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

Noise power spectrum (NPS) in computed tomography: Enabling local NPS measurement without stationarity and ergodicity assumptions

BACKGROUND

Conventional methods for estimating the noise power spectrum (NPS) often necessitate multiple computed tomography (CT) data acquisitions and are required to satisfy stringent stationarity and ergodicity conditions, which prove challenging in CT imaging systems.

PURPOSE

The aim was to revisit the conventional NPS estimation method, leading to a new framework that estimates local NPS without relying on stationarity or ergodicity, thus facilitating experimental NPS estimations.

METHODS

The scientific foundation of the conventional CT NPS measurement method, based on the Wiener-Khintchine theorem, was reexamined, emphasizing the critical conditions of stationarity and ergodicity. This work proposes an alternative framework, characterized by its independence from stationarity and ergodicity, and its ability to facilitate local NPS estimations. A spatial average of local NPS over a Region of Interest (ROI) yields the conventional NPS for that ROI. The connections and differences between the proposed alternative method and the conventional method are discussed. Experimental studies were conducted to validate the new method.

RESULTS

(1) The NPS estimated using the conventional method was demonstrated to correspond to the spatial average of pointwise NPS from the proposed NPS estimation framework. (2) The NPS estimated over an ROI with the conventional method was shown to be the sum of the NPS estimated from the proposed method and a contribution from measurement uncertainty. (3) Local NPS estimations from the proposed method in this work elucidate the impact of surrounding image content on local NPS variations.

CONCLUSION

The NPS estimation method proposed in this work allows for the estimation of local NPS without relying on stationarity and ergodicity conditions, offering local NPS estimations with significantly improved precision.

Experimental measurement of local noise power spectrum (NPS) in photon counting detector-CT (PCD-CT) using a single data acquisition

BACKGROUND

Accurate noise power spectra (NPS) measurement in clinical X-ray CT exams is challenging due to the need for repeated scans, which expose patients to high radiation risks. A reliable method for single CT acquisition NPS estimation is thus highly desirable.

PURPOSE

To develop a method for estimating local NPS from a single photon counting detector-CT (PCD-CT) acquisition.

METHODS

A novel nearly statistical bias-free estimator was constructed from the raw counts data of PCD-CT scan to estimate the variance of sinogram projection data. An analytical algorithm is employed to reconstruct point-wise covariancecov ( x i , x j ) $\text{cov}({\bf x}_i,{\bf x}_j)$ between any two image pixel/voxel locationsx i ${\bf x}_i$ andx j ${\bf x_j}$ . A Fourier transform is applied to obtain the desired point-wise NPS for any chosen locationx i ${\bf x}_i$ . The method was validated using experimental data acquired from a benchtop PCD-CT system with various physical phantoms, and the results were compared with the conventional local NPS measurement method using repeated scans and statistical ensemble averaging.

RESULTS

The experimental results demonstrate that (1) the proposed method can achieve pointwise/local NPS measurement for a region of interest (ROI) located at any chosen position, accurately characterizing the NPS with spatial structures resulting from image content heterogeneity; (2) the local NPS measured using the proposed method show a higher precision in the measured NPS compared to the conventional measurement method; (3) spatial averaging of the local NPS yields the conventional NPS for a given local ROI.

CONCLUSION

A new method was developed to enable local NPS from a single PCD-CT acquisition.

Analytical covariance estimation for iterative CT reconstruction methods

Covariance of reconstruction images are useful to analyze the magnitude and correlation of noise in the evaluation of systems and reconstruction algorithms. The covariance estimation requires a big number of image samples that are hard to acquire in reality. A covariance propagation method from projection by a few noisy realizations is studied in this work. Based on the property of convergent points of cost funtions, the proposed method is composed of three steps, (1) construct a relationship between the covariance of projection and corresponding reconstruction from cost functions at its convergent point, (2) simplify the covariance relationship constructed in (1) by introducing an approximate gradient of penalties, and (3) obtain an analytical covariance estimation according to the simplified relationship in (2). Three approximation methods for step (2) are studied: the linear approximation of the gradient of penalties (LAM), the Taylor apprximation (TAM), and the mixture of LAM and TAM (MAM). TV and qGGMRF penalized weighted least square methods are experimented on. Results from statistical methods are used as reference. Under the condition of unstable 2nd derivative of penalties such as TV, the covariance image estimated by LAM accords to reference well but of smaller values, while the covarianc estimation by TAM is quite off. Under the conditon of relatively stable 2nd derivative of penalties such as qGGMRF, TAM performs well and LAM is again with a negative bias in magnitude. MAM gives a best performance under both conditions by combining LAM and TAM. Results also show that only one noise realization is enough to obtain reasonable covariance estimation analytically, which is important for practical usage. This work suggests the necessity and a new way to estimate the covariance for non-quadratically penalized reconstructions. Currently, the proposed method is computationally expensive for large size reconstructions.Computational efficiency is our future work to focus.

Accelerated 3D image reconstruction with a morphological pyramid and noise-power convergence criterion

Model-based iterative reconstruction (MBIR) for cone-beam CT (CBCT) offers better noise-resolution tradeoff and image quality than analytical methods for acquisition protocols with low x-ray dose or limited data, but with increased computational burden that poses a drawback to routine application in clinical scenarios. This work develops a comprehensive framework for acceleration of MBIR in the form of penalized weighted least squares optimized with ordered subsets separable quadratic surrogates. The optimization was scheduled on a set of stages forming a morphological pyramid varying in voxel size. Transition between stages was controlled with a convergence criterion based on the deviation between the mid-band noise power spectrum (NPS) measured on a homogeneous region of the evolving reconstruction and that expected for the converged image, computed with an analytical model that used projection data and the reconstruction parameters. A stochastic backprojector was developed by introducing a random perturbation to the sampling position of each voxel for each ray in the reconstruction within a voxel-based backprojector, breaking the deterministic pattern of sampling artifacts when combined with an unmatched Siddon forward projector. This fast, forward and backprojector pair were included into a multi-resolution reconstruction strategy to provide support for objects partially outside of the field of view. Acceleration from ordered subsets was combined with momentum accumulation stabilized with an adaptive technique that automatically resets the accumulated momentum when it diverges noticeably from the current iteration update. The framework was evaluated with CBCT data of a realistic abdomen phantom acquired on an imaging x-ray bench and with clinical CBCT data from an angiography robotic C-arm (Artis Zeego, Siemens Healthineers, Forchheim, Germany) acquired during a liver embolization procedure. Image fidelity was assessed with the structural similarity index (SSIM) computed with a converged reconstruction. The accelerated framework provided accurate reconstructions in 60 s (SSIM = 0.97) and as little as 27 s (SSIM = 0.94) for soft-tissue evaluation. The use of simple forward and backprojectors resulted in 9.3× acceleration. Accumulation of momentum provided extra ∼3.5× acceleration with stable convergence for 6-30 subsets. The NPS-driven morphological pyramid resulted in initial faster convergence, achieving similar SSIM with 1.5× lower runtime than the single-stage optimization. Acceleration of MBIR to provide reconstruction time compatible with clinical applications is feasible via architectures that integrate algorithmic acceleration with approaches to provide stable convergence, and optimization schedules that maximize convergence speed.

Task-driven source-detector trajectories in cone-beam computed tomography: I. Theory and methods

We develop a mathematical framework for the design of orbital trajectories that are optimal to a particular imaging task (or tasks) in advanced cone-beam computed tomography systems that have the capability of general source-detector positioning. The framework allows various parameterizations of the orbit as well as constraints based on imaging system capabilities. To accommodate nonstandard system geometries, a model-based iterative reconstruction method is applied. Such algorithms generally complicate the assessment and prediction of reconstructed image properties; however, we leverage efficient implementations of analytical predictors of local noise and spatial resolution that incorporate dependencies of the reconstruction algorithm on patient anatomy, x-ray technique, and geometry. These image property predictors serve as inputs to a task-based performance metric defined by detectability index, which is optimized with respect to the orbital parameters of data acquisition. We investigate the framework of the task-driven trajectory design in several examples to examine the dependence of optimal source-detector trajectories on the imaging task (or tasks), including location and spatial-frequency dependence. A variety of multitask objectives are also investigated, and the advantages to imaging performance are quantified in simulation studies.

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.

Backprojection Wiener deconvolution for computed tomographic reconstruction

Analytical CT reconstruction is popular in practice because of its computational efficiency, but it suffers from low reconstruction quality when an insufficient number of projections are used. To address this issue, this paper presents a new analytical method of backprojection Wiener deconvolution (BPWD). BPWD executes backprojection first, and then applies a Wiener deconvolution to the whole backprojected image. The Wiener filter is derived from a ramp filter, enabling the proposed approach to perform reconstruction and denoising simultaneously. The use of a filter after backprojection does not differentiate between real sampled projections and interpolated ones, introducing reconstruction errors. Therefore a weighted ramp filter was applied to increase the contribution of real sampled projections in the reconstruction, thus improving reconstruction quality. Experiments on synthetic data and real phase-contrast x-ray images showed that the proposed approach yields better reconstruction quality compared to the classical filtered backprojection (FBP) method, with comparable reconstruction speed.

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.