Implicit neural representation (INR) characterizes the attributes of a signal as a function of corresponding coordinates which emerges as a sharp weapon for solving inverse problems. However, the expressive power of INR is limited by the spectral bias in the network training. In this paper, we find that such a frequency-related problem could be greatly solved by re-arranging the coordinates of the input signal, for which we propose the disorder-invariant implicit neural representation (DINER) by augmenting a hash-table to a traditional INR backbone. Given discrete signals sharing the same histogram of attributes and different arrangement orders, the hash-table could project the coordinates into the same distribution for which the mapped signal can be better modeled using the subsequent INR network, leading to significantly alleviated spectral bias. Furthermore, the expressive power of the DINER is determined by the width of the hash-table. Different width corresponds to different geometrical elements in the attribute space, \textit{e.g.}, 1D curve, 2D curved-plane and 3D curved-volume when the width is set as $1$, $2$ and $3$, respectively. More covered areas of the geometrical elements result in stronger expressive power. Experiments not only reveal the generalization of the DINER for different INR backbones (MLP vs. SIREN) and various tasks (image/video representation, phase retrieval, refractive index recovery, and neural radiance field optimization) but also show the superiority over the state-of-the-art algorithms both in quality and speed. \textit{Project page:} \url{https://ezio77.github.io/DINER-website/}

翻译：隐式神经表示通过将信号属性表示为对应坐标的函数，在解决逆问题中展现出强大能力。然而，网络训练中的频谱偏差限制了隐式神经表示的表达能力。本文发现，重新排列输入信号的坐标可显著缓解这一频率相关问题，为此我们提出无序不变隐式神经表示（DINER），通过在传统隐式神经表示主干中增加哈希表实现。对于具有相同属性直方图但排列顺序不同的离散信号，哈希表可将坐标投影至同一分布，使得后续隐式神经表示网络能更好地建模映射后的信号，从而显著缓解频谱偏差。此外，DINER的表达能力由哈希表的宽度决定。不同宽度对应属性空间中不同的几何元素，例如当宽度设置为1、2和3时，分别对应一维曲线、二维曲面和三维弯曲体。几何元素覆盖面积越大，表达能力越强。实验不仅揭示了DINER对不同隐式神经表示主干（MLP与SIREN）及多种任务（图像/视频表示、相位恢复、折射率重建和神经辐射场优化）的泛化能力，还展示了其在质量和速度上均优于现有最优算法。项目页面：\url{https://ezio77.github.io/DINER-website/}