Convolutional Analysis Operator Learning: Acceleration and Convergence

Quantitative MR Image Reconstruction Using Parameter-Specific Dictionary Learning With Adaptive Dictionary-Size and Sparsity-Level Choice

OBJECTIVE

We propose a method for the reconstruction of parameter-maps in Quantitative Magnetic Resonance Imaging (QMRI).

METHODS

Because different quantitative parameter-maps differ from each other in terms of local features, we propose a method where the employed dictionary learning (DL) and sparse coding (SC) algorithms automatically estimate the optimal dictionary-size and sparsity level separately for each parameter-map. We evaluated the method on a T1-mapping QMRI problem in the brain using the BrainWeb data as well as in-vivo brain images acquired on an ultra-high field 7 T scanner. We compared it to a model-based acceleration for parameter mapping (MAP) approach, other sparsity-based methods using total variation (TV), Wavelets (Wl), and Shearlets (Sh) to a method which uses DL and SC to reconstruct qualitative images, followed by a non-linear (DL+Fit).

RESULTS

Our algorithm surpasses MAP, TV, Wl, and Sh in terms of RMSE and PSNR. It yields better or comparable results to DL+Fit by additionally significantly accelerating the reconstruction by a factor of approximately seven.

CONCLUSION

The proposed method outperforms the reported methods of comparison and yields accurate T1-maps. Although presented for T1-mapping in the brain, our method's structure is general and thus most probably also applicable for the the reconstruction of other quantitative parameters in other organs.

SIGNIFICANCE

From a clinical perspective, the obtained T1-maps could be utilized to differentiate between healthy subjects and patients with Alzheimer's disease. From a technical perspective, the proposed unsupervised method could be employed to obtain ground-truth data for the development of data-driven methods based on supervised learning.

Systematic Review on Learning-based Spectral CT

Spectral computed tomography (CT) has recently emerged as an advanced version of medical CT and significantly improves conventional (single-energy) CT. Spectral CT has two main forms: dual-energy computed tomography (DECT) and photon-counting computed tomography (PCCT), which offer image improvement, material decomposition, and feature quantification relative to conventional CT. However, the inherent challenges of spectral CT, evidenced by data and image artifacts, remain a bottleneck for clinical applications. To address these problems, machine learning techniques have been widely applied to spectral CT. In this review, we present the state-of-the-art data-driven techniques for spectral CT.

Systematic Review on Learning-based Spectral CT

Spectral computed tomography (CT) has recently emerged as an advanced version of medical CT and significantly improves conventional (single-energy) CT. Spectral CT has two main forms: dual-energy computed tomography (DECT) and photon-counting computed tomography (PCCT), which offer image improvement, material decomposition, and feature quantification relative to conventional CT. However, the inherent challenges of spectral CT, evidenced by data and image artifacts, remain a bottleneck for clinical applications. To address these problems, machine learning techniques have been widely applied to spectral CT. In this review, we present the state-of-the-art data-driven techniques for spectral CT.

Momentum-Net: Fast and Convergent Iterative Neural Network for Inverse Problems

Iterative neural networks (INN) are rapidly gaining attention for solving inverse problems in imaging, image processing, and computer vision. INNs combine regression NNs and an iterative model-based image reconstruction (MBIR) algorithm, often leading to both good generalization capability and outperforming reconstruction quality over existing MBIR optimization models. This paper proposes the first fast and convergent INN architecture, Momentum-Net, by generalizing a block-wise MBIR algorithm that uses momentum and majorizers with regression NNs. For fast MBIR, Momentum-Net uses momentum terms in extrapolation modules, and noniterative MBIR modules at each iteration by using majorizers, where each iteration of Momentum-Net consists of three core modules: image refining, extrapolation, and MBIR. Momentum-Net guarantees convergence to a fixed-point for general differentiable (non)convex MBIR functions (or data-fit terms) and convex feasible sets, under two asymptomatic conditions. To consider data-fit variations across training and testing samples, we also propose a regularization parameter selection scheme based on the "spectral spread" of majorization matrices. Numerical experiments for light-field photography using a focal stack and sparse-view computational tomography demonstrate that, given identical regression NN architectures, Momentum-Net significantly improves MBIR speed and accuracy over several existing INNs; it significantly improves reconstruction quality compared to a state-of-the-art MBIR method in each application.

Accelerated Log-Regularized Convolutional Transform Learning and Its Convergence Guarantee

Convolutional transform learning (CTL), learning filters by minimizing the data fidelity loss function in an unsupervised way, is becoming very pervasive, resulting from keeping the best of both worlds: the benefit of unsupervised learning and the success of the convolutional neural network. There have been growing interests in developing efficient CTL algorithms. However, developing a convergent and accelerated CTL algorithm with accurate representations simultaneously with proper sparsity is an open problem. This article presents a new CTL framework with a log regularizer that can not only obtain accurate representations but also yield strong sparsity. To efficiently address our nonconvex composite optimization, we propose to employ the proximal difference of the convex algorithm (PDCA) which relies on decomposing the nonconvex regularizer into the difference of two convex parts and then optimizes the convex subproblems. Furthermore, we introduce the extrapolation technology to accelerate the algorithm, leading to a fast and efficient CTL algorithm. In particular, we provide a rigorous convergence analysis for the proposed algorithm under the accelerated PDCA. The experimental results demonstrate that the proposed algorithm can converge more stably to desirable solutions with lower approximation error and simultaneously with stronger sparsity and, thus, learn filters efficiently. Meanwhile, the convergence speed is faster than the existing CTL algorithms.

Content-Aware Scalable Deep Compressed Sensing

To more efficiently address image compressed sensing (CS) problems, we present a novel content-aware scalable network dubbed CASNet which collectively achieves adaptive sampling rate allocation, fine granular scalability and high-quality reconstruction. We first adopt a data-driven saliency detector to evaluate the importance of different image regions and propose a saliency-based block ratio aggregation (BRA) strategy for sampling rate allocation. A unified learnable generating matrix is then developed to produce sampling matrix of any CS ratio with an ordered structure. Being equipped with the optimization-inspired recovery subnet guided by saliency information and a multi-block training scheme preventing blocking artifacts, CASNet jointly reconstructs the image blocks sampled at various sampling rates with one single model. To accelerate training convergence and improve network robustness, we propose an SVD-based initialization scheme and a random transformation enhancement (RTE) strategy, which are extensible without introducing extra parameters. All the CASNet components can be combined and learned end-to-end. We further provide a four-stage implementation for evaluation and practical deployments. Experiments demonstrate that CASNet outperforms other CS networks by a large margin, validating the collaboration and mutual supports among its components and strategies. Codes are available at https://github.com/Guaishou74851/CASNet.

Revisiting [Formula: see text]-wavelet compressed-sensing MRI in the era of deep learning

Following their success in numerous imaging and computer vision applications, deep-learning (DL) techniques have emerged as one of the most prominent strategies for accelerated MRI reconstruction. These methods have been shown to outperform conventional regularized methods based on compressed sensing (CS). However, in most comparisons, CS is implemented with two or three hand-tuned parameters, while DL methods enjoy a plethora of advanced data science tools. In this work, we revisit [Formula: see text]-wavelet CS reconstruction using these modern tools. Using ideas such as algorithm unrolling and advanced optimization methods over large databases that DL algorithms utilize, along with conventional insights from wavelet representations and CS theory, we show that [Formula: see text]-wavelet CS can be fine-tuned to a level close to DL reconstruction for accelerated MRI. The optimized [Formula: see text]-wavelet CS method uses only 128 parameters compared to >500,000 for DL, employs a convex reconstruction at inference time, and performs within <1% of a DL approach that has been used in multiple studies in terms of quantitative quality metrics.

Noise Reduction in CT Using Learned Wavelet-Frame Shrinkage Networks

Encoding-decoding (ED) CNNs have demonstrated state-of-the-art performance for noise reduction over the past years. This has triggered the pursuit of better understanding the inner workings of such architectures, which has led to the theory of deep convolutional framelets (TDCF), revealing important links between signal processing and CNNs. Specifically, the TDCF demonstrates that ReLU CNNs induce low-rankness, since these models often do not satisfy the necessary redundancy to achieve perfect reconstruction (PR). In contrast, this paper explores CNNs that do meet the PR conditions. We demonstrate that in these type of CNNs soft shrinkage and PR can be assumed. Furthermore, based on our explorations we propose the learned wavelet-frame shrinkage network, or LWFSN and its residual counterpart, the rLWFSN. The ED path of the (r)LWFSN complies with the PR conditions, while the shrinkage stage is based on the linear expansion of thresholds proposed Blu and Luisier. In addition, the LWFSN has only a fraction of the training parameters (<1%) of conventional CNNs, very small inference times, low memory footprint, while still achieving performance close to state-of-the-art alternatives, such as the tight frame (TF) U-Net and FBPConvNet, in low-dose CT denoising.

An improved iterative neural network for high-quality image-domain material decomposition in dual-energy CT

PURPOSE

Dual-energy computed tomography (DECT) has been widely used in many applications that need material decomposition. Image-domain methods directly decompose material images from high- and low-energy attenuation images, and thus, are susceptible to noise and artifacts on attenuation images. The purpose of this study is to develop an improved iterative neural network (INN) for high-quality image-domain material decomposition in DECT, and to study its properties.

METHODS

We propose a new INN architecture for DECT material decomposition. The proposed INN architecture uses distinct cross-material convolutional neural network (CNN) in image refining modules, and uses image decomposition physics in image reconstruction modules. The distinct cross-material CNN refiners incorporate distinct encoding-decoding filters and cross-material model that captures correlations between different materials. We study the distinct cross-material CNN refiner with patch-based reformulation and tight-frame condition.

RESULTS

Numerical experiments with extended cardiac-torso phantom and clinical data show that the proposed INN significantly improves the image quality over several image-domain material decomposition methods, including a conventional model-based image decomposition (MBID) method using an edge-preserving regularizer, a recent MBID method using pre-learned material-wise sparsifying transforms, and a noniterative deep CNN method. Our study with patch-based reformulations reveals that learned filters of distinct cross-material CNN refiners can approximately satisfy the tight-frame condition.

CONCLUSIONS

The proposed INN architecture achieves high-quality material decompositions using iteration-wise refiners that exploit cross-material properties between different material images with distinct encoding-decoding filters. Our tight-frame study implies that cross-material CNN refiners in the proposed INN architecture are useful for noise suppression and signal restoration. This article is protected by copyright. All rights reserved.

Multi-channel convolutional analysis operator learning for dual-energy CT reconstruction

Objective. Dual-energy computed tomography (DECT) has the potential to improve contrast and reduce artifacts and the ability to perform material decomposition in advanced imaging applications. The increased number of measurements results in a higher radiation dose, and it is therefore essential to reduce either the number of projections for each energy or the source x-ray intensity, but this makes tomographic reconstruction more ill-posed. Approach. We developed the multi-channel convolutional analysis operator learning (MCAOL) method to exploit common spatial features within attenuation images at different energies and we propose an optimization method which jointly reconstructs the attenuation images at low and high energies with mixed norm regularization on the sparse features obtained by pre-trained convolutional filters through the convolutional analysis operator learning (CAOL) algorithm. Main results. Extensive experiments with simulated and real computed tomography data were performed to validate the effectiveness of the proposed methods, and we report increased reconstruction accuracy compared with CAOL and iterative methods with single and joint total variation regularization. Significance. Qualitative and quantitative results on sparse views and low-dose DECT demonstrate that the proposed MCAOL method outperforms both CAOL applied on each energy independently and several existing state-of-the-art model-based iterative reconstruction techniques, thus paving the way for dose reduction.

PALMNUT: An Enhanced Proximal Alternating Linearized Minimization Algorithm with Application to Separate Regularization of Magnitude and Phase

We introduce a new algorithm for complex image reconstruction with separate regularization of the image magnitude and phase. This optimization problem is interesting in many different image reconstruction contexts, although is nonconvex and can be difficult to solve. In this work, we first describe a novel implementation of the previous proximal alternating linearized minimization (PALM) algorithm to solve this optimization problem. We then make enhancements to PALM, leading to a new algorithm named PALMNUT that combines the PALM together with Nesterov's momentum and a novel approach that relies on uncoupled coordinatewise step sizes derived from coordinatewise Lipschitz-like bounds. Theoretically, we establish that a version of PALMNUT (without Nesterov's momentum) monotonically decreases the objective function, guaranteeing convergence of the cost function value. Empirical results obtained in the context of magnetic resonance imaging demonstrate that PALMNUT has computational advantages over common existing approaches like alternating minimization. Although our focus is on the application to separate magnitude and phase regularization, we expect that the same approach may also be useful in other nonconvex optimization problems with similar objective function structure.

Improved Low-Count Quantitative PET Reconstruction With an Iterative Neural Network

Image reconstruction in low-count PET is particularly challenging because gammas from natural radioactivity in Lu-based crystals cause high random fractions that lower the measurement signal-to-noise-ratio (SNR). In model-based image reconstruction (MBIR), using more iterations of an unregularized method may increase the noise, so incorporating regularization into the image reconstruction is desirable to control the noise. New regularization methods based on learned convolutional operators are emerging in MBIR. We modify the architecture of an iterative neural network, BCD-Net, for PET MBIR, and demonstrate the efficacy of the trained BCD-Net using XCAT phantom data that simulates the low true coincidence count-rates with high random fractions typical for Y-90 PET patient imaging after Y-90 microsphere radioembolization. Numerical results show that the proposed BCD-Net significantly improves CNR and RMSE of the reconstructed images compared to MBIR methods using non-trained regularizers, total variation (TV) and non-local means (NLM). Moreover, BCD-Net successfully generalizes to test data that differs from the training data. Improvements were also demonstrated for the clinically relevant phantom measurement data where we used training and testing datasets having very different activity distributions and count-levels.

Convolutional Analysis Operator Learning: Dependence on Training Data

Convolutional analysis operator learning (CAOL) enables the unsupervised training of (hierarchical) convolutional sparsifying operators or autoencoders from large datasets. One can use many training images for CAOL, but a precise understanding of the impact of doing so has remained an open question. This paper presents a series of results that lend insight into the impact of dataset size on the filter update in CAOL. The first result is a general deterministic bound on errors in the estimated filters, and is followed by a bound on the expected errors as the number of training samples increases. The second result provides a high probability analogue. The bounds depend on properties of the training data, and we investigate their empirical values with real data. Taken together, these results provide evidence for the potential benefit of using more training data in CAOL.