Structured adaptive mesh refinement (AMR), commonly implemented via quadtrees and octrees, underpins a wide range of applications including databases, computer graphics, physics simulations, and machine learning. However, octrees enforce isotropic refinement in regions of interest, which can be especially inefficient for problems that are intrinsically anisotropic--much resolution is spent where little information is gained. This paper presents omnitrees as an anisotropic generalization of octrees and related data structures. Omnitrees allow to refine only the locally most important dimensions, providing tree structures that are less deep than bintrees and less wide than octrees. As a result, the convergence of the AMR schemes can be increased by up to a factor of the dimensionality d for very anisotropic problems, quickly offsetting their modest increase in storage overhead. We validate this finding on the problem of binary shape representation across 4,166 three-dimensional objects: Omnitrees increase the mean convergence rate by 1.5x, require less storage to achieve equivalent error bounds, and maximize the information density of the stored function faster than octrees. These advantages are projected to be even stronger for higher-dimensional problems. We provide a first validation by introducing a time-dependent rotation to create four-dimensional representations, and discuss the properties of their 4-d octree and omnitree approximations. Overall, omnitree discretizations can make existing AMR approaches more efficient, and open up new possibilities for high-dimensional applications.

翻译：结构化自适应网格细化（AMR）通常通过四叉树和八叉树实现，广泛应用于数据库、计算机图形学、物理模拟和机器学习等领域。然而，八叉树强制在感兴趣区域进行各向同性细化，这对于本质各向异性问题尤其低效——大量分辨率被消耗在信息增益微小的区域。本文提出全能树作为八叉树及相关数据结构的各向异性推广。全能树允许仅细化局部最重要的维度，产生的树结构深度低于二叉树、宽度小于八叉树。因此，对于强各向异性问题，AMR方案的收敛速度最高可提升维度数d倍，迅速抵消其存储开销的适度增加。我们在4,166个三维物体的二值形状表示问题上验证了这一发现：全能树将平均收敛速度提升1.5倍，以更少存储达到相同误差界，且比八叉树更快实现存储函数的信息密度最大化。对于更高维问题，这些优势预计将更为显著。我们通过引入时间相关旋转构建四维表示进行初步验证，并分析其四维八叉树与全能树近似的特性。总体而言，全能树离散化能提升现有AMR方法的效率，并为高维应用开辟新可能。