Uncertainty Visualization via Low-Dimensional Posterior Projections

In ill-posed inverse problems, it is commonly desirable to obtain insight into the full spectrum of plausible solutions, 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 employing 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.

Uncertainty Quantification via Neural Posterior Principal Components

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. However, 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. Examples are available at https://eliasnehme.github.io/NPPC/

Contrastive Divergence Learning is a Time Reversal Adversarial Game

Contrastive divergence (CD) learning is a classical method for fitting unnormalized statistical models to data samples. Despite its wide-spread use, the convergence properties of this algorithm are still not well understood. The main source of difficulty is an unjustified approximation which has been used to derive the gradient of the loss. In this paper, we present an alternative derivation of CD that does not require any approximation and sheds new light on the objective that is actually being optimized by the algorithm. Specifically, we show that CD is an adversarial learning procedure, where a discriminator attempts to classify whether a Markov chain generated from the model has been time-reversed. Thus, although predating generative adversarial networks (GANs) by more than a decade, CD is, in fact, closely related to these techniques. Our derivation settles well with previous observations, which have concluded that CD's update steps cannot be expressed as the gradients of any fixed objective function. In addition, as a byproduct, our derivation reveals a simple correction that can be used as an alternative to Metropolis-Hastings rejection, which is required when the underlying Markov chain is inexact (e.g. when using Langevin dynamics with a large step).

Bayesian Time-of-Flight for Realtime Shape, Illumination and Albedo

We propose a computational model for shape, illumination and albedo inference in a pulsed time-of-flight (TOF) camera. In contrast to TOF cameras based on phase modulation, our camera enables general exposure profiles. This results in added flexibility and requires novel computational approaches. To address this challenge we propose a generative probabilistic model that accurately relates latent imaging conditions to observed camera responses. While principled, realtime inference in the model turns out to be infeasible, and we propose to employ efficient non-parametric regression trees to approximate the model outputs. As a result we are able to provide, for each pixel, at video frame rate, estimates and uncertainty for depth, effective albedo, and ambient light intensity. These results we present are state-of-the-art in depth imaging. The flexibility of our approach allows us to easily enrich our generative model. We demonstrate that by extending the original single-path model to a two-path model, capable of describing some multipath effects. The new model is seamlessly integrated in the system at no additional computational cost. Our work also addresses the important question of optimal exposure design in pulsed TOF systems. Finally, for benchmark purposes and to obtain realistic empirical priors of multipath and insights into this phenomena, we propose a physically accurate simulation of multipath phenomena.