We present a simple comparative framework for testing and developing uncertainty modeling in uncertain marching cubes implementations. The selection of a model to represent the probability distribution of uncertain values directly influences the memory use, run time, and accuracy of an uncertainty visualization algorithm. We use an entropy calculation directly on ensemble data to establish an expected result and then compare the entropy from various probability models, including uniform, Gaussian, histogram, and quantile models. Our results verify that models matching the distribution of the ensemble indeed match the entropy. We further show that fewer bins in nonparametric histogram models are more effective whereas large numbers of bins in quantile models approach data accuracy.
Uncertainty is inherent to most data, including vector field data, yet it is often omitted in visualizations and representations. Effective uncertainty visualization can enhance the understanding and interpretability of vector field data. For instance, in the context of severe weather events such as hurricanes and wildfires, effective uncertainty visualization can provide crucial insights about fire spread or hurricane behavior and aid in resource management and risk mitigation. Glyphs are commonly used for representing vector uncertainty but are often limited to 2D. In this work, we present a glyph-based technique for accurately representing 3D vector uncertainty and a comprehensive framework for visualization, exploration, and analysis using our new glyphs. We employ hurricane and wildfire examples to demonstrate the efficacy of our glyph design and visualization tool in conveying vector field uncertainty.
As the world increasingly relies on mathematical models for forecasts in different areas, effective communication of uncertainty in time series predictions is important for informed decision making. This study explores how users estimate probabilistic uncertainty in time series predictions under different variants of line charts depicting uncertainty. It examines the role of individual characteristics and the influence of user-reported metrics on uncertainty estimations. By addressing these aspects, this paper aims to enhance the understanding of uncertainty visualization and for improving communication in time series forecast visualizations and the design of prediction data dashboards.As the world increasingly relies on mathematical models for forecasts in different areas, effective communication of uncertainty in time series predictions is important for informed decision making. This study explores how users estimate probabilistic uncertainty in time series predictions under different variants of line charts depicting uncertainty. It examines the role of individual characteristics and the influence of user-reported metrics on uncertainty estimations. By addressing these aspects, this paper aims to enhance the understanding of uncertainty visualization and for improving communication in time series forecast visualizations and the design of prediction data dashboards.
In this paper, we visualize and quantify the predictive uncertainty of gradient-based post hoc visual explanations for neural networks. Predictive uncertainty refers to the variability in the network predictions under perturbations to the input. Visual post hoc explainability techniques highlight features within an image to justify a network's prediction. We theoretically show that existing evaluation strategies of visual explanatory techniques partially reduce the predictive uncertainty of neural networks. This analysis allows us to construct a plug in approach to visualize and quantify the remaining predictive uncertainty of any gradient-based explanatory technique. We show that every image, network, prediction, and explanatory technique has a unique uncertainty. The proposed uncertainty visualization and quantification yields two key observations. Firstly, oftentimes under incorrect predictions, explanatory techniques are uncertain about the same features that they are attributing the predictions to, thereby reducing the trustworthiness of the explanation. Secondly, objective metrics of an explanation's uncertainty, empirically behave similarly to epistemic uncertainty. We support these observations on two datasets, four explanatory techniques, and six neural network architectures. The code is available at https://github.com/olivesgatech/VOICE-Uncertainty.
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 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 at https://eliasnehme.github.io/NPPC/
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 (\ie 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 webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .
Visualizing the uncertainty of ensemble simulations is challenging due to the large size and multivariate and temporal features of ensemble data sets. One popular approach to studying the uncertainty of ensembles is analyzing the positional uncertainty of the level sets. Probabilistic marching cubes is a technique that performs Monte Carlo sampling of multivariate Gaussian noise distributions for positional uncertainty visualization of level sets. However, the technique suffers from high computational time, making interactive visualization and analysis impossible to achieve. This paper introduces a deep-learning-based approach to learning the level-set uncertainty for two-dimensional ensemble data with a multivariate Gaussian noise assumption. We train the model using the first few time steps from time-varying ensemble data in our workflow. We demonstrate that our trained model accurately infers uncertainty in level sets for new time steps and is up to 170X faster than that of the original probabilistic model with serial computation and 10X faster than that of the original parallel computation.
With the advent of sophisticated machine learning (ML) techniques and the promising results they yield, especially in medical applications, where they have been investigated for different tasks to enhance the decision-making process. Since visualization is such an effective tool for human comprehension, memorization, and judgment, we have presented a first-of-its-kind estimation approach we refer to as Visualized Learning for Machine Learning (VL4ML) that not only can serve to assist physicians and clinicians in making reasoned medical decisions, but it also allows to appreciate the uncertainty visualization, which could raise incertitude in making the appropriate classification or prediction. For the proof of concept, and to demonstrate the generalized nature of this visualized estimation approach, five different case studies are examined for different types of tasks including classification, regression, and longitudinal prediction. A survey analysis with more than 100 individuals is also conducted to assess users' feedback on this visualized estimation method. The experiments and the survey demonstrate the practical merits of the VL4ML that include: (1) appreciating visually clinical/medical estimations; (2) getting closer to the patients' preferences; (3) improving doctor-patient communication, and (4) visualizing the uncertainty introduced through the black box effect of the deployed ML algorithm. All the source codes are shared via a GitHub repository.
Optical coherence tomography (OCT) is commonly used to analyze retinal layers for assessment of ocular diseases. In this paper, we propose a method for retinal layer segmentation and quantification of uncertainty based on Bayesian deep learning. Our method not only performs end-to-end segmentation of retinal layers, but also gives the pixel wise uncertainty measure of the segmentation output. The generated uncertainty map can be used to identify erroneously segmented image regions which is useful in downstream analysis. We have validated our method on a dataset of 1487 images obtained from 15 subjects (OCT volumes) and compared it against the state-of-the-art segmentation algorithms that does not take uncertainty into account. The proposed uncertainty based segmentation method results in comparable or improved performance, and most importantly is more robust against noise.