Direct Reconstruction Algorithm Research Articles

Direct two-dimensional reconstruction algorithm for an inverse-geometry CT system

An inverse-geometry computed tomography (IGCT) system uses a large source array opposite a smaller detector array. A previously described IGCT reconstruction algorithm uses gridding, but this gridding step produces blurring in the reconstructed image. In this article, the authors describe a two-dimensional (2D) IGCT reconstruction algorithm without gridding. In the transverse direction, the raw data of the IGCT system can be viewed as being composed of many fan beams. Because the spacing between source spots is larger than the spot width, each fan beam has undersampled projection data, but the missing samples are effectively provided by other undersampled fan beam views. In the proposed method, a direct fan beam reconstruction algorithm is used to process each undersampled fan beam. Initial images with this method showed ring artifacts caused by nonuniform sampling in the radial direction as compared to an ideal fan beam. A new method for correcting this effect was developed. With this correction, high quality images were obtained. The noise performance of the proposed 2D IGCT reconstruction algorithm was investigated, and it was comparable to that of the fan beam system. A MTF study showed that the proposed method achieves better resolution than the gridding method.

Convergence and stability of the inverse scattering series for diffuse waves

We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the series.

Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors

A method for photoacoustic tomography is presented that uses line integrals over the acoustic wave field from a photoacoustic source for the reconstruction of a three-dimensional image. The line integrals are acquired with an optical line sensor based on a Mach–Zehnder interferometer. Image reconstruction is a two-step process. In the first step data from a scan of the line outside the object are used to reconstruct a linear projection of the source distribution. In the second step the inverse linear Radon transform is applied to multiple projections taken at different directions. This study focuses on the first step comparing two different open scan curves of the line detector around the object and corresponding two-dimensional reconstruction algorithms. An open curve, such as a 180° arc or an ‘L’ formed by two lines, establishes a limited view problem for which limitations exist concerning the stability of the reconstruction. Using experimental data from phantoms, time domain direct and iterative reconstruction algorithms as well as frequency domain (FD) algorithms are compared. The results indicate that although the image quality for all algorithms is similar, the FD algorithms offer much faster reconstruction speed, whereas the time domain algorithms allow for a refinement of image quality by including special correction procedures.

Direct reconstruction algorithm of current dipoles for vector magnetoencephalography and electroencephalography

This paper presents a novel algorithm to reconstruct parameters of a sufficient number of current dipoles that describe data (equivalent current dipoles, ECDs, hereafter) from radial/vector magnetoencephalography (MEG) with and without electroencephalography (EEG). We assume a three-compartment head model and arbitrary surfaces on which the MEG sensors and EEG electrodes are placed. Via the multipole expansion of the magnetic field, we obtain algebraic equations relating the dipole parameters to the vector MEG/EEG data. By solving them directly, without providing initial parameter guesses and computing forward solutions iteratively, the dipole positions and moments projected onto the xy-plane (equatorial plane) are reconstructed from a single time shot of the data. In addition, when the head layers and the sensor surfaces are spherically symmetric, we show that the required data reduce to radial MEG only. This clarifies the advantage of vector MEG/EEG measurements and algorithms for a generally-shaped head and sensor surfaces. In the numerical simulations, the centroids of the patch sources are well localized using vector/radial MEG measured on the upper hemisphere. By assuming the model order to be larger than the actual dipole number, the resultant spurious dipole is shown to have a much smaller strength magnetic moment (about 0.05 times smaller when the SNR = 16 dB), so that the number of ECDs is reasonably estimated. We consider that our direct method with greatly reduced computational cost can also be used to provide a good initial guess for conventional dipolar/multipolar fitting algorithms.

Identification of small inhomogeneities: Asymptotic factorization

We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral equation methods. It demonstrates the strong relation between factorization methods and MUSIC-type methods for the solution of this inverse problem.

Fast 3D spatial EPR imaging using spiral magnetic field gradient

Electron paramagnetic resonance imaging (EPRI) provides direct detection and mapping of free radicals. The continuous wave (CW) EPRI technique, in particular, has been widely used in a variety of applications in the fields of biology and medicine due to its high sensitivity and applicability to a wide range of free radicals and paramagnetic species. However, the technique requires long image acquisition periods, and this limits its use for many in vivo applications where relatively rapid changes occur in the magnitude and distribution of spins. Therefore, there has been a great need to develop fast EPRI techniques. We report the development of a fast 3D CW EPRI technique using spiral magnetic field gradient. By spiraling the magnetic field gradient and stepping the main magnetic field, this approach acquires a 3D image in one sweep of the main magnetic field, enabling significant reduction of the imaging time. A direct one-stage 3D image reconstruction algorithm, modified for reconstruction of the EPR images from the projections acquired with the spiral magnetic field gradient, was used. We demonstrated using a home-built L-band EPR system that the spiral magnetic field gradient technique enabled a 4–7-fold accelerated acquisition of projections. This technique has great potential for in vivo studies of free radicals and their metabolism.

Filtered Back-Projection in$4pi$Compton Imaging With a Single 3D Position Sensitive CdZnTe Detector

The Compton scattering camera can provide higher detection efficiency since the use of a mechanical collimator is not required. Several image reconstruction algorithms for Compton scattering cameras, such as maximum likelihood method and ART, can provide good imaging performance but are indirect and computational intensive. A direct reconstruction algorithm, such as filtered back-projection, is preferable if computational time is critical. Parra proposed an analytical inversion algorithm from cone-beam projections for Compton scattering cameras with a complete data set. In reality, a Compton camera is always limited by its configuration and can only provide an incomplete data set with a finite range of scattering angles. In this paper, we investigate a filtered back-projection algorithm applied to single 3-dimensional position sensitive CdZnTe detectors

Direct computed tomographic reconstruction for directional-derivative projections of computed tomography of diffraction enhanced imaging

X-ray diffraction enhanced imaging based on synchrotron radiation has extremely high sensitivity of weakly absorbing low-Z samples in medical and biological fields. This letter is dedicated to a direct reconstruction algorithm for directional-derivative projections of computed tomography of diffraction enhanced imaging. It is a “one-step” algorithm and does not require any restoration processing compared with the current “two-step” methods. The actual values of the sample’s refractive index decrement can be estimated from its reconstruction images directly. The algorithm is proven by the actual experiment at the Beijing Synchrotron Radiation Facility and the reconstructed images are described finally.

An orthogonal projection and regularization technique for magnetospheric radio tomography

A challenging problem in ill‐posed inverse problems is incorporating prior knowledge of the solution into reconstruction techniques. This problem is particularly important in magnetospheric radio tomography where the path integrated measurements of the target region may be sparse. We present in this paper an orthogonal projection and regularization (OPR) technique that incorporates prior knowledge of magnetospheric parameters from existing models or past measurements into a direct reconstruction algorithm. The OPR scheme extracts first an optimal orthonormal basis containing the main features of the unknowns from an ensemble of modeled or measured snapshots through the proper orthogonal decomposition (POD) and then projects the line‐of‐sight equations onto the subspace spanned by the empirical orthonormal basis. The resulting low‐dimensional model is well‐conditioned, its coordinates are uncorrelated, and it contains prior knowledge of the solution. The magnetospheric parameters in the transformed coordinate are reconstructed from the low‐dimensional model and quantities in the physical coordinate are easily recovered from the POD transformation. On the basis of magnetohydrodynamic (MHD) simulations and hypothetical satellite constellations, we demonstrate that the POD‐based method may perform significantly better than the regularized direct method with sparse path‐integrated measurements, combined with a few (5–10) model snapshots.

Direct cone‐beam cardiac reconstruction algorithm with cardiac banding artifact correction

Multislice helical computed tomography (CT) is a promising noninvasive technique for coronary artery imaging. Various factors can cause inconsistencies in cardiac CT data, which can result in degraded image quality. These inconsistencies may be the result of the patient physiology (e.g., heart rate variations), the nature of the data (e.g., cone-angle), or the reconstruction algorithm itself. An algorithm which provides the best temporal resolution for each slice, for example, often provides suboptimal image quality for the entire volume since the cardiac temporal resolution (TRc) changes from slice to slice. Such variations in TRc can generate strong banding artifacts in multiplanar reconstruction images or three-dimensional images. Discontinuous heart walls and coronary arteries may compromise the accuracy of the diagnosis. A beta-blocker is often used to reduce and stabilize patients' heart rate but cannot eliminate the variation. In order to obtain robust and optimal image quality, a software solution that increases the temporal resolution and decreases the effect of heart rate is highly desirable. This paper proposes an ECG-correlated direct cone-beam reconstruction algorithm (TCOT-EGR) with cardiac banding artifact correction (CBC) and disconnected projections redundancy compensation technique (DIRECT). First the theory and analytical model of the cardiac temporal resolution is outlined. Next, the performance of the proposed algorithms is evaluated by using computer simulations as well as patient data. It will be shown that the proposed algorithms enhance the robustness of the image quality against inconsistencies by guaranteeing smooth transition of heart cycles used in reconstruction.

Direct reconstruction of kinetic parameter images from dynamic PET data

Our goal in this paper is the estimation of kinetic model parameters for each voxel corresponding to a dense three-dimensional (3-D) positron emission tomography (PET) image. Typically, the activity images are first reconstructed from PET sinogram frames at each measurement time, and then the kinetic parameters are estimated by fitting a model to the reconstructed time-activity response of each voxel. However, this "indirect" approach to kinetic parameter estimation tends to reduce signal-to-noise ratio (SNR) because of the requirement that the sinogram data be divided into individual time frames. In 1985, Carson and Lange proposed, but did not implement, a method based on the expectation-maximization (EM) algorithm for direct parametric reconstruction. The approach is "direct" because it estimates the optimal kinetic parameters directly from the sinogram data, without an intermediate reconstruction step. However, direct voxel-wise parametric reconstruction remained a challenge due to the unsolved complexities of inversion and spatial regularization. In this paper, we demonstrate and evaluate a new and efficient method for direct voxel-wise reconstruction of kinetic parameter images using all frames of the PET data. The direct parametric image reconstruction is formulated in a Bayesian framework, and uses the parametric iterative coordinate descent (PICD) algorithm to solve the resulting optimization problem. The PICD algorithm is computationally efficient and is implemented with spatial regularization in the domain of the physiologically relevant parameters. Our experimental simulations of a rat head imaged in a working small animal scanner indicate that direct parametric reconstruction can substantially reduce root-mean-squared error (RMSE) in the estimation of kinetic parameters, as compared to indirect methods, without appreciably increasing computation.

Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics

A cone-beam image reconstruction algorithm using spherical harmonic expansions is proposed. The reconstruction algorithm is in the form of a summation of inner products of two discrete arrays of spherical harmonic expansion coefficients at each cone-beam point of acquisition. This form is different from the common filtered backprojection algorithm and the direct Fourier reconstruction algorithm. There is no re-sampling of the data, and spherical harmonic expansions are used instead of Fourier expansions. As a special case, a new fan-beam image reconstruction algorithm is also derived in terms of a circular harmonic expansion. Computer simulation results for both cone-beam and fan-beam algorithms are presented for circular planar orbit acquisitions. The algorithms give accurate reconstructions; however, the implementation of the cone-beam reconstruction algorithm is computationally intensive. A relatively efficient algorithm is proposed for reconstructing the central slice of the image when a circular scanning orbit is used.

A direct impedance tomography algorithm for locating small inhomogeneities

Impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface. In its most general form impedance tomography is quite ill-posed, but when additional a-priori information is admitted the situation changes dramatically. In this paper we consider the case where the goal is to find a number of small objects (inhomogeneities) inside an otherwise known conductor. Taking advantage of the smallness of the inhomogeneities, we can use asymptotic analysis to design a direct (i.e., non-iterative) reconstruction algorithm for the determination of their locations. The viability of this direct approach is documented by numerical examples.

A direct reconstruction algorithm for electrical impedance tomography.

A direct (noniterative) reconstruction algorithm for electrical impedance tomography in the two-dimensional (2-D), cross-sectional geometry is reviewed. New results of a reconstruction of a numerically simulated phantom chest are presented. The algorithm is based on the mathematical uniqueness proof by A. I. Nachman [1996] for the 2-D inverse conductivity problem. In this geometry, several of the clinical applications include monitoring heart and lung function, diagnosis of pulmonary embolus, diagnosis of pulmonary edema, monitoring for internal bleeding, and the early detection of breast cancer.

Direct space–time characterization of the electric fields of ultrashort optical pulses

We report the complete spatiotemporal characterization of ultrashort light pulses by use of a self-referencing device based on shearing interferometry in the space and frequency domains. The apparatus combines a spatially resolved spectral shearing interferometer with a spectrally resolved spatial shearing interferometer. The electric field as a function of one transverse spatial coordinate and time is obtained from a single experimental trace by means of a direct and fast algebraic phase reconstruction algorithm. The method has been tested in several common laboratory situations.

Reconstructing the discrete Wigner function and some properties of the measurement bases

We derive a direct reconstruction algorithm for the discrete Wigner function through different types of measurements. For a system described in a Hilbert space of dimension ${N=N}_{1}\dots{}{N}_{p},$ where the numbers ${N}_{i}$ are prime, the reconstruction is accomplished with ${(N}_{1}+1)\dots{}{(N}_{p}+1)$ factorable (local) von Neumann measurements. For the special case where the dimension is a power of a prime, the reconstruction can be performed in a much more efficient way using $N+1$ complementary measurements. If the system is composed of a number of smaller subsystems, these measurements will then in general be nonseparable.

Electrical Impedance Tomography

t. This paper surveys some of the work our group has done in electrical impedance tomography.

A new route for the interpretationof reconstructed wavefunctions in HRTEM

During the last years the renewed interest in focus variation wavefunction reconstruction algorithms has lead to a spectacular improvement of the obtainable resolution in FEG-HRTEM. Unfortunately, it was found that only for thin specimens the reconstructed wavefunction is directly interpretable in terms of the projected atomic potential. Hence there is a need for a direct structure reconstruction algorithm, starting from the reconstructed electron wavefunction. Here we propose a new channelling method in real space that only relies on the very basic concepts of dynamical diffraction in zone axis orientation, and which is very suitable to reveal the correspondence between the wavefunction and the column structure. In this way, a parametrised analytical expression can be obtained so as to reconstruct the projected structure of the object, requiring minimal prior knowledge.In order to develop a sensible structure reconstruction method, a good understanding of the multiple dynamical diffraction process is required.

Time-of-flight diffraction tomography for NDT applications

An ultrasonic imaging system based on a direct time-space reconstruction algorithm has been developed. The radio-frequency signals are collected along a linear synthetic aperture with point sources and receivers in separate positions. This allows the investigation of defects with different incident angles and the final image is less sensitive to the defect characteristics of size, shape and orientation. Theoretical formulae for the lateral and the axial resolution are derived and the actual system performance is evaluated with a computer testing program. Theoretical models of ultrasonic scattering from simple artificial defects embedded in metals, based on the Born approximation and the geometrical theory of diffraction, are employed to simulate signals from some simplified objects covering a broad ka range, with ka up to about 5. Comparisons between theoretical, simulated and experimental lateral and axial resolution are reported, with lateral resolutions found by experiment of 1.34γ, by simulation λ compared with the theoretical limit of 0.36λ.

A new three-dimensional reconstruction method using algebraic reconstruction techniques

Three-dimensional image reconstruction plays a very important role in noninvasive diagnosis of biological systems and nondestructive evaluation of manufactured work-pieces. A new direct three-dimensional reconstruction algorithm, called TART (Three-dimensional ART), is presented in this paper. Oblique projection data are used and an ART-based algorithm is introduced to compensate for the limiting constraints of incomplete projection and/or limited angular coverage. The fact that oblique projection gives useful information to the reconstruction algorithm is shown mathematically. The algorithm can be used to solve the reconstruction problem under the conditions of both complete data and incomplete data. The algorithm first maps geometric information and projection data from an oblique plane into a horizontal plane, then calculates the weighting factors for the voxels based on this horizontal plane, and finally performs a 3-D ART reconstruction. Two experimental results illustrate the superiority of the algorithm over the previous reconstruction methods.