separable haar reconstructions shows that improvements in terms of linearity shift invariance and aliasing can be obtained even for the case of a limited number of views considerations of projection degradations are then analyzed according to a specific imaging device to derive the optimum oversampling ratio

radon transforms on grassmann manifolds if n p q 1 the affine grassmann manifolds g p n of p planes and g q n of q planes in rn are dual homogeneous spaces of the euclidean motion group m n with g p n and g q n being incident if and only if meets at a right angle the resulting integral transform from functions on g p n to functions on g q n generalizes both the classical radon transform and its dual we show that this transform intertwines the actions of certain m n invariant differential operators on g p n and g q n and we prove an inversion formula for this transform when n is odd generalizing the radon inversion formula and obtaining in particular an inversion formula for the dual transform

failure of cancellation for direct sums of line bundles in answer to a question of murthy and wiegand examples are given of finitely generated projective modules l of rank 1 over a commutative ring r such that l l 1 is stably free but not free examples are also given of projective modules for which the determinant map det aut p r is not onto some related topological questions are also discussed

hierarchical reconstruction using geometry and sinogram restoration the authors describe and demonstrate a hierarchical reconstruction algorithm for use in noisy and limited angle or sparse angle tomography the algorithm estimates an object s mass center of mass and convex hull from the available projections and uses this information along with fundamental mathematical constraints to estimate a full set of smoothed projections the mass and center of mass estimates are made using a least squares estimator derived from the principles of consistency of the radon transform the convex hull estimate is produced by first estimating the positions of support lines of the object from each available projection and then estimating the overall convex hull using prior shape information estimating the position of two support lines from a single projection is accomplished using a generalized likelihood ratio technique for estimating jumps in linear systems results for simulated objects in a variety of measurement situations are shown and several possible extensions to this work are discussed

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

a geometric projection space reconstruction algorithm abstract we present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles the algorithm calculates the maximum a posteriori map estimate of the full sinogram which is an image of the 2 d radon transform of the object from the available data it is implemented using a primal dual constrained optimization procedure that solves a partial differential equation in the primal phase with an efficient local relaxation algorithm and uses a simple lagrange multiplier update in the dual phase the sinogram prior probability is given by a markov random field mrf that includes information about the mass center of mass and convex hull of the object and about the smoothness fundamental constraints and periodicity of the 2 d radon transform the object is reconstructed using convolution backprojection applied to the estimated sinogram we show several reconstructed objects which are obtained from simulated limited and sparse angle data using the described algorithm and compare these results with images obtained using convolution backprojection directly

limiting carleman weights and anisotropic inverse problems in this article we consider the anisotropic calderon problem and related inverse problems the approach is based on limiting carleman weights introduced in kenig et al ann math 165 567 591 2007 in the euclidean case we characterize those riemannian manifolds which admit limiting carleman weights and give a complex geometrical optics construction for a class of such manifolds this is used to prove uniqueness results for anisotropic inverse problems via the attenuated geodesic ray transform earlier results in dimension n 3 were restricted to real analytic metrics

finite representation type and direct sum cancellation abstract consider the notion of finite representation type frt for short an integral domain r has frt if there are only finitely many isomorphism classes of indecomposable finitely generated torsion free r modules now specialize let r be of the form d c o where d is a principal ideal domain whose residue fields are finite c d is a nonzero nonunit and o is the ring of integers of some finite separable field extension of the quotient field of d if the d rank of r is at least four then r does not have frt in this case we show that cancellation of finitely generated torsion free r modules is valid if and only if every unit of o c o is liftable to a unit of o we also give a complete analysis of cancellation for some rings of the form d c o having frt we include some examples which illustrate the difficult cubic case

determination of the spacetime from local time measurements we consider an inverse problem for a lorentzian spacetime m g and show that time measurements that is the knowledge of the lorentzian time separation function on a submanifold sigma subset m determine the c infty jet of the metric in the fermi coordinates associated to sigma we use this result to study the global determination of the spacetime m g when it has a real analytic structure or is stationary and satisfies the einstein scalar field equations in addition to this we require that m g is geodesically complete modulo scalar curvature singularities the results are lorentzian counterparts of extensively studied inverse problems in riemannian geometry the determination of the jet of the metric and the boundary rigidity problem we give also counterexamples in cases when the assumptions are not valid and discuss inverse problems in general relativity

the geodesic x ray transform with matrix weights author s paternain gp salo m uhlmann g zhou h abstract 2019 by johns hopkins university press consider a compact riemannian manifold of dimension 3 with strictly convex boundary such that the manifold admits a strictly convex function we show that the attenuated ray transform in the presence of an arbitrary connection and higgs field is injective modulo the natural obstruction for functions and one forms we also show that the connection and the higgs field are uniquely determined by the scattering relation modulo gauge transformations the proofs involve a reduction to a local result showing that the geodesic x ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions as a somewhat striking corollary we show that these integral geometry problems can be solved on strictly convex manifolds of dimension 3 having nonnegative sectional curvature similar results were known earlier in negative sectional curvature we also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography

the attenuated ray transform on simple surfaces we show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective moreover we give a stability estimate and develop a reconstruction procedure

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

time domain imaging of radar targets using algorithms for reconstruction from projections a time domain bistatic inverse scattering identity based on the radon transform and the space time magnetic integral equation is derived for real time use in short pulse radar systems it is shown that using a sine modulated exponential pulse siliep and algorithms for reconstruction from projections provides an edge enhanced image a simulation based on stepped frequency multi aspect measurements of aircraft models produces clear images with highly defined edges images using data from a restricted range of aspect angles are also demonstrated

a sharp stability estimate for tensor tomography in non positive curvature we consider the geodesic x ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non positive curvature we establish a stability estimate of the form l 2 mapsto h 1 2 t where the h 1 2 t space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions fan beam geometry only tangential derivatives at the boundary are used the proof is based on the pestov identity with boundary term localized in frequency

on emission tomography of inhomogeneous media the problem of finding the source distribution for particles or radiation in a bounded domain d from the emitting flow through the boundary of d is considered the particles are supposed to move with unit velocity along geodesics of a riemannian metric and can be absorbed by the medium the metric and the absorption are known uniqueness of the solution to this problem and a stability estimate are obtained under a certain assumption on the absorption and curvature of the metric

a characterization of fourier and radon transforms on euclidean space we show that a continuous operator behaving under rotations positive dilations and translations like the fourier or the radon transform on r must be a constant multiple of one of these transforms we prove this characterization for various function spaces e g we characterize the fourier transform as an operator acting on spaces between 6d r and r on the other hand a counterexample shows that the radon transform is not determined by its behaviour above as an operator from r to s x r but we can characterize the radon trans form as an operator acting between r and s 1 x r the space of integrable distributions on s x r in the special case n our methods sharpen results of j l b cooper and h kober who characterize the fourier transform as an operator from lp r into lp r 1 s p z 2 introduction several authors have been concerned with the question whether an operator on function spaces on r is determined up to a constant multiple by its behaviour under rotations dilations and translations for instance stein 15 showed that the translation invariant hubert and riesz transforms on l2 can be characterized in such a way the fourier and radon transforms on r also have a well known simple be haviour under rotations positive dilations and translations in this article we prove that such a behaviour is characteristic an operator acting under those operations like the fourier or the radon transform is a constant multiple of one of these transforms in the case of the fourier transform for n 1 our methods e g determining homogeneous distributions by euler s equation sharpen results of cooper 1 and kober 10 who characterized by different methods the fourier transform as an operator from lp r into lp r 1 p 2 we also obtain connections to hardy 7 and plancherel 12 who looked for operators on l2 r behaving under dilations like the fourier transform further our characterizations can be used to obtain corresponding ones for operators related to the fourier and radon transforms our result implies the result on hilbert and riesz transforms mentioned above stein 15 we also remark that such characteristic properties can be applied to verify inversion formulas for the corresponding operator in the case of the radon

correction to on testing trained vector quantizer codebooks this correspondence formally defines partial radon trans forms for functions of more than two dimensions it shows that a generalized projection slice theorem exists which connects planar and hy perplanar projections of a function to its fourier transform in addition a general theoretical framework is provided for carrying out dimensional backprojection reconstruction in a multistage fashion through the use of the partial radon transform

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

the initial boundary value mixed problems for elastic half plane with the conditions of contact friction the exact solutions of nonstationary contact problems of elastodynamics for a half plane with the dry and viscous friction in the contact zone having the contact edge point moving with arbitrary variable velocity along the boundary of a half plane are obtained in a closed form a new method of solution based on the use of radon transform is used

estimates of n widths of sobolev s classes on compact globally symmetric spaces of rank one estimates of kolmogorov s and linear n widths of sobolev s classes on compact globally symmetric spaces of rank 1 i e on sd pd r pd c pd h p16 cay are established it is shown that these estimates have sharp orders in different important cases new estimates for the p q norms of multiplier operators k k n are given we apply our results to get sharp orders of best polynomial approximation and n widths

exploitation of source nonstationarity in underdetermined blind source separation with advanced clustering techniques the problem of blind source separation bss is investigated following the assumption that the time frequency tf distributions of the input sources do not overlap quadratic tf representation is used to exploit the sparsity of the statistically nonstationary sources however separation performance is shown to be limited by the selection of a certain threshold in classifying the eigenvectors of the tf matrices drawn from the observation mixtures two methods are therefore proposed based on recently introduced advanced clustering techniques namely gap statistics and self splitting competitive learning sscl to mitigate the problem of eigenvector classification the novel integration of these two approaches successfully overcomes the problem of artificial sources induced by insufficient knowledge of the threshold and enables automatic determination of the number of active sources over the observation the separation performance is thereby greatly improved practical consequences of violating the tf orthogonality assumption in the current approach are also studied which motivates the proposal of a new solution robust to violation of orthogonality in this new method the tf plane is partitioned into appropriate blocks and source separation is thereby carried out in a block by block manner numerical experiments with linear chirp signals and gaussian minimum shift keying gmsk signals are included which support the improved performance of the proposed approaches

tensor tomography on surfaces we show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective this solves a long standing inverse problem in the two dimensional case

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

dictionary learning algorithms for sparse representation algorithms for data driven learning of domain specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of bayesian models with concave schur concave csc negative log priors such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen environmentally matched dictionary the elements of the dictionary can be interpreted as concepts features or words capable of succinct expression of events encountered in the environment the source of the measured signals this is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries the proverbial 25 words or less but not necessarily as succinct as one entry to learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment we develop algorithms that iterate between a representative set of sparse representations found by variants of focuss and an update of the dictionary using these sparse representations experiments were performed using synthetic data and natural images for complete dictionaries we demonstrate that our algorithms have improved performance over other independent component analysis ica methods measured in terms of signal to noise ratios of separated sources in the overcomplete case we show that the true underlying dictionary and sparse sources can be accurately recovered in tests with natural images learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries that is images encoded with an overcomplete dictionary have both higher compression fewer bits per pixel and higher accuracy lower mean square error

on two dimensional maximum entropy spectral estimation a novel method of two dimensional spectral estimation which the authors introduced recently 1987 using the radon transform and a one dimensional autoregressive model ar led them to investigate the maximization of entropy subject to the correlation matching constraints in the radon space instead of solving the 2d maximum entropy me spectral estimation problem the authors convert it into a problem which is easier to solve it is shown that a radial slice of the 2d me spectrum can be obtained by 1d ar modeling of the projections radon transform of a stationary random field srf the advantages and limitations of using this new duality relation to estimate the complete 2d me spectra on a polar raster are discussed

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

a characterization of the radon transform s range by a system of pdes abstract let g be a compactly supported function of d planes in r n we prove that then g is in the range of the radon transform if and only if g satisfies an ultrahyperbolic system of pdes we parameterize the d planes by d 1 points x0 x1 xd on them and get the pde 2 x i k x j 1 2 x i 1 x j k g x 0 x 1 x d vol x i x 0 i 1 d 0 where xik denotes the kth coordinate of x1 at the end we analyze in detail the case of d 1

invariant distributions beurling transforms and tensor tomography in higher dimensions in the recent articles paternain et al j differ geom 98 147 181 2014 invent math 193 229 247 2013 a number of tensor tomography results were proved on two dimensional manifolds the purpose of this paper is to extend some of these methods to manifolds of any dimension a central concept is the surjectivity of the adjoint of the geodesic ray transform or equivalently the existence of certain distributions that are invariant under the geodesic flow we prove that on any anosov manifold one can find invariant distributions with controlled first fourier coefficients the proof is based on subelliptic type estimates and a pestov identity we present an alternative construction valid on manifolds with nonpositive curvature based on the fact that a natural beurling transform on such manifolds turns out to be essentially a contraction finally we obtain uniqueness results in tensor tomography both on simple and anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified jacobi equation