# IW-NeRF: Using Implicit Watermarks to Protect the Copyright of Neural Radiation Fields
## 4 Proposed Method
### Watermark Information Embedding Stage  
This article is influenced by advancements in neural network watermarking, focusing on embedding watermark representations within carrier networks. The methodology leverages network unfolding techniques for watermark embedding, ensuring the preservation of watermark network parameters within a newly constructed neural network structure. By adopting implicit representations, especially through straightforward network architectures like MLP, embedding watermark parameters using a designated key is achieved without compromising performance. The new network design is centered on the watermark network, with inspiration drawn from models like the NeRF. Three key network deployment strategies are introduced, depicted in Figure 5, to optimize the embedding process efficiently while maintaining the integrity of the watermark information within the carrier network, as follows:  
(1) Horizontal expansion. This strategy entails the insertion of new layers subsequent to the watermark network, preserving the original network structure, to construct a larger network, as illustrated in Figure 5a. (2) Vertical expansion. An alternative approach to expanding the network is to maintain the layer count unchanged. Given that both the watermark and extended networks embody the NeRF model, aside from the input and output layers, we augment solely the count of neurons within the hidden layers, as depicted in Figure 5b. (3) Mixed expansion. Mixed expansion encompasses both horizontal and vertical expansion, concurrently increasing the layer count and the neuron count within existing hidden layers. This constitutes a versatile operation, as demonstrated in Figure 5c, where the entirety of the watermark network's content and parameters are encapsulated within a single network. Indeed, both horizontal and vertical expansion can be encapsulated under the umbrella of mixed expansion, and our objective is to construct  
Figure 4: The overall framework of our algorithm IW-NeRF.  
the carrier network's structure using expansion methodologies that are as simplistic as is feasible.  
Employing watermark networks and carrier networks to encapsulate watermark information and NeRF models, we naturally leverage mixed dilation to construct carrier networks. Within implicitly represented 3D scenes, upon the training of the watermark network, we can assemble a carrier network through the watermark network. Due to the necessity of embedding the parameters of the trained watermark network within the carrier network, we designate a shared key K, comprising a randomized binary sequence of zeros and ones, as presented in Figure 2. Within our scheme, the key is defined as \(\text{K}=\{k_{0},k_{1},k_{2},\dots,k_{m}\}\), wherein \(k_{0}\) represents the count of layers of the watermark network neurons within the carrier network, with a binary value of 1 indicating that the layer includes neurons of the watermark network and 0 indicating otherwise. The sequence \(\{k_{1},k_{2},\dots,k_{m}\}\) denotes the location of the watermark network neurons within the carrier network, where each layer corresponds to a binary stream \(k_{m}\) of length \(d_{k_{m}}\), specifying the neuron count in that layer. Each bit within this sequence represents a neuron, with 1 indicating it belongs to the watermark information and 0 indicating it belongs to the carrier information. For example, in Figure 5c, the corresponding key K is \(\{01110,00110,101101,01101\}\). This method significantly reduces the length of the key through \(k_{0}\).