Explicit Inversion Formula Research Articles

Monte Carlo Modeling of Compton-Scattering Angles in a Mildly Relativistic Plasma

Abstract If the inversion method is applied to random-number generation in a Monte Carlo simulation of Compton-scattering, we often encounter cubic equations. These may be solved using well-known numerical iterative methods, such as the Newton–Raphson, or bisection methods. These numerical methods, however, may lead to many iterations, which should be avoided for modeling efficiency. We present explicit inversion formulae of two cubic equations arising from the modeling of Compton-scattering angles in a mildly or weakly relativistic plasma. Explicit analytical methods are superior to any other rejection or numerical inversion methods in terms of the modeling efficiency, and will be useful for Monte Carlo simulations of X-ray reflection from cold material and the Sunyaev–Zel’dovich effect.

An inversion formula for the exponential radon transform in spatial domain with variable focal-length fan-beam collimation geometry.

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.

Higher-rank wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

Let M n , m be the space of real n × m matrices which can be identified with the Euclidean space R n m . We introduce continuous wavelet transforms on M n , m with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on M n , m and coincide with classical ones in the rank-one case m = 1 . We prove an analog of Calderón's reproducing formula for L 2 -functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on M n , m . We also introduce continuous ridgelet transforms associated to matrix planes in M n , m . An inversion formula for these transforms follows from that for the Radon transform. The new approach makes it possible to reconstruct a function on R n m from data on a set of planes of zero measure.

An Identification Problem for Multiterminal Networks: Solving for the Traffic Matrix from Input-Output Measurements

We consider the problem of determining the unknown characteristics of a _random routing strategy_ from _end-to-end_ measurements. More specifically, we construct aMarkov chain that models the traffic of messages in a multiterminal network consisting of _input, intermediate_, and _output_ terminals. The topology of the network is assumed to be known, but the Markovian routing strategy is not known. We solve the problem of determining the unknown one-step transition probability matrix of our random walk from input-output measurements of "travel time." We give explicit inversion formulas (up to a natural gauge) in a nontrivial example. The result holds for a large (but not arbitrary) class of multiterminal networks, many of which are indicated here. The networks that we display here are constructed in a canonical fashion from certain graphs. Some of these graphs as well as the way to go from graphs to networks are also displayed. One example of a graph for which our method works is the edge graph of a hypercube in any dimension.

An efficient reconstruction method for nonuniform attenuation compensation in nonparallel beam geometries based on Novikov's explicit inversion formula

This paper investigates an accurate reconstruction method to invert the attenuated Radon transform in nonparallel beam (NPB) geometries. The reconstruction method contains three major steps: 1) performing one-dimensional phase-shift rebinning; 2) implementing nonuniform Hilbert transform; and 3) applying Novikov's explicit inversion formula. The method seems to be adaptive to different settings of fan-beam geometry from very long to very short focal lengths without sacrificing reconstruction accuracy. Compared to the conventional bilinear rebinning technique, the presented method showed a better spatial resolution, as measured by modulation transfer function. Numerical experiments demonstrated its computational efficiency and stability to different levels of Poisson noise. Even with complicated geometries such as varying focal-length and asymmetrical fan-beam collimation, the presented method achieved nearly the same reconstruction quality of parallel-beam geometry. This effort can facilitate quantitative reconstruction of single photon emission computed tomography for cardiac imaging, which may need NPB collimation geometries and require high computational efficiency.

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator ( E + Λ ) , where Λ = ( − Δ ) 1 / 2 , Δ is the Laplacian and E is the identity operator.

Ray transforms in hyperbolic geometry

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.

Speed up of an analytical algorithm for nonuniform attenuation correction by using PC video/graphics card architecture

A major task in quantitative SPECT (single photon emission computed tomography) reconstruction is compensation for object-specific attenuation, which is usually nonuniform. Mathematically this task is expressed as the inversion of the attenuated Radon transform. Novikov had derived an explicit inversion formula for the attenuated Radon transform for parallel-beam collimation geometry. In our previous work, we extended his work to variable focusing fan-beam (VFF) collimators. A ray-driven analytical inversion formula for VFF reconstruction with nonuniform attenuation was derived. The drawback of ray-driven methods is that they are time consuming. In this work, we proposed a fast implementation method, which includes algorithm optimization and acceleration by the texture-mapping architecture of PC graphics/video card. We further investigated the noise properties and associated artifacts of the analytical inversion formula. The artifacts were remarkably reduced when more projections were sampled to mitigate the problem of wide bandwidth of the discrete Hilbert transform. The reconstruction from noisy data demonstrated the accuracy and robustness of the presented ray-driven analytical inversion formula with dramatic speed acceleration by the PC graphics/video card.

Confluent polynomial Vandermonde-like matrices: displacement structures, inversion formulas and fast algorithm*1

In the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial Vandermonde-like matrices, which generalize the ordinary simple nodes polynomial Vandermonde matrices studied earlier by different authors. The displacement structure approach leads to the explicit inversion formulas for confluent polynomial Vandermonde-like matrices and fast algorithms for inversion and for solving the associated linear systems.

Confluent polynomial Vandermonde-like matrices: displacement structures, inversion formulas and fast algorithm

In the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial Vandermonde-like matrices, which generalize the ordinary simple nodes polynomial Vandermonde matrices studied earlier by different authors. The displacement structure approach leads to the explicit inversion formulas for confluent polynomial Vandermonde-like matrices and fast algorithms for inversion and for solving the associated linear systems.

A noise property analysis of single-photon emission computed tomography data

We give formulae describing the simplest statistical properties of single-photon emissioncomputed tomography (SPECT) data modelled as the attenuated ray transform withPoisson noise. To obtain some of these formulae we obtain and use an inequality relatingthe even and odd parts of the attenuated ray transform without noise. Preciseequations relating the even and odd parts of this transform are also discussed.Using these results we propose new possibilities for improving the stability ofSPECT imaging based on the explicit inversion formula for the attenuated raytransformation with respect to the Poisson noise in the emission data. Numericalexamples illustrating some of the theoretical conclusions of the present work aregiven.

An analytical inversion of the nonuniformly attenuated Radon transform with variable focal-length fan-beam collimators

Single photon emission computed tomography (SPECT) is based on the measurement of radiation emitted by a radiotracer injected into the patient. Because of photoelectric absorption and Compton scatter, the gamma photons are attenuated inside the body before arriving at the detector. A quantitative reconstruction must consider the attenuation, which is usually nonuniform. Novikov has derived an explicit inversion formula for the nonuniformly attenuated Radon transform for SPECT reconstruction of parallel-beam collimated projections. In this paper, we extend his research to variable focal-length fan-beam VFF collimator geometry. A ray-driven analytical formula for VFF reconstruction with nonuniform attenuation was derived. As a unified framework, this formula can be used for parallel-beam, fan-beam, and VFF collimators. Its accuracy is demonstrated by computer simulation experiments.

Precise finite element error analysis by Yamamoto's explicit inversion formula for tridiagonal matrices - an extension of Babu?ka-Osborn's theorems

In this paper, we apply Yamamoto's explicit inversion formula for tridiagonal matrices to error analysis of the piecewise linear finite element method for two-point boundary value problems. Using the formula, we extend the theorems by Babuska-Osborn given in ``Can a finite element method perform arbitrarily badly?'' Math. Comp. 69 (2000) 443-462 to a general class of two-point boundary value problems.

Completely monotone fading memory relaxation modulii

In linear viscoelasticity, the fundamental model is the Boltzmann caual integral equation which defines how the stress σ(t) at time t depends on the earlier history of the shear rate via the relaxation modulus (kernel) G (t). Physical reality is achieved by requiring that the form of the relaxation modulus G (t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G (t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the rheological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the rheological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.

Parabolic Wavelet Transforms and Lebesgue Spaces of Parabolic Potentials

Parabolic wavelet transforms associated with heat operators -D x + ∂/∂t and I - D x + ∂/∂ t in R n+1 are introduced. A Calderon-type reproducing formula for functions f ∈ L p (R n+1 ) is proven. By making use of these transforms, new explicit inversion formulas for the Jones-Sampson parabolic potentials are obtained, and characterization of the corresponding anisotropic Lebesgue spaces is given.

A nonlinear inverse problem inspired by three‐dimensional diffuse tomography: Explicit formulas

AbstractWe use time of flight information to obtain explicit inversion formulas for a simple model of optical tomography. This can be considered as an instance of a more general nonlinear inversion problem for multi‐terminal networks. In this case the network is nonplanar with lots of cycles but the underlying physical model yields a highly structured situation. © 2003 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 198–203, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10031

A nonlinear inverse problem inspired by three-dimensional diffuse tomography

Optical, or diffuse tomography, refers to the use of low-energy probesto obtain images of highly scattering media. The inverse problem forone of the earliest and crudest models of optical tomography amounts toreconstructing the one-step transition probability matrix for a Markov chain(with three kinds of states) from boundary measurements. This modelis too simple and too general to faithfully reflect the physics of diffusetomography but could be of interest in other set-ups. It gives a difficult class ofnonlinear inverse problems for certain networks with a complex pattern ofconnections which are motivated by the diffuse tomography picture. Aremarkable feature of this simple model is that, at least for systems arisingfrom very coarse tomographic discretizations, it gives an exactly solvablesystem of nonlinear equations, i.e. a certain number of unknowns areexpressible in terms of the data and a number of free parameters. Theadvantages of this rather uncommon accident are clear: for instance, it ispossible to go beyond iterative methods of solution, which are very commonfor nonlinear problems. This opens the door for a careful study of theill-conditioned nature of the problem, a subject that is not touched uponhere.The lure of an explicit inversion formula, especially in a nonlinear problem is toomuch of a temptation to pass on, and this paper takes a step in that direction.What we get here is really a careful procedure which might eventually be puttogether as an explicit inversion formula.Existing recursive algorithms in the two-dimensional case hinge on a verycomplete study of a 2 × 2 system, for which it has been previouslyshown that using photon count and time-of-flight information all the unknownparameters are given analytically up to an explicitly described eight-dimensionalgauge.Here we consider, in detail, the three-dimensional situation and present thesolution of a very general 2 × 2 × 2 system.There are a total of 288 unknowns and if only photon count is used there are 576pieces of data. A total of 48 of the unknowns can be prescribed arbitrarily andthen the remaining 240 unknowns can be solved uniquely in terms of these freeparameters and the data. Most of this is done by writing down explicitformulae as in the two-dimensional case. When (the first moment of)time of flight is also used, we show that all parameters are determined upto a 24-dimensional gauge. We show that, unless physical assumptionsare made on the model, this freedom in picking 24 parameters cannotbe eliminated. The results given here should be the basis of a recursivealgorithm for larger systems. This would require using some good ideas frommultiresolution analysis, domain decomposition and/or multigrid methods.

Inverse scattering with diffusing waves.

We consider the problem of imaging the optical properties of a highly scattering medium probed by diffuse light. An analytic solution to this problem is derived from the singular value decomposition of the forward-scattering operator, which leads to explicit inversion formulas for the inverse scattering problem with diffusing waves. Computer simulations are used to illustrate these results in model systems.

Inverse scattering for the diffusion equation with general boundary conditions.

We consider the inverse scattering problem for the diffusion equation. A solution to this problem in the form of an explicit inversion formula is derived. Computer simulations are used illustrate our approach in model systems.

INVERSION FORMULAS FOR TRIDIAGONAL MATRICES WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS1*

This paper first reviews some results on the structure of inverses of nonsingular and irreducible tridiagonal matrices. Next, explicit inversion formulas are given for certain, not necessarily symmetric, tridiagonal matrices. The results are then applied to matrices arising from discretization of two-point boundary value problems of the Sturm-Liouville type. The results show harmonic relations between the Green functions and the discrete Green functions for the problems. Finally the results are extended to block tridiagonal matrices. This work was supported by Scientific Research Grant-in-Aid from JSPS.