README.md

Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains (PyTorch Implementation)

Mingyu Kim

This is PyTorch implementation on 1D_NTK, 2D_Image_regression and 3D_NeRF experiments described in the paper.

Simple 2D coordinate (No tranform)	Random cos and sin features with scale : 10	Random cos and sin features with scale : 100

Abstract

This paper illustrates that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

Code

We provide several pys for experiments in this paper. If you want to approach the original codes, please visit original repository.

       - The directory structure should be orgainzed as follows :

Fourier_feature_torch
├── 1D_ntk_opt
│   ├── 1d_ntk_opt_torch_v2.py
│   ├── ab_opt_dict.pt
│   └── utils_torch_ntk_v2.py
├── 2D_image_regression
│   ├── image_regression.py
│   └── Results
├── 3D_simple_nerf
│   ├── 3D_simple_nerf.py
│   ├── download_lego.sh
├── FFT_practice
│   ├── example_fft.py
│   └── Results
└── README.md

Prerequisite

Our code works on torch>=1.12 and cuda>=11.4. Install the required Python packages via

pip install -r ./requirements.txt

However, we prefer conda environment to pip installation. Please refer this environment via

conda env create -f ./torch112.yml

1D NTK experiments

For the 1D NTK experiment, this code can be executed as follows:

python 1d_ntk_opt_torch_v2.py

After finished this code, this code outputs two figures. supp_opt_torch.png provides function values of NTK kernels varying parameters of fourier feature.

│   ├── Results
│   │   ├── supp_opt_torch.png
│   │   └── torch_temp_opt_fam_p2.0.png

2D image regression experiments

For the 2D image regression, this code can be executed as follows:

cd 2D_image_regression
python image_regression.py

After finished this code, this code outputs both animated image and original image .

│   ├── Results
│       ├── MLP_10000_basic_1
│       │   ├── generated_img_1.mp4
│       │   └── original_img.png
│       ├── MLP_10000_gauss_1
│       │   ├── generated_img_1.mp4
│       │   └── original_img.png
│       ├── MLP_10000_gauss_10
│       │   ├── generated_img_10.mp4
│       │   └── original_img.png
│       ├── MLP_10000_gauss_100
│       │   ├── generated_img_100.mp4
│       │   └── original_img.png
│       └── MLP_10000_no_-1
│           ├── generated_img_-1.mp4
│           └── original_img.png

3D NeRF experiments

For the 3D NeRF, this code can be executed as follows:

cd 3D_simple_nerf
sh download_lego.sh
python image_regression.py

After finished this code, this code outputs bat_plot, validation_ground_truth_image and created_image by NeRF models. In this experiment, we choose one fourier feature among {"no_encoding", "basic", "position_enc", "position_enc_new" and "gaussian features"}

│   └── Result
│       ├── bar_plot.png
│       ├── test_image.png
│       └── test.png

Acknowledgements

This code is migrated from the jax code by Matthew Tancik et al., this repository.