PET Reconstruction

Penalized direct estimation using partial dynamic data

Direct parametric estimation in positron emission tomography (PET) has been developed to compute the voxel-based kinetic parameters in the reconstruction process, obtaining more accurate physiological information of tracer uptake. Although the direct parametric imaging can achieve accurate kinetic analysis, the long acquisition time is still painful, particularly for sick and old patients. To address this issue, we explore the feasibility to estimate voxel-based kinetic parameters using partial dynamic data, specifically the first and last 10 minutes of a typical dynamic scan. To improve the quality of the direct parametric imaging with partial dynamic data, we propose a novel penalized direct estimation method containing log-likelihood, ridge regression and patch-based joint similarity penalty of kinetic images, in which the structural similarity weight between kinetic images can be used for improving the features of binding potential image. In our optimization, the alternating direction method of multipliers (ADMM) with a separable quadratic surrogate (SQS) is exploited. We validate the proposed method using a brain phantom, and demonstrate that the proposed method outperforms the conventional direct estimation methods even using partial dynamic data.

(a) Ground-truth, and binding potential images (k3/k4) using (b) the NLS after reconstruction (OSEM with Gaussian smoothing FWHM 4 mm), (c) penalized direct estimation using partial dynamic data, (d) penalized direct estimation using full dynamic data, (e) the proposed method using partial dynamic data and (f) the proposed method using full dynamic data.

K. Kim, Y. D. Son, G. El Fakhri and Q. Li, Penalized direct estimation using joint similarity of kinetic images with partial dynamic data, Society of Nuclear Medicine and Molecular Imaging (SNMMI), June. 2017


K. Kim, G. El Fakhri and Q. Li, Direct Parametric Imaging using Partial Dynamic Data, IEEE Nuclear Science Symposium and Medical Imaging Conference (NSSMIC), Nov. 2016.

Penalized MLAA with MR prior

PET/MR scanner has been developed for both molecular and morphological assessment with great potentials. In the PET/MR scan, the attenuation correction is still a problem. One method is the MR-based attenuation correction that generates the synthetic CT images from MR images. However, the lack of bone signal and the bias from the synthetic CT image can degrade the PET image quality. Another method is a maximum likelihood reconstruction of activity and attenuation (MLAA) using the time-of-flight (TOF) PET emission data, however, the noise component is considerably high from TOFPET data. To address this issue, we propose a penalized MLAA using a spatially-encoded anatomic MR prior, which jointly use a patch-based spatially-encoded similarity weight of MR image to improve the attenuation image quality. In addition, we propose a non-divergence criteria using a consistency condition in the iterative process. We exploit an alternating direction method of multipliers (ADMM) algorithm to optimize the cost function. In real patient study, we demonstrate that the proposed method outperforms the conventional MLAA.

Bone cancer patient study: (a) (i) Dixon-based In-phase MR image, and attenuation images using (ii) the conventional MLAA, (iii) the Dixon-based attenuation mapping and (iiii) the proposed method. (b) Activity images (i) before and after attenuation correction using (ii) the conventional MLAA, (iii) Dixon-based synthetic CT (MRAC) and (iiii) the proposed method.

K. Kim, Y. D. Son, G. El Fakhri and Q. Li, Penalized direct estimation using joint similarity of kinetic images with partial dynamic data, Society of Nuclear Medicine and Molecular Imaging (SNMMI), June. 2017


K. Kim, G. El Fakhri and Q. Li, Direct Parametric Imaging using Partial Dynamic Data, IEEE Nuclear Science Symposium and Medical Imaging Conference (NSSMIC), Nov. 2016.

Non-uniform TOF PET reconstruction for uniform convergence

Time of fight (TOF) PET reconstruction statistically improves the image quality and the fast convergence speed, so-called the recovery rate. Although TOF PET can improve the overall signal to noise ratio (SNR) of the image compared to non-TOF PET, the SNR disparity between separate regions in the reconstructed image using TOF data becomes higher than that using non-TOF data. In particular, because of TOF bins having different photon statistics, the SNR for low activity or small regions is significantly lower than the SNR for high activity regions, which can degrade the overall image quality in practice when terminating TOF-PET reconstruction after a finite number of iterations because different SNRs of regions have different recovery rates. Achieving more uniform recovery rates across different SNR regions is crucial to improve the quality of the reconstructed image. In this project, we propose a TOF-PET reconstruction algorithm using the ordered subsets non-uniform separable quadratic surrogates (OS-NU-SQS) algorithm with Nesterov's momentum method and quadratic roughness regularization. In computer simulations, we demonstrate that the proposed method can improve the image quality and recovery rate uniformity compared to the TOF-based conventional ordered subset expectation maximization (OSEM) and conventional SQS algorithms with early stopping criterion. Furthermore, the proposed method is accelerated using our GPU implementation, making the algorithm more practical.

XCAT phantom simulation setup using (a) three different region of interests (ROIs) with high intensity components at (b) lung, (c) spine and (d) liver.

Recovery rate comparison by normalized root mean square difference (NRMSD) of (a) OS-SQS, (b) OSEM, (c) OS-NUSQS and (d) OS-NUSQS with momentum for three ROIs

K. Kim, J. C. Ye, L. Cheng, K. Ying, G. E. Fakhri and Q. Li, TOF-PET ordered subset reconstruction using non-uniform separable quadratic surrogates algorithm, IEEE International Symposium on Biomedical Imaging (ISBI), Beijing, China, April, 2014.

CT Reconstruction

Sparse-View Spectral CT Reconstruction Using Spectral Patch-Based Low-Rank Penalty

Spectral computed tomography (CT) is a promising technique with the potential for improving lesion detection, tissue characterization, and material decomposition. In this project, we are interested in kVp switching-based spectral CT that alternates distinct kVp X-ray transmissions during gantry rotation. This system can acquire multiple X-ray energy transmissions without additional radiation dose. However, only sparse views are generated for each spectral measurement; and the spectra themselves are limited in number. To address these limitations, we propose a penalized maximum likelihood method using spectral patch-based low-rank penalty, which exploits the self-similarity of patches that are collected at the same position in spectral images. The main advantage is that the relatively small number of materials within each patch allows us to employ the low-rank penalty that is less sensitive to intensity changes while preserving edge directions. In our optimization formulation, the cost function consists of the Poisson log-likelihood for X-ray transmission and the nonconvex patch-based low-rank penalty. Since the original cost function is difficult to minimize directly, we propose an optimization method using separable quadratic surrogate and concave convex procedure algorithms for the log-likelihood and penalty terms, which results in an alternating minimization that provides a computational advantage because each subproblem can be solved independently. We performed computer simulations and a real experiment using a kVp switching-based spectral CT with sparse-view measurements, and compared the proposed method with conventional algorithms. We confirmed that the proposed method improves spectral images both qualitatively and quantitatively. Furthermore, our GPU implementation significantly reduces the computational cost.

(a) Source kVp switching-based spectral CT and (b) the decomposition into spectral bands.

Patch grouping in spectral images and its principal component analysis, which justifies the low-rank penalty.

(a) Ground truth, and reconstruction images by (b) FBP, (c) SQS, (d) TV, (e) RPCA, and (f) the proposed method. Reconstruction energies are (1) 80 kVp, (2) 100 kVp, and (3) 120 kVp. Red box in (1a) is an ROI.

Low-dose CT reconstruction using spatially encoded nonlocal penalty

(Winner of 2016 Low Dose CT Grand Challenge supported by AAPM and Mayo clinic)

Computed tomography (CT) is one of the most used imaging modalities for imaging both symptomatic and asymptomatic patients. However, because of the high demand for lower radiation dose during CT scans, the reconstructed image can suffer from noise and artifacts due to the trade-off between the image quality and the radiation dose. The purpose of this project is to improve the image quality of quarter dose images and to select the best hyper-parameters using the regular dose image as ground truth. We first generated the axially stacked 2D sinograms from the multislice raw projections with flying focal spots using a single slice rebinning method, which is an axially approximate method to provide simple implementation and efficient memory usage. To improve the image quality, a cost function containing the Poisson log-likelihood and spatially-encoded non-local penalty is proposed. Specifically, an ordered subsets separable quadratic surrogates (OS-SQS) method for the log-likelihood is exploited and the patch-based similarity constraint with a spatially variant factor is developed to reduce the noise significantly while preserving features. Furthermore, we applied the Nesterov's momentum method for acceleration and the diminishing number of subsets strategy for noise consistency. Fast non-local weight calculation is also utilized to reduce the computational cost. Datasets given by the Low Dose CT Grand Challenge were used for the validation, exploiting the training datasets with the regular and quarter dose data. The most important step in this project was to fine-tune the hyper-parameters to provide the best image for diagnosis. Using the regular dose filtered back-projection (FBP) image as ground truth, we could carefully select the hyper-parameters by conducting a bias and standard deviation study, and we obtained the best images in a fixed number of iterations. We demonstrated that the proposed method with well selected hyper-parameters improved the image quality using quarter dose data. The quarter dose proposed method was compared with regular dose FBP, quarter dose FBP, quarter dose L1-based 3-D TV method. We confirmed that the quarter dose proposed image was comparable to the regular dose FBP image and was better than images using other quarter dose methods. The 20 patient data were evaluated by radiologists at the Mayo clinic, and this method was awarded first place in the Low Dose CT Grand Challenge. We proposed the iterative CT reconstruction method using a spatially-encoded non-local penalty and ordered subsets separable quadratic surrogates with the Nesterov's momentum and diminishing number of subsets. The results demonstrated that the proposed method with fine-tuned hyper-parameters can significantly improve the image quality and provide accurate diagnostic features at quarter dose. The performance of the proposed method should be further improved for small lesions, and a more thorough evaluation using additional clinical data is required in the future.

Metastatic images of 4 patients (a-d), and (i) regular dose FBP, (ii) quarter dose FBP, (iii) quarter dose using total variation penalty and (iiii) quarter dose proposed method. Red arrows indicate metastasis.

In press

Duel Energy CT reconstruction

Dual-energy computed tomography (CT), which uses two X-ray spectra enabling material differentiation by analyzing material-dependent photo-electric and Compton effects, has become more widely used in various medical applications such as material decomposition and k-edge subtraction imaging. In most image reconstruction methods using Gaussian filtering, total variation (TV) for dual-energy CT, two CT images of high and low energies are separately reconstructed and utilized in applications. To improve the image quality by considering redundant information of both the high and low energy images, we propose a novel dual-energy CT reconstruction method using the guided image filtering algorithm, which jointly reconstructs two images of different energy spectra simultaneously. In our computer simulation, we demonstrate that the image quality can be significantly improved, which can be potentially exploited in material decomposition and radiation dose reduction. The proposed method will be validated using real experimental data of dual-energy CT.

(a) Ground truth and the reconstructed images of (b) FBP, (c) Gaussian filtering (FWHM=5mm), (d) SQS with TV and (e) the proposed method with (i) 80 kVp and (ii) 140 kVp, respectively. Here, color ranges are [0, 0.07] and [0, 0.035] for (i) and (ii).

H. Yang, K. Kim, G. El Fakhri, K. Kang, Y. Xing and Q. Li, Dual-energy CT Reconstruction using Guided Image Filtering, IEEE Nuclear Science Symposium and Medical Imaging Conference (NSSMIC), Nov. 2016.

Multi-Materials Decomposition using clinical Dual-energy CT

Dual energy computed tomography (DECT) can provide the information of the attenuation coefficient function of the scanned object. The classic projection-domain decomposition methods have been proposed to achieve this goal in which two components to delineate the attenuation coefficient function are used to decompose the projection data. In medical applications, the scanned body is typically composed of many types of materials, such as bone, soft tissue, fat and blood. Thus the attenuation coefficient function can be naturally decomposed with three or more components. However, as the number of the components is more than two, the two projection datasets of DECT are theoretically insufficient to perform the decomposition. Some works have proposed the multi-materials decomposition method using DECT in which each pixel of scanned object was decomposed with a group of three basis materials picked from a material library. However, this method requires a projection-domain decomposition firstly to estimate the attenuation coefficient function of the scanned object. The problem of the projection-domain method is that the spectrum of the x-ray source or additional calibration data is necessary. However, these two requirements may not be satisfied in practice. To address this issue, we propose an image-domain multi-materials decomposition method using DECT. Specifically, the dual energy images are jointly denoised by guided filtering with separable quadratic surrogate and then we calculate the adaptive attenuation function from the reconstruction images to perform the multi-materials decomposition.

1) Low-energy and 2) High-energy reconstructed images of the clinical DECT data

The multi-materials decomposed images

T. Zhao, K. Kim, D. Wu, Q Li, Multi-Materials Decomposition using clinical Dual-energy CT, IEEE Nuclear Science Symposium and Medical Imaging Conference (NSSMIC), Oct. 2017.

Metal artifact reduction using L1 and non-local penalties with iterative sinogram correction

Metal artifact reduction is a challenging issue in CT reconstruction. Due to insufficient measurements after passing through the metal object, the break down of the inconsistency in attenuation sinogram results in severe steak artifacts in the reconstructed image. In this project, we propose a metal artifact reduction method using l1 norma and non-local penalties with iterative sinogram correction, where the 3D in-painting algorithm is iteratively used to estimate the sinogram in the iteratively updated metal regions. Metal and non-metal images using l1 norm and non-local penalties are reconstructed separately. The split Bregmann algorithm and the generalized non-local formula were applied to solve the optimization problems associated with l1 norm and non-local penalties. Both body phantom simulations and real dental CT experiments verify that the proposed method can significantly reduce the metal artifacts and provide more clear details of the image structure.

Flowchart of the iterative sinogram correction.

Iterative reconstruction using (a) l1 and (b) non-local penalties for metal and non-metal images, and (c) final image is the sum of both images.

K. Kim, J. C. Ye, G. E. Fakhri and Q. Li, Metal artifact reduction using L1 and non-local penalties with iterative sinogram correction, The Third International Conference on Image Formation in X-Ray Computed Tomography, Salt Lake City, Utah, USA, June, 2014.


We are developing a 4D CT angiographic reconstruction method using 3D registration-based low rank minimization. Data was provided by MICCAI 2017 Coronary Artery Reconstruction Challenge. Main developer of this method is Kiwan Jeon (from National Institute for Mathematical Sciences South Korea) and Kyungsang Kim.