on the jacobians of plane cubics abstract let s be a scheme and f a ternary cubic form whose ten coefficients are sections of o s without common zero the equation f 0 defines a family of plane cubic curves x p s 2 s parametrized by s we prove that the family of generalized jacobians of those cubic curves is a group scheme j s which is the locus of smoothness of a scheme f 0 where f is a weierstrass cubic form f f x y z y 2 z a 1 xyz a 2 yz 2 x 3 a 2 x 2 z a 4 xz 2 a 6 z 3 in which the coefficient a i is a homogeneous polynomial with integral coefficients of degree i in the ten coefficients of f which we give explicitly a key ingredint of the proof is a characterization over sufficiently nice bases of group algebraic spaces which can be described by such a weierstrass equation

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

discretization of the radon transform and of its inverse by spline convolutions we present an explicit formula for b spline convolution kernels these are defined as the convolution of several b splines of variable widths h sub i and degrees n sub i we apply our results to derive spline convolution based algorithms for two closely related problems the computation of the radon transform and of its inverse first we present an efficient discrete implementation of the radon transform that is optimal in the least squares sense we then consider the reverse problem and introduce a new spline convolution version of the filtered back projection algorithm for tomographic reconstruction in both cases our explicit kernel formula allows for the use of high degree splines these offer better approximation performance than the conventional lower degree formulations e g piecewise constant or piecewise linear models we present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity in particular we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

multiresolution tomographic reconstruction using wavelets shows how the separable two dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm this algorithm is similar to the conventional filtered backprojection algorithm except that the filters are now angle dependent and the backprojection gives the wavelet coefficients of the reconstruction which are then used to synthesize the reconstruction at various resolution levels by reconstructing only a small localized region at high resolution the authors show how radiation exposure and computation can be significantly reduced compared to a standard reconstruction

on some algebras associated to genus one curves abstract haile han and kuo have studied certain non commutative algebras associated to a binary quartic or ternary cubic form we extend their construction to pairs of quadratic forms in four variables and conjecture a further generalisation to genus one curves of arbitrary degree these constructions give an explicit realisation of an isomorphism relating the weil ch telet and brauer groups of an elliptic curve

local algorithms in exterior tomography exterior tomographic data are taken over lines outside a central region and such data occur in the industrial nondestructive evaluation of large objects such as rockets we explain using microlocal analysis which singularities are well reconstructed from exterior data and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform e t quinto singular value decompositions and inversion methods for the exterior radon transform and a spherical transform j math anal appl 95 1983 437 448 we extend lambda tomography to exterior data and to limited angle exterior data the algorithm is tested on industrial data from perceptics inc

wavelet based multiresolution local tomography we develop an algorithm to reconstruct the wavelet coefficients of an image from the radon transform data the proposed method uses the properties of wavelets to localize the radon transform and can be used to reconstruct a local region of the cross section of a body using almost completely local data that significantly reduces the amount of exposure and computations in x ray tomography the property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments the ramp filtered version of the scaling function as well as the wavelet function has extremely rapid decay we show that the variance of the elements of the null space is negligible in the locally reconstructed image also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm to reconstruct a local region 16 pixels in radius in a 256 spl times 256 image we require 22 of full exposure data

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

a method for implementing dirichlet and third type boundary conditions in ptrw simulations we present an efficient and accurate numerical method for implementing dirichlet boundary conditions in particle tracking random walk ptrw simulations of advective dispersive transport this is a challenge because defining concentrations for dirichlet boundary conditions requires invoking control volumes of some kind which are not natural to the lagrangian based ptrw concept our method performs a galerkin projection of ptrw based particle densities onto control volumes that discretize the boundary thus we obtain concentration values at the boundary condition and can control the particle release rates such that the prescribed boundary values are met this allows for complex shaped internal and external boundaries where concentration values are fixed to prescribed values third type boundary conditions can be addressed as well we test and illustrate the properties and behavior of our method in a series of test cases the results are benchmarked against the conceptually related semianalytical method masst multiple analytical source superposition technique and to those of a finite element method fem while masst is restricted to uniform velocity fields due to the underlying analytical solutions fem is limited in heterogeneous velocity fields at large peclet numbers by numerical dispersion in the feasible discretization range the results demonstrate that our proposed method performs better than the other methods in both regimes

efficient framework for model based tomographic image reconstruction using wavelet packets the use of model based algorithms in tomographic imaging offers many advantages over analytical inversion methods however the relatively high computational complexity of model based approaches often restricts their efficient implementation in practice many modern imaging modalities such as computed tomography positron emission tomography or optoacoustic tomography normally use a very large number of pixels voxels for image reconstruction consequently the size of the forward model matrix hinders the use of many inversion algorithms in this paper we present a new framework for model based tomographic reconstructions which is based on a wavelet packet representation of the imaged object and the acquired projection data the frequency localization property of the wavelet packet base leads to an approximately separable model matrix for which reconstruction at each spatial frequency band is independent and requires only a fraction of the projection data thus the large model matrix is effectively separated into a set of smaller matrices facilitating the use of inversion schemes whose complexity is highly nonlinear with respect to matrix size the performance of the new methodology is demonstrated for the case of 2 d optoacoustic tomography for both numerically generated and experimental data

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

local tomography ii standard or global tomography involves the reconstruction of a function f from line integrals local tomography in this paper involves the reconstruction of a related function lf alpha lambda f mu lambda 1 f where lambda is the square root of the positive laplacian delta this article is a sequel to the article local tomography siam j appl math 52 1992 pp 459 484 1193 1198 by faridani ritman and smith the principal new results are 1 good bounds for lambda f and lambda 1 f outside the support of f particularly when f has 0 moments up to some order 2 identification and reduction of global effects in local tomography i e identification and reduction of the dependence of lf x on the values of f at points at an intermediate distance from x 3 an algorithm for computing approximate density jumps from lambda f when f is a linear combination of characteristic functions and a smooth background several examples are given some from real x ray da

explicit n descent on elliptic curves i algebra this is the first in a series of papers in which we study the n selmer group of an elliptic curve with the aim of representing its elements as genus one normal curves of degree n the methods we describe are practical in the case n 3 for elliptic curves over the rationals and have been implemented in magma

on an algebra determined by a quartic curve of genus one abstract let k be a field of characteristic not two to each irreducible quartic f x over k we associate a certain algebra a f given by explicit generators and relations we prove a f is an azumaya algebra of rank four over its center and that its center is the coordinate ring of an affine piece of an elliptic curve the jacobian of the curve c y 2 f x its simple images are quaternion algebras and the resulting function from the group of k rational points on the jacobian to the brauer group of k is a group homomorphism whose image is the relative brauer group of central simple k algebra split by the function field of c we also show that the algebra a f is split if and only if c has a k rational point

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

tomographic reconstruction using nonseparable wavelets in this paper the use of nonseparable wavelets for tomographic reconstruction is investigated local tomography is also presented the algorithm computes both the quincunx approximation and detail coefficients of a function from its projections simulation results showed that nonseparable wavelets provide a reconstruction improvement versus separable wavelets

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

on an algebra associated to a ternary cubic curve abstract in this paper we construct an algebra associated to a cubic curve c defined over a field f of characteristic not two or three we prove that this algebra is an azumaya algebra of rank nine its center is the affine coordinate ring of an elliptic curve the jacobian of the cubic curve c the induced function from the group of f rational points on the jacobian into the brauer group of f is a group homomorphism with image precisely the relative brauer group of classes of central simple f algebras split by the function field of c we also prove that this algebra is split if and only if the cubic curve c has an f rational point these results generalize haile s work on the clifford algebra of a binary cubic form

relative brauer groups of torsors of period two abstract we consider the problem of computing the relative brauer group of a torsor of period two under an elliptic curve we show how this problem can be reduced to finding a set of generators for the group of rational points on the elliptic curve this extends work of haile and han to the case of torsors with unequal period and index our results also apply to torsors under higher dimensional abelian varieties several numerical examples are given

feldkamp type cone beam tomography in the wavelet framework x ray computed tomography ct is in transition from fan beam to cone beam geometry for cone beam volumetric imaging reduction of radiation exposure remains an important issue because the wavelet approach was shown to be effective and flexible for two dimensional 2 d local region reconstruction the authors are motivated to perform wavelet local ct in cone beam geometry in this paper they formulate the feldkamp cone beam reconstruction from the wavelet perspective derive both full scan and half scan feldkamp type formulas for either global or local reconstruction and demonstrate the feasibility and utility in synthetic and real data it is found that using the wavelet feldkamp approach a three dimensional 3 d region of interest roi can be reconstructed with neither severe image artifacts nor any significant constant bias in the authors simulation and experiments

wavelet based reconstruction for limited angle x ray tomography the aim of x ray tomography is to reconstruct an unknown physical body from a collection of projection images when the projection images are only available from a limited angle of view the reconstruction problem is a severely ill posed inverse problem statistical inversion allows stable solution of the limited angle tomography problem by complementing the measurement data by a priori information in this work the unknown attenuation distribution inside the body is represented as a wavelet expansion and a besov space prior distribution together with positivity constraint is used the wavelet expansion is thresholded before reconstruction to reduce the dimension of the computational problem feasibility of the method is demonstrated by numerical examples using in vitro data from mammography and dental radiology

generalized sampling a variational approach ii applications for pt i see ibid vol 50 no 8 p 1965 76 2000 the variational reconstruction theory from a companion paper finds a solution consistent with some linear constraints and minimizing a quadratic plausibility criterion it is suitable for treating vector and multidimensional signals here we apply the theory to a generalized sampling system consisting of a multichannel filterbank followed by a nonuniform sampling we provide ready made formulas which should permit application of the technique directly to problems at hand we comment on the practical aspects of the method such as numerical stability and speed we show the reconstruction formula and apply it to several practical examples including new variational formulation of derivative sampling landmark warping and tomographic reconstruction

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