Exact Inversion Formula Research Articles

Universal Inversion Formulas for Recovering a Function from Spherical Means

The problem of reconstruction a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary an arbitrarily shaped smooth convex domain. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed smoothing integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.

Numerical inversion of the Funk transform on the rotation group

The reconstruction of a function on the rotation group from mean values along all geodesics is an overdetermined problem, i.e. it is sufficient to know the mean values for a three-dimensional subset of all geodesics on the rotation group. In this paper we give a Fourier slice theorem for the restricted problem. Based on the Fourier slice theorem and fast Fourier transforms on the rotation group and the sphere we introduce a fast algorithm for the forward transform. Analyzing the inverse problem we come up with an exact inversion formula for bandlimited functions on the rotation group. Unfortunately this inversion formula turns out to be extremely ill conditioned. Therefore we introduce an iterative approach which makes use of regularization and the fast algorithm for the forward transform. Numerical experiments indicate the applicability of our algorithms.

A numerical solver based on B-splines for 2D vector field tomography in a refracting medium

Efficient and stable numerical solvers for vector tomography problems taking refractions into account are subject of current research. This article is concerned with the problem of reconstructing a 2D-vector field in a refracting medium from its known longitudinal or transverse ray transform. The refraction is modeled using a Riemannian metric in the domain under consideration. We propose a numerical solver that is based on the least squares method where we use a finite basis consisting of B-splines as basis functions. In that sense the method can be seen as a projection method for minimizing an L2-data fitting term. Numerical simulations show a good performance of that method, also compared to methods relying on exact inversion formulas.

An exact inversion formula for cone beam vector tomography

In this paper, we present an exact inversion formula for the cone beam transform of vector fields. This transform is the mathematical model for vector tomography with cone beam data. The cone beam data consist of line integrals of a vector field along rays which emanate from points on a given source trajectory. It is known that an exact reconstruction of the solenoidal part of the searched for vector field is possible if the trajectory satisfies the Tuy condition of order 3. We derive an inversion formula under the assumption that the source trajectory satisfies Tuy’s condition of order 3 and the vector field is smooth and compactly supported. This formula thus represents an analogue of the known inversion formulas for the scalar cone beam transform.

Inversion of circular means and the wave equation on convex planar domains

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary ∂Ω of a smooth convex bounded domain Ω⊂R2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in Ω modulo an explicitly computed smoothing integral operator KΩ. For circular and elliptical domains the operator KΩ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on ∂Ω. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.

Inversion of the V-line Radon transform in a disc and its applications in imaging

Several novel imaging modalities proposed during the last couple of years are based on a mathematical model, which uses the V-line Radon transform (VRT). This transform, sometimes called broken-ray Radon transform, integrates a function along V-shaped piecewise linear trajectories composed of two intervals in the plane with a common endpoint. Image reconstruction problems in these modalities require inversion of the VRT. While there are ample results about inversion of the regular Radon transform integrating along straight lines, very little is known for the case of the V-line Radon transform. In this paper, we derive an exact inversion formula for the VRT of functions supported in a disc of arbitrary radius. The formula uses a two-dimensional restriction of VRT data, namely the incident ray is normal to the boundary of the disc, and the breaking angle is fixed. Our method is based on the classical filtered back-projection inversion formula of the Radon transform, and has similar features in terms of stability, speed, and accuracy.

An inversion formula for a Prandtl–Ishlinskii operator with time dependent thresholds

We extend the well-known inversion result for Prandtl–Ishlinskii operators to the case of time dependent thresholds. The exact inversion formula is shown to hold under the condition that the distances between the thresholds do not decrease in time. Examples show that this threshold dilation condition cannot be completely omitted.

On the Ill-Posed Character of the Lorentz Integral Transform

An exact inversion formula for the Lorentz integral transform (LIT) is provided together with the spectrum of the LIT kernel. The exponential increase of the inverse Fourier transform of the LIT kernel entering the inversion formula explains the ill-posed character of the LIT approach. Also the continuous spectrum of the LIT kernel, which approaches zero points necessarily to the same defect. A possible cure is discussed and numerically illustrated.

Computing Reconstruction Kernels for Circular 3-D Cone Beam Tomography

In this paper, we present techniques for deriving inversion algorithms in 3-D computer tomography. To this end, we introduce the mathematical model and apply a general strategy, the so-called approximate inverse, for deriving both exact and numerical inversion formulas. Using further approximations, we derive a 2-D shift-invariant filter for circular-orbit cone-beam imaging. Results from real data are presented.

An exact inversion formula from determining a planar fault from boundary measurements

This paper considers the inverse problem of determining a time-dependent slip fault which releases shear stress elastically. The input data are the accelerations measured on the free external surface. A new formulae for determining explicitly the geometry of the planar fault is proposed. Potential applications to real earthquakes are discussed.

An exact inversion formula from determining a planar fault from boundary measurements

An exact inversion formula from determining a planar fault from boundary measurements

Phonon spectrum of YBCO obtained by specific heat inversion method for realdata

In this paper, the phonon spectrum of YBCO is obtained from experimentalspecific heat data by an exact inversion formula with a parameter for eliminatingdivergences. The results can be compared to those of neutron inelasticscattering, which can only be carried out in a few laboratories. Somekey points of specific heat-phonon spectrum inversion (SPI) theory anda method of asymptotic behaviour control are discussed. An improvedunique existence theorem is presented, and a universal function set fornumerical calculation of SPI is calculated with high accuracy, which makes theinversion method applicable and convenient in practice. This is the firsttime specific heat-phonon SPI has been realized for a concrete system.

Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms

Linear diffusion problems defined within irregular multidimensional regions are analytically solved through integral transforms, requiring numerical routines only for integration purposes, when a general functional boundary representation is considered. Auxiliary one-dimensional eigenvalue problems mapping the irregular region are applied with an integral transformation procedure so that the original differential Sturm–Liouville system gives place to an algebraic eigenvalue problem. The exact analytical inversion formula is then employed to yield the desired potential, explicitly, at any point within the domain. To allow for improved flexibility and further applicability, the related integration is simplified through an approximate boundary representation using lines connecting user provided points instead of the former exact representation of the irregular bounds, which is particularly advantageous when a functional description of the boundaries is not available. A cylindrical region test case with known exact solution is considered, and treated as an irregular region in the Cartesian coordinates system. Convergence behavior and error analysis are carefully undertaken and illustrated.

Inversion ofthe attenuated Radon transform

We derive an exact inversion formula for the attenuated Radontransform. The formula is closely related to Novikov's inversionformula, but our derivation is completelydifferent. We also give an implementation of the inversion formula very similar to the filtered backprojectionalgorithm of x-ray tomography and present numerical results.

An approximate short time Laplace transform inversion method

The “standard” numerical methods used for inverting the Laplace transform are based on a regularization of an exact inversion formula. They are very sensitive to noise in the Laplace transformed function. In this article we suggest a different strategy. The inversion formula we use is an approximate one, but it is stable with respect to noise. The new approximate expression is obtained from a short time expansion of the Bromwich inversion formula. We show that this approximate result can be significantly improved when iterated, while remaining stable with respect to noise. The iterated method is exact for the class of functions of type EmeaE. The method is applied to a harmonic model of the stilbene molecule, to a truncated exponent series, and to the flux–flux correlation function for the parabolic barrier. These examples demonstrate the utility of the method for application to problems of interest in molecular dynamics.

A concrete realization of specific heat-phonon spectrum inversion for YBCO

In this Letter, a phonon spectrum of YBCO is obtained from its experimental specific heat data by an exact inversion formula with eliminating divergence parameter [Dai Xianxi, Xu Xinwen and Dai Jiqiong, Proceedings of Beijing International Conference on High T c Superconductivity, Sept. 4–8 Beijing, China, (1989) 521], [Dai Xianxi, Xu Xinwen and Dai Jiqiong, Phys. Lett. A 147 (1990) 445]. The results are comparable to that from neutron inelastic scattering. Some key points of specific heat-phonon spectrum inversion (SPI) theory as well as a method of asymptotic behavior control are discussed. An improved unique existence theorem is presented. A universal function set for the numerical calculation in SPI is obtained, which will make the inversion method applicable and convenient in practice. This is the first time to realize the specific heat-phonon spectrum inversion in a concrete system.

Inversion of the Van der Monde matrix

We deduce an analytical formula for inversion of a complex Van der Monde matrix. It has applications in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. Previously the inversion of such a matrix has been performed by approximations; we further deduce an exact inversion formula.

Discrete signed mixtures of Exponentials

In this paper we consider the problem of recovering the parameters in a mixture of exponential functions, where the weights may be of different sign. Both the case where the resulting mixture is completely known and the case where it has to be estimated from a finite number of noisy data are considered. In the first case an exact inversion formula can be derived. The recovery is derived by exploiting the resemblance of the mixture to a convolution on the multiplicative group of the positive numbers. These mixtures are the derivatives of the generalized hyperexponential distribution functions that have well-known applications in stochastic modeling.

A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection

An exact inversion formula written in the form of shift-variant filtered-backprojection (FBP) is given for reconstruction from cone-beam data taken from any orbit satisfying H.K. Tuy's (1983) sufficiency conditions. The method is based on a result of P. Grangeat (1987), involving the derivative of the three-dimensional (3D) Radon transform, but unlike Grangeat's algorithm, no 3D rebinning step is required. Data redundancy, which occurs when several cone-beam projections supply the same values in the Radon domain, is handled using an elegant weighting function and without discarding data. The algorithm is expressed in a convenient cone-beam detector reference frame, and a specific example for the case of a dual orthogonal circular orbit is presented. When the method is applied to a single circular orbit (even though Tuy's condition is not satisfied), it is shown to be equivalent to the well-known algorithm of L.A. Feldkamp et al. (1984).

Stability of the inversion of 3D fixed-frequency scattering data

Numerical methods for solving 3D inverse scattering problems with fixed-energy data are described. Exact inversion formulas are given. Stability of the inversion methods based on these formulas is investigated. An explicit formula is given for construction of the Dirichlet-to-Neumann map from the scattering amplitude. This formula is used as a basis for another numerical approach to the inverse scattering problem. Several open problems are formulated.