We study the possibility of imitating the functionality of an image-to-image translation model by observing input-output pairs. We focus on cases where training the model from scratch is impossible, either because training data are unavailable or because the model architecture is unknown. This is the case, for example, with commercial models for biological applications. Since the development of these models requires large investments, their owners commonly keep them confidential and reveal only a few input-output examples on the company’s website or in an academic paper. Surprisingly, we find that even a single example typically suffices for learning to imitate the model’s functionality, and that this can be achieved using a simple distillation approach. We present an extensive ablation study encompassing a wide variety of model architectures, datasets and tasks, to characterize the factors affecting vulnerability to functionality imitation, and provide a preliminary theoretical discussion on the reasons for this unwanted behavior.

Editing real images using a pre-trained text-to-image (T2I) diffusion/flow model often involves inverting the image into its corresponding noise map. However, inversion by itself is typically insufficient for obtaining satisfactory results, and therefore many methods additionally intervene in the sampling process. Such methods achieve improved results but are not seamlessly transferable between model architectures. Here, we introduce FlowEdit, a text-based editing method for pre-trained T2I flow models, which is inversion-free, optimization-free and model agnostic. Our method constructs an ODE that directly maps between the source and target distributions (corresponding to the source and target text prompts) and achieves a lower transport cost than the inversion approach. This leads to state-of-the-art results, as we illustrate with Stable Diffusion 3 and FLUX. Code and examples are available on the project's webpage.

Photo-realistic image restoration algorithms are typically evaluated by distortion measures (e.g., PSNR, SSIM) and by perceptual quality measures (e.g., FID, NIQE), where the desire is to attain the lowest possible distortion without compromising on perceptual quality. To achieve this goal, current methods commonly attempt to sample from the posterior distribution, or to optimize a weighted sum of a distortion loss (e.g., MSE) and a perceptual quality loss (e.g., GAN). Unlike previous works, this paper is concerned specifically with the *optimal* estimator that minimizes the MSE under a constraint of perfect perceptual index, namely where the distribution of the reconstructed images is equal to that of the ground-truth ones. A recent theoretical result shows that such an estimator can be constructed by optimally transporting the posterior mean prediction (MMSE estimate) to the distribution of the ground-truth images. Inspired by this result, we introduce Posterior-Mean Rectified Flow (PMRF), a simple yet highly effective algorithm that approximates this optimal estimator. In particular, PMRF first predicts the posterior mean, and then transports the result to a high-quality image using a rectified flow model that approximates the desired optimal transport map. We investigate the theoretical utility of PMRF and demonstrate that it consistently outperforms previous methods on a variety of image restoration tasks.

We present a novel generative approach based on Denoising Diffusion Models (DDMs), which produces high-quality image samples along with their losslessly compressed bit-stream representations. This is obtained by replacing the standard Gaussian noise sampling in the reverse diffusion with a selection of noise samples from pre-defined codebooks of fixed iid Gaussian vectors. Surprisingly, we find that our method, termed Denoising Diffusion Codebook Model (DDCM), retains sample quality and diversity of standard DDMs, even for extremely small codebooks. We leverage DDCM and pick the noises from the codebooks that best match a given image, converting our generative model into a highly effective lossy image codec achieving state-of-the-art perceptual image compression results. More generally, by setting other noise selections rules, we extend our compression method to any conditional image generation task (e.g., image restoration), where the generated images are produced jointly with their condensed bit-stream representations. Our work is accompanied by a mathematical interpretation of the proposed compressed conditional generation schemes, establishing a connection with score-based approximations of posterior samplers for the tasks considered. Code and demo are available on our project’s website.

While diffusion models generate high-quality images via probability flow, the theoretical understanding of this process remains incomplete. A key question is when probability flow converges to training samples or more general points on the data manifold. We analyze this by studying the probability flow of shallow ReLU neural network denoisers trained with minimal $\ell^2$ norm. For intuition, we introduce a simpler score flow and show that for orthogonal datasets, both flows follow similar trajectories, converging to a training point or a sum of training points. However, early stopping by the diffusion time scheduler allows probability flow to reach more general manifold points. This reflects the tendency of diffusion models to both memorize training samples and generate novel points that combine aspects of multiple samples, motivating our study of such behavior in simplified settings. We extend these results to obtuse simplex data and, through simulations in the orthogonal case, confirm that probability flow converges to a training point, a sum of training points, or a manifold point. Moreover, memorization decreases when the number of training samples grows, as fewer samples accumulate near training points.

Diffusion Models have demonstrated remarkable capabilities in handling inverse problems, offering high-quality posterior-sampling-based solutions. Despite significant advances, a fundamental trade-off persists regarding the way the conditioned synthesis is employed: Zero-shot approaches can accommodate any linear degradation but rely on approximations that reduce accuracy. In contrast, training-based methods model the posterior correctly, but cannot adapt to the degradation at test-time. Here we introduce InvFusion, the first training-based degradation-aware posterior sampler. InvFusion combines the best of both worlds - the strong performance of supervised approaches and the flexibility of zero-shot methods. This is achieved through a novel architectural design that seamlessly integrates the degradation operator directly into the diffusion denoiser. We compare InvFusion against existing general-purpose posterior samplers, both degradation-aware zero-shot techniques and blind training-based methods. Experiments on the FFHQ and ImageNet datasets demonstrate state-of-the-art performance. Beyond posterior sampling, we further demonstrate the applicability of our architecture, operating as a general Minimum Mean Square Error predictor, and as a Neural Posterior Principal Component estimator.

We present TokenVerse - a method for multi-concept personalization, leveraging a pre-trained text-to-image diffusion model. Our framework can disentangle complex visual elements and attributes from as little as a single image, while enabling seamless plug-and-play generation of combinations of concepts extracted from multiple images. As opposed to existing works, TokenVerse can handle multiple images with multiple concepts each, and supports a wide-range of concepts, including objects, accessories, materials, pose, and lighting. Our work exploits a DiT-based text-to-image model, in which the input text affects the generation through both attention and modulation (shift and scale). We observe that the modulation space is semantic and enables localized control over complex concepts. Building on this insight, we devise an optimization-based framework that takes as input an image and a text description, and finds for each word a distinct direction in the modulation space. These directions can then be used to generate new images that combine the learned concepts in a desired configuration. We demonstrate the effectiveness of TokenVerse in challenging personalization settings, and showcase its advantages over existing methods.

Langevin dynamics (LD) is widely used for sampling from distributions and for optimization. In this work, we derive a closed-form expression for the expected loss of preconditioned LD near stationary points of the objective function. We use the fact that at the vicinity of such points, LD reduces to an Ornstein–Uhlenbeck process, which is amenable to convenient mathematical treatment. Our analysis reveals that when the preconditioning matrix satisfies a particular relation with respect to the noise covariance, LD's expected loss becomes proportional to the rank of the objective's Hessian. We illustrate the applicability of this result in the context of neural networks, where the Hessian rank has been shown to capture the complexity of the predictor function but is usually computationally hard to probe. Finally, we use our analysis to compare SGD-like and Adam-like preconditioners and identify the regimes under which each of them leads to a lower expected loss.

The dynamical stability of optimization methods at the vicinity of minima of the loss has recently attracted significant attention. For gradient descent (GD), stable convergence is possible only to minima that are sufficiently flat w.r.t. the step size, and those have been linked with favorable properties of the trained model. However, while the stability threshold of GD is well-known, to date, no explicit expression has been derived for the exact threshold of stochastic GD (SGD). In this paper, we derive such a closed-form expression. Specifically, we provide an explicit condition on the step size that is both necessary and sufficient for the linear stability of SGD in the mean square sense. Our analysis sheds light on the precise role of the batch size B. In particular, we show that the stability threshold is monotonically non-decreasing in the batch size, which means that reducing the batch size can only decrease stability. Furthermore, we show that SGD’s stability threshold is equivalent to that of a mixture process which takes in each iteration a full batch gradient step w.p. 1-p, and a single sample gradient step w.p. $p$, where $p \approx 1/B$. This indicates that even with moderate batch sizes, SGD’s stability threshold is very close to that of GD’s. We also prove simple necessary conditions for linear stability, which depend on the batch size, and are easier to compute than the precise threshold. Finally, we derive the asymptotic covariance of the dynamics around the minimum, and discuss its dependence on the learning rate. We validate our theoretical findings through experiments on the MNIST dataset.

Denoising diffusion probabilistic models (DDPMs) employ a sequence of white Gaussian noise samples to generate an image. In analogy with GANs, those noise maps could be considered as the latent code associated with the generated image. However, this native noise space does not possess a convenient structure, and is thus challenging to work with in editing tasks. Here, we propose an alternative latent noise space for DDPM that enables a wide range of editing operations via simple means, and present an inversion method for extracting these edit-friendly noise maps for any given image (real or synthetically generated). As opposed to the native DDPM noise space, the edit-friendly noise maps do not have a standard normal distribution and are not statistically independent across timesteps. However, they allow perfect reconstruction of any desired image, and simple transformations on them translate into meaningful manipulations of the output image (e.g. shifting, color edits). Moreover, in text-conditional models, fixing those noise maps while changing the text prompt, modifies semantics while retaining structure. We illustrate how this property enables text-based editing of real images via the diverse DDPM sampling scheme (in contrast to the popular non-diverse DDIM inversion). We also show how it can be used within existing diffusion-based editing methods to improve their quality and diversity. The code of the method is attached to this submission.

In ill-posed inverse problems, it is commonly desirable to obtain insight into the full spectrum of plausible so-lutions, rather than extracting only a single reconstruction. Information about the plausible solutions and their likelihoods is encoded in the posterior distribution. However, for high-dimensional data, this distribution is challenging to visualize. In this work, we introduce a new approach for estimating and visualizing posteriors by em-ploying energy-based models (EBMs) over low-dimensional subspaces. Specifically, we train a conditional EBM that receives an input measurement and a set of directions that span some low-dimensional subspace of solutions, and outputs the probability density function of the posterior within that space. We demonstrate the effectiveness of our method across a diverse range of datasets and image restoration problems, showcasing its strength in uncertainty quantification and visualization. As we show, our method outperforms a baseline that projects samples from a diffusion-based posterior sampler, while being orders of magnitude faster. Furthermore, it is more accurate than a baseline that assumes a Gaussian posterior. Code is available at https://github.com/yairomer/PPDE.

Denoising Diffusion Models (DDMs) have emerged as a strong competitor to Generative Adversarial Networks (GANs). However, despite their widespread use in image synthesis and editing applications, their latent space is still not as well understood. Recently, a semantic latent space for DDMs, coined ‘$h$-space’, was shown to facilitate semantic image editing in a way reminiscent of GANs. The$h$-space is comprised of the bottleneck activations in the DDM's denoiser across all timesteps of the diffusion process. In this paper, we explore the properties of$h$-space and propose several novel methods for finding meaningful semantic directions within it. We start by studying unsupervised methods for revealing interpretable semantic directions in pretrained DDMs. Specifically, we show that interpretable directions emerge as the principal components in the latent space. Additionally, we provide a novel method for discovering image-specific semantic directions by spectral analysis of the Jacobian of the denoiser w.r.t. the latent code. Next, we extend the analysis by finding directions in a supervised fashion in unconditional DDMs. We demonstrate how such directions can be found by annotating generated samples with a domain-specific attribute classifier. We further show how to semantically disentangle the found directions by simple linear projection. Our approaches are applicable without requiring any architectural modifications, text-based guidance, CLIP-based optimization, or model fine-tuning.

Image restoration problems are typically ill-posed in the sense that each degraded image can be restored in infinitely many valid ways. To accommodate this, many works generate a diverse set of outputs by attempting to randomly sample from the posterior distribution of natural images given the degraded input. Here we argue that this strategy is commonly of limited practical value because of the heavy tail of the posterior distribution. Consider for example inpainting a missing region of the sky in an image. Since there is a high probability that the missing region contains no object but clouds, any set of samples from the posterior would be entirely dominated by (practically identical) completions of sky. However, arguably, presenting users with only one clear sky completion, along with several alternative solutions such as airships, birds, and balloons, would better outline the set of possibilities. In this paper, we initiate the study of **meaningfully diverse** image restoration. We explore several post-processing approaches that can be combined with any diverse image restoration method to yield semantically meaningful diversity. Moreover, we propose a practical approach for allowing diffusion based image restoration methods to generate meaningfully diverse outputs, while incurring only negligent computational overhead. We conduct extensive user studies to analyze the proposed techniques, and find the strategy of reducing similarity between outputs to be significantly favorable over posterior sampling. Code and examples are available on the [project's webpage](https://noa-cohen.github.io/MeaningfulDiversityInIR/).

Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (*i*.*e*., the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project [website](https://hilamanor.github.io/GaussianDenoisingPosterior/).

Text-to-image (T2I) diffusion models achieve state-of-the-art results in image synthesis and editing. However, leveraging such pre-trained models for video editing is considered a major challenge. Many existing works attempt to enforce temporal consistency in the edited video through explicit correspondence mechanisms, either in pixel space or between deep features. These methods, however, struggle with strong nonrigid motion. In this paper, we introduce a fundamentally different approach, which is based on the observation that spatiotemporal slices of natural videos exhibit similar characteristics to natural images. Thus, the same T2I diffusion model that is normally used only as a prior on video frames, can also serve as a strong prior for enhancing temporal consistency by applying it on spatiotemporal slices. Based on this observation, we present Slicedit, a method for text-based video editing that utilizes a pre-trained T2I diffusion model to process both spatial and spatiotemporal slices. Our method generates videos that retain the structure and motion of the original video while adhering to the target text. Through extensive experiments, we demonstrate Slicedit's ability to edit a wide range of real-world videos, confirming its clear advantages compared to existing baselines.

Editing signals using large pre-trained models, in a zero-shot manner, has recently seen rapid advancements in the image domain. However, this wave has yet to reach the audio domain. In this paper, we explore two zero-shot editing techniques for audio signals, which use DDPM inversion with pre-trained diffusion models. The first, which we coin ZEro-shot Text-based Audio (ZETA) editing, is adopted from the image domain. The second, named ZEro-shot UnSupervized (ZEUS) editing, is a novel approach for discovering semantically meaningful editing directions without supervision. When applied to music signals, this method exposes a range of musically interesting modifications, from controlling the participation of specific instruments to improvisations on the melody. Samples and code can be found on our examples page.

We study the behavior of deterministic methods for solving inverse problems in imaging. These methods are commonly designed to achieve two goals: (1) attaining high perceptual quality, and (2) generating reconstructions that are consistent with the measurements. We provide a rigorous proof that the better a predictor satisfies these two requirements, the larger its Lipschitz constant must be, regardless of the nature of the degradation involved. In particular, to approach perfect perceptual quality and perfect consistency, the Lipschitz constant of the model must grow to infinity. This implies that such methods are necessarily more susceptible to adversarial attacks. We demonstrate our theory on single image super-resolution algorithms, addressing both noisy and noiseless settings. We also show how this undesired behavior can be leveraged to explore the posterior distribution, thereby allowing the deterministic model to imitate stochastic methods.

When solving ill-posed inverse problems, one often desires to explore the space of potential solutions rather than be presented with a single plausible reconstruction. Valuable insights into these feasible solutions and their associated probabilities are embedded in the posterior distribution. However, when confronted with data of high dimensionality (such as images), visualizing this distribution becomes a formidable challenge, necessitating the application of effective summarization techniques before user examination. In this work, we introduce a new approach for visualizing posteriors across multiple levels of granularity using *tree*-valued predictions. Our method predicts a tree-valued hierarchical summarization of the posterior distribution for any input measurement, in a single forward pass of a neural network. We showcase the efficacy of our approach across diverse datasets and image restoration challenges, highlighting its prowess in uncertainty quantification and visualization. Our findings reveal that our method performs comparably to a baseline that hierarchically clusters samples from a diffusion-based posterior sampler, yet achieves this with orders of magnitude greater speed. Code and examples are available at our [webpage](https://eliasnehme.github.io/PosteriorTrees/).

Fairness in image restoration tasks is the desire to treat different sub-groups of images equally well. Existing definitions of fairness in image restoration are highly restrictive. They consider a reconstruction to be a correct outcome for a group (e.g., women) *only* if it falls within the group's set of ground truth images (e.g., natural images of women); otherwise, it is considered *entirely* incorrect. Consequently, such definitions are prone to controversy, as errors in image restoration can manifest in various ways. In this work we offer an alternative approach towards fairness in image restoration, by considering the *Group Perceptual Index* (GPI), which we define as the statistical distance between the distribution of the group's ground truth images and the distribution of their reconstructions. We assess the fairness of an algorithm by comparing the GPI of different groups, and say that it achieves perfect *Perceptual Fairness* (PF) if the GPIs of all groups are identical. We motivate and theoretically study our new notion of fairness, draw its connection to previous ones, and demonstrate its utility on state-of-the-art face image restoration algorithms.

A prominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to *classify* between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years. In this work we present *classification diffusion models* (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-of-the-art negative log likelihood (NLL) among methods with this property.

Diffusion models are the current state-of-the-art in image generation, synthesizing high-quality images by breaking down the generation process into many fine-grained denoising steps. Despite their good performance, diffusion models are computationally expensive, requiring many neural function evaluations (NFEs). In this work, we propose an anytime diffusion-based method that can generate viable images when stopped at arbitrary times before completion. Using existing pretrained diffusion models, we show that the generation scheme can be recomposed as two nested diffusion processes, enabling fast iterative refinement of a generated image. In experiments on ImageNet and Stable Diffusion-based text-to-image generation, we show, both qualitatively and quantitatively, that our method’s intermediate generation quality greatly exceeds that of the original diffusion model, while the final generation result remains comparable. We illustrate the applicability of Nested Diffusion in several settings, including for solving inverse problems, and for rapid text-based content creation by allowing user intervention throughout the sampling process.1

Customizing text-to-image (T2I) models has seen tremendous progress recently, particularly in areas such as personalization, stylization, and conditional generation. However, expanding this progress to video generation is still in its infancy, primarily due to the lack of customized video data. In this work, we introduce Still-Moving, a novel generic framework for customizing a text-to-video (T2V) model, without requiring any customized video data. The framework applies to the prominent T2V design where the video model is built over a T2I model (e.g., via inflation). We assume access to a customized version of the T2I model, trained only on still image data (e.g., using DreamBooth). Naively plugging in the weights of the customized T2I model into the T2V model often leads to significant artifacts or insufficient adherence to the customization data. To overcome this issue, we train lightweight Spatial Adapters that adjust the features produced by the injected T2I layers. Importantly, our adapters are trained on "frozen videos" (i.e., repeated images), constructed from image samples generated by the customized T2I model. This training is facilitated by a novel Motion Adapter module, which allows us to train on such static videos while preserving the motion prior of the video model. At test time, we remove the Motion Adapter modules and leave in only the trained Spatial Adapters. This restores the motion prior of the T2V model while adhering to the spatial prior of the customized T2I model. We demonstrate the effectiveness of our approach on diverse tasks including personalized, stylized, and conditional generation. In all evaluated scenarios, our method seamlessly integrates the spatial prior of the customized T2I model with a motion prior supplied by the T2V model.

Although CNNs are believed to be invariant to translations, recent works have shown this is not the case due to aliasing effects that stem from down-sampling layers. The existing architectural solutions to prevent the aliasing effects are partial since they do not solve those effects that originate in non-linearities. We propose an extended antialiasing method that tackles both down-sampling and nonlinear layers, thus creating truly alias-free, shift-invariant CNNs11Our code is available at github.com/hmichaeli/alias_free_convnets/.. We show that the presented model is invariant to integer as well as fractional (i.e., sub-pixel) translations, thus outperforming other shift-invariant methods in terms of robustness to adversarial translations.

Denoising diffusion models (DDMs) have led to staggering performance leaps in image generation, editing and restoration. However, existing DDMs use very large datasets for training. Here, we introduce a framework for training a DDM on a single image. Our method, which we coin SinDDM, learns the internal statistics of the training image by using a multi-scale diffusion process. To drive the reverse diffusion process, we use a fully-convolutional light-weight denoiser, which is conditioned on both the noise level and the scale. This architecture allows generating samples of arbitrary dimensions, in a coarse-to-fine manner. As we illustrate, SinDDM generates diverse high-quality samples, and is applicable in a wide array of tasks, including style transfer and harmonization. Furthermore, it can be easily guided by external supervision. Particularly, we demonstrate text-guided generation from a single image using a pre-trained CLIP model.

Stochastic restoration algorithms allow to explore the space of solutions that correspond to the degraded input. In this paper we reveal additional fundamental advantages of stochastic methods over deterministic ones, which further motivate their use. First, we prove that any restoration algorithm that attains perfect perceptual quality and whose outputs are consistent with the input must be a posterior sampler, and is thus required to be stochastic. Second, we illustrate that while deterministic restoration algorithms may attain high perceptual quality, this can be achieved only by filling up the space of all possible source images using an extremely sensitive mapping, which makes them highly vulnerable to adversarial attacks. Indeed, we show that enforcing deterministic models to be robust to such attacks profoundly hinders their perceptual quality, while robustifying stochastic models hardly influences their perceptual quality, and improves their output variability. These findings provide a motivation to foster progress in stochastic restoration methods, paving the way to better recovery algorithms.

We propose an image restoration algorithm that can control the perceptual quality and/or the mean square error (MSE) of any pre-trained model, trading one over the other at test time. Our algorithm is few-shot: Given about a dozen images restored by the model, it can significantly improve the perceptual quality and/or the MSE of the model for newly restored images without further training. Our approach is motivated by a recent theoretical result that links between the minimum MSE (MMSE) predictor and the predictor that minimizes the MSE under a perfect perceptual quality constraint. Specifically, it has been shown that the latter can be obtained by optimally transporting the output of the former, such that its distribution matches that of the source data. Thus, to improve the perceptual quality of a predictor that was originally trained to minimize MSE, we approximate the optimal transport by a linear transformation in the latent space of a variational auto-encoder, which we compute in closed-form using empirical means and covariances. Going beyond the theory, we find that applying the same procedure on models that were initially trained to achieve high perceptual quality, typically improves their perceptual quality even further. And by interpolating the results with the original output of the model, we can improve their MSE on the expense of perceptual quality. We illustrate our method on a variety of degradations applied to general content images with arbitrary dimensions.

Many practical settings call for the reconstruction of temporal signals from corrupted or missing data. Classic examples include decoding, tracking, signal enhancement and denoising. Since the reconstructed signals are ultimately viewed by humans, it is desirable to achieve reconstructions that are pleasing to human perception. Mathematically, perfect perceptual-quality is achieved when the distribution of restored signals is the same as that of natural signals, a requirement which has been heavily researched in static estimation settings (i.e. when a whole signal is processed at once). Here, we study the problem of optimal causal filtering under a perfect perceptual-quality constraint, which is a task of fundamentally different nature. Specifically, we analyze a Gaussian Markov signal observed through a linear noisy transformation. In the absence of perceptual constraints, the Kalman filter is known to be optimal in the MSE sense for this setting. Here, we show that adding the perfect perceptual quality constraint (i.e. the requirement of temporal consistency), introduces a fundamental dilemma whereby the filter may have to ``knowingly'' ignore new information revealed by the observations in order to conform to its past decisions. This often comes at the cost of a significant increase in the MSE (beyond that encountered in static settings). Our analysis goes beyond the classic innovation process of the Kalman filter, and introduces the novel concept of an unutilized information process. Using this tool, we present a recursive formula for perceptual filters, and demonstrate the qualitative effects of perfect perceptual-quality estimation on a video reconstruction problem.

Uncertainty quantification is crucial for the deployment of image restoration models in safety-critical domains, like autonomous driving and biological imaging. To date, methods for uncertainty visualization have mainly focused on per-pixel estimates. Yet, a heatmap of per-pixel variances is typically of little practical use, as it does not capture the strong correlations between pixels. A more natural measure of uncertainty corresponds to the variances along the principal components (PCs) of the posterior distribution. Theoretically, the PCs can be computed by applying PCA on samples generated from a conditional generative model for the input image. However, this requires generating a very large number of samples at test time, which is painfully slow with the current state-of-the-art (diffusion) models. In this work, we present a method for predicting the PCs of the posterior distribution for any input image, in a single forward pass of a neural network. Our method can either wrap around a pre-trained model that was trained to minimize the mean square error (MSE), or can be trained from scratch to output both a predicted image and the posterior PCs. We showcase our method on multiple inverse problems in imaging, including denoising, inpainting, super-resolution, and biological image-to-image translation. Our method reliably conveys instance-adaptive uncertainty directions, achieving uncertainty quantification comparable with posterior samplers while being orders of magnitude faster. Code and examples are available on our [webpage](https://eliasnehme.github.io/NPPC/).

The ever-growing amounts of visual contents captured on a daily basis necessitate the use of lossy compression methods in order to save storage space and transmission bandwidth. While extensive research efforts are devoted to improving compression techniques, every method inevitably discards information. Especially at low bit rates, this information often corresponds to semantically meaningful visual cues, so that decompression involves significant ambiguity. In spite of this fact, existing decompression algorithms typically produce only a single output, and do not allow the viewer to explore the set of images that map to the given compressed code. In this work we propose the first image decompression method to facilitate user-exploration of the diverse set of natural images that could have given rise to the compressed input code, thus granting users the ability to determine what could and what could not have been there in the original scene. Specifically, we develop a novel deep-network based decoder architecture for the ubiquitous JPEG standard, which allows traversing the set of decompressed images that are consistent with the compressed JPEG file. To allow for simple user interaction, we develop a graphical user interface comprising several intuitive exploration tools, including an automatic tool for examining specific solutions of interest. We exemplify our framework on graphical, medical and forensic use cases, demonstrating its wide range of potential applications.

We introduce a new generator architecture, aimed at fast and efficient high-resolution image-to-image translation. We design the generator to be an extremely lightweight function of the full-resolution image. In fact, we use pixel-wise networks; that is, each pixel is processed independently of others, through a composition of simple affine transformations and nonlinearities. We take three important steps to equip such a seemingly simple function with adequate expressivity. First, the parameters of the pixel-wise networks are spatially varying, so they can represent a broader function class than simple 1 × 1 convolutions. Second, these parameters are predicted by a fast convolutional network that processes an aggressively low-resolution representation of the input. Third, we augment the input image by concatenating a sinusoidal encoding of spatial coordinates, which provides an effective inductive bias for generating realistic novel high-frequency image content. As a result, our model is up to 18× faster than state-of-the-art baselines. We achieve this speedup while generating comparable visual quality across different image resolutions and translation domains.

The lower the distortion of an estimator, the more the distribution of its outputs generally deviates from the distribution of the signals it attempts to estimate. This phenomenon, known as the perception-distortion tradeoff, has captured significant attention in image restoration, where it implies that fidelity to ground truth images comes on the expense of perceptual quality (deviation from statistics of natural images). However, despite the increasing popularity of performing comparisons on the perception-distortion plane, there remains an important open question: what is the minimal distortion that can be achieved under a given perception constraint? In this paper, we derive a closed form expression for this distortion-perception (DP) function for the mean squared-error (MSE) distortion and Wasserstein-2 perception index. We prove that the DP function is always quadratic, regardless of the underlying distribution. This stems from the fact that estimators on the DP curve form a geodesic in Wasserstein space. In the Gaussian setting, we further provide a closed form expression for such estimators. For general distributions, we show how these estimators can be constructed from the estimators at the two extremes of the tradeoff: The global MSE minimizer, and a minimizer of the MSE under a perfect perceptual quality constraint. The latter can be obtained as a stochastic transformation of the former.

Models for audio generation are typically trained on hours of recordings. Here, we illustrate that capturing the essence of an audio source is typically possible from as little as a few tens of seconds from a single training signal. Specifically, we present a GAN-based generative model that can be trained on one short audio signal from any domain (e.g. speech, music, etc.) and does not require pre-training or any other form of external supervision. Once trained, our model can generate random samples of arbitrary duration that maintain semantic similarity to the training waveform, yet exhibit new compositions of its audio primitives. This enables a long line of interesting applications, including generating new jazz improvisations or new a-cappella rap variants based on a single short example, producing coherent modifications to famous songs (e.g. adding a new verse to a Beatles song based solely on the original recording), filling-in of missing parts (inpainting), extending the bandwidth of a speech signal (super-resolution), and enhancing old recordings without access to any clean training example. We show that in all cases, no more than 20 seconds of training audio commonly suffice for our model to achieve state-of-the-art results. This is despite its complete lack of prior knowledge about the nature of audio signals in general.

Features extracted from deep layers of classification networks are widely used as image descriptors. Here, we exploit an unexplored property of these features: their internal dissimilarity. While small image patches are known to have similar statistics across image scales, it turns out that the internal distribution of deep features varies distinctively between scales. We show how this deep self dissimilarity (DSD) property can be used as a powerful visual fingerprint. Particularly, we illustrate that full-reference and no-reference image quality measures derived from DSD are highly correlated with human preference. In addition, incorporating DSD as a loss function in training of image restoration networks, leads to results that are at least as photo-realistic as those obtained by GAN based methods, while not requiring adversarial training.

The loss terrains of over-parameterized neural networks have multiple global minima. However, it is well known that stochastic gradient descent (SGD) can stably converge only to minima that are sufficiently flat w.r.t. SGD's step size. In this paper we study the effect that this mechanism has on the function implemented by the trained model. First, we extend the existing knowledge on minima stability to non-differentiable minima, which are common in ReLU nets. We then use our stability results to study a single hidden layer univariate ReLU network. In this setting, we show that SGD is biased towards functions whose second derivative (w.r.t the input) has a bounded weighted $L_1$ norm, and this is regardless of the initialization. In particular, we show that the function implemented by the network upon convergence gets smoother as the learning rate increases. The weight multiplying the second derivative is larger around the center of the support of the training distribution, and smaller towards its boundaries, suggesting that a trained model tends to be smoother at the center of the training distribution.

Fast acquisition of depth information is crucial for accurate 3D tracking of moving objects. Snapshot depth sensing can be achieved by wavefront coding, in which the point-spread function (PSF) is engineered to vary distinctively with scene depth by altering the detection optics. In low-light applications, such as 3D localization microscopy, the prevailing approach is to condense signal photons into a single imaging channel with phase-only wavefront modulation to achieve a high pixel-wise signal to noise ratio. Here we show that this paradigm is generally suboptimal and can be significantly improved upon by employing multi-channel wavefront coding, even in low-light applications. We demonstrate our multi-channel optimization scheme on 3D localization microscopy in densely labelled live cells where detectability is limited by overlap of modulated PSFs. At extreme densities, we show that a split-signal system, with end-to-end learned phase masks, doubles the detection rate and reaches improved precision compared to the current state-of-the-art, single-channel design. We implement our method using a bifurcated optical system, experimentally validating our approach by snapshot volumetric imaging and 3D tracking of fluorescently labelled subcellular elements in dense environments.

Single image super resolution (SR) has seen major performance leaps in recent years. However, existing methods do not allow exploring the infinitely many plausible reconstructions that might have given rise to the observed low-resolution (LR) image. These different explanations to the LR image may dramatically vary in their textures and fine details, and may often encode completely different semantic information. In this paper, we introduce the task of explorable super resolution. We propose a framework comprising a graphical user interface with a neural network backend, allowing editing the SR output so as to explore the abundance of plausible HR explanations to the LR input. At the heart of our method is a novel module that can wrap any existing SR network, analytically guaranteeing that its SR outputs would precisely match the LR input, when down- sampled. Besides its importance in our setting, this module is guaranteed to decrease the reconstruction error of any SR network it wraps, and can be used to cope with blur kernels that are different from the one the network was trained for. We illustrate our approach in a variety of use cases, ranging from medical imaging and forensics, to graphics.

It is well known that (stochastic) gradient descent has an implicit bias towards flat minima. In deep neural network training, this mechanism serves to screen out minima. However, the precise effect that this has on the trained network is not yet fully understood. In this paper, we characterize the flat minima in linear neural networks trained with a quadratic loss. First, we show that linear ResNets with zero initialization necessarily converge to the flattest of all minima. We then prove that these minima correspond to nearly balanced networks whereby the gain from the input to any intermediate representation does not change drastically from one layer to the next. Finally, we show that consecutive layers in flat minima solutions are coupled. That is, one of the left singular vectors of each weight matrix, equals one of the right singular vectors of the next matrix. This forms a distinct path from input to output, that, as we show, is dedicated to the signal that experiences the largest gain end-to-end. Experiments indicate that these properties are characteristic of both linear and nonlinear models trained in practice.

We introduce SinGAN, an unconditional generative model that can be learned from a single natural image. Our model is trained to capture the internal distribution of patches within the image, and is then able to generate high quality, diverse samples that carry the same visual content as the image. SinGAN contains a pyramid of fully convolutional GANs, each responsible for learning the patch distribution at a different scale of the image. This allows generating new samples of arbitrary size and aspect ratio, that have significant variability, yet maintain both the global structure and the fine textures of the training image. In contrast to previous single image GAN schemes, our approach is not limited to texture images, and is not conditional (i.e. it generates samples from noise). User studies confirm that the generated samples are commonly confused to be real images. We illustrate the utility of SinGAN in a wide range of image manipulation tasks.

Lossy compression algorithms are typically designed and analyzed through the lens of Shannon’s rate-distortion theory, where the goal is to achieve the lowest possible distortion (e.g., low MSE or high SSIM) at any given bit rate. However, in recent years, it has become increasingly accepted that "low distortion" is not a synonym for "high perceptual quality", and in fact optimization of one often comes at the expense of the other. In light of this understanding, it is natural to seek for a generalization of rate-distortion theory which takes perceptual quality into account. In this paper, we adopt the mathematical definition of perceptual quality recently proposed by Blau & Michaeli (2018), and use it to study the three-way tradeoff between rate, distortion, and perception. We show that restricting the perceptual quality to be high, generally leads to an elevation of the rate-distortion curve, thus necessitating a sacrifice in either rate or distortion. We prove several fundamental properties of this triple-tradeoff, calculate it in closed form for a Bernoulli source, and illustrate it visually on a toy MNIST example.

Sparse representation over redundant dictionaries constitute a good model for many classes of signals (e.g., patches of natural images, segments of speech signals, and so on). However, despite its popularity, a very little are known about the representation capacity of this model. In this paper, we study how redundant a dictionary must be so as to allow any vector to admit a sparse approximation with a prescribed sparsity and a prescribed level of accuracy. We address this problem, both in a worst-case setting and in an average-case one. For each scenario, we derive lower and upper bounds on the minimal required overcompleteness. Our bounds have simple closed-form expressions that allow to easily deduce the asymptotic behavior in large dimensions. In particular, we find that the required overcompleteness grows exponentially with the sparsity level and polynomially with the allowed representation error. This implies that universal sparse representation is practical only at moderate sparsity levels, but can be achieved with a relatively high accuracy. As a side effect of our analysis, we obtain a tight lower bound on the regularized incomplete beta function, which may be interesting in its own right. We illustrate the validity of our results through numerical simulations, which support our findings.

Image restoration algorithms are typically evaluated by some distortion measure (e.g. PSNR, SSIM, IFC, VIF) or by human opinion scores that quantify perceived perceptual quality. In this paper, we prove mathematically that distortion and perceptual quality are at odds with each other. Specifically, we study the optimal probability for correctly discriminating the outputs of an image restoration algorithm from real images. We show that as the mean distortion decreases, this probability must increase (indicating worse perceptual quality). As opposed to the common belief, this result holds true for any distortion measure, and is not only a problem of the PSNR or SSIM criteria. However, as we show experimentally, for some measures it is less severe (e.g. distance between VGG features). We also show that generative-adversarial-nets (GANs) provide a principled way to approach the perception-distortion bound. This constitutes theoretical support to their observed success in low-level vision tasks. Based on our analysis, we propose a new methodology for evaluating image restoration methods, and use it to perform an extensive comparison between recent super-resolution algorithms.

In recent years, deep neural networks (DNNs) achieved unprecedented performance in many low-level vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of millions of parameters. To make DNNs implementable on platforms with limited resources, it is necessary to weaken the tradeoff between performance and efficiency. In this paper, we propose a new activation unit, which is particularly suitable for image restoration problems. In contrast to the widespread per-pixel activation units, like ReLUs and sigmoids, our unit implements a learnable nonlinear function with spatial connections. This enables the net to capture much more complex features, thus requiring a significantly smaller number of layers in order to reach the same performance. We illustrate the effectiveness of our units through experiments with state-of-the-art nets for denoising, deraining, and super resolution, which are already considered to be very small. With our approach, we are able to further reduce these models by nearly 50% without incurring any degradation in performance.

We present an algorithm for modifying small non-local variations between repeating structures and patterns in multiple images of the same scene. The modification is consistent across views, even-though the images could have been photographed from different view points and under different lighting conditions. We show that when modifying each image independently the correspondence between them breaks and the geometric structure of the scene gets distorted. Our approach modifies the views while maintaining correspondence, hence, we succeed in modifying appearance and structure variations consistently. We demonstrate our methods on a number of challenging examples, photographed in different lighting, scales and view points.

A prominent property of natural images is that groups of similar patches within them tend to lie on low-dimensional subspaces. This property has been previously used for image denoising, with particularly notable success via weighted nuclear norm minimization (WNNM). In this paper, we extend the WNNM method into a general image restoration algorithm, capable of handling arbitrary degradations (e.g. blur, missing pixels, etc.). Our approach is based on a novel regularization term which simultaneously penalizes for high weighted nuclear norm values of all the patch groups in the image. Our regularizer is isolated from the data-term, thus enabling convenient treatment of arbitrary degradations. Furthermore, it exploits the fractal property of natural images, by accounting for patch similarities also across different scales of the image. We propose a variable splitting method for solving the resulting optimization problem. This leads to an algorithm that is quite different from "plug-and-play" techniques, which solve image-restoration problems using a sequence of denoising steps. As we verify through extensive experiments, our algorithm achieves state of the art results in deblurring and inpainting, outperforming even the recent deep net based methods.

In many areas of neuroscience and biological data analysis, it is desired to reveal common patterns among a group of subjects. Such analyses play important roles e.g., in detecting functional brain networks from fMRI scans and in identifying brain regions which show increased activity in response to certain stimuli. Group level techniques usually assume that all subjects in the group behave according to a single statistical model, or that deviations from the common model have simple parametric forms. Therefore, complex subject-specific deviations from the common model severely impair the performance of such methods. In this paper, we propose nonparametric algorithms for estimating the common covariance matrix and the common density function of several variables in a heterogeneous group of subjects. Our estimates converge to the true model as the number of subjects tends to infinity, under very mild conditions. We illustrate the effectiveness of our methods through extensive simulations as well as on real-data from fMRI scans and from arterial blood pressure and photoplethysmogram measurements.

Canonical correlation analysis (CCA) is a classical representation learning technique for finding correlated variables in multi-view data. Several nonlinear extensions of the original linear CCA have been proposed, including kernel and deep neural network methods. These approaches seek maximally correlated projections among families of functions, which the user specifies (by choosing a kernel or neural network structure), and are computationally demanding. Interestingly, the theory of nonlinear CCA, without functional restrictions, had been studied in the population setting by Lancaster already in the 1950s, but these results have not inspired practical algorithms. We revisit Lancaster’s theory to devise a practical algorithm for nonparametric CCA (NCCA). Specifically, we show that the solution can be expressed in terms of the singular value decomposition of a certain operator associated with the joint density of the views. Thus, by estimating the population density from data, NCCA reduces to solving an eigenvalue system, superficially like kernel CCA but, importantly, without requiring the inversion of any kernel matrix. We also derive a partially linear CCA (PLCCA) variant in which one of the views undergoes a linear projection while the other is nonparametric. Using a kernel density estimate based on a small number of nearest neighbors, our NCCA and PLCCA algorithms are memory-efficient, often run much faster, and perform better than kernel CCA and comparable to deep CCA.

We present an algorithm for automatically detecting and visualizing small non-local variations between repeating structures in a single image. Our method allows to automatically correct these variations, thus producing an 'idealized' version of the image in which the resemblance between recurring structures is stronger. Alternatively, it can be used to magnify these variations, thus producing an exaggerated image which highlights the various variations that are difficult to spot in the input image. We formulate the estimation of deviations from perfect recurrence as a general optimization problem, and demonstrate it in the particular cases of geometric deformations and color variations.

Super resolution (SR) algorithms typically assume that the blur kernel is known (either the Point Spread Function 'PSF' of the camera, or some default low-pass filter, e.g. a Gaussian). However, the performance of SR methods significantly deteriorates when the assumed blur kernel deviates from the true one. We propose a general framework for "blind" super resolution. In particular, we show that: (i) Unlike the common belief, the PSF of the camera is the wrong blur kernel to use in SR algorithms. (ii) We show how the correct SR blur kernel can be recovered directly from the low-resolution image. This is done by exploiting the inherent recurrence property of small natural image patches (either internally within the same image, or externally in a collection of other natural images). In particular, we show that recurrence of small patches across scales of the low-res image (which forms the basis for single-image SR), can also be used for estimating the optimal blur kernel. This leads to significant improvement in SR results.

We address the problem of recovering signals from samples taken at their rate of innovation. Our only assumption is that the sampling system is such that the parameters defining the signal can be stably determined from the samples. As such, our analysis subsumes previously studied nonlinear acquisition devices and nonlinear signal classes. Our strategy relies on minimizing a least-squares (LS) objective, which is generally non-convex and might possess many local minima. We show, though, that under the stability hypothesis, any optimization method designed to trap a stationary point necessarily converges to the true solution. We demonstrate the usefulness of our approach in recovering finite-duration and periodic pulse streams.

We address the problems of multi-domain and single-domain regression based on distinct labeled training sets for each of the domains and a large unlabeled training set from all domains. We formulate these problems as ones of Bayesian estimation with partial knowledge of statistical relations. We propose a worst-case design strategy and study the resulting estimators. Our analysis explicitly accounts for the cardinality of the labeled sets and includes the special cases in which one of the labeled sets is very large or, in the other extreme, completely missing. We demonstrate our estimators in the context of audio-visual word recognition and provide comparisons to several recently proposed multi-modal learning algorithms.

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels for linear reconstruction, based on a generalization of the Karhunen-Loève transform. The results are illustrated for several types of time-delay estimation problems.

Causal processing of a signal's samples is crucial in on-line applications such as audio rate conversion, compression, tracking and more. This paper addresses the problem of causally reconstructing continuous-time signals from their samples. We treat a rich variety of sampling mechanisms encountered in practice, namely in which each sampling function is obtained by applying a unitary operator on its predecessor. Examples include pointwise sampling at the output of an anti-aliasing filter and magnetic resonance imaging, which correspond respectively to the translation and modulation operators. Such sequences of functions were studied extensively in the context of stationary random processes. We thus utilize powerful tools from this discipline, to derive a causal interpolation method that best approximates the commonly used non-causal reconstruction formula.

In this paper, we address the problem of enhancement of a noisy GARCH process using a particle filter. We compare our approach experimentally to a previously developed recursive estimation scheme. Simulations indicate that a significant gain in performance is obtained, at the cost of higher sensitivity to errors in the GARCH parameters. The proposed method allows tackling arbitrary driving noise distributions as well as arbitrary fidelity criteria.

We address the problem of Bayesian estimation where the statistical relation between the signal and measurements is only partially known. We propose modeling partial Baysian knowledge by using an auxiliary random vector called instrument. The joint probability distributions of the instrument and the signal, and of the instrument and the measurements, are known. However, the joint probability function of the signal and measurements is unknown. Our model generalizes that underlying the method of instrumental variables from statistics, in that the instrument does not have to satisfy any requirements and no parametric form for the optimal regressor needs to be available. We begin by deriving an estimator for this scenario, via a worst-case design strategy. We then propose a non-parametric method for learning this estimator from a set of examples. Finally, we demonstrate our approach in the context of enhancement of facial images that have undergone an unknown degradation.

We revisit the problem of recovering a continuous-time signal lying within a known shift-invariant subspace from nonlinear and nonideal samples. Recently, an iterative algorithm for perfect recovery of such signals was proposed. This method requires operations which are not linear time-invariant (LTI), rendering it impractical due to its polynomial dependency on the data length. We describe an alternative iterative algorithm for recovering the signal, which involves only LTI operations. In the revised method, each iteration is much faster and implementation is simpler. Furthermore, the overall running time of our approach depends linearly on the number of samples.

We address the problem of minimum mean-squared error (MMSE) estimation where the estimator is constrained to belong to a pre-defined set of functions. We derive a simple closed form formula that reveals the structure of the restricted estimator for a wide class of constraints. Using this formula we study various types of constrained estimation problems that arise commonly in the fields of signal processing and communication.

We address the problem of minimum mean-squared error (MMSE) estimation under convex constraints. The familiar orthogonality principle, developed for linear constraints, is generalized to include convex restrictions. Using the extended principle, we study two types of convex constraints: constraints on the estimated vector (e.g. bounded norm), and constraints on the structure of the estimator (e.g. filter with bounded coefficients). It is shown that in both cases there exists a simple closed form expression for the constrained MMSE estimator. As an application of our approach, we develop Wiener type filters under certain restrictions, which allow for practical implementations.