Michael Lustig

Ke Wang, Mariya Doneva, Jakob Meineke et al.

Magnetic Resonance Fingerprinting (MRF) is an efficient quantitative MRI technique that can extract important tissue and system parameters such as T1, T2, B0, and B1 from a single scan. This property also makes it attractive for retrospectively synthesizing contrast-weighted images. In general, contrast-weighted images like T1-weighted, T2-weighted, etc., can be synthesized directly from parameter maps through spin-dynamics simulation (i.e., Bloch or Extended Phase Graph models). However, these approaches often exhibit artifacts due to imperfections in the mapping, the sequence modeling, and the data acquisition. Here we propose a supervised learning-based method that directly synthesizes contrast-weighted images from the MRF data without going through the quantitative mapping and spin-dynamics simulation. To implement our direct contrast synthesis (DCS) method, we deploy a conditional Generative Adversarial Network (GAN) framework and propose a multi-branch U-Net as the generator. The input MRF data are used to directly synthesize T1-weighted, T2-weighted, and fluid-attenuated inversion recovery (FLAIR) images through supervised training on paired MRF and target spin echo-based contrast-weighted scans. In-vivo experiments demonstrate excellent image quality compared to simulation-based contrast synthesis and previous DCS methods, both visually as well as by quantitative metrics. We also demonstrate cases where our trained model is able to mitigate in-flow and spiral off-resonance artifacts that are typically seen in MRF reconstructions and thus more faithfully represent conventional spin echo-based contrast-weighted images.

Vivek Athalye, Michael Lustig, Martin Uecker

In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional g-factor noise analysis to both noise amplification and approximation errors. This is demonstrated with numerical examples.

Frederic Wang, Han Qi, Alfredo De Goyeneche et al.

Although deep learning (DL) methods are powerful for solving inverse problems, their reliance on high-quality training data is a major hurdle. This is significant in high-dimensional (dynamic/volumetric) magnetic resonance imaging (MRI), where acquisition of high-resolution fully sampled k-space data is impractical. We introduce a novel mathematical framework, dubbed k-band, that enables training DL models using only partial, limited-resolution k-space data. Specifically, we introduce training with stochastic gradient descent (SGD) over k-space subsets. In each training iteration, rather than using the fully sampled k-space for computing gradients, we use only a small k-space portion. This concept is compatible with different sampling strategies; here we demonstrate the method for k-space "bands", which have limited resolution in one dimension and can hence be acquired rapidly. We prove analytically that our method stochastically approximates the gradients computed in a fully-supervised setup, when two simple conditions are met: (i) the limited-resolution axis is chosen randomly-uniformly for every new scan, hence k-space is fully covered across the entire training set, and (ii) the loss function is weighed with a mask, derived here analytically, which facilitates accurate reconstruction of high-resolution details. Numerical experiments with raw MRI data indicate that k-band outperforms two other methods trained on limited-resolution data and performs comparably to state-of-the-art (SoTA) methods trained on high-resolution data. k-band hence obtains SoTA performance, with the advantage of training using only limited-resolution data. This work hence introduces a practical, easy-to-implement, self-supervised training framework, which involves fast acquisition and self-supervised reconstruction and offers theoretical guarantees.

Michael Kellman, Michael Lustig, Laura Waller

The goal of this tutorial is to explain step-by-step how to implement physics-based learning for the rapid prototyping of a computational imaging system. We provide a basic overview of physics-based learning, the construction of a physics-based network, and its reduction to practice. Specifically, we advocate exploiting the auto-differentiation functionality twice, once to build a physics-based network and again to perform physics-based learning. Thus, the user need only implement the forward model process for their system, speeding up prototyping time. We provide an open-source Pytorch implementation of a physics-based network and training procedure for a generic sparse recovery problem

Han Cui, Alfredo De Goyeneche, Efrat Shimron et al.

Image Quality Assessment (IQA) is essential in various Computer Vision tasks such as image deblurring and super-resolution. However, most IQA methods require reference images, which are not always available. While there are some reference-free IQA metrics, they have limitations in simulating human perception and discerning subtle image quality variations. We hypothesize that the JPEG quality factor is representatives of image quality measurement, and a well-trained neural network can learn to accurately evaluate image quality without requiring a clean reference, as it can recognize image degradation artifacts based on prior knowledge. Thus, we developed a reference-free quality evaluation network, dubbed "Quality Factor (QF) Predictor", which does not require any reference. Our QF Predictor is a lightweight, fully convolutional network comprising seven layers. The model is trained in a self-supervised manner: it receives JPEG compressed image patch with a random QF as input, is trained to accurately predict the corresponding QF. We demonstrate the versatility of the model by applying it to various tasks. First, our QF Predictor can generalize to measure the severity of various image artifacts, such as Gaussian Blur and Gaussian noise. Second, we show that the QF Predictor can be trained to predict the undersampling rate of images reconstructed from Magnetic Resonance Imaging (MRI) data.

Efrat Shimron, Jonathan I. Tamir, Ke Wang et al.

While open databases are an important resource in the Deep Learning (DL) era, they are sometimes used "off-label": data published for one task are used for training algorithms for a different one. This work aims to highlight that in some cases, this common practice may lead to biased, overly-optimistic results. We demonstrate this phenomenon for inverse problem solvers and show how their biased performance stems from hidden data preprocessing pipelines. We describe two preprocessing pipelines typical of open-access databases and study their effects on three well-established algorithms developed for Magnetic Resonance Imaging (MRI) reconstruction: Compressed Sensing (CS), Dictionary Learning (DictL), and DL. In this large-scale study we performed extensive computations. Our results demonstrate that the CS, DictL and DL algorithms yield systematically biased results when naively trained on seemingly-appropriate data: the Normalized Root Mean Square Error (NRMSE) improves consistently with the preprocessing extent, showing an artificial increase of 25%-48% in some cases. Since this phenomenon is generally unknown, biased results are sometimes published as state-of-the-art; we refer to that as subtle data crimes. This work hence raises a red flag regarding naive off-label usage of Big Data and reveals the vulnerability of modern inverse problem solvers to the resulting bias.

Ke Wang, Jonathan I Tamir, Alfredo De Goyeneche et al.

Purpose: To improve reconstruction fidelity of fine structures and textures in deep learning (DL) based reconstructions. Methods: A novel patch-based Unsupervised Feature Loss (UFLoss) is proposed and incorporated into the training of DL-based reconstruction frameworks in order to preserve perceptual similarity and high-order statistics. The UFLoss provides instance-level discrimination by mapping similar instances to similar low-dimensional feature vectors and is trained without any human annotation. By adding an additional loss function on the low-dimensional feature space during training, the reconstruction frameworks from under-sampled or corrupted data can reproduce more realistic images that are closer to the original with finer textures, sharper edges, and improved overall image quality. The performance of the proposed UFLoss is demonstrated on unrolled networks for accelerated 2D and 3D knee MRI reconstruction with retrospective under-sampling. Quantitative metrics including NRMSE, SSIM, and our proposed UFLoss were used to evaluate the performance of the proposed method and compare it with others. Results: In-vivo experiments indicate that adding the UFLoss encourages sharper edges and more faithful contrasts compared to traditional and learning-based methods with pure l2 loss. More detailed textures can be seen in both 2D and 3D knee MR images. Quantitative results indicate that reconstruction with UFLoss can provide comparable NRMSE and a higher SSIM while achieving a much lower UFLoss value. Conclusion: We present UFLoss, a patch-based unsupervised learned feature loss, which allows the training of DL-based reconstruction to obtain more detailed texture, finer features, and sharper edges with higher overall image quality under DL-based reconstruction frameworks.

Ke Wang, Michael Kellman, Christopher M. Sandino et al.

Deep learning (DL) based unrolled reconstructions have shown state-of-the-art performance for under-sampled magnetic resonance imaging (MRI). Similar to compressed sensing, DL can leverage high-dimensional data (e.g. 3D, 2D+time, 3D+time) to further improve performance. However, network size and depth are currently limited by the GPU memory required for backpropagation. Here we use a memory-efficient learning (MEL) framework which favorably trades off storage with a manageable increase in computation during training. Using MEL with multi-dimensional data, we demonstrate improved image reconstruction performance for in-vivo 3D MRI and 2D+time cardiac cine MRI. MEL uses far less GPU memory while marginally increasing the training time, which enables new applications of DL to high-dimensional MRI.

Michael Kellman, Kevin Zhang, Jon Tamir et al.

Critical aspects of computational imaging systems, such as experimental design and image priors, can be optimized through deep networks formed by the unrolled iterations of classical model-based reconstructions (termed physics-based networks). However, for real-world large-scale inverse problems, computing gradients via backpropagation is infeasible due to memory limitations of graphics processing units. In this work, we propose a memory-efficient learning procedure that exploits the reversibility of the network's layers to enable data-driven design for large-scale computational imaging systems. We demonstrate our method on a small-scale compressed sensing example, as well as two large-scale real-world systems: multi-channel magnetic resonance imaging and super-resolution optical microscopy.

Michael Kellman, Jon Tamir, Emrah Boston et al.

Computational imaging systems jointly design computation and hardware to retrieve information which is not traditionally accessible with standard imaging systems. Recently, critical aspects such as experimental design and image priors are optimized through deep neural networks formed by the unrolled iterations of classical physics-based reconstructions (termed physics-based networks). However, for real-world large-scale systems, computing gradients via backpropagation restricts learning due to memory limitations of graphical processing units. In this work, we propose a memory-efficient learning procedure that exploits the reversibility of the network's layers to enable data-driven design for large-scale computational imaging. We demonstrate our methods practicality on two large-scale systems: super-resolution optical microscopy and multi-channel magnetic resonance imaging.

Michael J. Anderson, Jonathan I. Tamir, Javier S. Turek et al.

Magnetic resonance imaging is capable of producing volumetric images without ionizing radiation. Nonetheless, long acquisitions lead to prohibitively long exams. Compressed sensing (CS) can enable faster scanning via sub-sampling with reduced artifacts. However, CS requires significantly higher reconstruction computation, limiting current clinical applications to 2D/3D or limited-resolution dynamic imaging. Here we analyze the practical limitations to T2 Shuffling, a four-dimensional CS-based acquisition, which provides sharp 3D-isotropic-resolution and multi-contrast images in a single scan. Our improvements to the pipeline on a single machine provide a 3x overall reconstruction speedup, which allowed us to add algorithmic changes improving image quality. Using four machines, we achieved additional 2.1x improvement through distributed parallelization. Our solution reduced the reconstruction time in the hospital to 90 seconds on a 4-node cluster, enabling its use clinically. To understand the implications of scaling this application, we simulated running our reconstructions with a multiple scanner setup typical in hospitals.

Frank Ong, Joseph Cheng, Michael Lustig

Purpose: To develop a general phase regularized image reconstruction method, with applications to partial Fourier imaging, water-fat imaging and flow imaging. Theory and Methods: The problem of enforcing phase constraints in reconstruction was studied under a regularized inverse problem framework. A general phase regularized reconstruction algorithm was proposed to enable various joint reconstruction of partial Fourier imaging, water-fat imaging and flow imaging, along with parallel imaging (PI) and compressed sensing (CS). Since phase regularized reconstruction is inherently non-convex and sensitive to phase wraps in the initial solution, a reconstruction technique, named phase cycling, was proposed to render the overall algorithm invariant to phase wraps. The proposed method was applied to retrospectively under-sampled in vivo datasets and compared with state of the art reconstruction methods. Results: Phase cycling reconstructions showed reduction of artifacts compared to reconstructions with- out phase cycling and achieved similar performances as state of the art results in partial Fourier, water-fat and divergence-free regularized flow reconstruction. Joint reconstruction of partial Fourier + water-fat imaging + PI + CS, and partial Fourier + divergence-free regularized flow imaging + PI + CS were demonstrated. Conclusion: The proposed phase cycling reconstruction provides an alternative way to perform phase regularized reconstruction, without the need to perform phase unwrapping. It is robust to the choice of initial solutions and encourages the joint reconstruction of phase imaging applications.

Patrick Virtue, Stella X. Yu, Michael Lustig

The task of MRI fingerprinting is to identify tissue parameters from complex-valued MRI signals. The prevalent approach is dictionary based, where a test MRI signal is compared to stored MRI signals with known tissue parameters and the most similar signals and tissue parameters retrieved. Such an approach does not scale with the number of parameters and is rather slow when the tissue parameter space is large. Our first novel contribution is to use deep learning as an efficient nonlinear inverse mapping approach. We generate synthetic (tissue, MRI) data from an MRI simulator, and use them to train a deep net to map the MRI signal to the tissue parameters directly. Our second novel contribution is to develop a complex-valued neural network with new cardioid activation functions. Our results demonstrate that complex-valued neural nets could be much more accurate than real-valued neural nets at complex-valued MRI fingerprinting.

Patrick Virtue, Michael Lustig

In Fourier-based medical imaging, sampling below the Nyquist rate results in an underdetermined system, in which linear reconstructions will exhibit artifacts. Another consequence of under-sampling is lower signal to noise ratio (SNR) due to fewer acquired measurements. Even if an oracle provided the information to perfectly disambiguate the underdetermined system, the reconstructed image could still have lower image quality than a corresponding fully sampled acquisition because of the reduced measurement time. The effects of lower SNR and the underdetermined system are coupled during reconstruction, making it difficult to isolate the impact of lower SNR on image quality. To this end, we present an image quality prediction process that reconstructs fully sampled, fully determined data with noise added to simulate the loss of SNR induced by a given under-sampling pattern. The resulting prediction image empirically shows the effect of noise in under-sampled image reconstruction without any effect from an underdetermined system. We discuss how our image quality prediction process can simulate the distribution of noise for a given under-sampling pattern, including variable density sampling that produces colored noise in the measurement data. An interesting consequence of our prediction model is that we can show that recovery from underdetermined non-uniform sampling is equivalent to a weighted least squares optimization that accounts for heterogeneous noise levels across measurements. Through a series of experiments with synthetic and in vivo datasets, we demonstrate the efficacy of the image quality prediction process and show that it provides a better estimation of reconstruction image quality than the corresponding fully-sampled reference image.

Frank Ong, Michael Lustig

We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often exhibit local correlations in multiple scales. Concretely, we propose a multi-scale low rank modeling that represents a data matrix as a sum of block-wise low rank matrices with increasing scales of block sizes. We then consider the inverse problem of decomposing the data matrix into its multi-scale low rank components and approach the problem via a convex formulation. Theoretically, we show that under various incoherence conditions, the convex program recovers the multi-scale low rank components \revised{either exactly or approximately}. Practically, we provide guidance on selecting the regularization parameters and incorporate cycle spinning to reduce blocking artifacts. Experimentally, we show that the multi-scale low rank decomposition provides a more intuitive decomposition than conventional low rank methods and demonstrate its effectiveness in four applications, including illumination normalization for face images, motion separation for surveillance videos, multi-scale modeling of the dynamic contrast enhanced magnetic resonance imaging and collaborative filtering exploiting age information.

Martin Uecker, Michael Lustig

Purpose: To develop an ESPIRiT-based method to estimate coil sensitivities with image phase as a building block for efficient and robust image reconstruction with phase constraints. Theory and Methods: ESPIRiT is a new framework for calibration of the coil sensitivities and reconstruction in parallel Magnetic Resonance Imaging (MRI). Applying ESPIRiT to a combined set of physical and virtual conjugate coils (VCC-ESPIRiT) implicitly exploits conjugate symmetry in k-space similar to VCC-GRAPPA. Based on this method, a new post-processing step is proposed for the explicit computation of coil sensitivities that include the absolute phase of the image. The accuracy of the computed maps is directly validated using a test based on projection onto fully sampled coil images and also indirectly in phase-constrained parallel-imaging reconstructions. Results: The proposed method can estimate accurate sensitivities which include low-resolution image phase. In case of high-frequency phase variations VCC-ESPIRiT yields an additional set of maps that indicates the existence of a high-frequency phase component. Taking this additional set of maps into account can improve the robustness of phase-constrained parallel imaging. Conclusion: The extended VCC-ESPIRiT is a useful tool for phase-constrained imaging.