Penalized weighted least-squares image reconstruction for dual energy X-ray transmission tomography

We present a dual-energy (DE) transmission computed tomography (CT) reconstruction method. It is statistically motivated and features nonnegativity constraints in the density domain. A penalized weighted least squares (PWLS) objective function has been chosen to handle the non-Poisson noise added by amorphous silicon (aSi:H) detectors. A Gauss-Seidel algorithm has been used to minimize the objective function. The behavior of the method in terms of bias/standard deviation tradeoff has been compared to that of a DE method that is based on filtered back projection (FBP). The advantages of the DE PWLS method are largest for high noise and/or low flux cases. Qualitative results suggest this as well. Also, the reconstructed images of an object with opaque regions are presented. Possible applications of the method are: attenuation correction for positron emission tomography (PET) images, various quantitative computed tomography (QCT) methods such as bone mineral densitometry (BMD), and the removal of metal streak artifacts.     

An iterative maximum-likelihood polychromatic algorithm for CT

A new iterative maximum-likelihood reconstruction algorithm for X-ray computed tomography is presented. The algorithm prevents beam hardening artifacts by incorporating a polychromatic acquisition model. The continuous spectrum of the X-ray tube is modeled as a number of discrete energies. The energy dependence of the attenuation is taken into account by decomposing the linear attenuation coefficient into a photoelectric component and a Compton scatter component. The relative weight of these components is constrained based on prior material assumptions. Excellent results are obtained for simulations and for phantom measurements. Beam-hardening artifacts are effectively eliminated. The relation with existing algorithms is discussed. The results confirm that improving the acquisition model assumed by the reconstruction algorithm results in reduced artifacts. Preliminary results indicate that metal artifact reduction is a very promising application for this new algorithm.     

Attenuation compensation of cone beam SPECT images using maximum likelihood reconstruction

Attenuation compensation for cone beam single-photon emission computed tomography (SPECT) imaging is performed by cone beam maximum likelihood reconstruction with attenuation included in the transition matrix. Since the transition matrix is too large to be stored in conventional computers, the E-M maximum likelihood estimator is implemented with a ray-tracing algorithm, efficiently recalculating each matrix element as needed. The method was applied and tested in both uniform and nonuniform density phantoms. Test projections sets were obtained from Monte Carlo simulations and experiments using a commercially available cone beam collimator. For representative regions of interest. reconstruction of a uniform sphere is accurate to within 3% throughout, in comparison to a reference image simulated and reconstructed without attenuation. High- and low-activity regions in a uniform density are reconstructed accurately, except that low-activity regions in a more active background have a small error. This error is explainable by the nonnegativity constraints of the E-M estimator and the image statistical noise     

Statistical image reconstruction for polyenergetic X-ray computed tomography

This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energy-dependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energy-dependent mass attenuation coefficient. We formulate a penalized-likelihood function for this polyenergetic model and develop an ordered-subsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated X-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts.     

Reconstructing the 3D shape and bone mineral density distribution of the proximal femur from dual-energy X-ray absorptiometry

The accurate diagnosis of osteoporosis has gained increasing importance due to the aging of our society. Areal bone mineral density (BMD) measured by dual-energy X-ray absorptiometry (DXA) is an established criterion in the diagnosis of osteoporosis. This measure, however, is limited by its two-dimensionality. This work presents a method to reconstruct both the 3D bone shape and 3D BMD distribution of the proximal femur from a single DXA image used in clinical routine. A statistical model of the combined shape and BMD distribution is presented, together with a method for its construction from a set of quantitative computed tomography (QCT) scans. A reconstruction is acquired in an intensity based 3D-2D registration process whereby an instance of the model is found that maximizes the similarity between its projection and the DXA image. Reconstruction experiments were performed on the DXA images of 30 subjects, with a model constructed from a database of QCT scans of 85 subjects. The accuracy was evaluated by comparing the reconstructions with the same subject QCT scans. The method presented here can potentially improve the diagnosis of osteoporosis and fracture risk assessment from the low radiation dose and low cost DXA devices currently used in clinical routine.     

Monotonic penalized-likelihood image reconstruction for X-ray fluorescence computed tomography

In this paper, we derive a monotonic penalized-likelihood algorithm for image reconstruction in X-ray fluorescence computed tomography (XFCT) when the attenuation maps at the energies of the fluorescence X-rays are unknown. In XFCT, a sample is irradiated with pencil beams of monochromatic synchrotron radiation that stimulate the emission of fluorescence X-rays from atoms of elements whose K- or L-edges lie below the energy of the stimulating beam. Scanning and rotating the object through the beam allows for acquisition of a tomographic dataset that can be used to reconstruct images of the distribution of the elements in question. XFCT is a stimulated emission tomography modality, and it is thus necessary to correct for attenuation of the incident and fluorescence photons. The attenuation map is, however, generally known only at the stimulating beam energy and not at the energies of the various fluorescence X-rays of interest. We have developed a penalized-likelihood image reconstruction strategy for this problem. The approach alternates between updating the distribution of a given element and updating the attenuation map for that element's fluorescence X-rays. The approach is guaranteed to increase the penalized likelihood at each iteration. Because the joint objective function is not necessarily concave, the approach may drive the solution to a local maximum. To encourage the algorithm to seek out a reasonable local maximum, we include in the objective function a prior that encourages a relationship, based on physical considerations, between the fluorescence attenuation map and the distribution of the element being reconstructed.     

The GEM MAP algorithm with 3-D SPECT system response

In single photon emission computed tomography (SPECT), every reconstruction algorithm must use some model for the response of the gamma camera to emitted gamma-rays. The true camera response is both spatially variant and object dependent. These two properties result from the effects of scatter, septal penetration, and attenuation, and they forestall determination of the true response with any precision. This motivates the investigation of the performance of reconstruction algorithms when there are errors between the camera response used in the reconstruction algorithm and the true response of the gamma camera. In this regard, the authors compare the filtered backprojection algorithm, the expectation-maximization maximum likelihood algorithm, and the generalized expectation maximization (GEM) maximum a posteriori (MAP) algorithm, a Bayesian algorithm which uses a Markov random field prior.     

A cone-beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography

A filtered backprojection reconstruction algorithm was developed for cardiac single photon emission computed tomography with cone-beam geometry. The algorithm reconstructs cone-beam projections collected from ;short scan' acquisitions of a detector traversing a noncircular planar orbit. Since the algorithm does not correct for photon attenuation, it is designed to reconstruct data collected over an angular range of slightly more than 180 degrees with the assumption that the range of angles is oriented so as not to acquire the highly attenuated posterior projections of emissions from cardiac radiopharmaceuticals. This sampling scheme is performed to minimize the attenuation artifacts that result from reconstructing posterior projections. From computer simulations, it is found that reconstruction of attenuated projections has a greater effect upon quantitation and image quality than any potential cone-beam reconstruction artifacts resulting from insufficient sampling of cone-beam projections. With nonattenuated projection data, cone beam reconstruction errors in the heart are shown to be small (errors of at most 2%).     

The Use of Filtering Methods to Compensate for Constant Attenuation in Single-Photon Emission Computed Tomography

A back projection of filtered projection (BKFIL) reconstruction algorithm is presented that is applicable to single-photon emission computed tomography (ECT) in the presence of a constant attenuating medium such as the brain. The filters used in transmission computed tomography (TCT)-comprised of a ramp multiplied by window functions?are modified so that the single-photon ECT filter is a function of the constant attenuation coefficient. The filters give good reconstruction results with sufficient angular and lateral sampling. With continuous samples the BKFIL algorithm has a point spread function that is the Hankel transform of the window function. The resolution and statistical properties of the filters are demonstrated by various simulations which assume an ideal detector response. Statistical formulas for the reconstructed image show that the square of the percent-root-mean-square (percent-rms) uncertainty of the reconstruction is inversely proportional to the total measured counts. The results indicate that constant attenuation can be compensated for by using an attenuation-dependent filter that reconstructs the transverse section reliably. Computer time requirements are two times that of conventional TCT or positron ECT and there is no increase in memory requirements.     

Variable pitch reconstruction using John's equation

We present an algorithm to reconstruct helical cone beam computed tomography (CT) data acquired at variable pitch. The algorithm extracts a halfscan segment of projections using an extended version of the advanced single slice rebinning (ASSR) algorithm. ASSR rebins constant pitch cone beam data to fan beam projections that approximately lie on a plane that is tilted to optimally fit the source helix. For variable pitch, the error between the tilted plane chosen by ASSR and the source helix increases, resulting in increased image artifacts. To reduce the artifacts, we choose a reconstruction plane, which is tilted and shifted relative to the source trajectory. We then correct rebinned fan beam data using John's equation to virtually move the source into the tilted and shifted reconstruction plane. Results obtained from simulated phantom images and scanner images demonstrate the applicability of the proposed algorithm.      

Reconstruction Algorithm for Fan Beam with a Displaced Center-of-Rotation

A convolutional backprojection algorithm is derived for a fan beam geometry that has its center-of-rotation displaced from the midline of the fan beam. In single photon emission computed tomography (SPECT), where a transaxial converging collimator is used with a rotating gamma camera, it is difficult to precisely align the collimator so that the mechanical center-of-rotation is colinear with the midline of the fan beam. A displacement of the center-of-rotation can also occur in X-ray CT when the X-ray source is mispositioned. Standard reconstruction algorithms which directly filter and backproject the fan beam data without rebinning into parallel beam geometry have been derived for a geometry having its center-of-rotation at the midline of the fan beam. However, in the case of a misalignment of the center-of-rotation, if these conventional reconstruction algorithms are used to reconstruct the fan beam projections, structured artifacts and a loss of resolution will result. We illustrate these artifacts with simulations and demonstrate how the news algorithm corrects for this misalignment. We also show a method to estimate the parameters of the fan beam geometry including the shift in the center-of-rotation.     

Anatomical data fusion for quantitative reconstruction inmyocardial tomoscintigraphy using a spline model of the thorax organs

The authors present the fusion of anatomical data as a method for improving the reconstruction in single photon emission computed tomography (SPECT). Anatomical data is used to deduce a parameterized model of organs in a reconstructed slice using spline curves. This model allows the authors to define the imaging process, i.e., the direct problem, more adequately, and furthermore to restrict the reconstruction to the emitting zones. Instead of the usual square pixels, the authors use a new kind of discretization pixel, which fits to the contour in the region of interest. In the reconstruction phase, the authors estimate the activity in the emitting zones and also the optimum parameters of their model. Concentrating on the left ventricular (LV) wall activity, the simulation and phantom results show an accurate estimation of both the myocardial shape and the radioactive emission     

Modeling the axial extension of a transmission line source within iterative reconstruction via multiple transmission sources

Reconstruction algorithms for transmission tomography have generally assumed that the photons reaching a particular detector bin at a particular angle originate from a single point source. In this paper, we highlight several cases of extended transmission sources, in which it may be useful to approach the estimation of attenuation coefficients as a problem involving multiple transmission point sources. Examined in detail is the case of a fixed transmission line source with a fan-beam collimator. This geometry can result in attenuation images that have significant axial blur. Herein it is also shown, empirically, that extended transmission sources can result in biased estimates of the average attenuation, and an explanation is proposed. The finite axial resolution of the transmission line source configuration is modeled within iterative reconstruction using an expectation-maximization algorithm that was previously derived for estimating attenuation coefficients from single photon emission computed tomography (SPECT) emission data. The same algorithm is applicable to both problems because both can be thought of as involving multiple transmission sources. It is shown that modeling axial blur within reconstruction removes the bias in the average estimated attenuation and substantially improves the axial resolution of attenuation images.     

Choice of initial conditions in the ML reconstruction of fan-beamtransmission with truncated projection data

The authors investigate the effects of initial conditions in the iterative maximum-likelihood (ML) reconstruction of fan-beam transmission projection data with truncation. In an iterative ML reconstruction, the estimate of the transmission reconstructed image in the previous iteration is multiplied by some factors to obtain the current estimate. Normally, a flat initial condition (FIC) or an image with equal positive pixel values is used as initial condition for an ML reconstruction. Usage of FIC has also been perceived as a way of preventing any bias on the reconstruction which may have come from the initial condition. When projection data have truncation, the authors show that using are FIC in an ML iterative reconstruction can introduce a bias to the reconstruction inside the densely sampled region (DSR), whose projection data have no truncation at any angle. To reduce this bias, the authors propose to use the largest right singular vector (LRSV) of the system matrix as the initial condition, and demonstrate that the bias can be reduced with the LRSV. When data truncation is reduced, the LRSV approaches the FIC. This result does not contradict to the use of FIC when projection data are not truncated. The authors also demonstrate that the reconstructed transmission image using LRSV as initial condition provides a more accurate attenuation coefficient distribution than that using FIC. However, the improvement is mostly in the area outside the DSR     

An approximation formula for the variance of PET region-of-interest values

An approximation formula for the variance of positron emission tomography (PET) region-of-interest (ROI) values has been developed, implemented, and evaluated. This formula does not require access to the original projection data and is therefore convenient for routine use. The formula was derived by applying successive approximations to the filtered-backprojection reconstruction algorithm. ROI variance is estimated from the product of mean pixel variance within the region and a term accounting for the intercorrelation of all pixel pairs inside the region. The formula accounts for radioactivity distribution, attenuation, randoms, scatter, deadtime, detector normalization, scan length, decay, and reconstruction filter. The algorithm was tested by comparison to the exact ROI variance as calculated with Huesman's algorithm. Tests with scan data from phantoms, animals, and humans obtained on the Scanditronix PC2048-15B tomograph showed the approximation formula to be accurate to within +/-10%     

Algorithm for X-ray Scatter, Beam-Hardening, and Beam Profile Correction in Diagnostic (Kilovoltage) and Treatment (Megavoltage) Cone Beam CT

Quantitative reconstruction of cone beam X-ray computed tomography (CT) datasets requires accurate modeling of scatter, beam-hardening, beam profile, and detector response. Typically, commercial imaging systems use fast empirical corrections that are designed to reduce visible artifacts due to incomplete modeling of the image formation process. In contrast, Monte Carlo (MC) methods are much more accurate but are relatively slow. Scatter kernel superposition (SKS) methods offer a balance between accuracy and computational practicality. We show how a single SKS algorithm can be employed to correct both kilovoltage (kV) energy (diagnostic) and megavoltage (MV) energy (treatment) X-ray images. Using MC models of kV and MV imaging systems, we map intensities recorded on an amorphous silicon flat panel detector to water-equivalent thicknesses (WETs). Scattergrams are derived from acquired projection images using scatter kernels indexed by the local WET values and are then iteratively refined using a scatter magnitude bounding scheme that allows the algorithm to accommodate the very high scatter-to-primary ratios encountered in kV imaging. The algorithm recovers radiological thicknesses to within 9% of the true value at both kV and megavolt energies. Nonuniformity in CT reconstructions of homogeneous phantoms is reduced by an average of 76% over a wide range of beam energies and phantom geometries.      

Reconstruction From Uniformly Attenuated SPECT Projection Data Using the DBH Method

An algorithm was developed for the 2-D reconstruction of truncated and nontruncated uniformly attenuated data acquired from single photon emission computed tomography (SPECT). The algorithm is able to reconstruct data from half-scan (180$^{circ}$ ) and short-scan (180$^{circ}$ +fan angle) acquisitions for parallel- and fan-beam geometries, respectively, as well as data from full-scan (360 $^{circ}$) acquisitions. The algorithm is a derivative, backprojection, and Hilbert transform (DBH) method, which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted Hilbert transform. The kernel of the inverse weighted Hilbert transform is solved numerically using matrix inversion. Numerical simulations confirm that the DBH method provides accurate reconstructions from half-scan and short-scan data, even when there is truncation. However, as the attenuation increases, finer data sampling is required.      

Simultaneous MR-Compatible Emission and Transmission Imaging for PET Using Time-of-Flight Information

Quantitative positron emission tomography (PET) imaging relies on accurate attenuation correction. Predicting attenuation values from magnetic resonance (MR) images is difficult because MR signals are related to proton density and relaxation properties of tissues. Here, we propose a method to derive the attenuation map from a transmission scan. An annulus transmission source is positioned inside the field-of-view of the PET scanner. First a blank scan is acquired. The patient is injected with FDG and placed inside the scanner. 511-keV photons coming from the patient and the transmission source are acquired simultaneously. Time-of-flight information is used to extract the coincident photons originating from the annulus. The blank and transmission data are compared in an iterative reconstruction method to derive the attenuation map. Simulations with a digital phantom were performed to validate the method. The reconstructed attenuation coefficients differ less than 5% in volumes of interest inside the lungs, bone, and soft tissue. When applying attenuation correction in the reconstruction of the emission data a standardized uptake value error smaller than 9% was obtained for all tissues. In conclusion, our method can reconstruct the attenuation map and the emission data from a simultaneous scan without prior knowledge about the anatomy or the attenuation coefficients of the tissues.     

Evaluation of an efficient compensation method for quantitative fan-beam brain SPECT reconstruction

Fan-beam collimators are designed to improve the system sensitivity and resolution for imaging small objects such as the human brain and breasts in single photon emission computed tomography (SPECT). Many reconstruction algorithms have been studied and applied to this geometry to deal with every kind of degradation factor. This paper presents a new reconstruction approach for SPECT with circular orbit, which demonstrated good performance in terms of both accuracy and efficiency. The new approach compensates for degradation factors including noise, scatter, attenuation, and spatially variant detector response. Its uniform attenuation approximation strategy avoids the additional transmission scan for the brain attenuation map, hence reducing the patient radiation dose and furthermore simplifying the imaging procedure. We evaluate and compare this new approach with the well-established ordered-subset expectation-maximization iterative method, using Monte Carlo simulations. We perform quantitative analysis with regional bias-variance, receiver operating characteristics, and Hotelling trace studies for both methods. The results demonstrate that our reconstruction strategy has comparable performance with a significant reduction of computing time.