Neural Radiance Fields (NeRFs) have emerged as a groundbreaking technique in the field of computer vision and graphics, enabling the creation of highly detailed 3D scenes from a set of 2D images. This blog post delves into the principles behind NeRFs, their implementation, and their potential applications.
Understanding NeRFs
NeRF, introduced by Mildenhall et al. in 2020, represents a scene as a continuous 5D function that maps a 3D location and 2D viewing direction to a color and density. This function is approximated using a neural network, typically a multi-layer perceptron (MLP).
Key components of NeRF:
- Input: 3D coordinates (x, y, z) and viewing direction (θ, φ)
- Output: RGB color and density σ
- Rendering: Volume rendering techniques to project the 3D representation onto 2D images
How NeRFs Work
- Scene Representation: The scene is represented as a continuous 5D function.
- Neural Network: An MLP is trained to approximate this function.
- Ray Marching: For each pixel, rays are cast into the scene.
- Sampling: Points are sampled along each ray.
- Density Prediction: The network predicts density and color at each sample point.
- Volume Rendering: The predictions are integrated along the ray to produce the final pixel color.
Implementing a Basic NeRF
Here's a simplified PyTorch implementation of a NeRF model:
import torch
import torch.nn as nn
class NeRF(nn.Module):
def __init__(self, D=8, W=256, input_ch=3, input_ch_views=3, output_ch=4, skips=[4]):
super(NeRF, self).__init__()
self.D = D
self.W = W
self.input_ch = input_ch
self.input_ch_views = input_ch_views
self.skips = skips
self.pts_linears = nn.ModuleList(
[nn.Linear(input_ch, W)] +
[nn.Linear(W, W) if i not in skips else nn.Linear(W + input_ch, W) for i in range(D-1)]
)
self.views_linears = nn.ModuleList([nn.Linear(input_ch_views + W, W//2)])
self.feature_linear = nn.Linear(W, W)
self.alpha_linear = nn.Linear(W, 1)
self.rgb_linear = nn.Linear(W//2, 3)
def forward(self, x):
input_pts, input_views = torch.split(x, [self.input_ch, self.input_ch_views], dim=-1)
h = input_pts
for i, l in enumerate(self.pts_linears):
h = self.pts_linears[i](h)
h = nn.functional.relu(h)
if i in self.skips:
h = torch.cat([input_pts, h], -1)
alpha = self.alpha_linear(h)
feature = self.feature_linear(h)
h = torch.cat([feature, input_views], -1)
for i, l in enumerate(self.views_linears):
h = self.views_linears[i](h)
h = nn.functional.relu(h)
rgb = self.rgb_linear(h)
return torch.cat([rgb, alpha], -1)
# Initialize the model
model = NeRF()
# Example input
batch_size = 1024
input_tensor = torch.rand(batch_size, 6) # 3D coordinates + 3D viewing direction
# Forward pass
output = model(input_tensor)
print(output.shape) # Should be (1024, 4) - RGB + densityTraining a NeRF
Training a NeRF involves the following steps:
- Data Preparation: Collect a set of images of the scene from different viewpoints, along with their camera parameters.
- Ray Generation: Generate rays for each pixel in the training images.
- Sampling: Sample points along each ray.
- Forward Pass: Pass the sampled points through the NeRF model.
- Volume Rendering: Use the predicted colors and densities to render the image.
- Loss Calculation: Compare the rendered image with the ground truth.
- Optimization: Update the model parameters to minimize the loss.
Here's a simplified training loop:
import torch
import torch.optim as optim
def train_nerf(model, train_data, num_epochs, lr=5e-4):
optimizer = optim.Adam(model.parameters(), lr=lr)
mse_loss = nn.MSELoss()
for epoch in range(num_epochs):
for batch in train_data:
rays_o, rays_d, target_rgb = batch
# Sample points along the rays
t_vals = torch.linspace(0., 1., steps=64)
z_vals = near * (1.-t_vals) + far * t_vals
pts = rays_o[...,None,:] + rays_d[...,None,:] * z_vals[...,:,None]
# Flatten the points and add viewing directions
pts_flat = pts.reshape(-1, 3)
dirs_flat = rays_d[:,None].expand_as(pts).reshape(-1, 3)
inputs = torch.cat([pts_flat, dirs_flat], -1)
# Forward pass
raw = model(inputs)
raw = raw.reshape(list(pts.shape[:-1]) + [4])
# Volume rendering (simplified)
rgb_map, depth_map, acc_map = volume_render(raw, z_vals, rays_d)
# Compute loss
loss = mse_loss(rgb_map, target_rgb)
# Optimization step
optimizer.zero_grad()
loss.backward()
optimizer.step()
print(f"Epoch {epoch+1}, Loss: {loss.item()}")
# Assuming we have a train_data loader and a volume_render function
train_nerf(model, train_data, num_epochs=100)Advanced Techniques and Improvements
Several improvements have been proposed to enhance NeRF's performance:
- Positional Encoding: Mapping input coordinates to higher-dimensional space improves the model's ability to capture high-frequency details.
def positional_encoding(x, L=10):
freq = 2.**torch.linspace(0., L-1, L)
spectrum = x[...,None] * freq
sin = torch.sin(spectrum)
cos = torch.cos(spectrum)
return torch.cat([sin, cos], dim=-1)
# Usage in the NeRF model
input_pts_encoded = positional_encoding(input_pts)
input_views_encoded = positional_encoding(input_views, L=4)
x = torch.cat([input_pts_encoded, input_views_encoded], dim=-1)- Hierarchical Sampling: Using a coarse-to-fine sampling strategy to allocate more samples to regions of high expected density.
def hierarchical_sampling(rays_o, rays_d, coarse_model, fine_model):
# Coarse sampling
t_vals_coarse = torch.linspace(0., 1., steps=64)
z_vals_coarse = near * (1.-t_vals_coarse) + far * t_vals_coarse
pts_coarse = rays_o[...,None,:] + rays_d[...,None,:] * z_vals_coarse[...,:,None]
raw_coarse = coarse_model(pts_coarse)
weights_coarse = compute_weights(raw_coarse, z_vals_coarse, rays_d)
# Fine sampling
z_vals_fine = sample_pdf(z_vals_coarse, weights_coarse, 128)
pts_fine = rays_o[...,None,:] + rays_d[...,None,:] * z_vals_fine[...,:,None]
raw_fine = fine_model(pts_fine)
# Combine coarse and fine samples
z_vals_combined = torch.sort(torch.cat([z_vals_coarse, z_vals_fine], -1), -1)[0]
raw_combined = torch.cat([raw_coarse, raw_fine], -2)
return z_vals_combined, raw_combined
# Assuming compute_weights and sample_pdf functions are implementedView-Dependent Effects: Incorporating viewing direction to capture specular reflections and other view-dependent phenomena.
Faster Rendering: Techniques like NeRF in the Wild (NeRF-W) and Instant-NGP for accelerated training and rendering.
Applications of NeRFs
Virtual Reality and Augmented Reality: Creating immersive 3D environments from real-world imagery.
Film and Visual Effects: Generating novel viewpoints for scenes without the need for extensive 3D modeling.
Cultural Heritage Preservation: Digitally preserving historical sites and artifacts in 3D.
E-commerce: Creating 3D product visualizations from a set of product photos.
Robotics and Autonomous Navigation: Improving 3D scene understanding for better navigation and interaction.
Challenges and Future Directions
While NeRFs have shown impressive results, several challenges remain:
Computational Complexity: NeRFs are computationally expensive to train and render, limiting real-time applications.
Dynamic Scenes: Extending NeRFs to handle dynamic or deformable objects is an active area of research.
Generalization: Current NeRFs are typically trained for a single scene. Generalizing to unseen scenes is challenging.
Incomplete or Noisy Data: Improving robustness to handle imperfect input data, such as sparse or noisy images.
Combining with Other 3D Representations: Integrating NeRFs with other 3D representations like meshes or point clouds for hybrid approaches.
Recent Advancements
1. Instant-NGP (Neural Graphics Primitives)
Instant-NGP, introduced by Müller et al., significantly accelerates NeRF training and rendering using multi-resolution hash encoding and fully-fused CUDA kernels.
# Pseudo-code for Instant-NGP's multi-resolution hash encoding
class HashEncoding(nn.Module):
def __init__(self, num_levels, features_per_level, log2_hashmap_size):
super().__init__()
self.num_levels = num_levels
self.features_per_level = features_per_level
self.log2_hashmap_size = log2_hashmap_size
self.hash_tables = nn.ParameterList([
nn.Parameter(torch.rand(2**log2_hashmap_size, features_per_level))
for _ in range(num_levels)
])
def forward(self, x):
encoded = []
for i, table in enumerate(self.hash_tables):
x_scaled = x * (2**i)
x_hashed = hash_function(x_scaled) % (2**self.log2_hashmap_size)
encoded.append(table[x_hashed])
return torch.cat(encoded, dim=-1)
# Usage in the NeRF model
hash_encoding = HashEncoding(num_levels=16, features_per_level=2, log2_hashmap_size=19)
input_encoded = hash_encoding(input_pts)2. NeRF in the Wild (NeRF-W)
NeRF-W extends NeRF to handle unconstrained photo collections, accounting for varying illumination, transient objects, and camera uncertainty.
class NeRFW(nn.Module):
def __init__(self, ...):
super().__init__()
self.static_nerf = NeRF(...)
self.transient_nerf = NeRF(...)
self.appearance_encoding = nn.Embedding(num_images, 256)
def forward(self, x, image_idx):
static_output = self.static_nerf(x)
transient_output = self.transient_nerf(x)
appearance_embedding = self.appearance_encoding(image_idx)
# Combine static and transient outputs with appearance
# ...
# Usage
model = NeRFW()
output = model(input_tensor, image_idx)3. Mip-NeRF 360
Mip-NeRF 360 improves NeRF's ability to handle large-scale scenes with varying scales and unbounded backgrounds.
def integrated_positional_encoding(means, covs, max_freq=10):
# Compute integrated positional encoding
freqs = 2.**torch.linspace(0, max_freq-1, max_freq)
terms = []
for freq in freqs:
for func in [torch.sin, torch.cos]:
term = func(2 * np.pi * freq * means) * torch.exp(-2 * (np.pi * freq)**2 * covs)
terms.append(term)
return torch.cat(terms, dim=-1)
# Usage in Mip-NeRF 360
means, covs = compute_ray_samples(rays_o, rays_d)
encoded_samples = integrated_positional_encoding(means, covs)Conclusion
Neural Radiance Fields have revolutionized the field of 3D scene reconstruction from 2D images, offering a powerful and flexible approach to representing complex 3D environments. As research in this area continues to advance, we can expect to see even more impressive applications and improvements in performance.
The combination of NeRFs with other deep learning techniques and traditional computer graphics methods is likely to yield exciting new capabilities in virtual and augmented reality, computer vision, and beyond. As computational efficiency improves and the ability to handle dynamic scenes advances, NeRFs may become a standard tool in various industries, from entertainment to robotics.
Researchers and practitioners in the field should stay attuned to the rapid developments in NeRF technology, as it continues to push the boundaries of what's possible in 3D scene representation and rendering.
References
- Mildenhall, B., et al. (2020). NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis. ECCV 2020.
- Martin-Brualla, R., et al. (2021). NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections. CVPR 2021.
- Müller, T., et al. (2022). Instant Neural Graphics Primitives with a Multiresolution Hash Encoding. ACM Trans. Graph. 41, 4, Article 102.
- Barron, J. T., et al. (2021). Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields. ICCV 2021.
- Tancik, M., et al. (2022). Block-NeRF: Scalable Large Scene Neural View Synthesis. CVPR 2022.
- Park, K., et al. (2021). Nerfies: Deformable Neural Radiance Fields. ICCV 2021.
- Yu, A., et al. (2021). PlenOctrees for Real-time Rendering of Neural Radiance Fields. ICCV 2021.