Fast predictions of variance images for fan-beam transmission tomography with quadratic regularization

Predicting image properties in penalized-likelihood reconstructions of flat-panel CBCT

PURPOSE

Model-based iterative reconstruction (MBIR) algorithms such as penalized-likelihood (PL) methods exhibit data-dependent and shift-variant properties. Image quality predictors have been derived to prospectively estimate local noise and spatial resolution, facilitating both system hardware design and tuning of reconstruction methods. However, current MBIR image quality predictors rely on idealized system models, ignoring physical blurring effects and noise correlations found in real systems. In this work, we develop and validate a new set of predictors using a physical system model specific to flat-panel cone-beam CT (FP-CBCT).

METHODS

Physical models appropriate for integration with MBIR analysis are developed and parameterized to represent nonidealities in FP projection data including focal spot blur, scintillator blur, detector aperture effect, and noise correlations. Flat-panel-specific predictors for local spatial resolution and local noise properties in PL reconstructions are developed based on these realistic physical models. Estimation accuracy of conventional (idealized) and FP-specific predictors is investigated and validated against experimental CBCT measurements using specialized phantoms.

RESULTS

Validation studies show that flat-panel-specific predictors can accurately estimate the local spatial resolution and noise properties, while conventional predictors show significant deviations in the magnitude and scale of the spatial resolution and local noise. The proposed predictors show accurate estimations over a range of imaging conditions including varying x-ray technique and regularization strength. The conventional spatial resolution prediction is sharper than ground truth. Using conventional spatial resolution predictor, the full width at half maximum (FWHM) of local point spread function (PSF) is underestimated by 0.2 mm. This mismatch is mostly eliminated in FP-specific prediction. The general shape and amplitude of local noise power spectrum (NPS) FP-specific predictions are consistent with measurement, while the conventional predictions underestimated the noise level by 70%.

CONCLUSION

The proposed image quality predictors permit accurate estimation of local spatial resolution and noise properties for PL reconstruction, accounting for dependencies on the system geometry, x-ray technique, and patient-specific anatomy in real FP-CBCT. Such tools enable prospective analysis of image quality for a range of goals including novel system and acquisition design, adaptive and task-driven imaging, and tuning of MBIR for robust and reliable behavior.

Image quality and dose characteristics for an O-arm intraoperative imaging system with model-based image reconstruction

PURPOSE

To assess the imaging performance and radiation dose characteristics of the O-arm CBCT imaging system (Medtronic Inc., Littleton MA) and demonstrate the potential for improved image quality and reduced dose via model-based image reconstruction (MBIR).

METHODS

Two main studies were performed to investigate previously unreported characteristics of the O-arm system. First is an investigation of dose and 3D image quality achieved with filtered back-projection (FBP) - including enhancements in geometric calibration, handling of lateral truncation and detector saturation, and incorporation of an isotropic apodization filter. Second is implementation of an MBIR algorithm based on Huber-penalized likelihood estimation (PLH) and investigation of image quality improvement at reduced dose. Each study involved measurements in quantitative phantoms as a basis for analysis of contrast-to-noise ratio and spatial resolution as well as imaging of a human cadaver to test the findings under realistic imaging conditions.

RESULTS

View-dependent calibration of system geometry improved the accuracy of reconstruction as quantified by the full-width at half maximum of the point-spread function - from 0.80 to 0.65 mm - and yielded subtle but perceptible improvement in high-contrast detail of bone (e.g., temporal bone). Standard technique protocols for the head and body imparted absorbed dose of 16 and 18 mGy, respectively. For low-to-medium contrast (<100 HU) imaging at fixed spatial resolution (1.3 mm edge-spread function) and fixed dose (6.7 mGy), PLH improved CNR over FBP by +48% in the head and +35% in the body. Evaluation at different dose levels demonstrated 30% increase in CNR at 62% of the dose in the head and 90% increase in CNR at 50% dose in the body.

CONCLUSIONS

A variety of improvements in FBP implementation (geometric calibration, truncation and saturation effects, and isotropic apodization) offer the potential for improved image quality and reduced radiation dose on the O-arm system. Further gains are possible with MBIR, including improved soft-tissue visualization, low-dose imaging protocols, and extension to methods that naturally incorporate prior information of patient anatomy and/or surgical instrumentation.

Image-domain multimaterial decomposition for dual-energy CT based on prior information of material images

PURPOSE

Dual-Energy Computed Tomography (DECT) is of great interest in medical imaging, security inspection, and nondestructive testing. Most DECT reconstruction methods focus on producing two material images with different linear attenuation coefficients. However, the ability to reconstruct three or more basis materials is clinically and industrially important. Under the assumption that there are at most three materials in each pixel, there are a few methods that estimate multiple material images from DECT measurements by enforcing sum-to-one and a box constraint ([0 1]) derived from both the volume and mass conservation assumption. The recently proposed image-domain multimaterial decomposition (MMD) method introduces edge-preserving regularization for each material image. It enforces the assumption that there are at most three materials in each pixel using a time-consuming loop over all possible material triplets. However, this method neglects relations among material images. We propose a new image-domain MMD model for DECT that considers the prior information that different material images have common or complementary edges and encourages sparsity of material composition in each pixel using regularization.

METHOD

The proposed PWLS-TNV-ℓ0 method uses penalized weighted least-square (PWLS) reconstruction with three regularization terms. The first term is total nuclear variation (TNV) that accounts for the image property that basis material images share common or complementary boundaries and each material image is piecewise constant. The second term is an ℓ0 norm that encourages each pixel containing a small subset of material types out of several possible materials. The third term is a characteristic function based on sum-to-one and a box constraint derived from the volume and mass conservation assumption. We apply the Alternating Direction Method of Multipliers (ADMM) to optimize the cost function of the PWLS-TNV-ℓ0 method.

RESULT

We evaluated the proposed method on a simulated digital phantom, Catphan©600 phantom and patient's pelvis data. We implemented two existing image-domain MMD methods for DECT, the Direct Inversion and the PWLS-EP-LOOP method. We initialized the PWLS-TNV-ℓ0 method and the PWLS-EP-LOOP method with the results of the Direct Inversion method and compared performance of the proposed method with that of the PWLS-EP-LOOP method. The proposed method lowers the bias of decomposed material fractions by 84.47% in the digital phantom study, by 99.50% in the Catphan©600 phantom study, and by 99.64% in the pelvis patient study, respectively, compared to the PWLS-EP-LOOP method. The proposed method reduces noise standard deviation (STD) by 52.21% in the Catphan©600 phantom study, and by 16.74% in the patient's pelvis study, compared to the PWLS-EP-LOOP method. The proposed method increases volume fraction accuracy by 6.04%,20.55%, and 13.46% for the digital phantom, the Catphan©600 phantom, and the patient's pelvis study, respectively, compared to the PWLS-EP-LOOP method. Compared with the PWLS-EP-LOOP method, the root mean square percentage error [RMSE(%)] of electron densities in the Catphan©600 phantom is decreased by about 7.39%.

CONCLUSIONS

We proposed an image-domain MMD method, PWLS-TNV-ℓ0 , for DECT. The PWLS-TNV-ℓ0 method takes low rank property of material image gradients, sparsity of material composition and mass and volume conservation into consideration. The proposed method suppresses noise, reduces cross contamination, and improves accuracy in the decomposed material images, compared to the PWLS-EP-LOOP method.

Task-Driven Optimization of Fluence Field and Regularization for Model-Based Iterative Reconstruction in Computed Tomography

This paper presents a joint optimization of dynamic fluence field modulation (FFM) and regularization in quadratic penalized-likelihood reconstruction that maximizes a task-based imaging performance metric. We adopted a task-driven imaging framework for prospective designs of the imaging parameters. A maxi-min objective function was adopted to maximize the minimum detectability index ( ) throughout the image. The optimization algorithm alternates between FFM (represented by low-dimensional basis functions) and local regularization (including the regularization strength and directional penalty weights). The task-driven approach was compared with three FFM strategies commonly proposed for FBP reconstruction (as well as a task-driven TCM strategy) for a discrimination task in an abdomen phantom. The task-driven FFM assigned more fluence to less attenuating anteroposterior views and yielded approximately constant fluence behind the object. The optimal regularization was almost uniform throughout image. Furthermore, the task-driven FFM strategy redistribute fluence across detector elements in order to prescribe more fluence to the more attenuating central region of the phantom. Compared with all strategies, the task-driven FFM strategy not only improved minimum by at least 17.8%, but yielded higher over a large area inside the object. The optimal FFM was highly dependent on the amount of regularization, indicating the importance of a joint optimization. Sample reconstructions of simulated data generally support the performance estimates based on computed . The improvements in detectability show the potential of the task-driven imaging framework to improve imaging performance at a fixed dose, or, equivalently, to provide a similar level of performance at reduced dose.

Task-driven optimization of CT tube current modulation and regularization in model-based iterative reconstruction

Tube current modulation (TCM) is routinely adopted on diagnostic CT scanners for dose reduction. Conventional TCM strategies are generally designed for filtered-backprojection (FBP) reconstruction to satisfy simple image quality requirements based on noise. This work investigates TCM designs for model-based iterative reconstruction (MBIR) to achieve optimal imaging performance as determined by a task-based image quality metric. Additionally, regularization is an important aspect of MBIR that is jointly optimized with TCM, and includes both the regularization strength that controls overall smoothness as well as directional weights that permits control of the isotropy/anisotropy of the local noise and resolution properties. Initial investigations focus on a known imaging task at a single location in the image volume. The framework adopts Fourier and analytical approximations for fast estimation of the local noise power spectrum (NPS) and modulation transfer function (MTF)-each carrying dependencies on TCM and regularization. For the single location optimization, the local detectability index (d') of the specific task was directly adopted as the objective function. A covariance matrix adaptation evolution strategy (CMA-ES) algorithm was employed to identify the optimal combination of imaging parameters. Evaluations of both conventional and task-driven approaches were performed in an abdomen phantom for a mid-frequency discrimination task in the kidney. Among the conventional strategies, the TCM pattern optimal for FBP using a minimum variance criterion yielded a worse task-based performance compared to an unmodulated strategy when applied to MBIR. Moreover, task-driven TCM designs for MBIR were found to have the opposite behavior from conventional designs for FBP, with greater fluence assigned to the less attenuating views of the abdomen and less fluence to the more attenuating lateral views. Such TCM patterns exaggerate the intrinsic anisotropy of the MTF and NPS as a result of the data weighting in MBIR. Directional penalty design was found to reinforce the same trend. The task-driven approaches outperform conventional approaches, with the maximum improvement in d' of 13% given by the joint optimization of TCM and regularization. This work demonstrates that the TCM optimal for MBIR is distinct from conventional strategies proposed for FBP reconstruction and strategies optimal for FBP are suboptimal and may even reduce performance when applied to MBIR. The task-driven imaging framework offers a promising approach for optimizing acquisition and reconstruction for MBIR that can improve imaging performance and/or dose utilization beyond conventional imaging strategies.

Pixel-wise estimation of noise statistics on iterative CT reconstruction from a single scan

PURPOSE

As iterative CT reconstruction continues to advance, the spatial distribution of noise standard deviation (STD) and accurate noise power spectrum (NPS) on the reconstructed CT images become important for method evaluation as well as optimization of algorithm parameters. Using a single CT scan, we propose a practical method for pixel-wise calculation of noise statistics on an iteratively reconstructed CT image, which enables accurate calculation of noise STD for each pixel and NPS.

METHOD

We first derive the noise propagation from measured projections to an iteratively reconstructed CT image provided that the projection noise is known. We then show that the model of noise propagation remains approximately unchanged for extra simulated noise added on the measured projections. To compute the noise STD map and the NPS map on an iteratively reconstructed CT image from a single scan, we first iteratively reconstruct the CT image from the measured projections using an existing reconstruction algorithm. The same measured projections are added by different sets (a total of 32 sets in our implementation) of projection noise simulated from an estimated projection noise model, and are then used to iteratively reconstruct different CT images. The calculations of the noise STD map and the NPS map are finally performed on the entire stack of these different reconstruction images.

RESULTS

We evaluate our method on an anthropomorphic head phantom, and demonstrate the clinical utility on a set of head and neck patient CT data, using two iterative CT reconstruction algorithms: the penalized weighted least-square (PWLS) algorithm and the total-variation (TV) regularization. In the head phantom case, repeated scans are acquired to generate the ground truths of noise STD and NPS maps. Using only one single scan, the proposed method accurately calculates the noise STD maps with a root-mean-square error (RMSE) of less than 5HU. In the NPS map estimation, we compare the result of our proposed method with that of the conventional method which calculates the NPS maps on a uniform region of interest on one CT image. Our method outperforms the conventional method on the NPS map estimation with RMSE reduced by 92%. The implementation of the proposed method on the patient data successfully provides the noise STD values around complex structures and a high-quality NPS map.

CONCLUSION

The proposed method accurately calculates noise STD for each pixel and NPS on an iteratively reconstructed CT image, with no requirement of repeated CT scans. It provides a detailed evaluation of imaging performance of different iterative reconstruction methods on the same CT dataset.

Task-driven optimization of CT tube current modulation and regularization in model-based iterative reconstruction

Tube current modulation (TCM) is routinely adopted on diagnostic CT scanners for dose reduction. Conventional TCM strategies are generally designed for filtered-backprojection (FBP) reconstruction to satisfy simple image quality requirements based on noise. This work investigates TCM designs for model-based iterative reconstruction (MBIR) to achieve optimal imaging performance as determined by a task-based image quality metric. Additionally, regularization is an important aspect of MBIR that is jointly optimized with TCM, and includes both the regularization strength that controls overall smoothness as well as directional weights that permits control of the isotropy/anisotropy of the local noise and resolution properties. Initial investigations focus on a known imaging task at a single location in the image volume. The framework adopts Fourier and analytical approximations for fast estimation of the local noise power spectrum (NPS) and modulation transfer function (MTF)-each carrying dependencies on TCM and regularization. For the single location optimization, the local detectability index (d') of the specific task was directly adopted as the objective function. A covariance matrix adaptation evolution strategy (CMA-ES) algorithm was employed to identify the optimal combination of imaging parameters. Evaluations of both conventional and task-driven approaches were performed in an abdomen phantom for a mid-frequency discrimination task in the kidney. Among the conventional strategies, the TCM pattern optimal for FBP using a minimum variance criterion yielded a worse task-based performance compared to an unmodulated strategy when applied to MBIR. Moreover, task-driven TCM designs for MBIR were found to have the opposite behavior from conventional designs for FBP, with greater fluence assigned to the less attenuating views of the abdomen and less fluence to the more attenuating lateral views. Such TCM patterns exaggerate the intrinsic anisotropy of the MTF and NPS as a result of the data weighting in MBIR. Directional penalty design was found to reinforce the same trend. The task-driven approaches outperform conventional approaches, with the maximum improvement in d' of 13% given by the joint optimization of TCM and regularization. This work demonstrates that the TCM optimal for MBIR is distinct from conventional strategies proposed for FBP reconstruction and strategies optimal for FBP are suboptimal and may even reduce performance when applied to MBIR. The task-driven imaging framework offers a promising approach for optimizing acquisition and reconstruction for MBIR that can improve imaging performance and/or dose utilization beyond conventional imaging strategies.

Fast Variance Prediction for Iteratively Reconstructed CT Images With Locally Quadratic Regularization

Predicting noise properties of iteratively reconstructed CT images is useful for analyzing reconstruction methods; for example, local noise power spectrum (NPS) predictions may be used to quantify the detectability of an image feature, to design regularization methods, or to determine dynamic tube current adjustment during a CT scan. This paper presents a method for fast prediction of reconstructed image variance and local NPS for statistical reconstruction methods using quadratic or locally quadratic regularization. Previous methods either require impractical computation times to generate an approximate map of the variance of each reconstructed voxel, or are restricted to specific CT geometries. Our method can produce a variance map of the entire image, for locally shift-invariant CT geometries with sufficiently fine angular sampling, using a computation time comparable to a single back-projection. The method requires only the projection data to be used in the reconstruction, not a reconstruction itself, and is reasonably accurate except near image edges where edge-preserving regularization behaves highly nonlinearly. We evaluate the accuracy of our method using reconstructions of both simulated CT data and real CT scans of a thorax phantom.

Fast analytical approach of application specific dose efficient spectrum selection for diagnostic CT imaging and PET attenuation correction

Computed tomography (CT) has been used for a variety of applications, two of which include diagnostic imaging and attenuation correction for PET or SPECT imaging. Ideally, the x-ray tube spectrum should be optimized for the specific application to minimize the patient radiation dose while still providing the necessary information. In this study, we proposed a projection-based analytic approach for the analysis of contrast, noise, and bias. Dose normalized contrast to noise ratio (CNRD), inverse noise normalized by dose (IND) and bias are used as evaluation metrics to determine the optimal x-ray spectrum. Our simulation investigated the dose efficiency of the x-ray spectrum ranging from 40 kVp to 200 kVp. Water cylinders with diameters of 15 cm, 24 cm, and 35 cm were used in the simulation to cover a variety of patient sizes. The effects of electronic noise and pre-patient copper filtration were also evaluated. A customized 24 cm CTDI-like phantom with 13 mm diameter inserts filled with iodine (10 mg ml^-1), tantalum (10 mg ml^-1), water, and PMMA was measured with both standard (1.5 mGy) and ultra-low (0.2 mGy) dose to verify the simulation results at tube voltages of 80, 100, 120, and 140 kVp. For contrast-enhanced diagnostic imaging, the simulation results indicated that for high dose without filtration, the optimal kVp for water contrast is approximately 100 kVp for a 15 cm water cylinder. However, the 60 kVp spectrum produces the highest CNRD for bone and iodine. The optimal kVp for tantalum has two selections: approximately 50 and 100 kVp. The kVp that maximizes CNRD increases when the object size increases. The trend in the CTDI phantom measurements agrees with the simulation results, which also agrees with previous studies. Copper filtration improved the dose efficiency for water and tantalum, but reduced the iodine and bone dose efficiency in a clinically-relevant range (70-140 kVp). Our study also shows that for CT-based attenuation correction applications for PET or SPECT, a higher-kVp spectrum with copper filtration is preferable. This method is developed based on filter back projection and does not require image reconstruction or Monte Carlo dose estimates; thus, it could potentially be used for patient-specific and task-based on-the-fly protocol optimization.

Mixed Confidence Estimation for Iterative CT Reconstruction

Dynamic (4D) CT imaging is used in a variety of applications, but the two major drawbacks of the technique are its increased radiation dose and longer reconstruction time. Here we present a statistical analysis of our previously proposed Mixed Confidence Estimation (MCE) method that addresses both these issues. This method, where framed iterative reconstruction is only performed on the dynamic regions of each frame while static regions are fixed across frames to a composite image, was proposed to reduce computation time. In this work, we generalize the previous method to describe any application where a portion of the image is known with higher confidence (static, composite, lower-frequency content, etc.) and a portion of the image is known with lower confidence (dynamic, targeted, etc). We show that by splitting the image space into higher and lower confidence components, MCE can lower the estimator variance in both regions compared to conventional reconstruction. We present a theoretical argument for this reduction in estimator variance and verify this argument with proof-of-principle simulations. We also propose a fast approximation of the variance of images reconstructed with MCE and confirm that this approximation is accurate compared to analytic calculations of and multi-realization image variance. This MCE method requires less computation time and provides reduced image variance for imaging scenarios where portions of the image are known with more certainty than others allowing for potentially reduced radiation dose and/or improved dynamic imaging.

Task-Based Design of Fluence Field Modulation in CT for Model-Based Iterative Reconstruction

A task-driven imaging framework for prospective fluence field modulation (FFM) is developed in this paper. The design approach uses a system model that includes a parameterized FFM acquisition and model-based iterative reconstruction (MBIR) for image formation. Using prior anatomical knowledge (e.g. from a low-dose 3D scout image), accurate predictions of spatial resolution and noise as a function of FFM are integrated into a task-based objective function. Specifically, detectability index (d'), a common metric for task-based image quality assessment, is computed for a specific formulation of the imaging task. To optimize imaging performance in across an image volume, a maximin objective function was adopted to maximize the minimum detectability index for many locations sampled throughout the volume. To reduce the dimensionality, FFM patterns were represented using wavelet bases, the coefficients of which were optimized using the covariance matrix adaptation evolutionary strategy (CMA-ES) algorithm. The optimization was performed for a mid-frequency discrimination task involving a cluster of micro-calcifications in an abdomen phantom. The task-driven design yielded FFM patterns that were significantly different from traditional strategies proposed for FBP reconstruction. In addition to a higher minimum d' consistent with the objective function, the task-driven approach also improved d' to a greater extent over a larger area of the phantom. Results from this work suggests that FFM strategies suitable for FBP reconstruction need to be reevaluated in the context of MBIR and that a task-driven imaging framework provides a promising approach for such optimization.

Estimation of noise properties for TV-regularized image reconstruction in computed tomography

A method for predicting the image covariance resulting from total-variation-penalized iterative image reconstruction (TV-penalized IIR) is presented and demonstrated in a variety of contexts. The method is validated against the sample covariance from statistical noise realizations for a small image using a variety of comparison metrics. Potential applications for the covariance approximation include investigation of image properties such as object- and signal-dependence of noise, and noise stationarity. These applications are demonstrated, along with the construction of image pixel variance maps for two-dimensional 128 × 128 pixel images. Methods for extending the proposed covariance approximation to larger images and improving computational efficiency are discussed. Future work will apply the developed methodology to the construction of task-based image quality metrics such as the Hotelling observer detectability for TV-based IIR.

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).

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).

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.

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.

Fisher information-based evaluation of image quality for time-of-flight PET

The use of time-of-flight (TOF) information during positron emission tomography (PET) reconstruction has been found to improve the image quality. In this work we quantified this improvement using two existing methods: 1) a very simple analytical expression only valid for a central point in a large uniform disk source and 2) efficient analytical approximations for postfiltered maximum likelihood expectation maximization (MLEM) reconstruction with a fixed target resolution, predicting the image quality in a pixel or in a small region of interest based on the Fisher information matrix. Using this latter method the weighting function for filtered backprojection reconstruction of TOF PET data proposed by C. Watson can be derived. The image quality was investigated at different locations in various software phantoms. Simplified as well as realistic phantoms, measured both with TOF PET systems and with a conventional PET system, were simulated. Since the time resolution of the system is not always accurately known, the effect on the image quality of using an inaccurate kernel during reconstruction was also examined with the Fisher information-based method. First, we confirmed with this method that the variance improvement in the center of a large uniform disk source is proportional to the disk diameter and inversely proportional to the time resolution. Next, image quality improvement was observed in all pixels, but in eccentric and high-count regions the contrast-to-noise ratio (CNR) increased less than in central and low- or medium-count regions. Finally, the CNR was seen to decrease when the time resolution was inaccurately modeled (too narrow or too wide) during reconstruction. Although the maximum CNR is not very sensitive to the time resolution error, using an inaccurate TOF kernel tends to introduce artifacts in the reconstructed image.

Analysis of Resolution and Noise Properties of Nonquadratically Regularized Image Reconstruction Methods for PET

We present accurate and efficient methods for estimating the spatial resolution and noise properties of nonquadratically regularized image reconstruction for positron emission tomography (PET). It is well known that quadratic regularization tends to over-smooth sharp edges. Many types of edge-preserving nonquadratic penalties have been proposed to overcome this problem. However, there has been little research on the quantitative analysis of nonquadratic regularization due to its nonlinearity. In contrast, quadratically regularized estimators are approximately linear and are well understood in terms of resolution and variance properties. We derive new approximate expressions for the linearized local perturbation response (LLPR) and variance using the Taylor expansion with the remainder term. Although the expressions are implicit, we can use them to accurately predict resolution and variance for nonquadratic regularization where the conventional expressions based on the first-order Taylor truncation fail. They also motivate us to extend the use of a certainty-based modified penalty to nonquadratic regularization cases in order to achieve spatially uniform perturbation responses, analogous to uniform spatial resolution in quadratic regularization. Finally, we develop computationally efficient methods for predicting resolution and variance of nonquadratically regularized reconstruction and present simulations that illustrate the validity of these methods.

Single and multipinhole collimator design evaluation method for small animal SPECT

High-resolution functional imaging of small animals is often obtained by single pinhole SPECT with circular orbit acquisition. Multipinhole SPECT adds information due to its improved sampling, and can improve the trade-off between resolution and sensitivity. To evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality. In this paper, we propose a fast, approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization (MLEM) reconstruction by studying their linearized local impulse response (LLIR) and (co)variance for a predefined target resolution. For validation, the contrast-to-noise ratios (CNRs) in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs. A good agreement was observed between the CNRs obtained with the approximate method and those obtained with postsmoothed MLEM reconstructions of simulated noisy projections. This good agreement was quantified by a least squares fit through these results, which yielded a line with slope 1.02 (1.00 expected) and a y-intercept close to zero (0 expected). 95.4% of the validation points lie within three standard deviations from that line. Using the approximate method, the influence on the CNR of varying a parameter in realistic single and multipinhole designs was examined. The investigated parameters were the aperture diameter, the distance between the apertures and the axis-of-rotation, the focal distance, the acceptance angle, the position of the apertures, the focusing distance, and the number of pinholes. The results can generally be explained by the change in sensitivity, the amount of postsmoothing, and the amount of overlap in the projections. The method was applied to multipinhole designs with apertures focusing at a single point, but is also applicable to other designs.