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Kang Lin, Reinhard Heckel

Deep learning based methods for image reconstruction are state-of-the-art for a variety of imaging tasks. However, neural networks often perform worse if the training data differs significantly from the data they are applied to. For example, a network trained for accelerated magnetic resonance imaging (MRI) on one scanner performs worse on another scanner. In this work, we investigate the impact of the training data on the model's performance and robustness for accelerated MRI. We find that models trained on the combination of various data distributions, such as those obtained from different MRI scanners and anatomies, exhibit robustness equal or superior to models trained on the best single distribution for a specific target distribution. Thus training on diverse data tends to improve robustness. Furthermore, training on diverse data does not compromise in-distribution performance, i.e., a model trained on diverse data yields in-distribution performance at least as good as models trained on the more narrow individual distributions. Our results suggest that training a model for imaging on a variety of distributions tends to yield a more effective and robust model than maintaining separate models for individual distributions.

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Simon Wiedemann, Reinhard Heckel

Cryogenic electron tomography (cryo-ET) is a technique for imaging biological samples such as viruses, cells, and proteins in 3D. A microscope collects a series of 2D projections of the sample, and the goal is to reconstruct the 3D density of the sample called the tomogram. This is difficult as the 2D projections have a missing wedge of information and are noisy. Tomograms reconstructed with conventional methods, such as filtered back-projection, suffer from the noise, and from artifacts and anisotropic resolution due to the missing wedge of information. To improve the visual quality and resolution of such tomograms, we propose a deep-learning approach for simultaneous denoising and missing wedge reconstruction called DeepDeWedge. DeepDeWedge is based on fitting a neural network to the 2D projections with a self-supervised loss inspired by noise2noise-like methods. The algorithm requires no training or ground truth data. Experiments on synthetic and real cryo-ET data show that DeepDeWedge achieves competitive performance for deep learning-based denoising and missing wedge reconstruction of cryo-ET tomograms.

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Frederic Wang, Han Qi, Alfredo De Goyeneche, Reinhard Heckel, Michael Lustig, Efrat Shimron

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.

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Stefan Bamberger, Reinhard Heckel, Felix Krahmer

We investigate to what extent it is possible to solve linear inverse problems with $ReLu$ networks. Due to the scaling invariance arising from the linearity, an optimal reconstruction function $f$ for such a problem is positive homogeneous, i.e., satisfies $f(\lambda x) = \lambda f(x)$ for all non-negative $\lambda$. In a $ReLu$ network, this condition translates to considering networks without bias terms. We first consider recovery of sparse vectors from few linear measurements. We prove that $ReLu$- networks with only one hidden layer cannot even recover $1$-sparse vectors, not even approximately, and regardless of the width of the network. However, with two hidden layers, approximate recovery with arbitrary precision and arbitrary sparsity level $s$ is possible in a stable way. We then extend our results to a wider class of recovery problems including low-rank matrix recovery and phase retrieval. Furthermore, we also consider the approximation of general positive homogeneous functions with neural networks. Extending previous work, we establish new results explaining under which conditions such functions can be approximated with neural networks. Our results also shed some light on the seeming contradiction between previous works showing that neural networks for inverse problems typically have very large Lipschitz constants, but still perform very well also for adversarial noise. Namely, the error bounds in our expressivity results include a combination of a small constant term and a term that is linear in the noise level, indicating that robustness issues may occur only for very small noise levels.

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Anselm Krainovic, Mahdi Soltanolkotabi, Reinhard Heckel

Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal $\ell_2$-worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing.

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Tobit Klug, Dogukan Atik, Reinhard Heckel

Supervised training of deep neural networks on pairs of clean image and noisy measurement achieves state-of-the-art performance for many image reconstruction tasks, but such training pairs are usually difficult to collect. A variety of self-supervised methods enable training based on noisy measurements only, without clean images. In this work, we investigate the cost of self-supervised training by characterizing its sample complexity. We focus on a class of self-supervised methods that enable the computation of unbiased estimates of gradients of the supervised loss, including noise2noise methods. We first analytically show that a model trained with such self-supervised training is as good as the same model trained in a supervised fashion, but self-supervised training requires more examples than supervised training. We then study self-supervised denoising and accelerated MRI empirically and characterize the cost of self-supervised training in terms of the number of additional samples required, and find that the performance gap between self-supervised and supervised training vanishes as a function of the training examples, at a problem-dependent rate, as predicted by our theory.

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Johannes F. Kunz, Stefan Ruschke, Reinhard Heckel

In this paper, we propose an approach for cardiac magnetic resonance imaging (MRI), which aims to reconstruct a real-time video of a beating heart from continuous highly under-sampled measurements. This task is challenging since the object to be reconstructed (the heart) is continuously changing during signal acquisition. To address this challenge, we represent the beating heart with an implicit neural network and fit the network so that the representation of the heart is consistent with the measurements. The network in the form of a multi-layer perceptron with Fourier-feature inputs acts as an effective signal prior and enables adjusting the regularization strength in both the spatial and temporal dimensions of the signal. We examine the proposed approach for 2D free-breathing cardiac real-time MRI in different operating regimes, i.e., for different image resolutions, slice thicknesses, and acquisition lengths. Our method achieves reconstruction quality on par with or slightly better than state-of-the-art untrained convolutional neural networks and superior image quality compared to a recent method that fits an implicit representation directly to Fourier-domain measurements. However, this comes at a higher computational cost. Our approach does not require any additional patient data or biosensors including electrocardiography, making it potentially applicable in a wide range of clinical scenarios.

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Youssef Mansour, Reinhard Heckel

Recently, self-supervised neural networks have shown excellent image denoising performance. However, current dataset free methods are either computationally expensive, require a noise model, or have inadequate image quality. In this work we show that a simple 2-layer network, without any training data or knowledge of the noise distribution, can enable high-quality image denoising at low computational cost. Our approach is motivated by Noise2Noise and Neighbor2Neighbor and works well for denoising pixel-wise independent noise. Our experiments on artificial, real-world camera, and microscope noise show that our method termed ZS-N2N (Zero Shot Noise2Noise) often outperforms existing dataset-free methods at a reduced cost, making it suitable for use cases with scarce data availability and limited compute resources. A demo of our implementation including our code and hyperparameters can be found in the following colab notebook: https://colab.research.google.com/drive/1i82nyizTdszyHkaHBuKPbWnTzao8HF9b

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Fatih Furkan Yilmaz, Reinhard Heckel

Modern image classifiers achieve high predictive accuracy, but the predictions typically come without reliable uncertainty estimates. Conformal prediction algorithms provide uncertainty estimates by predicting a set of classes based on the probability estimates of the classifier (for example, the softmax scores). To provide such sets, conformal prediction algorithms often rely on estimating a cutoff threshold for the probability estimates, and this threshold is chosen based on a calibration set. Conformal prediction methods guarantee reliability only when the calibration set is from the same distribution as the test set. Therefore, the methods need to be recalibrated for new distributions. However, in practice, labeled data from new distributions is rarely available, making calibration infeasible. In this work, we consider the problem of predicting the cutoff threshold for a new distribution based on unlabeled examples only. While it is impossible in general to guarantee reliability when calibrating based on unlabeled examples, we show that our method provides excellent uncertainty estimates under natural distribution shifts, and provably works for a specific model of a distribution shift.

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Tobit Klug, Reinhard Heckel

Deep neural networks trained end-to-end to map a measurement of a (noisy) image to a clean image perform excellent for a variety of linear inverse problems. Current methods are only trained on a few hundreds or thousands of images as opposed to the millions of examples deep networks are trained on in other domains. In this work, we study whether major performance gains are expected from scaling up the training set size. We consider image denoising, accelerated magnetic resonance imaging, and super-resolution and empirically determine the reconstruction quality as a function of training set size, while optimally scaling the network size. For all three tasks we find that an initially steep power-law scaling slows significantly already at moderate training set sizes. Interpolating those scaling laws suggests that even training on millions of images would not significantly improve performance. To understand the expected behavior, we analytically characterize the performance of a linear estimator learned with early stopped gradient descent. The result formalizes the intuition that once the error induced by learning the signal model is small relative to the error floor, more training examples do not improve performance.

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