NeRF
Introduction
Neural Radiance Fields (NeRF) is a computer vision technique used to generate high-quality 3D reconstruction models. It uses deep learning technology to extract the geometric shape and texture information of objects from images from multiple perspectives, and then uses this information to generate a continuous 3D radiation field, which can present a highly realistic 3D model at any angle and distance. NeRF technology has broad application prospects in computer graphics, virtual reality, augmented reality and other fields.
input Output
The input of NeRF (Neural Radiation Field) is images and camera parameters from multiple viewpoints, and the output is a continuous 3D radiation field. Specifically, the input is a set of 2D images and corresponding camera parameters (including camera position and orientation), and the output is a function representing the color and density of each point in the 3D scene.
Advantages and disadvantages
The advantage of NeRF technology is that the generated 3D model has high quality and high fidelity, and can present real object surface and texture details at any angle and distance. Furthermore, it can generate 3D models from any number of input images without specific processing or labeling of the inputs. However, NeRF technology also has some disadvantages, such as requiring a large amount of computing resources and time for training, and it is difficult to deal with large-scale scenes and complex lighting conditions. In addition, since NeRF is a technology based on view synthesis, it is necessary to ensure that the input viewing angle is wide enough and sufficient when generating the model, otherwise it may cause occlusion and holes in the model.
principle
NeRF technology builds a multi-layer perceptron (MLP) model to represent the color and density of each point in the scene by combining the input images of multiple viewing angles and camera parameters. In the training phase, NeRF uses image rendering technology to render the generated 3D scene into a 2D image, and compares it with the real image to optimize the model parameters. The key to the NeRF model is how to represent the color and density of each point in the scene. To do this, NeRF uses a technique called volume rendering. Volume rendering divides 3D space into many small voxels and interpolates the color and density within each voxel. By interpolating all voxels, the color and density values of the entire scene can be obtained.
structure
The implementation structure of NeRF (Neural Radiation Field) consists of two main components: encoder and decoder.
The encoder usually consists of a convolutional neural network (CNN), which is responsible for extracting the spatial position and view characteristics of each point in the scene from the input multiple view images and camera parameters. Each convolutional layer in the encoder can map input data from a low-dimensional space to a high-dimensional space and extract more complex feature representations.
The decoder usually consists of a multi-layer perceptron (MLP) responsible for generating a continuous 3D radiation field from the features extracted by the encoder. Specifically, the decoder takes as input the spatial position and view features of each point from the encoder, and outputs the color and density values of the point. Each MLP layer in the decoder can map the input data to another high-dimensional space and extract more complex feature representations.
During the training phase, NeRF takes a set of 2D images and corresponding camera parameters as input, and uses a rendering equation to transform the input data into a 3D scene. Rendering equations are used to calculate the intersection points of rays from the camera position and direction with objects in the scene, and to determine the color and density values for each point. NeRF then uses a neural network to approximate the rendering equation to minimize the difference between the generated scene and the real image.
pytorch implementation
Third-party library for NeRF implementation
PyTorch does not directly integrate NeRF, but third-party libraries can be used for NeRF implementation. One of the more popular libraries is nerf-pytorch, which you can install with the following command:
pip install nerf-pytorch
Once installed, you can implement a simple NeRF model with the following line of code:
import torch
import nerf
model = nerf.models.NeRF()
This line of code will create a NeRF model with default parameters. You can use the model object to perform a forward pass on the input data, for example:
x = torch.randn(10, 3) # 10 samples with 3 features each
y = model(x)
print(y.shape) # output shape: (10, 4)
Here we generate an input data x that contains 10 samples, each with 3 features. We feed the input data into the model and print the output y. It can be seen that the shape of the output y is (10, 4), where the first dimension indicates that there are 10 samples, and the second dimension indicates that there are 4 features. In NeRF, the last dimension is usually used to represent information such as color or transparency.
Self-fulfillment
import torch
from torch import nn
from torch.nn import functional as F
class NeRF(nn.Module):
def __init__(self, input_dims, output_dims, hidden_dims=256, num_layers=8):
super().__init__()
self.input_dims = input_dims
self.output_dims = output_dims
# MLP layers
layers = []
for i in range(num_layers):
layers.append(nn.Linear(input_dims, hidden_dims))
layers.append(nn.ReLU(inplace=True))
input_dims = hidden_dims
layers.append(nn.Linear(hidden_dims, output_dims))
self.mlp = nn.Sequential(*layers)
def forward(self, x):
x = self.mlp(x)
return x
A PyTorch model class called NeRF is defined here, which accepts parameters such as input dimension input_dims, output dimension output_dims, hidden layer dimension hidden_dims and layer number num_layers. In the initialization function, we define an MLP model with multiple linear layers and a ReLU activation function. In the forward function, we use the MLP model to process the input data x and return the output.
To use this model, we can first create an instance object, and then input the input data into the model, for example:
# create model instance
model = NeRF(input_dims=3, output_dims=3)
# generate input data
x = torch.randn(10, 3)
# forward pass
y = model(x)
print(y.shape) # (10, 3)
In this example, we create an input data x with 10 samples, each with 3 features. We feed the input data into the NeRF model and print the output y. It can be seen that the shape of the output y is (10, 3), which means that there are 10 samples, and each sample has 3 features.
Explanation of related terms
radiation field
Radiance Field describes the propagation behavior of light. In three-dimensional space, for any ray (that is, the starting point and direction), each point in the scene can calculate the radiance of the ray at that point. For each point, the radiation field can be represented by a color value and a radiosity value. The color value refers to the surface color of the point, while the radiosity value refers to how light or dark the point is under light. High-quality rendered images can be produced by calculating the radiance of light rays throughout a 3D scene.
In NeRF (Neural Radiation Field), the concept of radiation field is extended to calculate the color and density of each point in the scene in the direction of the ray for any ray in three-dimensional space. Therefore, the radiation field of NeRF can be used to represent the surface color and density information of objects in a 3D scene. Using this information, highly realistic 3D models can be presented at any angle and distance.
Multilayer Perceptron (MLP)
Multilayer Perceptron (MLP) is a feedforward neural network (Feedforward Neural Network) structure. It consists of multiple layers of fully connected neurons, where neurons in each layer are connected to all neurons in the previous layer. The main idea is to map input data into high-dimensional space through nonlinear mapping, and combine these mapping results into output through multi-layer nonlinear transformation.
It can be said that the activation function is a nonlinear mapping, and the nonlinear mapping can introduce new features. When we map the data from the original low-dimensional space to a higher-dimensional space, we can consider that the features of the original data have been expanded and more features have been introduced. This makes the input data more separated in the new space, and this separation allows us to better model and classify the data, while also improving the expressive power and performance of the model.
In machine learning and deep learning, we typically view data in a high-dimensional space as a set of features that describe different aspects of the input data. By mapping raw data into a high-dimensional space, and modeling and analyzing the data in this space, we can better understand the meaning of the data and discover the relationship between the data, thereby improving our model performance.
density of dots
In computer graphics, point density usually refers to the number of points contained in a given area. For example, if we have a scene and need to render it, we can divide the scene into small regions and calculate the density of points within each region. If the density of points in an area is high, it means that there are more geometric details in the area, and a higher sampling rate is required to capture the details; on the contrary, if the density of points in an area is low, it means that in the area The geometry within is relatively simple, requiring a low sampling rate to capture geometric information.
In NeRF, the density of points is also a very important concept. We want to be able to uniformly sample points in the scene when rendering an image to capture more comprehensive geometric information. Therefore, when training the NeRF model, we need to sample the scene and make the density of the sampling points as uniform as possible. Usually, we randomly generate some points in the scene, and calculate the visibility of each point according to the camera position and orientation, and use the visible points for training the model. In this process, we need to balance the number and density of sampling points to obtain the best results.
volume rendering
Volume rendering (Volume Rendering) is a technology that converts 3D data (such as medical images, CT scans, geological survey data, etc.) into visual images. In 3D data, each pixel not only contains color information, but also various physical quantity information, such as density, temperature, velocity, etc. Volume rendering technology can visualize these physical quantity information, so that people can better understand and analyze 3D data.
The core idea of volume rendering is to sample each pixel in 3D data, and calculate its color and transparency based on the physical quantity information of each sampling point. Specifically, we can represent 3D data as a volume texture, and then use a ray tracing algorithm to sample points in the volume texture, calculate the color and transparency of each sample point, and synthesize them into the final image.
In NeRF, volume rendering techniques are used to convert 3D scenes into visualized images. We can represent a 3D scene as a dense point cloud and use the NeRF model to predict the color and transparency of each point. Then, when rendering the image, we can use volume rendering techniques to sample the point cloud, calculate the color and transparency of each sampled point, and composite it into the final image. In this way, the 3D scene can be visualized as a 2D image, allowing people to better understand and analyze the geometric and optical information in the scene.
volumetric texture
Volume texture (volume texture) refers to a technology that stores 3D image data in 3D texture (3D texture), which can realize fast rendering and manipulation of volume data in 3D space.
Typically, volumetric textures are composed of discrete volumetric pixels, each of which represents a small cube in 3D space. Each voxel stores the physical quantity information in the cube, such as density, color, texture, etc. By interpolating voxel data, continuous volumetric data can be generated in three-dimensional space to achieve rendering effects.
Volumetric textures are often used in medical image processing, computer-aided design, simulation and other fields. In nerf, the volume texture is used to store the density information and color information in the scene, so as to realize the rendering of the scene.
rendering equation
During the training phase of NeRF, the model needs to learn the color and transparency of each point in each scene. In order to achieve this goal, NeRF introduces the rendering equation (Rendering Equation), which is used to describe the physical laws of the interaction between all rays and objects on the path of light from the camera to a point in the scene.
The rendering equation is an important concept in computer graphics. It describes the path of light from the camera through various materials in the scene, and finally reaches the pixel, and calculates the color value of the pixel. The rendering equation is usually expressed as an integral equation that includes the physics of the various light propagation and interactions in the scene.
The core of the rendering equation is the reflection model under the integral symbol, which describes the distribution of light reflected from a point, and is usually called BRDF (Bidirectional Reflectance Distribution Function). BRDF describes the relationship between the direction of incoming light and the direction of outgoing light at a point, and depends on the material and surface state of the object.
The calculation process of the rendering equation is divided into two stages: the first stage is to solve the intersection point of the camera ray and the object in the scene; the second stage is to calculate the integral of the rendering equation. In the first stage, a ray tracing algorithm is used to calculate the intersection point of the camera ray with the objects in the scene and determine the propagation path of the ray in the scene. In the second stage, it is necessary to calculate the color and brightness of the light at the intersection point, and add it to the color and brightness of all rays emitted from other directions to obtain the final rendering result.
For the NeRF model, in the training phase, the model needs to be optimized according to the rendering equation so that it can generate images that are as similar as possible to the real scene. In the testing phase, it is necessary to use the rendering equation to render the scenes shot by the camera from different angles to obtain high-quality 3D rendered images.
During the training phase, our goal is to optimize the weights in the neural network by minimizing the difference between real and rendered images. Therefore, the rendering equation is mainly used in the training phase to generate the rendered image, calculate the error between the rendered image and the real image, and optimize the neural network.