We explore sinusoidal neural networks to represent periodic tileable textures. Our approach leverages the Fourier series by initializing the first layer of a sinusoidal neural network with integer frequencies with a period $P$. We prove that the compositions of sinusoidal layers generate only integer frequencies with period $P$. As a result, our network learns a continuous representation of a periodic pattern, enabling direct evaluation at any spatial coordinate without the need for interpolation. To enforce the resulting pattern to be tileable, we add a regularization term, based on the Poisson equation, to the loss function. Our proposed neural implicit representation is compact and enables efficient reconstruction of high-resolution textures with high visual fidelity and sharpness across multiple levels of detail. We present applications of our approach in the domain of anti-aliased surface.

翻译：我们探索了利用正弦神经网络来表示周期性可拼贴纹理。该方法通过将正弦神经网络的第一层初始化为具有周期$P$的整数频率，从而利用傅里叶级数。我们证明了正弦层的复合运算仅会产生周期为$P$的整数频率。因此，网络能够学习到周期性模式的连续表示，使得无需插值即可直接评估任意空间坐标下的值。为使生成的模式具有可拼贴性，我们在损失函数中引入基于泊松方程的正则化项。所提出的神经隐式表示具有紧凑性，能够高效重建高分辨率纹理，并在多个细节层次上保持高视觉保真度与锐利度。我们展示了该方法在抗锯齿表面领域的应用。