Image reconstruction by domain-transform manifold learning


Image reconstruction is essential for imaging applications across the physical and life sciences, including optical and radar systems, magnetic resonance imaging, X-ray computed tomography, positron emission tomography, ultrasound imaging and radio astronomy1,2,3. During image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the exact inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain4,5, the composition of which depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. Here we present a unified framework for image reconstruction—automated transform by manifold approximation (AUTOMAP)—which recasts image reconstruction as a data-driven supervised learning task that allows a mapping between the sensor and the image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for various magnetic resonance imaging acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate that manifold learning during training results in sparse representations of domain transforms along low-dimensional data manifolds, and observe superior immunity to noise and a reduction in reconstruction artefacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate that AUTOMAP and other learned reconstruction approaches will accelerate the development of new acquisition strategies across imaging modalities.

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Figure 1: Schematic representations of AUTOMAP image reconstruction.
Figure 2: Reconstruction performance of AUTOMAP compared with conventional techniques.
Figure 3: Analysis of AUTOMAP neural networks.
Figure 4: Learning reconstruction of phase for in vivo data.
Figure 5: Performance of AUTOMAP in real-world acquisitions.


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We acknowledge M. Michalski and the computational resources and assistance provided by the Massachusetts General Hospital (MGH) and the Brigham and Women’s Hospital (BWH) Center for Clinical Data Science (CCDS). The CCDS is supported by MGH, BWH, the MGH Department of Radiology, the BWH Department of Radiology, and through industry partnership with NVIDIA. We also acknowledge the Center for Machine Learning at Martinos. We also thank J. Stockmann, J. Polimeni, D. E. J. Waddington and R. L. Walsworth for their comments on this manuscript, and B. Bilgic and C. Liao for their assistance in human subject data acquisition. We acknowledge C. Catana for providing raw PET data and for filtered back projection and OSEM reconstructions. We also thank M. Haskell for providing the MRI motion encoding model. B.Z. was supported by the National Institutes of Health/National Institute of Biomedical Imaging and Bioengineering F32 Fellowship (EB022390). Data were provided in part by the HCP, MGH-USC Consortium (Principal Investigators: Bruce R. Rosen, Arthur W. Toga and Van Wedeen; U01MH093765), which was funded by the NIH Blueprint Initiative for Neuroscience Research grant; the National Institutes of Health grant P41EB015896; and the Instrumentation Grants S10RR023043, 1S10RR023401, 1S10RR019307.

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B.Z., J.Z.L., S.F.C., B.R.R. and M.S.R. conceptualized the problem and contributed to experimental design. B.Z. developed, implemented and tested the technical framework. J.Z.L. and B.Z. constructed the theoretical description. B.Z., J.Z.L., S.F.C., B.R.R. and M.S.R. wrote the manuscript.

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Correspondence to Matthew S. Rosen.

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Extended data figures and tables

Extended Data Figure 1 Reconstruction performance of AUTOMAP in low-signal-to-noise-ratio regimes.

Reference brain images were encoded into sensor-domain sampling strategies with high levels of additive white Gaussian noise and reconstructed using both AUTOMAP and conventional approaches: ae, spiral k-space encoding compared with conjugate-gradient SENSE reconstruction with NUFFT regridding; fj, Radon projection encoding compared with model-based iterative reconstruction. Image magnitude signal-to-noise ratios (SNRs) and error maps (with root mean squared error calculations) with respect to reference ground truth images are also shown. For each encoding experiment, both error maps are windowed to the same level.

Extended Data Figure 2 Effect of training corpus on image reconstruction.

ac, AUTOMAP was trained using sensor-image pairs of Cartesian Fourier encoded corpora derived from either ImageNet, HCP brain images, or random-valued Gaussian noise without any real-world image structure. Each trained network was then used to reconstruct a noise-corrupted Cartesian k-space brain dataset. The signal-to-noise ratio (SNR) of the reconstructed images is shown. The apparent intensity discontinuity in the region above the eyes is due to the masking process used to de-identify the data in the HCP protocol (see Methods for more details).

Extended Data Figure 3 Training curves of optimizer loss convergence.

Mean squared error (MSE) loss was minimized with stochastic gradient descent using the RMSProp algorithm and plotted here against training epoch count for: a, Cartesian Fourier encoding on IMAGENET corpus; b, spiral Fourier encoding on IMAGENET corpus; and c, Cartesian undersampled Fourier encoding on HCP brain corpus. The validation error tracks the training error without upward divergence, demonstrating a stable training regime with good bias-variance tradeoff.

Extended Data Figure 4 Reconstruction of motion-corrupted MRI.

a, T2-weighted reference image acquired at 3 T with a turbo spin-echo sequence. b, Three-dimensional motion trajectories measured during an Alzheimer’s patient study. c, d, These motion trajectories were used to corrupt the k-space of this reference image, and it was reconstructed without motion compensation using inverse Fourier transform (c) and AUTOMAP (d). Both images show comparable artefact level and structure, demonstrating the stability of AUTOMAP reconstruction in the presence of unanticipated subject motion. A/P refers to anterior and posterior translational motion, L/R refers to left and right translational motion.

Extended Data Figure 5 Reconstruction of PET scanner data.

ad, Human FDG PET sinogram data (a) was reconstructed using (b) filtered back projection (FBP), (c) OP-OSEM and (d) AUTOMAP.

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Zhu, B., Liu, J., Cauley, S. et al. Image reconstruction by domain-transform manifold learning. Nature 555, 487–492 (2018).

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