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Self-supervised methods have recently proved to be nearly as effective as supervised methods in various imaging inverse problems, paving the way for learning-based methods in scientific and medical imaging applications where ground truth data is hard or expensive to obtain. This is the case in magnetic resonance imaging and computed tomography. These methods critically rely on invariance to translations and/or rotations of the image distribution to learn from incomplete measurement data alone. However, existing approaches fail to obtain competitive performances in the problems of image super-resolution and deblurring, which play a key role in most imaging systems. In this work, we show that invariance to translations and rotations is insufficient to learn from measurements that only contain low-frequency information. Instead, we propose a new self-supervised approach that leverages the fact that many image distributions are approximately scale-invariant, and that can be applied to any inverse problem where high-frequency information is lost in the measurement process. We demonstrate throughout a series of experiments on real datasets that the proposed method outperforms other self-supervised approaches, and obtains performances on par with fully supervised learning.

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Recent advances in unsupervised learning have highlighted the possibility of learning to reconstruct signals from noisy and incomplete linear measurements alone. These methods play a key role in medical and scientific imaging and sensing, where ground truth data is often scarce or difficult to obtain. However, in practice, measurements are not only noisy and incomplete but also quantized. Here we explore the extreme case of learning from binary observations and provide necessary and sufficient conditions on the number of measurements required for identifying a set of signals from incomplete binary data. Our results are complementary to existing bounds on signal recovery from binary measurements. Furthermore, we introduce a novel self-supervised learning approach, which we name SSBM, that only requires binary data for training. We demonstrate in a series of experiments with real datasets that SSBM performs on par with supervised learning and outperforms sparse reconstruction methods with a fixed wavelet basis by a large margin.

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Dongdong Chen, Mike Davies, Matthias J. Ehrhardt, Carola-Bibiane Schönlieb, Ferdia Sherry, Julián Tachella

From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry. Here symmetry refers to the invariance property of signal sets to transformations such as translation, rotation or scaling. Symmetry can also be incorporated into deep neural networks in the form of equivariance, allowing for more data-efficient learning. While there has been important advances in the design of end-to-end equivariant networks for image classification in recent years, computational imaging introduces unique challenges for equivariant network solutions since we typically only observe the image through some noisy ill-conditioned forward operator that itself may not be equivariant. We review the emerging field of equivariant imaging and show how it can provide improved generalization and new imaging opportunities. Along the way we show the interplay between the acquisition physics and group actions and links to iterative reconstruction, blind compressed sensing and self-supervised learning.

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Solving a linear inverse problem requires knowledge about the underlying signal model. In many applications, this model is a priori unknown and has to be learned from data. However, it is impossible to learn the model using observations obtained via a single incomplete measurement operator, as there is no information outside the range of the inverse operator, resulting in a chicken-and-egg problem: to learn the model we need reconstructed signals, but to reconstruct the signals we need to know the model. Two ways to overcome this limitation are using multiple measurement operators or assuming that the signal model is invariant to a certain group action. In this paper, we present necessary and sufficient sampling conditions for learning the signal model from partial measurements which only depend on the dimension of the model, and the number of operators or properties of the group action that the model is invariant to. As our results are agnostic of the learning algorithm, they shed light into the fundamental limitations of learning from incomplete data and have implications in a wide range set of practical algorithms, such as dictionary learning, matrix completion and deep neural networks.

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Single-photon light detection and ranging (lidar) captures depth and intensity information of a 3D scene. Reconstructing a scene from observed photons is a challenging task due to spurious detections associated with background illumination sources. To tackle this problem, there is a plethora of 3D reconstruction algorithms which exploit spatial regularity of natural scenes to provide stable reconstructions. However, most existing algorithms have computational and memory complexity proportional to the number of recorded photons. This complexity hinders their real-time deployment on modern lidar arrays which acquire billions of photons per second. Leveraging a recent lidar sketching framework, we show that it is possible to modify existing reconstruction algorithms such that they only require a small sketch of the photon information. In particular, we propose a sketched version of a recent state-of-the-art algorithm which uses point cloud denoisers to provide spatially regularized reconstructions. A series of experiments performed on real lidar datasets demonstrates a significant reduction of execution time and memory requirements, while achieving the same reconstruction performance than in the full data case.

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In many real-world settings, only incomplete measurement data are available which can pose a problem for learning. Unsupervised learning of the signal model using a fixed incomplete measurement process is impossible in general, as there is no information in the nullspace of the measurement operator. This limitation can be overcome by using measurements from multiple operators. While this idea has been successfully applied in various applications, a precise characterization of the conditions for learning is still lacking. In this paper, we fill this gap by presenting necessary and sufficient conditions for learning the signal model which indicate the interplay between the number of distinct measurement operators $G$, the number of measurements per operator $m$, the dimension of the model $k$ and the dimension of the signals $n$. In particular, we show that generically unsupervised learning is possible if each operator obtains at least $m>k+n/G$ measurements. Our results are agnostic of the learning algorithm and have implications in a wide range of practical algorithms, from low-rank matrix recovery to deep neural networks.

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Deep networks provide state-of-the-art performance in multiple imaging inverse problems ranging from medical imaging to computational photography. However, most existing networks are trained with clean signals which are often hard or impossible to obtain. Equivariant imaging (EI) is a recent self-supervised learning framework that exploits the group invariance present in signal distributions to learn a reconstruction function from partial measurement data alone. While EI results are impressive, its performance degrades with increasing noise. In this paper, we propose a Robust Equivariant Imaging (REI) framework which can learn to image from noisy partial measurements alone. The proposed method uses Stein's Unbiased Risk Estimator (SURE) to obtain a fully unsupervised training loss that is robust to noise. We show that REI leads to considerable performance gains on linear and nonlinear inverse problems, thereby paving the way for robust unsupervised imaging with deep networks. Code will be available at: https://github.com/edongdongchen/REI.

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Single-photon lidar devices are able to collect an ever-increasing amount of time-stamped photons in small time periods due to increasingly larger arrays, generating a memory and computational bottleneck on the data processing side. Recently, a sketching technique was introduced to overcome this bottleneck which compresses the amount of information to be stored and processed. The size of the sketch scales with the number of underlying parameters of the time delay distribution and not, fundamentally, with either the number of detected photons or the time-stamp resolution. In this paper, we propose a detection algorithm based solely on a small sketch that determines if there are surfaces or objects in the scene or not. If a surface is detected, the depth and intensity of a single object can be computed in closed-form directly from the sketch. The computational load of the proposed detection algorithm depends solely on the size of the sketch, in contrast to previous algorithms that depend at least linearly in the number of collected photons or histogram bins, paving the way for fast, accurate and memory efficient lidar estimation. Our experiments demonstrate the memory and statistical efficiency of the proposed algorithm both on synthetic and real lidar datasets.

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In various imaging problems, we only have access to compressed measurements of the underlying signals, hindering most learning-based strategies which usually require pairs of signals and associated measurements for training. Learning only from compressed measurements is impossible in general, as the compressed observations do not contain information outside the range of the forward sensing operator. We propose a new end-to-end self-supervised framework that overcomes this limitation by exploiting the equivariances present in natural signals. Our proposed learning strategy performs as well as fully supervised methods. Experiments demonstrate the potential of this framework on inverse problems including sparse-view X-ray computed tomography on real clinical data and image inpainting on natural images. Code will be released.

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Single-photon lidar has become a prominent tool for depth imaging in recent years. At the core of the technique, the depth of a target is measured by constructing a histogram of time delays between emitted light pulses and detected photon arrivals. A major data processing bottleneck arises on the device when either the number of photons per pixel is large or the resolution of the time stamp is fine, as both the space requirement and the complexity of the image reconstruction algorithms scale with these parameters. We solve this limiting bottleneck of existing lidar techniques by sampling the characteristic function of the time of flight (ToF) model to build a compressive statistic, a so-called sketch of the time delay distribution, which is sufficient to infer the spatial distance and intensity of the object. The size of the sketch scales with the degrees of freedom of the ToF model (number of objects) and not, fundamentally, with the number of photons or the time stamp resolution. Moreover, the sketch is highly amenable for on-chip online processing. We show theoretically that the loss of information for compression is controlled and the mean squared error of the inference quickly converges towards the optimal Cram\'er-Rao bound (i.e. no loss of information) for modest sketch sizes. The proposed compressed single-photon lidar framework is tested and evaluated on real life datasets of complex scenes where it is shown that a compression rate of up-to 1/150 is achievable in practice without sacrificing the overall resolution of the reconstructed image.

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