Deep Neural Networks (DNNs) are well-known to act as over-parameterized deep image priors (DIP) that regularize various image inverse problems. Meanwhile, researchers also proposed extremely compact, under-parameterized image priors (e.g., deep decoder) that are strikingly competent for image restoration too, despite a loss of accuracy. These two extremes push us to think whether there exists a better solution in the middle: between over- and under-parameterized image priors, can one identify "intermediate" parameterized image priors that achieve better trade-offs between performance, efficiency, and even preserving strong transferability? Drawing inspirations from the lottery ticket hypothesis (LTH), we conjecture and study a novel "lottery image prior" (LIP) by exploiting DNN inherent sparsity, stated as: given an over-parameterized DNN-based image prior, it will contain a sparse subnetwork that can be trained in isolation, to match the original DNN's performance when being applied as a prior to various image inverse problems. Our results validate the superiority of LIPs: we can successfully locate the LIP subnetworks from over-parameterized DIPs at substantial sparsity ranges. Those LIP subnetworks significantly outperform deep decoders under comparably compact model sizes (by often fully preserving the effectiveness of their over-parameterized counterparts), and they also possess high transferability across different images as well as restoration task types. Besides, we also extend LIP to compressive sensing image reconstruction, where a pre-trained GAN generator is used as the prior (in contrast to untrained DIP or deep decoder), and confirm its validity in this setting too. To our best knowledge, this is the first time that LTH is demonstrated to be relevant in the context of inverse problems or image priors.

We introduce Optimal Eye Surgeon (OES), a framework for pruning and training deep image generator networks. Typically, untrained deep convolutional networks, which include image sampling operations, serve as effective image priors (Ulyanov et al., 2018). However, they tend to overfit to noise in image restoration tasks due to being overparameterized. OES addresses this by adaptively pruning networks at random initialization to a level of underparameterization. This process effectively captures low-frequency image components even without training, by just masking. When trained to fit noisy images, these pruned subnetworks, which we term Sparse-DIP, resist overfitting to noise. This benefit arises from underparameterization and the regularization effect of masking, constraining them in the manifold of image priors. We demonstrate that subnetworks pruned through OES surpass other leading pruning methods, such as the Lottery Ticket Hypothesis, which is known to be suboptimal for image recovery tasks (Wu et al., 2023). Our extensive experiments demonstrate the transferability of OES-masks and the characteristics of sparse-subnetworks for image generation. Code is available at https://github.com/Avra98/Optimal-Eye-Surgeon.git.

Phase retrieval is the inverse problem of recovering a signal from magnitude-only Fourier measurements, and underlies numerous imaging modalities, such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as holography, includes a reference object that is placed adjacent to the specimen of interest before measurements are collected. The resulting inverse problem, known as holographic phase retrieval, is well-known to have improved problem conditioning relative to the original. This innovation, i.e. Holographic CDI, becomes crucial at the nanoscale, where imaging specimens such as viruses, proteins, and crystals require low-photon measurements. This data is highly corrupted by Poisson shot noise, and often lacks low-frequency content as well. In this work, we introduce a dataset-free deep learning framework for holographic phase retrieval adapted to these challenges. The key ingredients of our approach are the explicit and flexible incorporation of the physical forward model into an automatic differentiation procedure, the Poisson log-likelihood objective function, and an optional untrained deep image prior. We perform extensive evaluation under realistic conditions. Compared to competing classical methods, our method recovers signal from higher noise levels and is more resilient to suboptimal reference design, as well as to large missing regions of low frequencies in the observations. To the best of our knowledge, this is the first work to consider a dataset-free machine learning approach for holographic phase retrieval.

Convolutional Neural Networks (CNNs) are highly effective for image reconstruction problems. Typically, CNNs are trained on large amounts of training images. Recently, however, un-trained neural networks such as the Deep Image Prior and Deep Decoder have achieved excellent image reconstruction performance for standard image reconstruction problems such as image denoising and image inpainting, without using any training data. This success raises the question whether un-trained neural networks can compete with trained ones for practical imaging tasks. To address this question, we consider accelerated magnetic resonance imaging (MRI), an important medical imaging problem, which has received significant attention from the deep-learning community, and for which a dedicated training set exists. We study and optimize un-trained architectures, and as a result, propose a variation of the architectures of the deep image prior and deep decoder. We show that the resulting convolutional decoder out-performs other un-trained methods and---most importantly---achieves on-par performance with a standard trained baseline, the U-net, on the FastMRI dataset, a new dataset for benchmarking deep learning based reconstruction methods. Besides achieving on-par reconstruction performance relative to trained methods, we demonstrate that a key advantage over trained methods is robustness to out-of-distribution examples.

Untrained convolutional neural networks have emerged as highly successful tools for image recovery and restoration, for a variety of problems including denoising, compressive sensing, and inpainting [Uly 18; Jin 19; Vee 18; JH19; Hec19; HH19; Bos 20; Wan 20; HA20; Aro 20]. As opposed to trained convolutional neural networks, that learn an image prior from training data, untrained convolutional networks act as an image prior without any training and solely based on the architecture of the network and the optimization procedure used to fit them. The benefit of untrained networks was first observed in the Deep Image Prior (DIP) paper [Uly 18]. The key observation of Ulyanov et al. [Uly 18] is that fitting a standard overparameterized convolutional autoencoder (specifically, the U-net [Ron 15] or variations thereoff) to a single noisy/corrupted image, when combined with early stopping, yields excellent denoising, inpainting, and super-resolution performance. Subsequent literature has demonstrated that many elements of the architecture of a convolutional autoencoder--such as the encoder part--are irrelevant for this behavior to emerge.

Generative models, such as GANs, learn an explicit low-dimensional representation of a particular class of images, and so they may be used as natural image priors for solving inverse problems such as image restoration and compressive sensing. GAN priors have demonstrated impressive performance on these tasks, but they can exhibit substantial representation error for both in-distribution and out-of-distribution images, because of the mismatch between the learned, approximate image distribution and the data generating distribution. In this paper, we demonstrate a method for reducing the representation error of GAN priors by modeling images as the linear combination of a GAN prior with a Deep Decoder. The deep decoder is an underparameterized and most importantly unlearned natural signal model similar to the Deep Image Prior. No knowledge of the specific inverse problem is needed in the training of the GAN underlying our method. For compressive sensing and image superresolution, our hybrid model exhibits consistently higher PSNRs than both the GAN priors and Deep Decoder separately, both on in-distribution and out-of-distribution images. This model provides a method for extensibly and cheaply leveraging both the benefits of learned and unlearned image recovery priors in inverse problems.

Deep convolutional neural networks trained on large datsets have emerged as an intriguing alternative for compressing images and solving inverse problems such as denoising and compressive sensing. However, it has only recently been realized that even without training, convolutional networks can function as concise image models, and thus regularize inverse problems. In this paper, we provide further evidence for this finding by studying variations of convolutional neural networks that map few weight parameters to an image. The networks we consider only consist of convolutional operations, with either fixed or parameterized filters followed by ReLU non-linearities. We demonstrate that with both fixed and parameterized convolutional filters those networks enable representing images with few coefficients. What is more, the underparameterization enables regularization of inverse problems, in particular recovering an image from few observations. We show that, similar to standard compressive sensing guarantees, on the order of the number of model parameters many measurements suffice for recovering an image from compressive measurements. Finally, we demonstrate that signal recovery with a un-trained convolutional network outperforms standard l1 and total variation minimization for magnetic resonance imaging (MRI).

Deep neural networks, in particular convolutional neural networks, have become highly effective tools for compressing images and solving inverse problems including denoising, inpainting, and reconstruction from few and noisy measurements. This success can be attributed in part to their ability to represent and generate natural images well. Contrary to classical tools such as wavelets, image-generating deep neural networks have a large number of parameters---typically a multiple of their output dimension---and need to be trained on large datasets. In this paper, we propose an untrained simple image model, called the deep decoder, which is a deep neural network that can generate natural images from very few weight parameters. The deep decoder has a simple architecture with no convolutions and fewer weight parameters than the output dimensionality. This underparameterization enables the deep decoder to compress images into a concise set of network weights, which we show is on par with wavelet-based thresholding. Further, underparameterization provides a barrier to overfitting, allowing the deep decoder to have state-of-the-art performance for denoising. The deep decoder is simple in the sense that each layer has an identical structure that consists of only one upsampling unit, pixel-wise linear combination of channels, ReLU activation, and channelwise normalization. This simplicity makes the network amenable to theoretical analysis, and it sheds light on the aspects of neural networks that enable them to form effective signal representations.