# IW-NeRF: Using Implicit Watermarks to Protect the Copyright of Neural Radiation Fields
## 1 Introduction  
Digital watermarking is a copyright protection technology that embeds copyright identifiers into digital media using algorithms [1; 2; 3]. In the event of copyright disputes, copyright owners can extract copyright information from digital media using the inverse operation of the embedding algorithm to confirm ownership. The concept of neural radiance fields (NeRFs), introduced by [PERSON] et al. [4], utilizes multilayer perceptrons (MLPs) to implicitly represent 3D scenes and render 2D images by synthesizing new perspectives. Due to the NeRF's robust representational capabilities, high generalization, and ease of learning, it is poised to become a mainstream technique in digital media representation. However, training NeRF models is challenging, and the theft of such models can lead to significant losses for their owners. Therefore, safeguarding NeRF model copyrights is of paramount importance. Traditional watermarking algorithms [5; 6; 7] typically rely on specific mathematical functions to embed watermarks into media by altering them. However, these methods often struggle to strike a balance between imperceptibility, robustness, and watermark capacity. The integration of deep learning in watermarking has revolutionized the field, eliminating the need for manually designing complex mathematical functions and show-casing superior performance. In deep learning watermarking, copyright holders encode watermark information into carrier images using an encoder and extract the watermark information from noisy images through a decoder [8], simulating a black-box scenario. While existing deep learning-based watermarking algorithms excel in robustness, imperceptibility, and embedding capacity, they are primarily tailored for multimedia data, likeimages [9], audio [10], and videos [11], with limited focus on watermarking implicit data such as NeRF. [PERSON] et al. [12] pioneered embedding watermarks in deep neural network models, proposing a model watermarking scheme that embeds watermarks in model parameters using a regularization technique. Model watermarking has since evolved rapidly, proving effective in safeguarding network model copyrights. Leveraging the idea of model watermarking, researchers are exploring solutions to protect NeRF copyrights. For instance, [PERSON] et al. [13] introduced the StegaNeRF approach for information hiding, while [PERSON] et al. [14] developed the CopyRNeRF scheme, aiming to safeguard NeRF model copyrights while preserving rendering quality and bit accuracy. However, these techniques may suffer from limited watermark capacity when using bit strings or images as watermark information, necessitating comprehensive representation of copyright holders' logos in embedded watermark data. To overcome the low watermark capacity and security vulnerabilities in existing NeRF watermarking techniques, we propose an implicit representation-based watermarking algorithm. This method encodes watermark information as continuous functions using implicit representation neurons, embedding these data into carrier networks via a key. By representing watermark information as neural network models instead of bit strings or images, we significantly enhance the watermark capacity. Our approach involves representing watermark information as continuous functions, embedding these parameters into an untrained network using a key, training the carrier network to encapsulate carrier information while sharing the trained watermark-containing carrier model across the network, and enabling copyright owners to extract watermark information using the key if model theft is suspected. In summary, this work contributes the following:  
1. The first application of implicit representations in NeRF copyright protection to address existing NeRF watermarking challenges. 2. A key-based carrier network construction method for lossless watermark information extraction. 3. Validation of our method across various datasets, ensuring model quality and successful watermark extraction. 4.