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Iterative PET Image Reconstruction Using Convolutional Neural Network Representation
Kuang Gong, Jiahui Guan, Kyungsang Kim, Xuezhu Zhang, Georges El Fakhri, Jinyi Qi, Quanzheng Li
TL;DR
PET reconstruction is challenged by an ill-posed inverse problem, low resolution and SNR, and limited detected photons. The paper represents the unknown image with a pretrained residual CNN inside an iterative, constrained reconstruction framework and evaluates it on simulated and hybrid real data. Quantitative results show better contrast-recovery versus noise trade-offs than CNN denoising, Gaussian filtering, and penalized reconstruction, while optimization remains sensitive to initialization.
Problem
PET reconstruction has low resolution and SNR because of physical degradation and limited detected photons, motivating improved image quality.
Method
A pretrained residual CNN represents the unknown PET image within an iterative reconstruction framework formulated as a constrained optimization problem.
Results
Quantitative results show better contrast recovery versus noise trade-offs than CNN denoising, Gaussian filtering, and penalized reconstruction methods.
Takeaways & Limitations
Measured-data constraints help the iterative CNN recover small features removed by image denoising, with higher lesion contrast recovery in simulation and real data.
Takeaways & Limitations
The nonlinear optimization can become trapped in local minima, and results depend on using a suitable initialization strategy.
Abstract
from arXiv · showhide
PET image reconstruction is challenging due to the ill-poseness of the inverse problem and limited number of detected photons. Recently deep neural networks have been widely and successfully used in computer vision tasks and attracted growing interests in medical imaging. In this work, we trained a deep residual convolutional neural network to improve PET image quality by using the existing inter-patient information. An innovative feature of the proposed method is that we embed the neural network in the iterative reconstruction framework for image representation, rather than using it as a post-processing tool. We formulate the objective function as a constraint optimization problem and solve it using the alternating direction method of multipliers (ADMM) algorithm. Both simulation data and hybrid real data are used to evaluate the proposed method. Quantification results show that our proposed iterative neural network method can outperform the neural network denoising and conventional penalized maximum likelihood methods.
I. INTRODUCTION
PET provides molecular-level imaging but suffers from low resolution and SNR because of physical degradation and limited detected photon counts. This paper uses inter-patient information through a deep CNN integrated into iterative reconstruction rather than post-processing.
- Motivation: PET supports molecular-level observation in oncology, neurology, and cardiology, but its resolution and SNR remain low.The stated causes are physical degradation factors and low coincident-photon counts.
- Motivation: Improving PET image quality is especially important for small lesion detection, brain imaging, and longitudinal studies.
- Related work: Existing approaches include improved system instrumentation, denoising methods, and regularized reconstruction using anatomical, temporal, or statistical priors.Examples include TOF, depth-of-interaction capability, local patch statistics, and gradient-based penalties.
- Contribution: The paper proposes integrating a deep CNN into PET iterative reconstruction to exploit existing inter-patient information.The network combines U-net and residual-network structures and represents the unknown image during reconstruction.
- Contribution: The main contributions are training a network with dynamic prior-patient data and incorporating it into iterative reconstruction, with better performance than CNN denoising.
B. Representing PET images using neural network
The paper represents PET images with a pretrained neural network so inter-patient and intra-patient information can enter iterative reconstruction. It combines this representation with a modified residual U-net architecture and reformulates the resulting nonlinear optimization as a constrained problem.
- Neural-network representation: The method adapts kernel-based image representation by using a neural network representation to incorporate prior information into reconstruction.The earlier kernel form embeds temporal or anatomical information in K, while the proposed approach is motivated by that representation strategy.
- Neural-network representation: Pretraining the neural network on existing data allows inter-patient and intra-patient information to be included in the iterative reconstruction framework.
- Network architecture: The network uses a U-net structure with batch normalization, repeated convolutions, down-sampling, up-sampling, and identity mappings.Its architecture is summarized in Fig. 1.
- Network architecture: Three modifications create a fully convolutional residual architecture: stride-2 convolution replaces max pooling, feature maps are added rather than concatenated, and the input connects directly to the output.These changes respectively avoid max pooling, reduce training parameters, and construct a residual network.
- Optimization formulation: Substituting the neural representation into the PET system model rewrites the reconstruction model, but the resulting objective is difficult to solve because of network nonlinearity.The paper therefore transfers the objective into constrained form; maximum-likelihood estimation is introduced before this reformulation.
C. Optimization
The constrained PET reconstruction problem is solved iteratively with ADMM, alternating between penalized image reconstruction and nonlinear neural-network input optimization. The neural-network subproblem uses first-order gradients and neighboring-slice information.
- ADMM solves the constrained optimization problem iteratively in three steps.
- The first subproblem is a penalized PET reconstruction problem solved using the optimization transfer method.
- The neural-network subproblem is a nonlinear least-squares problem requiring gradients with respect to the input α.
- Because Jacobian and Hessian calculations are difficult in the network platform, the method uses a first-order update with step size L.
- The update incorporates first-order gradients from four neighboring axial slices because the network input has five channels.
- Each ADMM iteration runs once for the image subproblem and five times for the neural-network subproblem, initializing α from a 30-iteration MLEM reconstruction.
D. Implementation details and reference methods
The network uses a TensorFlow implementation of a residual U-net-style model, while evaluation compares the proposed methods with Gaussian filtering and penalized reconstruction using a fair penalty.
- The network uses a 128 × 128 × 5 input, 128 × 128 output, Adam optimization, and an L2 training cost.
- The first-order gradient for the neural-network subproblem is implemented with TensorFlow's tf.gradient function.
- The proposed methods are compared with postreconstruction Gaussian filtering and penalized reconstruction.
- The fair penalty approaches an L-1 penalty for σ ≪|t| and resembles a quadratic penalty for σ ≫|t|.
- The reconstruction procedure is summarized in an algorithm for incorporating a convolutional neural network into iterative PET reconstruction.
A. Simulation study
The simulation study models a GE 690 scanner with XCAT phantoms, generates low-count training and testing data, and evaluates reconstruction using contrast-recovery and background-noise measurements.
- The simulation models a GE 690 scanner and uses nineteen XCAT phantoms with inserted lung lesions for training and testing.
- The simulated scanner has 13,824 LYSO crystals arranged in an 81 cm diameter ring with a 157 mm axial field of view.
- Poisson noise is added to match a 1-hour FDG scan with 5 mCi injection, while random and scatter events account for 60% of noise-free prompt data.
- Training uses high-count labels and one-tenth-count inputs reconstructed at ML EM iterations 20, 40, and 60, producing 49 × 18 × 3 training pairs.
- Twenty low-count realizations of the testing phantom are reconstructed for evaluation.
- Performance is quantified with contrast recovery from lung lesions and background standard deviation across repeated realizations and regions of interest.
B. Hybrid real data
The hybrid real-data study uses six one-hour FDG patient scans, creates training data from five patients, and evaluates noise and lesion contrast using inserted lesions in held-out data.
- Six patient data sets are acquired from one-hour FDG dynamic scans on a GE 690 scanner with 5 mCi injection.
- Five patient data sets provide training data, while the sixth is reserved for validation.
- Hybrid quantitative data are created by inserting 27 lesions into training scans and five lesions into testing scans.
- Training includes five low-dose realizations per patient and ML EM reconstructions at iterations 20, 40, and 60.
- Twenty testing realizations are used for noise evaluation, with 47 liver background ROIs used to calculate standard deviation.
- Lesion contrast recovery is measured by subtracting reconstructions without lesions from those with inserted lesions and comparing the result with ground truth.
A. Simulation results
In simulation data, iterative CNN reconstruction produced the strongest bias-variance trade-off, preserving uptake and details better than CNN denoising and other compared methods.
- The iterative CNN method generated higher lung-lesion uptake and more vessel detail than CNN denoising.CNN denoising was criticized for over-smoothing and losing small structures because of its L2-norm cost function.
- Both CNN approaches reduced noise while preserving detailed features better than traditional Gaussian post-filtering.The preserved details included thin myocardium regions.
- The penalized reconstruction preserved high lesion uptake but introduced noise spots in different regions.
- The iterative CNN method achieved the best bias-variance trade-off among all evaluated methods.This observation was consistent with the quantitative results in Fig. 6.
B. Real data results
For real data, iterative CNN reconstruction produced higher lesion uptake and clearer details than CNN denoising, while CNN methods achieved roughly two-fold STD reduction versus Gaussian filtering.
- The iterative CNN method produced higher inserted-lesion uptake than CNN denoising and the clearest spinal-region details among the compared methods.
- CNN methods achieved about two-fold STD reduction compared with Gaussian filtering.
- Iterative CNN reconstruction embeds CNN image representation within PET iterative reconstruction, unlike post-processing denoising.The measured-data constraint can help recover small features removed by image denoising methods.
- Higher lesion contrast recovery in both simulation and real data demonstrated the benefit of the measured-data constraint.
- CNN representation can incorporate generalized inter-patient prior information without explicitly specifying kernel basis functions.Multiple information sources can be aggregated through input channels, with the network learning their combination during training.
- Optimization is difficult because Subproblem (11) is nonlinear, and the first-order Nesterov method can become trapped in local minima.Using EM results after 30 iterations as input made the results more stable than a uniform-image initialization.
- The study used a modified fully convolutional U-net structure, while acknowledging that CNNs can remove small structures.The iterative framework can overcome this issue, but better feature-preserving network structures may improve performance.
VI. CONCLUSION
The paper proposes iterative PET reconstruction using convolutional neural network representation and evaluates it on simulated and real data.
- The proposed iterative CNN method performed better than CNN denoising, Gaussian filtering, and penalized reconstruction for contrast-recovery-versus-noise trade-offs.Future work will evaluate the method on more clinical data sets.