Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs

Task-based detectability in CT image reconstruction by filtered backprojection and penalized likelihood estimation

PURPOSE

Nonstationarity is an important aspect of imaging performance in CT and cone-beam CT (CBCT), especially for systems employing iterative reconstruction. This work presents a theoretical framework for both filtered-backprojection (FBP) and penalized-likelihood (PL) reconstruction that includes explicit descriptions of nonstationary noise, spatial resolution, and task-based detectability index. Potential utility of the model was demonstrated in the optimal selection of regularization parameters in PL reconstruction.

METHODS

Analytical models for local modulation transfer function (MTF) and noise-power spectrum (NPS) were investigated for both FBP and PL reconstruction, including explicit dependence on the object and spatial location. For FBP, a cascaded systems analysis framework was adapted to account for nonstationarity by separately calculating fluence and system gains for each ray passing through any given voxel. For PL, the point-spread function and covariance were derived using the implicit function theorem and first-order Taylor expansion according to Fessler ["Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography," IEEE Trans. Image Process. 5(3), 493-506 (1996)]. Detectability index was calculated for a variety of simple tasks. The model for PL was used in selecting the regularization strength parameter to optimize task-based performance, with both a constant and a spatially varying regularization map.

RESULTS

Theoretical models of FBP and PL were validated in 2D simulated fan-beam data and found to yield accurate predictions of local MTF and NPS as a function of the object and the spatial location. The NPS for both FBP and PL exhibit similar anisotropic nature depending on the pathlength (and therefore, the object and spatial location within the object) traversed by each ray, with the PL NPS experiencing greater smoothing along directions with higher noise. The MTF of FBP is isotropic and independent of location to a first order approximation, whereas the MTF of PL is anisotropic in a manner complementary to the NPS. Task-based detectability demonstrates dependence on the task, object, spatial location, and smoothing parameters. A spatially varying regularization "map" designed from locally optimal regularization can improve overall detectability beyond that achievable with the commonly used constant regularization parameter.

CONCLUSIONS

Analytical models for task-based FBP and PL reconstruction are predictive of nonstationary noise and resolution characteristics, providing a valuable framework for understanding and optimizing system performance in CT and CBCT.

Proximal ADMM for multi-channel image reconstruction in spectral X-ray CT

The development of spectral X-ray computed tomography (CT) using binned photon-counting detectors has received great attention in recent years and has enabled selective imaging of contrast agents loaded with K-edge materials. A practical issue in implementing this technique is the mitigation of the high-noise levels often present in material-decomposed sinogram data. In this work, the spectral X-ray CT reconstruction problem is formulated within a multi-channel (MC) framework in which statistical correlations between the decomposed material sinograms can be exploited to improve image quality. Specifically, a MC penalized weighted least squares (PWLS) estimator is formulated in which the data fidelity term is weighted by the MC covariance matrix and sparsity-promoting penalties are employed. This allows the use of any number of basis materials and is therefore applicable to photon-counting systems and K-edge imaging. To overcome numerical challenges associated with use of the full covariance matrix as a data fidelity weight, a proximal variant of the alternating direction method of multipliers is employed to minimize the MC PWLS objective function. Computer-simulation and experimental phantom studies are conducted to quantitatively evaluate the proposed reconstruction method.

Multi-material decomposition using statistical image reconstruction for spectral CT

Spectral computed tomography (CT) provides information on material characterization and quantification because of its ability to separate different basis materials. Dual-energy (DE) CT provides two sets of measurements at two different source energies. In principle, two materials can be accurately decomposed from DECT measurements. However, many clinical and industrial applications require three or more material images. For triple-material decomposition, a third constraint, such as volume conservation, mass conservation or both, is required to solve three sets of unknowns from two sets of measurements. The recently proposed flexible image-domain (ID) multi-material decomposition) method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel. We propose a penalized-likelihood (PL) method with edge-preserving regularizers for each material to reconstruct multi-material images using a similar constraint from sinogram data. We develop an optimization transfer method with a series of pixel-wise separable quadratic surrogate (PWSQS) functions to monotonically decrease the complicated PL cost function. The PWSQS algorithm separates pixels to allow simultaneous update of all pixels, but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material- and pixel-wise SQS algorithms. Comparing with the ID method using 2-D fan-beam simulations, the PL method greatly reduced noise, streak and cross-talk artifacts in the reconstructed basis component images, and achieved much smaller root mean square errors.

Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography

The model-based image reconstruction approaches in photoacoustic tomography have a distinct advantage compared to traditional analytical methods for cases where limited data is available. These methods typically deploy Tikhonov based regularization scheme to reconstruct the initial pressure from the boundary acoustic data. The model-resolution for these cases represents the blur induced by the regularization scheme. A method that utilizes this blurring model and performs the basis pursuit deconvolution to improve the quantitative accuracy of the reconstructed photoacoustic image is proposed and shown to be superior compared to other traditional methods via three numerical experiments. Moreover, this deconvolution including the building of an approximate blur matrix is achieved via the Lanczos bidagonalization (least-squares QR) making this approach attractive in real-time.

Patlak image estimation from dual time-point list-mode PET data

We investigate using dual time-point PET data to perform Patlak modeling. This approach can be used for whole body dynamic PET studies in which we compute voxel-wise estimates of Patlak parameters using two frames of data for each bed position. Our approach directly uses list-mode arrival times for each event to estimate the Patlak parametric image. We use a penalized likelihood method in which the penalty function uses spatially variant weighting to ensure a count independent local impulse response. We evaluate performance of the method in comparison to fractional changes in SUV values (%DSUV) between the two frames using Cramer Rao analysis and Monte Carlo simulation. Receiver operating characteristic (ROC) curves are used to compare performance in differentiating tumors relative to background based on the dynamic data sets. Using area under the ROC curve as a performance metric, we show superior performance of Patlak relative to %DSUV over a range of dynamic data sets and parameters. These results suggest that Patlak analysis may be appropriate for analysis of dual time-point whole body PET data and could lead to superior detection of tumors relative to %DSUV metrics.

Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging

The image reconstruction problem encountered in diffuse optical tomographic imaging is ill-posed in nature, necessitating the usage of regularization to result in stable solutions. This regularization also results in loss of resolution in the reconstructed images. A frame work, that is attributed by model-resolution, to improve the reconstructed image characteristics using the basis pursuit deconvolution method is proposed here. The proposed method performs this deconvolution as an additional step in the image reconstruction scheme. It is shown, both in numerical and experimental gelatin phantom cases, that the proposed method yields better recovery of the target shapes compared to traditional method, without the loss of quantitativeness of the results.

Efficient determination of the uncertainty for the optimization of SPECT system design: a subsampled fisher information matrix

System designs in single photon emission tomography (SPECT) can be evaluated based on the fundamental trade-off between bias and variance that can be achieved in the reconstruction of emission tomograms. This trade off can be derived analytically using the Cramer-Rao type bounds, which imply the calculation and the inversion of the Fisher information matrix (FIM). The inverse of the FIM expresses the uncertainty associated to the tomogram, enabling the comparison of system designs. However, computing, storing and inverting the FIM is not practical with 3-D imaging systems. In order to tackle the problem of the computational load in calculating the inverse of the FIM, a method based on the calculation of the local impulse response and the variance, in a single point, from a single row of the FIM, has been previously proposed for system design. However this approximation (circulant approximation) does not capture the global interdependence between the variables in shift-variant systems such as SPECT, and cannot account e.g., for data truncation or missing data. Our new formulation relies on subsampling the FIM. The FIM is calculated over a subset of voxels arranged in a grid that covers the whole volume. Every element of the FIM at the grid points is calculated exactly, accounting for the acquisition geometry and for the object. This new formulation reduces the computational complexity in estimating the uncertainty, but nevertheless accounts for the global interdependence between the variables, enabling the exploration of design spaces hindered by the circulant approximation. The graphics processing unit accelerated implementation of the algorithm reduces further the computation times, making the algorithm a good candidate for real-time optimization of adaptive imaging systems. This paper describes the subsampled FIM formulation and implementation details. The advantages and limitations of the new approximation are explored, in comparison with the circulant approximation, in the context of design optimization of a parallel-hole collimator SPECT system and of an adaptive imaging system (similar to the commercially available D-SPECT).

MAP reconstruction for Fourier rebinned TOF-PET data

Time-of-flight (TOF) information improves the signal-to-noise ratio in positron emission tomography (PET). The computation cost in processing TOF-PET sinograms is substantially higher than for nonTOF data because the data in each line of response is divided among multiple TOF bins. This additional cost has motivated research into methods for rebinning TOF data into lower dimensional representations that exploit redundancies inherent in TOF data. We have previously developed approximate Fourier methods that rebin TOF data into either three-dimensional (3D) nonTOF or 2D nonTOF formats. We refer to these methods respectively as FORET-3D and FORET-2D. Here we describe maximum a posteriori (MAP) estimators for use with FORET rebinned data. We first derive approximate expressions for the variance of the rebinned data. We then use these results to rescale the data so that the variance and mean are approximately equal allowing us to use the Poisson likelihood model for MAP reconstruction. MAP reconstruction from these rebinned data uses a system matrix in which the detector response model accounts for the effects of rebinning. Using these methods we compare the performance of FORET-2D and 3D with TOF and nonTOF reconstructions using phantom and clinical data. Our phantom results show a small loss in contrast recovery at matched noise levels using FORET compared to reconstruction from the original TOF data. Clinical examples show FORET images that are qualitatively similar to those obtained from the original TOF-PET data but with a small increase in variance at matched resolution. Reconstruction time is reduced by a factor of 5 and 30 using FORET3D+MAP and FORET2D+MAP respectively compared to 3D TOF MAP, which makes these methods attractive for clinical applications.

Regularization design in penalized maximum-likelihood image reconstruction for lesion detection in 3D PET

Detecting cancerous lesions is a major clinical application in emission tomography. In previous work, we have studied penalized maximum-likelihood (PML) image reconstruction for the detection task and proposed a method to design a shift-invariant quadratic penalty function to maximize detectability of a lesion at a known location in a two dimensional image. Here we extend the regularization design to maximize detectability of lesions at unknown locations in fully 3D PET. We used a multiview channelized Hotelling observer (mvCHO) to assess the lesion detectability in 3D images to mimic the condition where a human observer examines three orthogonal views of a 3D image for lesion detection. We derived simplified theoretical expressions that allow fast prediction of the detectability of a 3D lesion. The theoretical results were used to design the regularization in PML reconstruction to improve lesion detectability. We conducted computer-based Monte Carlo simulations to compare the optimized penalty with the conventional penalty for detecting lesions of various sizes. Only true coincidence events were simulated. Lesion detectability was also assessed by two human observers, whose performances agree well with that of the mvCHO. Both the numerical observer and human observer results showed a statistically significant improvement in lesion detection by using the proposed penalty function compared to using the conventional penalty function.

Sparsity Constrained Mixture Modeling for the Estimation of Kinetic Parameters in Dynamic PET

The estimation and analysis of kinetic parameters in dynamic positron emission tomography (PET) is frequently confounded by tissue heterogeneity and partial volume effects. We propose a new constrained model of dynamic PET to address these limitations. The proposed formulation incorporates an explicit mixture model in which each image voxel is represented as a mixture of different pure tissue types with distinct temporal dynamics. We use Cramér-Rao lower bounds to demonstrate that the use of prior information is important to stabilize parameter estimation with this model. As a result, we propose a constrained formulation of the estimation problem that we solve using a two-stage algorithm. In the first stage, a sparse signal processing method is applied to estimate the rate parameters for the different tissue compartments from the noisy PET time series. In the second stage, tissue fractions and the linear parameters of different time activity curves are estimated using a combination of spatial-regularity and fractional mixture constraints. A block coordinate descent algorithm is combined with a manifold search to robustly estimate these parameters. The method is evaluated with both simulated and experimental dynamic PET data.

Preclinical evaluation of parametric image reconstruction of [18F]FMISO PET: correlation with ex vivo immunohistochemistry

Compared to indirect methods, direct parametric image reconstruction (PIR) has the advantage of high quality and low statistical errors. However, it is not yet clear if this improvement in quality is beneficial for physiological quantification. This study aimed to evaluate direct PIR for the quantification of tumor hypoxia using the hypoxic fraction (HF) assessed from immunohistological data as a physiological reference. Sixteen mice with xenografted human squamous cell carcinomas were scanned with dynamic [18F]FMISO PET. Afterward, tumors were sliced and stained with H&E and the hypoxia marker pimonidazole. The hypoxic signal was segmented using k-means clustering and HF was specified as the ratio of the hypoxic area over the viable tumor area. The parametric Patlak slope images were obtained by indirect voxel-wise modeling on reconstructed images using filtered back projection and ordered-subset expectation maximization (OSEM) and by direct PIR (e.g., parametric-OSEM, POSEM). The mean and maximum Patlak slopes of the tumor area were investigated and compared with HF. POSEM resulted in generally higher correlations between slope and HF among the investigated methods. A strategy for the delineation of the hypoxic tumor volume based on thresholding parametric images at half maximum of the slope is recommended based on the results of this study.

Accelerating ordered subsets image reconstruction for X-ray CT using spatially nonuniform optimization transfer

Statistical image reconstruction algorithms in X-ray computed tomography (CT) provide improved image quality for reduced dose levels but require substantial computation time. Iterative algorithms that converge in few iterations and that are amenable to massive parallelization are favorable in multiprocessor implementations. The separable quadratic surrogate (SQS) algorithm is desirable as it is simple and updates all voxels simultaneously. However, the standard SQS algorithm requires many iterations to converge. This paper proposes an extension of the SQS algorithm that leads to spatially nonuniform updates. The nonuniform (NU) SQS encourages larger step sizes for the voxels that are expected to change more between the current and the final image, accelerating convergence, while the derivation of NU-SQS guarantees monotonic descent. Ordered subsets (OS) algorithms can also accelerate SQS, provided suitable "subset balance" conditions hold. These conditions can fail in 3-D helical cone-beam CT due to incomplete sampling outside the axial region-of-interest (ROI). This paper proposes a modified OS algorithm that is more stable outside the ROI in helical CT. We use CT scans to demonstrate that the proposed NU-OS-SQS algorithm handles the helical geometry better than the conventional OS methods and "converges" in less than half the time of ordinary OS-SQS.

Design and performance evaluation of a 20-aperture multipinhole collimator for myocardial perfusion imaging applications

Single photon emission computed tomography (SPECT) myocardial perfusion imaging remains a critical tool in the diagnosis of coronary artery disease. However, after more than three decades of use, photon detection efficiency remains poor and unchanged. This is due to the continued reliance on parallel-hole collimators first introduced in 1964. These collimators possess poor geometric efficiency. Here we present the performance evaluation results of a newly designed multipinhole collimator with 20 pinhole apertures (PH20) for commercial SPECT systems. Computer simulations and numerical observer studies were used to assess the noise, bias and diagnostic imaging performance of a PH20 collimator in comparison with those of a low energy high resolution (LEHR) parallel-hole collimator. Ray-driven projector/backprojector pairs were used to model SPECT imaging acquisitions, including simulation of noiseless projection data and performing MLEM/OSEM image reconstructions. Poisson noise was added to noiseless projections for realistic projection data. Noise and bias performance were investigated for five mathematical cardiac and torso (MCAT) phantom anatomies imaged at two gantry orbit positions (19.5 and 25.0 cm). PH20 and LEHR images were reconstructed with 300 MLEM iterations and 30 OSEM iterations (ten subsets), respectively. Diagnostic imaging performance was assessed by a receiver operating characteristic (ROC) analysis performed on a single MCAT phantom; however, in this case PH20 images were reconstructed with 75 pixel-based OSEM iterations (four subsets). Four PH20 projection views from two positions of a dual-head camera acquisition and 60 LEHR projections were simulated for all studies. At uniformly-imposed resolution of 12.5 mm, significant improvements in SNR and diagnostic sensitivity (represented by the area under the ROC curve, or AUC) were realized when PH20 collimators are substituted for LEHR parallel-hole collimators. SNR improves by factors of 1.94-2.34 for the five patient anatomies and two orbital positions studied. For the ROC analysis the PH20 AUC is larger than the LEHR AUC with a p-value of 0.0067. Bias performance, however, decreases with the use of PH20 collimators. Systematic analyses showed PH20 collimators present improved diagnostic imaging performance over LEHR collimators, requiring only collimator exchange on existing SPECT cameras for their use.

Quantitative statistical methods for image quality assessment

Quantitative measures of image quality and reliability are critical for both qualitative interpretation and quantitative analysis of medical images. While, in theory, it is possible to analyze reconstructed images by means of Monte Carlo simulations using a large number of noise realizations, the associated computational burden makes this approach impractical. Additionally, this approach is less meaningful in clinical scenarios, where multiple noise realizations are generally unavailable. The practical alternative is to compute closed-form analytical expressions for image quality measures. The objective of this paper is to review statistical analysis techniques that enable us to compute two key metrics: resolution (determined from the local impulse response) and covariance. The underlying methods include fixed-point approaches, which compute these metrics at a fixed point (the unique and stable solution) independent of the iterative algorithm employed, and iteration-based approaches, which yield results that are dependent on the algorithm, initialization, and number of iterations. We also explore extensions of some of these methods to a range of special contexts, including dynamic and motion-compensated image reconstruction. While most of the discussed techniques were developed for emission tomography, the general methods are extensible to other imaging modalities as well. In addition to enabling image characterization, these analysis techniques allow us to control and enhance imaging system performance. We review practical applications where performance improvement is achieved by applying these ideas to the contexts of both hardware (optimizing scanner design) and image reconstruction (designing regularization functions that produce uniform resolution or maximize task-specific figures of merit).

Model observers in medical imaging research

Model observers play an important role in the optimization and assessment of imaging devices. In this review paper, we first discuss the basic concepts of model observers, which include the mathematical foundations and psychophysical considerations in designing both optimal observers for optimizing imaging systems and anthropomorphic observers for modeling human observers. Second, we survey a few state-of-the-art computational techniques for estimating model observers and the principles of implementing these techniques. Finally, we review a few applications of model observers in medical imaging research.

Patchwork reconstruction with resolution modeling for digital breast tomosynthesis

PURPOSE

Digital breast tomosynthesis is a relatively new diagnostic x-ray modality that allows high resolution breast imaging while suppressing interference from overlapping anatomical structures. However, proper visualization of microcalcifications remains a challenge. For the subset of systems considered by the authors, the main cause of deterioration is movement of the x-ray source during exposures. They propose a modified grouped coordinate ascent algorithm that includes a specific acquisition model to compensate for this deterioration.

METHODS

A resolution model based on the movement of the x-ray source during image acquisition is created and combined with a grouped coordinate ascent algorithm. Choosing planes parallel to the detector surface as the groups enables efficient implementation of the position dependent resolution model. In the current implementation, the resolution model is approximated by a Gaussian smoothing kernel. The effect of the resolution model on the iterative reconstruction is evaluated by measuring contrast to noise ratio (CNR) of spherical microcalcifications in a homogeneous background. After this, the new reconstruction method is compared to the optimized filtered backprojection method for the considered system, by performing two observer studies: the first study simulates clusters of spherical microcalcifications in a power law background for a free search task; the second study simulates smooth or irregular microcalcifications in the same type of backgrounds for a classification task.

RESULTS

Including the resolution model in the iterative reconstruction methods increases the CNR of microcalcifications. The first observer study shows a significant improvement in detection of microcalcifications (p = 0.029), while the second study shows that performance on a classification task remains the same (p = 0.935) compared to the filtered backprojection method.

CONCLUSIONS

The new method shows higher CNR and improved visualization of microcalcifications in an observer experiment on synthetic data. Further study of the negative results of the classification task showed performance variations throughout the volume linked to the changing noise structure introduced by the combination of the resolution model and the smoothing prior.

Magnetic resonance-guided positron emission tomography image reconstruction

The resolution of positron emission tomography (PET) images is limited by the physics of positron-electron annihilation and instrumentation for photon coincidence detection. Model-based methods that incorporate accurate physical and statistical models have produced significant improvements in reconstructed image quality when compared with filtered backprojection reconstruction methods. However, it has often been suggested that by incorporating anatomical information, the resolution and noise properties of PET images could be further improved, leading to better quantitation or lesion detection. With the recent development of combined MR-PET scanners, we can now collect intrinsically coregistered magnetic resonance images. It is therefore possible to routinely make use of anatomical information in PET reconstruction, provided appropriate methods are available. In this article, we review research efforts over the past 20 years to develop these methods. We discuss approaches based on the use of both Markov random field priors and joint information or entropy measures. The general framework for these methods is described, and their performance and longer-term potential and limitations are discussed.

Characterization of statistical prior image constrained compressed sensing (PICCS): II. Application to dose reduction

PURPOSE

The ionizing radiation imparted to patients during computed tomography exams is raising concerns. This paper studies the performance of a scheme called dose reduction using prior image constrained compressed sensing (DR-PICCS). The purpose of this study is to characterize the effects of a statistical model of x-ray detection in the DR-PICCS framework and its impact on spatial resolution.

METHODS

Both numerical simulations with known ground truth and in vivo animal dataset were used in this study. In numerical simulations, a phantom was simulated with Poisson noise and with varying levels of eccentricity. Both the conventional filtered backprojection (FBP) and the PICCS algorithms were used to reconstruct images. In PICCS reconstructions, the prior image was generated using two different denoising methods: a simple Gaussian blur and a more advanced diffusion filter. Due to the lack of shift-invariance in nonlinear image reconstruction such as the one studied in this paper, the concept of local spatial resolution was used to study the sharpness of a reconstructed image. Specifically, a directional metric of image sharpness, the so-called pseudopoint spread function (pseudo-PSF), was employed to investigate local spatial resolution.

RESULTS

In the numerical studies, the pseudo-PSF was reduced from twice the voxel width in the prior image down to less than 1.1 times the voxel width in DR-PICCS reconstructions when the statistical model was not included. At the same noise level, when statistical weighting was used, the pseudo-PSF width in DR-PICCS reconstructed images varied between 1.5 and 0.75 times the voxel width depending on the direction along which it was measured. However, this anisotropy was largely eliminated when the prior image was generated using diffusion filtering; the pseudo-PSF width was reduced to below one voxel width in that case. In the in vivo study, a fourfold improvement in CNR was achieved while qualitatively maintaining sharpness; images also had a qualitatively more uniform noise spatial distribution when including a statistical model.

CONCLUSIONS

DR-PICCS enables to reconstruct CT images with lower noise than FBP and the loss of spatial resolution can be mitigated to a large extent. The introduction of statistical modeling in DR-PICCS may improve some noise characteristics, but it also leads to anisotropic spatial resolution properties. A denoising method, such as the directional diffusion filtering, has been demonstrated to reduce anisotropy in spatial resolution effectively when it was combined with DR-PICCS with statistical modeling.

Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals

We investigated the penalized maximum likelihood estimation of lifetime and amplitude images for fluorescence lifetime imaging microscopy. The proposed method penalizes large variations in the lifetimes and amplitudes in the spatial domain to reduces noise in the images, which is a serious problem in the conventional maximum likelihood estimation method. For an effective optimization of the objective function, we applied an optimization transfer method that is based on a separable surrogate function. Simulations show that the proposed method outperforms the conventional MLE method in terms of the estimation accuracy, and the proposed method yielded less noisy images in real experiments.

Repeatable noninvasive measurement of mouse myocardial glucose uptake with 18F-FDG: evaluation of tracer kinetics in a type 1 diabetes model

A noninvasive and repeatable method for assessing mouse myocardial glucose uptake with (18)F-FDG PET and Patlak kinetic analysis was systematically assessed using the vena cava image-derived blood input function (IDIF).

METHODS

Contrast CT and computer modeling was used to determine the vena cava recovery coefficient. Vena cava IDIF (n = 7) was compared with the left ventricular cavity IDIF, with blood and liver activity measured ex vivo at 60 min. The test-retest repeatability (n = 9) of Patlak influx constant K(i) at 10-40 min was assessed quantitatively using Bland-Altman analysis. Myocardial glucose uptake rates (rMGU) using the vena cava IDIF were calculated at baseline (n = 8), after induction of type 1 diabetes (streptozotocin [50 mg/kg] intraperitoneally, 5 d), and after acute insulin stimulation (0.08 mU/kg of body weight intraperitoneally). These changes were analyzed with a standardized uptake value calculation at 20 and 40 min after injection to correlate to the Patlak time interval.

RESULTS

The proximal mouse vena cava diameter was 2.54 ± 0.30 mm. The estimated recovery coefficient, calculated using nonlinear image reconstruction, decreased from 0.76 initially (time 0 to peak activity) to 0.61 for the duration of the scan. There was a 17% difference in the image-derived vena cava blood activity at 60 min, compared with the ex vivo blood activity measured in the γ-counter. The coefficient of variability for Patlak K(i) values between mice was found to be 23% with the proposed method, compared with 51% when using the left ventricular cavity IDIF (P < 0.05). No significant bias in K(i) was found between repeated scans with a coefficient of repeatability of 0.16 mL/min/g. Calculated rMGU values were reduced by 60% in type 1 diabetic mice from baseline scans (P < 0.03, ANOVA), with a subsequent increase of 40% to a level not significantly different from baseline after acute insulin treatment. These results were confirmed with a standardized uptake value measured at 20 and 40 min.

CONCLUSION

The mouse vena cava IDIF provides repeatable assessment of the blood time-activity curve for Patlak kinetic modeling of rMGU. An expected significant reduction in myocardial glucose uptake was demonstrated in a type 1 diabetic mouse model, with significant recovery after acute insulin treatment, using a mouse vena cava IDIF approach.

3D spectral imaging with synchrotron Fourier transform infrared spectro-microtomography

We report Fourier transform infrared spectro-microtomography, a nondestructive three-dimensional imaging approach that reveals the distribution of distinctive chemical compositions throughout an intact biological or materials sample. The method combines mid-infrared absorption contrast with computed tomographic data acquisition and reconstruction to enhance chemical and morphological localization by determining a complete infrared spectrum for every voxel (millions of spectra determined per sample).

Statistical reconstruction of material decomposed data in spectral CT

Photon-counting detector technology has enabled the first experimental investigations of energy-resolved computed tomography (CT) imaging and the potential use for K-edge imaging. However, limitations in regards to detecter technology have been imposing a limit to effective count rates. As a consequence, this has resulted in high noise levels in the obtained images given scan time limitations in CT imaging applications. It has been well recognized in the area of low-dose imaging with conventional CT that iterative image reconstruction provides a superior signal to noise ratio compared to traditional filtered backprojection techniques. Furthermore, iterative reconstruction methods also allow for incorporation of a roughness penalty function in order to make a trade-off between noise and spatial resolution in the reconstructed images. In this work, we investigate statistically-principled iterative image reconstruction from material-decomposed sinograms in spectral CT. The proposed reconstruction algorithm seeks to minimize a penalized likelihood-based cost functional, where the parameters of the likelihood function are estimated by computing the Fisher information matrix associated with the material decomposition step. The performance of the proposed reconstruction method is quantitatively investigated by use of computer-simulated and experimental phantom data. The potential for improved K-edge imaging is also demonstrated in an animal experiment.

Statistical reconstruction algorithms for continuous wave electron spin resonance imaging

Electron spin resonance imaging (ESRI) is an important branch of ESR that deals with heterogeneous samples ranging from semiconductor materials to small live animals and even humans. ESRI can produce either spatial images (providing information about the spatially dependent radical concentration) or spectral-spatial images, where an extra dimension is added to describe the absorption spectrum of the sample (which can also be spatially dependent). The mapping of oxygen in biological samples, often referred to as oximetry, is a prime example of an ESRI application. ESRI suffers frequently from a low signal-to-noise ratio (SNR), which results in long acquisition times and poor image quality. A broader use of ESRI is hampered by this slow acquisition, which can also be an obstacle for many biological applications where conditions may change relatively quickly over time. The objective of this work is to develop an image reconstruction scheme for continuous wave (CW) ESRI that would make it possible to reduce the data acquisition time without degrading the reconstruction quality. This is achieved by adapting the so-called "statistical reconstruction" method, recently developed for other medical imaging modalities, to the specific case of CW ESRI. Our new algorithm accounts for unique ESRI aspects such as field modulation, spectral-spatial imaging, and possible limitation on the gradient magnitude (the so-called "limited angle" problem). The reconstruction method shows improved SNR and contrast recovery vs. commonly used back-projection-based methods, for a variety of simulated synthetic samples as well as in actual CW ESRI experiments.

SPECT system optimization against a discrete parameter space

In this paper, we present an analytical approach for optimizing the design of a static SPECT system or optimizing the sampling strategy with a variable/adaptive SPECT imaging hardware against an arbitrarily given set of system parameters. This approach has three key aspects. First, it is designed to operate over a discretized system parameter space. Second, we have introduced an artificial concept of virtual detector as the basic building block of an imaging system. With a SPECT system described as a collection of the virtual detectors, one can convert the task of system optimization into a process of finding the optimum imaging time distribution (ITD) across all virtual detectors. Thirdly, the optimization problem (finding the optimum ITD) could be solved with a block-iterative approach or other nonlinear optimization algorithms. In essence, the resultant optimum ITD could provide a quantitative measure of the relative importance (or effectiveness) of the virtual detectors and help to identify the system configuration or sampling strategy that leads to an optimum imaging performance. Although we are using SPECT imaging as a platform to demonstrate the system optimization strategy, this development also provides a useful framework for system optimization problems in other modalities, such as positron emission tomography and x-ray computed tomography (Moore et al (2009 IEEE Nucl. Sci. Symp. Conf. Rec. pp 4154-7), Freed et al (2008 Med. Phys. 35 1912-25)).

Modeling and Control of Nonstationary Noise Characteristics in Filtered-Backprojection and Penalized Likelihood Image Reconstruction

Purpose

Nonstationarity of CT noise presents a major challenge to the assessment of image quality. This work presents models for imaging performance in both filtered backprojection (FBP) and penalized likelihood (PL) reconstruction that describe not only the dependence on the imaging chain but also the dependence on the object as well as the nonstationary characteristics of the signal and noise. The work furthermore demonstrates the ability to impart control over the imaging process by adjusting reconstruction parameters to exploit nonstationarity in a manner advantageous to a particular imaging task.

Methods

A cascaded systems analysis model was used to model the local noise-power spectrum (NPS) and modulation transfer function (MTF) for FBP reconstruction, with locality achieved by separate calculation of fluence and system gain for each view as a function of detector location. The covariance and impulse response function for PL reconstruction (quadratic penalty) were computed using the implicit function theorem and Taylor expansion. Detectability index was calculated under the assumption of local stationarity to show the variation in task-dependent image quality throughout the image for simple and complex, heterogeneous objects. Control of noise magnitude and correlation was achieved by applying a spatially varying roughness penalty in PL reconstruction in a manner that improved overall detectability.

Results

The models provide a foundation for task-based imaging performance assessment in FBP and PL image reconstruction. For both FBP and PL, noise is anisotropic and varies in a manner dependent on the path length of each view traversing the object. The anisotropy in turn affects task performance, where detectability is enhanced or diminished depending on the frequency content of the task relative to that of the NPS. Spatial variation of the roughness penalty can be exploited to control noise magnitude and correlation (and hence detectability).

Conclusions

Nonstationarity of image noise is a significant effect that can be modeled in both FBP and PL image reconstruction. Prevalent spatial-frequency-dependent metrics of spatial resolution and noise can be analyzed under assumptions of local stationarity, providing a means to analyze imaging performance as a function of location throughout the image. Knowledgeable selection of a spatially-varying roughness penalty in PL can potentially improve local noise and spatial resolution in a manner tuned to a particular imaging task.

Introduction to the analysis of PET data in oncology

Several reviews on specific topics related to positron emission tomography (PET) ranging in complexity from introductory to highly technical have already been published. This introduction to the analysis of PET data was written as a simple guide of the different phases of analysis of a given PET dataset, from acquisition to preprocessing, to the final data analysis. Although sometimes issues specific to PET in neuroimaging will be mentioned for comparison, most of the examples and applications provided will refer to oncology. Due to the limitations of space we couldn't address each issue comprehensively but, rather, we provided a general overview of each topic together with the references that the interested reader should consult. We will assume a familiarity with the basic principles of PET imaging.

Noise properties of motion-compensated tomographic image reconstruction methods

Motion-compensated image reconstruction (MCIR) methods incorporate motion models to improve image quality in the presence of motion. MCIR methods differ in terms of how they use motion information and they have been well studied separately. However, there have been less theoretical comparisions of different MCIR methods. This paper compares the theoretical noise properties of three popular MCIR methods assuming known nonrigid motion. We show the relationship among three MCIR methods-motion-compensated temporal regularization (MTR), the parametric motion model (PMM), and post-reconstruction motion correction (PMC)-for penalized weighted least square cases. These analyses show that PMM and MTR are matrix-weighted sums of all registered image frames, while PMC is a scalar-weighted sum. We further investigate the noise properties of MCIR methods with Poisson models and quadratic regularizers by deriving accurate and fast variance prediction formulas using an "analytical approach." These theoretical noise analyses show that the variances of PMM and MTR are lower than or comparable to the variance of PMC due to the statistical weighting. These analyses also facilitate comparisons of the noise properties of different MCIR methods, including the effects of different quadratic regularizers, the influence of the motion through its Jacobian determinant, and the effect of assuming that total activity is preserved. Two-dimensional positron emission tomography simulations demonstrate the theoretical results.

Correction for collimator-detector response in SPECT using point spread function template

Compensating for the collimator-detector response (CDR) in SPECT is important for accurate quantification. The CDR consists of both a geometric response and a septal penetration and collimator scatter response. The geometric response can be modeled analytically and is often used for modeling the whole CDR if the geometric response dominates. However, for radionuclides that emit medium or high-energy photons such as I-131, the septal penetration and collimator scatter response is significant and its modeling in the CDR correction is important for accurate quantification. There are two main methods for modeling the depth-dependent CDR so as to include both the geometric response and the septal penetration and collimator scatter response. One is to fit a Gaussian plus exponential function that is rotationally invariant to the measured point source response at several source-detector distances. However, a rotationally-invariant exponential function cannot represent the star-shaped septal penetration tails in detail. Another is to perform Monte-Carlo (MC) simulations to generate the depth-dependent point spread functions (PSFs) for all necessary distances. However, MC simulations, which require careful modeling of the SPECT detector components, can be challenging and accurate results may not be available for all of the different SPECT scanners in clinics. In this paper, we propose an alternative approach to CDR modeling. We use a Gaussian function plus a 2-D B-spline PSF template and fit the model to measurements of an I-131 point source at several distances. The proposed PSF-template-based approach is nearly non-parametric, captures the characteristics of the septal penetration tails, and minimizes the difference between the fitted and measured CDR at the distances of interest. The new model is applied to I-131 SPECT reconstructions of experimental phantom measurements, a patient study, and a MC patient simulation study employing the XCAT phantom. The proposed model yields up to a 16.5 and 10.8% higher recovery coefficient compared to the results with the conventional Gaussian model and the Gaussian plus exponential model, respectively.

Characterization of statistical prior image constrained compressed sensing. I. Applications to time-resolved contrast-enhanced CT

PURPOSE

Prior image constrained compressed sensing (PICCS) is an image reconstruction framework that takes advantage of a prior image to improve the image quality of CT reconstructions. An interesting question that remains to be investigated is whether or not the introduction of a statistical model of the photon detection in the PICCS reconstruction framework can improve the performance of the algorithm when dealing with high noise projection datasets. The goal of the research presented in this paper is to characterize the noise properties of images reconstructed using PICCS with and without statistical modeling. This paper investigates these properties in the clinical context of time-resolved contrast-enhanced CT.

METHODS

Both numerical phantom studies and an Institutional Review Board approved human subject study were used in this research. The conventional filtered backprojection (FBP), and PICCS with and without the statistical model were applied to each dataset. The prior image used in PICCS was generated by averaging over FBP reconstructions from different time frames of the time-resolved CT exam, thus reducing the noise level. Numerical studies were used to evaluate if the noise characteristics are altered for varying levels of noise, as well as for different object shapes. The dataset acquired in vivo was used to verify that the conclusions reached from numerical studies translate adequately to a clinical case. The results were analyzed using a variety of qualitative and quantitative metrics such as the universal image quality index, spatial maps of the noise standard deviations, the noise uniformity, the noise power spectrum, and the model-observer detectability.

RESULTS

The noise characteristics of PICCS were shown to depend on the noise level contained in the data, the level of eccentricity of the object, and whether or not the statistical model was applied. Most differences in the characteristics were observed in the regime of low incident x-ray fluence. No substantial difference was observed between PICCS with and without statistics in the high fluence domain. Objects with a semi-major axis ratio below 0.85 were more accurately reconstructed with lower noise using the statistical implementation. Above that range, for mostly circular objects, the PICCS implementation without the statistical model yielded more accurate images and a lower noise level. At all levels of eccentricity, the noise spatial distribution was more uniform and the model-observer detectability was greater for PICCS with the statistical model. The human subject study was consistent with the results obtained using numerical simulations.

CONCLUSIONS

For mildly eccentric objects in the low noise regime, PICCS without the noise model yielded equal or better noise level and image quality than the statistical formulation. However, in a vast majority of cases, images reconstructed using statistical PICCS have a noise power spectrum that facilitated the detection of model lesions. The inclusion of a statistical model in the PICCS framework does not always result in improved noise characteristics.

Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies

Fluorescence tomography (FT) is depth-resolved three-dimensional (3D) localization and quantification of fluorescence distribution in biological tissue and entails a highly ill-conditioned problem as depth information must be extracted from boundary measurements. Conventionally, L2 regularization schemes that penalize the euclidean norm of the solution and possess smoothing effects are used for FT reconstruction. Oversmooth, continuous reconstructions lack high-frequency edge-type features of the original distribution and yield poor resolution. We propose an alternative regularization method for FT that penalizes the total variation (TV) norm of the solution to preserve sharp transitions in the reconstructed fluorescence map while overcoming ill-posedness. We have developed two iterative methods for fast 3D reconstruction in FT based on TV regularization inspired by Rudin-Osher-Fatemi and split Bregman algorithms. The performance of the proposed method is studied in a phantom-based experiment using a noncontact constant-wave trans-illumination FT system. It is observed that the proposed method performs better in resolving fluorescence inclusions at different depths.

Simultaneous reconstruction of activity and attenuation in time-of-flight PET

In positron emission tomography (PET) and single photon emission tomography (SPECT), attenuation correction is necessary for quantitative reconstruction of the tracer distribution. Previously, several attempts have been made to estimate the attenuation coefficients from emission data only. These attempts had limited success, because the problem does not have a unique solution, and severe and persistent "cross-talk" between the estimated activity and attenuation distributions was observed. In this paper, we show that the availability of time-of-flight (TOF) information eliminates the cross-talk problem by destroying symmetries in the associated Fisher information matrix. We propose a maximum-a-posteriori reconstruction algorithm for jointly estimating the attenuation and activity distributions from TOF PET data. The performance of the algorithm is studied with 2-D simulations, and further illustrated with phantom experiments and with a patient scan. The estimated attenuation image is robust to noise, and does not suffer from the cross-talk that was observed in non-TOF PET. However, some constraining is still mandatory, because the TOF data determine the attenuation sinogram only up to a constant offset.

Spatial resolution properties of motion-compensated tomographic image reconstruction methods

Many motion-compensated image reconstruction (MCIR) methods have been proposed to correct for subject motion in medical imaging. MCIR methods incorporate motion models to improve image quality by reducing motion artifacts and noise. This paper analyzes the spatial resolution properties of MCIR methods and shows that nonrigid local motion can lead to nonuniform and anisotropic spatial resolution for conventional quadratic regularizers. This undesirable property is akin to the known effects of interactions between heteroscedastic log-likelihoods (e.g., Poisson likelihood) and quadratic regularizers. This effect may lead to quantification errors in small or narrow structures (such as small lesions or rings) of reconstructed images. This paper proposes novel spatial regularization design methods for three different MCIR methods that account for known nonrigid motion. We develop MCIR regularization designs that provide approximately uniform and isotropic spatial resolution and that match a user-specified target spatial resolution. Two-dimensional PET simulations demonstrate the performance and benefits of the proposed spatial regularization design methods.

A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction

Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and nonsmooth sparsity-promoting ones based on the l(1)-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT.

X-ray Fluorescence Emission Tomography (XFET) with Novel Imaging Geometries - A Monte Carlo Study

This paper presents a feasibility study for using two new imaging geometries for synchrotron X-ray fluorescence emission tomography (XFET) applications. In the proposed approaches, the object is illuminated with synchrotron X-ray beams of various cross-sectional dimensions. The resultant fluorescence photons are detected by high-resolution imaging-spectrometers coupled to collimation apertures. To verify the performance benefits of the proposed methods over the conventional line-by-line scanning approach, we have used both Monte Carlo simulations and an analytical system performance index to compare several different imaging geometries. This study has demonstrated that the proposed XFET approach could lead to a greatly improved imaging speed, which is critical for making XFET a practical imaging modality for a wide range of applications.

Image reconstruction in intravascular photoacoustic imaging

Intravascular photoacoustic (IVPA) imaging is a technique for visualizing atherosclerotic plaques with differential composition. Unlike conventional photoacoustic tomography scanning, where the scanning device rotates around the subject, the scanning aperture in IVPA imaging is enclosed within the imaged object. The display of the intravascular structure is typically obtained by converting detected photoacoustic waves into Cartesian coordinates, which can produce images with severe artifacts. Because the acquired data are highly limited, there does not exist a stable reconstruction algorithm for such imaging geometry. The purpose of this work was to apply image reconstruction concepts to explore the feasibility and efficacy of image reconstruction algorithms in IVPA imaging using traditional analytical formulas, such as a filtered back-projection (FBP) and the lambda-tomography method. Although the closed-form formulas are not exact for the IVPA system, a general picture of and interface information about objects are provided. To improve the quality of the reconstructed image, the iterative expectation maximization and penalized least-squares methods were adopted to minimize the difference between the measured signals and those generated by a reconstructed image. In this work, we considered both the ideal point detector and the acoustic transducers with finite- size aperture. The transducer effects including the spatial response of aperture and acoustoelectrical impulse responses were incorporated in the system matrix to reduce the aroused distortion in the IVPA reconstruction. Computer simulations and experiments were carried out to validate the methods. The applicability and the limitation of the reconstruction method were also discussed.

Comparison of sinogram- and image-domain penalized-likelihood image reconstruction estimators

PURPOSE

In recent years, the authors and others have been exploring the use of penalized-likelihood sinogram-domain smoothing and restoration approaches for emission and transmission tomography. The motivation for this strategy was initially pragmatic: to provide a more computationally feasible alternative to fully iterative penalized-likelihood image reconstruction involving expensive backprojections and reprojections, while still obtaining some of the benefits of the statistical modeling employed in penalized-likelihood approaches. In this work, the authors seek to compare the two approaches in greater detail.

METHODS

The sinogram-domain strategy entails estimating the "ideal" line integrals needed for reconstruction of an activity or attenuation distribution from the set of noisy, potentially degraded tomographic measurements by maximizing a penalized-likelihood objective function. The objective function models the data statistics as well as any degradation that can be represented in the sinogram domain. The estimated line integrals can then be input to analytic reconstruction algorithms such as filtered backprojection (FBP). The authors compare this to fully iterative approaches maximizing similar objective functions.

RESULTS

The authors present mathematical analyses based on so-called equivalent optimization problems that establish that the approaches can be made precisely equivalent under certain restrictive conditions. More significantly, by use of resolution-variance tradeoff studies, the authors show that they can yield very similar performance under more relaxed, realistic conditions.

CONCLUSIONS

The sinogram- and image-domain approaches are equivalent under certain restrictive conditions and can perform very similarly under more relaxed conditions. The match is particularly good for fully sampled, high-resolution CT geometries. One limitation of the sinogram-domain approach relative to the image-domain approach is the difficulty of imposing additional constraints, such as image non-negativity.

Sparsity and low-contrast object detectability

The application of sparsity-driven reconstruction methods to MRI to date has largely focused on situations where high-contrast features (e.g., gadolinium-enhanced vessels) are of primary interest. In clinical practice, however, low contrast features such as subtle lesions are often of equal or greater interest. Using an American College of Radiology MR quality assurance phantom and test, we describe a novel framework for systematically and automatically evaluating the low-contrast object detectability performance of different undersampled image reconstruction methods. This platform is used to evaluate three such methods, two based on classic Tikhonov regularization and one sparsity-driven method based on ℓ(1) -norm minimization (which is commonly used in compressive sensing, also known as compressed sensing, applications), across a wide range of sampling rates and parameterizations. Both the automated evaluation system and a manual evaluation of anatomical images with numerically-generated low contrast inserts demonstrate that sparse reconstructions exhibit superior low-contrast object detectability performance compared to both Tikhonov-regularized reconstructions. The implications of this result, and potential applications of both the described low-contrast object detectability platform and generalizations of it are then discussed.

Improved image fusion in PET/CT using hybrid image reconstruction and super-resolution

Purpose. To provide PET/CT image fusion with an improved PET resolution and better contrast ratios than standard reconstructions. Method. Using a super-resolution algorithm, several PET acquisitions were combined to improve the resolution. In addition, functional PET data was smoothed with a hybrid computed tomography algorithm (HCT), in which anatomical edge information taken from the CT was employed to retain sharper edges. The combined HCT and super-resolution technique were evaluated in phantom and patient studies using a clinical PET scanner. Results. In the phantom studies, 3 mm¹⁸F-FDG sources were resolved. PET contrast ratios improved (average: 54%, range: 45%–69%) relative to the standard reconstructions. In the patient study, target-to-background ratios also improved (average: 34%, range: 17%–47%). Given corresponding anatomical borders, sharper edges were depicted. Conclusion. A new method incorporating super-resolution and HCT for fusing PET and CT images has been developed and shown to provide higher-resolution metabolic images.

Noise-resolution tradeoffs in x-ray CT imaging: a comparison of penalized alternating minimization and filtered backprojection algorithms

PURPOSE

In comparison with conventional filtered backprojection (FBP) algorithms for x-ray computed tomography (CT) image reconstruction, statistical algorithms directly incorporate the random nature of the data and do not assume CT data are linear, noiseless functions of the attenuation line integral. Thus, it has been hypothesized that statistical image reconstruction may support a more favorable tradeoff than FBP between image noise and spatial resolution in dose-limited applications. The purpose of this study is to evaluate the noise-resolution tradeoff for the alternating minimization (AM) algorithm regularized using a nonquadratic penalty function.

METHODS

Idealized monoenergetic CT projection data with Poisson noise were simulated for two phantoms with inserts of varying contrast (7%-238%) and distance from the field-of-view (FOV) center (2-6.5 cm). Images were reconstructed for the simulated projection data by the FBP algorithm and two penalty function parameter values of the penalized AM algorithm. Each algorithm was run with a range of smoothing strengths to allow quantification of the noise-resolution tradeoff curve. Image noise is quantified as the standard deviation in the water background around each contrast insert. Modulation transfer functions (MTFs) were calculated from six-parameter model fits to oversampled edge-spread functions defined by the circular contrast-insert edges as a metric of local resolution. The integral of the MTF up to 0.5 1p/mm was adopted as a single-parameter measure of local spatial resolution.

RESULTS

The penalized AM algorithm noise-resolution tradeoff curve was always more favorable than that of the FBP algorithm. While resolution and noise are found to vary as a function of distance from the FOV center differently for the two algorithms, the ratio of noises when matching the resolution metric is relatively uniform over the image. The ratio of AM-to-FBP image variances, a predictor of dose-reduction potential, was strongly dependent on the shape of the AM's nonquadratic penalty function and was also strongly influenced by the contrast of the insert for which resolution is quantified. Dose-reduction potential, reported here as the fraction (%) of FBP dose necessary for AM to reconstruct an image with comparable noise and resolution, for one penalty parameter value of the AM algorithm was found to vary from 70% to 50% for low-contrast and high-contrast structures, respectively, and from 70% to 10% for the second AM penalty parameter value. However, the second penalty, AM-700, was found to suffer from poor low-contrast resolution when matching the high-contrast resolution metric with FBP.

CONCLUSIONS

The results of this simulation study imply that penalized AM has the potential to reconstruct images with similar noise and resolution using a fraction (10%-70%) of the FBP dose. However, this dose-reduction potential depends strongly on the AM penalty parameter and the contrast magnitude of the structures of interest. In addition, the authors' results imply that the advantage of AM can be maximized by optimizing the nonquadratic penalty function to the specific imaging task of interest. Future work will extend the methods used here to quantify noise and resolution in images reconstructed from real CT data.

An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography

Optoacoustic tomography (OAT) is a hybrid imaging modality that combines the advantages of optical and ultrasound imaging. Most existing reconstruction algorithms for OAT assume that the ultrasound transducers employed to record the measurement data are point-like. When transducers with large detecting areas and/or compact measurement geometries are utilized, this assumption can result in conspicuous image blurring and distortions in the reconstructed images. In this work, a new OAT imaging model that incorporates the spatial and temporal responses of an ultrasound transducer is introduced. A discrete form of the imaging model is implemented and its numerical properties are investigated. We demonstrate that use of the imaging model in an iterative reconstruction method can improve the spatial resolution of the optoacoustic images as compared to those reconstructed assuming point-like ultrasound transducers.

Noise propagation for iterative penalized-likelihood image reconstruction based on Fisher information

Iterative reconstruction algorithms have been widely used in PET and SPECT emission tomography. Accurate modeling of photon noise propagation is crucial for quantitative tomography applications. Iteration-based noise propagation methods have been developed for only a few algorithms that have explicit multiplicative update equations. And there are discrepancies between the iteration-based methods and Fessler's fixed-point method because of improper approximations. In this paper, we present a unified theoretical prediction of noise propagation for any penalized expectation maximization (EM) algorithm where the EM approach incorporates a penalty term. The proposed method does not require an explicit update equation. The update equation is assumed to be implicitly defined by a differential equation of a surrogate function. We derive the expressions using the implicit function theorem, Taylor series and the chain rule from vector calculus. We also derive the fixed-point expressions when iterative algorithms converge and show the consistency between the proposed method and the fixed-point method. These expressions are solely defined in terms of the partial derivatives of the surrogate function and the Fisher information matrices. We also apply the theoretical noise predictions for iterative reconstruction algorithms in emission tomography. Finally, we validate the theoretical predictions for MAP-EM and OSEM algorithms using Monte Carlo simulations with Jaszczak-like and XCAT phantoms, respectively.

Iterative reconstruction of Fourier-rebinned PET data using sinogram blurring function estimated from point source scans

PURPOSE

The accuracy of the system model that governs the transformation from the image space to the projection space in positron emission tomography (PET) greatly affects the quality of reconstructed images. For efficient computation in iterative reconstructions, the system model in PET can be factored into a product of geometric projection and sinogram blurring function. To further speed up reconstruction, fully 3D PET data can be rebinned into a stack of 2D sinograms and then be reconstructed using 2D iterative algorithms. The purpose of this work is to develop a method to estimate the sinogram blurring function to be used in reconstruction of Fourier-rebinned data.

METHODS

In a previous work, the authors developed an approach to estimating the sinogram blurring function of nonrebinned PET data from experimental scans of point sources. In this study, the authors extend this method to the estimation of sinogram blurring function for Fourier-rebinned PET data. A point source was scanned at a set of sampled positions in the microPET II scanner. The sinogram blurring function is considered to be separable between the transaxial and axial directions. A radially and angularly variant 2D blurring function is estimated from Fourier-rebinned point source scans to model the transaxial blurring with consideration of the detector block structure of the scanner; a space-variant 1D blurring kernel along the axial direction is estimated separately to model the correlation between neighboring planes due to detector intrinsic blurring and Fourier rebinning. The estimated sinogram blurring function is incorporated in a 2D maximum a posteriori (MAP) reconstruction algorithm for image reconstruction.

RESULTS

Physical phantom experiments were performed on the microPET II scanner to validate the proposed method. The authors compared the proposed method to 2D MAP reconstruction without sinogram blurring model and 2D MAP reconstruction with a Monte Carlo based blurring model. The results show that the proposed method produces images with improved contrast and spatial resolution. The reconstruction time is unaffected by the new method since the blurring component takes a relatively negligible part of the overall reconstruction time.

CONCLUSIONS

The proposed method can estimate sinogram blurring matrix for Fourier-rebinned PET data and can be used to improve contrast and spatial resolution of reconstructed images. The method can be applied to other human and animal scanners.

Comparison between parallel hole and rotating slat collimation: analytical noise propagation models

We have previously proposed a method to compare tomographic systems. It is assumed that each system acquires a tomographic scan of a certain tracer distribution in the same acquisition time. From this scan, each system is forced to reconstruct an image with a predefined spatial resolution. The system that can perform this task with the "most favorable" noise propagation is considered as the best system. The variance on pixel values or region-of-interest (ROI) values is used to assess the noise in the reconstructed image. In this paper, we extend this idea to compare the performance of parallel hole (PH) and rotating slat (RS) collimations. Two different analytical approaches were used to analyze the variance of the reconstructed pixel/ROI values. The first method is based on the filtered-backprojection (FBP) theory, and was applied to the central point of a uniform symmetrical phantom. It yields analytical expressions for the optimal collimator aperture and the corresponding variance of the reconstructed pixel values, but it can only be applied to highly symmetrical configurations. The second method is based on approximations for the Fisher information matrix. It provides numerical results, and it is more general and can be applied to nonsymmetrical objects and shift-variant tomographic systems. The collimations were compared for both planar imaging and volume imaging. The main results are as follows. 1) For cases where both methods are valid, they are in excellent agreement. 2a) The optimal collimator aperture varies linearly with the target resolution. 2b) For a fixed target resolution, the optimal collimator aperture depends on the collimator type and the imaging mode (planar or volume). 2c) The optimal aperture of PH is a factor of √2 larger than that of RS. 3a) The relative performance of the two collimators is determined by both the object size and the object-to-detector distance. 3b) Pixel variance and variances of ROIs with varying sizes yield very similar relative performance for RS versus PH.