Instant Neural Graphics Primitives with a Multiresolution Hash Encoding (2201.05989v2)

Abstract: Neural graphics primitives, parameterized by fully connected neural networks, can be costly to train and evaluate. We reduce this cost with a versatile new input encoding that permits the use of a smaller network without sacrificing quality, thus significantly reducing the number of floating point and memory access operations: a small neural network is augmented by a multiresolution hash table of trainable feature vectors whose values are optimized through stochastic gradient descent. The multiresolution structure allows the network to disambiguate hash collisions, making for a simple architecture that is trivial to parallelize on modern GPUs. We leverage this parallelism by implementing the whole system using fully-fused CUDA kernels with a focus on minimizing wasted bandwidth and compute operations. We achieve a combined speedup of several orders of magnitude, enabling training of high-quality neural graphics primitives in a matter of seconds, and rendering in tens of milliseconds at a resolution of ${1920!\times!1080}$.

• The paper presents a new parameterization using rotational impulses and a rotational Adam optimizer for stable, efficient camera matrix optimization.
• It leverages standard vector operations like dot and cross products to overcome numerical instability in traditional log-space rotation methods.
• The approach significantly enhances inverse rendering applications, improving 3D scene reconstruction, virtual reality, and computer vision performance.

Optimizing Camera Matrices through Rigid-Body Physics

The paper "Optimizing Camera Matrices through Rigid-Body Physics" by Thomas Müller from NVIDIA addresses crucial issues in the domain of inverse rendering, specifically related to optimizing camera parameters for scene reconstruction.

Introduction and Problem Description

The demand for accurate scene reconstruction has necessitated advancements not only in improving reconstruction algorithms given a set of camera parameters but also in determining optimal camera parameters for a specified scene. Traditional approaches to this problem often encounter complications due to malformed rotational components when performing automatic differentiation on camera matrices. Conventional methods have employed matrix-logarithm-space parameterizations and screw transforms to tackle these issues, but these methods come with their own complexities and computational inefficiencies.

The author proposes a novel parameterization approach utilizing rotational impulses in conjunction with a rotational Adam optimizer. This method retains the computational equivalence to log-space rotation generators while ensuring efficient, stable computations using standard vector operations such as dot and cross products.

Methodology

Camera matrices are formally defined as $C \in \mathbb{R}^{3 \times 4}$ with a rotation matrix $R \in \mathbb{R}^{3 \times 3}$ and a translation vector $t \in \mathbb{R}^3$. The goal is to translate the gradients derived from differentiable rendering algorithms (e.g., NeRF) into meaningful updates to the camera matrix $C$. While updating the camera position $t$ can be trivially handled by gradient-based optimizers like Adam, updating the rotation matrix $R$ presents significant challenges.

Naïve Gradient Descent:

Using naive gradient descent to update $R$ by computing $\partial L / \partial R$ leads to a non-valid rotation matrix. This is because the gradient descent might utilize all 9 degrees of freedom of the linear transform rather than the intended 3 for rotation.

Log-space Rotations:

Prior work has leveraged log-space rotation matrices to enforce rotational constraints. However, these approaches rely on auto-differentiation, suffering from numerical instability due to the intricacies of matrix logarithms and exponentials.

Impulse Vectors:

The paper's critical contribution is recognizing that the 3 values in a log-space rotation vector can be interpreted as the axis of rotation, with the vector's norm representing the rotation angle. This interpretation allows for the computationally efficient determination of rotational impulses via a single cross product. Averaging these impulses over a batch corresponds to averaging the impacts of individual gradient pushes on the camera, akin to treating the camera as a rigid body.

Rotational Adam:

The author suggests directly applying the Adam optimizer to these impulse vectors, noting its physical relevance to angular momentum. Care must be taken in representing $\partial R$ to ensure the computations remain valid and efficient.

Implications and Future Directions

This work offers significant practical implications for the field of computer graphics and inverse rendering. By adopting rotational impulses and a rotational variant of the Adam optimizer, the proposed method allows for more stable and efficient optimization of camera matrices. This can lead to enhanced performance in applications such as 3D scene reconstruction, virtual reality, and computer vision, where camera calibration and parameter optimization are critical.

Theoretically, the proposed method provides a robust framework for handling rotational parameterizations, opening doors for further research in optimization algorithms that might leverage similar principles. Future advancements may explore extending this method to accommodate more complex camera models or integrating it with other state-of-the-art rendering techniques.

Conclusion

The proposed approach of optimizing camera matrices through rigid-body physics, specifically using rotational impulses and a rotational Adam optimizer, presents a meaningful advancement in inverse rendering. It combines computational efficiency with theoretical robustness, demonstrating potential for wide-ranging applications in the field of computer graphics. Further exploration and experimentation could yield even more insights and improvements in camera parameter optimization.