Discrete tomography deals with problems of determining shape of a discrete object from a set of projections. In this paper, we deal with a fundamental problem in discreet tomography: reconstructing a discrete object in \(\mathbb {R}^3\) from its orthogonal projections, which we call three-dimensional discrete tomography. This problem has been mostly studied under the assumption that complete data of the projections are available. However, in practice, there might be missing data in the projections, which come from, e.g., the lack of precision in the measurements. In this paper, we consider the three-dimensional discrete tomography with missing data. Specifically, we consider the following three fundamental problems in discrete tomography: the consistency, counting, and uniqueness problems, and classify the computational complexities of these problems in terms of the length of one dimension. We also generalize these results to higher-dimensional discrete tomography, which has applications in operations research and statistics.

离散层析成像技术涉及从一组投影确定离散对象的形状的问题。在本文中，我们处理了离散层析成像中的一个基本问题：根据正交投影重建\（\ mathbb {R} ^ 3 \）中的离散物体，我们称其为三维离散层析成像。这个问题主要是在可以得到完整的投影数据的假设下进行的。但是，实际上，可能会缺少数据投影中的误差，例如由于测量精度不足。在本文中，我们考虑了缺少数据的三维离散层析成像。具体来说，我们考虑离散层析成像中的以下三个基本问题：一致性，计数和唯一性问题，并根据一维长度对这些问题的计算复杂性进行分类。我们还将这些结果推广到高维离散层析成像中，该技术已在运筹学和统计学中得到应用。