Summed Volume Tables (SVTs) allow one to compute integrals over the data values in any cubical area of a three-dimensional orthogonal grid in constant time, and they are especially interesting for building spatial search structures for sparse volumes. However, SVTs become extremely memory consuming due to the large values they need to store; for a dataset of n values an SVT requires 𝒪(n log n) bits. The 3D Fenwick tree allows recovering the integral values in 𝒪(log³ n) time, at a memory consumption of 𝒪(n) bits. We propose an algorithm that generates SVT representations that can flexibly trade speed for memory: From similar characteristics as SVTs, over equal memory consumption as 3D Fenwick trees at significantly lower computational complexity, to even further reduced memory consumption at the cost of raising computational complexity. For a 641 × 9601 × 9601 binary dataset, the algorithm can generate an SVT representation that requires 27.0GB and 46 · 8 data fetch operations to retrieve an integral value, compared to 27.5GB and 1521·8 fetches by 3D Fenwick trees, a decrease in fetches of 97%. A full SVT requires 247.6GB and 8 fetches per integral value. We present a novel hierarchical approach to compute and store intermediate prefix sums of SVTs, so that any prescribed memory consumption between 𝒪(n) bits and 𝒪(n log n) bits is achieved. We evaluate the performance of the proposed algorithm in a number of examples considering large volume data, and we perform comparisons to existing alternatives.

中文翻译：

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总体积表的参数化拆分

总体积表 (SVT) 允许人们在恒定时间内计算三维正交网格的任何立方区域中的数据值的积分，它们对于构建稀疏体积的空间搜索结构特别有趣。但是，由于需要存储大量值，SVT 变得非常消耗内存；对于包含 n 个值的数据集，SVT 需要 𝒪(n log n) 位。3D Fenwick 树允许恢复 𝒪( log ³n) 时间，内存消耗为 𝒪(n) 位。我们提出了一种生成 SVT 表示的算法，可以灵活地换取内存的速度：从与 SVT 相似的特性，在显着降低计算复杂度的情况下超过与 3D Fenwick 树相等的内存消耗，到以提高计算复杂度为代价进一步减少内存消耗。对于641 × 9601 × 9601二进制数据集，该算法可以生成需要 27.0GB 和46·8 次数据获取操作来检索整数值的 SVT 表示，而 27.5GB 和1521·8通过 3D Fenwick 树获取，获取减少了 97%。完整的 SVT 需要 247.6GB 和每个整数值 8 次提取。我们提出了一种新颖的分层方法来计算和存储 SVT 的中间前缀和，从而实现𝒪(n) 位和 𝒪(n log n) 位之间的任何规定的内存消耗。我们在考虑大量数据的多个示例中评估了所提出算法的性能，并与现有替代方案进行了比较。