Camera relocalization is a crucial problem in computer vision and robotics. Recent advancements in neural radiance fields (NeRFs) have shown promise in synthesizing photo-realistic images. Several works have utilized NeRFs for refining camera poses, but they do not account for lighting changes that can affect scene appearance and shadow regions, causing a degraded pose optimization process. In this paper, we propose a two-staged pipeline that normalizes images with varying lighting and shadow conditions to improve camera relocalization. We implement our scene representation upon a hash-encoded NeRF which significantly boosts up the pose optimization process. To account for the noisy image gradient computing problem in grid-based NeRFs, we further propose a re-devised truncated dynamic low-pass filter (TDLF) and a numerical gradient averaging technique to smoothen the process. Experimental results on several datasets with varying lighting conditions demonstrate that our method achieves state-of-the-art results in camera relocalization under varying lighting conditions. Code and data will be made publicly available.

When sampling for Bayesian inference, one popular approach is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS) which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density, and can prove slow in practice. We propose Hamiltonian neural networks (HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, HNNs do not require numerical gradients of the target density during sampling. Moreover, they satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose an HNN extension called latent HNNs (L-HNNs), which are capable of predicting latent variable outputs. Compared to HNNs, L-HNNs offer improved expressivity and reduced integration errors. Finally, we employ L-HNNs in NUTS with an online error monitoring scheme to prevent sample degeneracy in regions of low probability density. We demonstrate L-HNNs in NUTS with online error monitoring on several examples involving complex, heavy-tailed, and high-local-curvature probability densities. Overall, L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. Compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1--2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient by an order of magnitude.

In the previous sections we've discussed the static parts of a Neural Networks: how we can set up the network connectivity, the data, and the loss function. This section is devoted to the dynamics, or in other words, the process of learning the parameters and finding good hyperparameters. In theory, performing a gradient check is as simple as comparing the analytic gradient to the numerical gradient. In practice, the process is much more involved and error prone. This requires you to evaluate the loss function twice to check every single dimension of the gradient (so it is about 2 times as expensive), but the gradient approximation turns out to be much more precise. To see this, you can use Taylor expansion of \(f(x h)\) and \(f(x-h)\) and verify that the first formula has an error on order of \(O(h)\), while the second formula only has error terms on order of \(O(h 2)\) (i.e. it is a second order approximation). What are the details of comparing the numerical gradient \(f'_n\) and analytic gradient \(f'_a\)? That is, how do we know if the two are not compatible? You might be temped to keep track of the difference \(\mid f'_a - f'_n \mid \) or its square and define the gradient check as failed if that difference is above a threshold.

In the previous post, Coding Neural Network - Forward Propagation and Backpropagation, we implemented both forward propagation and backpropagation in numpy. However, implementing backpropagation from scratch is usually more prune to bugs/errors. Therefore, it's necessary before running the neural network on training data to check if our implementation of backpropagation is correct. Before we start, let's revisit what back-propagation is: We loop over the nodes in reverse topological order starting at the final node to compute the derivative of the cost with respect to each edge's node tail. In other words, we compute the derivative of cost function with respect to all parameters, i.e $\frac{\partial J}{\partial \theta}$ where $\theta$ represents the parameters of the model.

I've worked on Deep Learning for a few years as part of my research and among several of my related pet projects is ConvNetJS - a Javascript library for training Neural Networks. Javascript allows one to nicely visualize what's going on and to play around with the various hyperparameter settings, but I still regularly hear from people who ask for a more thorough treatment of the topic. This article (which I plan to slowly expand out to lengths of a few book chapters) is my humble attempt. It's on web instead of PDF because all books should be, and eventually it will hopefully include animations/demos etc. My personal experience with Neural Networks is that everything became much clearer when I started ignoring full-page, dense derivations of backpropagation equations and just started writing code. Thus, this tutorial will contain very little math (I don't believe it is necessary and it can sometimes even obfuscate simple concepts). Since my background is in Computer Science and Physics, I will instead develop the topic from what I refer to as hackers's perspective. Basically, I will strive to present the algorithms in a way that I wish I had come across when I was starting out. "…everything became much clearer when I started writing code." You might be eager to jump right in and learn about Neural Networks, backpropagation, how they can be applied to datasets in practice, etc. But before we get there, I'd like us to first forget about all that. Let's take a step back and understand what is really going on at the core. Update note: I suspended my work on this guide a while ago and redirected a lot of my energy to teaching CS231n (Convolutional Neural Networks) class at Stanford. The notes are on cs231.github.io These materials are highly related to material here, but more comprehensive and sometimes more polished. In my opinion, the best way to think of Neural Networks is as real-valued circuits, where real values (instead of boolean values {0,1}) "flow" along edges and interact in gates. However, instead of gates such as AND, OR, NOT, etc, we have binary gates such as * (multiply), (add), max or unary gates such as exp, etc.