三维照片的三次上同调环
Cubical Cohomology Ring of 3D Photographs
摘要：
三维物体的上同调和上同调环是刻画洞及其关系的拓扑不变量。上同调环传统上是在单纯复形上计算的。然而，立体复合体直接处理3D图像中的体素，不需要额外的三角剖分，便于在图像上下文中计算拓扑不变量的有效算法。在本文中，我们提出的公式，直接计算三维立体复形的上同调环，而不使用任何额外的三角剖分。从一个立方体复杂的$Q$，代表一个三维的二进制值的数字图片，其前景有一个连接的组件，我们首先计算上同调信息的边界上的对象，$\partial Q$的增量技术，然后，使用一个面减少算法，我们计算它的整个对象，最后，应用上述公式，上同调环计算从这样的信息。
英文摘要：
Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological invariants in the image context. In this paper, we present formulas to directly compute the cohomology ring of 3D cubical complexes without making use of any additional triangulation. Starting from a cubical complex $Q$ that represents a 3D binary-valued digital picture whose foreground has one connected component, we compute first the cohomological information on the boundary of the object, $\partial Q$ by an incremental technique; then, using a face reduction algorithm, we compute it on the whole object; finally, applying the mentioned formulas, the cohomology ring is computed from such information.
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