Formulating Event-based Image Reconstruction as a Linear Inverse Problem with Deep Regularization using Optical Flow

TL;DR

This work shows, for the first time, how tackling the combined problem of motion and brightness estimation leads to formulate event-based image reconstruction as a linear inverse problem that can be solved without training an image reconstruction RNN.

Abstract

Event cameras are novel bio-inspired sensors that measure per-pixel brightness differences asynchronously. Recovering brightness from events is appealing since the reconstructed images inherit the high dynamic range (HDR) and high-speed properties of events; hence they can be used in many robotic vision applications and to generate slow-motion HDR videos. However, state-of-the-art methods tackle this problem by training an event-to-image Recurrent Neural Network (RNN), which lacks explainability and is difficult to tune. In this work we show, for the first time, how tackling the combined problem of motion and brightness estimation leads us to formulate event-based image reconstruction as a linear inverse problem that can be solved without training an image reconstruction RNN. Instead, classical and learning-based regularizers are used to solve the problem and remove artifacts from the reconstructed images. The experiments show that the proposed approach generates images with visual quality on par with state-of-the-art methods despite only using data from a short time interval. State-of-the-art results are achieved using an image denoising Convolutional Neural Network (CNN) as the regularization function. The proposed regularized formulation and solvers have a unifying character because they can be applied also to reconstruct brightness from the second derivative. Additionally, the formulation is attractive because it can be naturally combined with super-resolution, motion-segmentation and color demosaicing. Code is available at https://github.com/tub-rip/event_based_image_rec_inverse_problem

Figures (26)

• Figure 1: Overview of the proposed method (bottom) in comparison with state-of-the-art event-to-image recurrent neural networks (RNNs) (top). End-to-end image reconstruction methods can be replaced by an explainable system that recovers both optical flow and image brightness. We show that, given optical flow, image reconstruction is a linear problem in the unknown brightness, and exploit this knowledge to estimate brightness by means of several classical and recent learning-based solvers.
• Figure 1: Quantitative evaluation of our method and the state of the art on sequences from Mueggler17ijrr. We report median values (since they are more robust to outliers than the mean) of MSE, SSIM and LPIPS quality metrics over all reconstructed images. Images are equalized before computing the metrics.
• Figure 2: From events to brightness. Paths followed by image reconstruction methods in terms of the visual quantities estimated. The two predominant categories of methods are either end-to-end or estimate the Laplacian $\nabla^2 L$ and subsequently solve Poisson's equation. We show ($i$) a new path based on the image of warped events (IWE), and ($ii$) how to improve Poisson's path by incorporating image priors. Paths are blue; visual quantities are black.
• Figure 2: Quantitative evaluation like \ref{['tab:imgrec:ijrr:equalized']}, but without histogram equalization.
• Figure 3: The events $\mathcal{E}$ in (a) are motion-compensated using optical flow to create the image of warped events (IWE) (c). The sharp IWE approximates the $x$-derivative of the ground truth frame better than the uncompensated image (b). Our method reconstructs brightness image (d) from the IWE (c) by solving a linear system of equations with regularization (CNN-based in this example). Texture details smaller than the contrast sensitivity $C$ in \ref{['eq:generativeEventCondition']} cannot be recovered since they do not trigger events. The event data in the figure is from the slider_far sequence in Mueggler17ijrr, consisting of rocks and tree-like textures.