In object detection, the Intersection over Union (\({\mathrm{IoU}}\)) is the most popular criterion used to validate the performance of an object detector on the testing object dataset, or to compare the performances of various object detectors on a common object dataset. The calculation of this criterion requires the determination of the overlapping area between two bounding boxes. If these latter are axis-aligned (or horizontal), then the exact calculation of their overlapping area is simple. But if these bounding boxes are rotated (or oriented), then the exact calculation of their overlapping area is laborious. Many rotated objects detectors have been developed using heuristics to approximate \({\mathrm{IoU}}\) between two rotated bounding boxes. We have shown, through counterexamples, that these heuristics are not efficient in the sense that they can lead to false positive or false negative detection, which can bias the performance of comparative studies between object detectors. In this paper, we develop a method to calculate exact value of \({{\mathrm{IoU}}}\) between two rotated bounding boxes. Moreover, we present an \((\epsilon ,\alpha )\)-estimator \(\widehat{{\mathrm{IoU}}}\) of \({{\mathrm{IoU}}}\) that satisfies \({\mathbf {Pr}} (|\widehat{{\mathrm{IoU}}} -{\mathrm{IoU}}| \le {\mathrm{IoU}}\epsilon )\ge 1-\alpha \). We also generalize the exact computing method and the \((\epsilon ,\alpha )\)-estimator of \({{\mathrm{IoU}}}\), to three-dimensional bounding boxes. Finally, we carry out many numerical experiments in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), in order to test the exact method of calculating the \({{\mathrm{IoU}}}\), and to compare the efficiency of the \((\epsilon ,\alpha )\)-estimator with respect to heuristic estimates of \({{\mathrm{IoU}}}\). Numerical study shows that the \((\epsilon ,\alpha )\)-estimator is distinguished by both precision and simplicity of implementation, while the exact calculation method is distinguished by both precision and speed.

在对象检测中，联合的交集（\({\mathrm{IoU}}\)）是最流行的标准，用于在测试对象数据集上验证对象检测器的性能，或比较各种对象检测器的性能在公共对象数据集上。该准则的计算需要确定两个边界框之间的重叠区域。如果后者是轴对齐（或水平），则它们重叠区域的精确计算很简单。但是如果这些边界框被旋转（或定向），那么它们重叠区域的精确计算是费力的。许多旋转物体检测器已经使用启发式算法来近似\({\mathrm{IoU}}\)在两个旋转的边界框之间。我们已经通过反例表明，这些启发式方法在可能导致误报或漏报检测的意义上是无效的，这可能会影响目标检测器之间比较研究的性能。在本文中，我们开发了一种方法来计算两个旋转边界框之间\({{\mathrm{IoU}}}\) 的精确值。此外，我们提出了一个\((\epsilon ,\alpha )\) -estimator \(\widehat{{\mathrm{IoU}}}\)的\({{\mathrm{IoU}}}\)满足\ ({\mathbf {Pr}} (|\widehat{{\mathrm{IoU}}} -{\mathrm{IoU}}| \le {\mathrm{IoU}}\epsilon )\ge 1-\alpha \) . 我们还概括了精确的计算方法和\((\epsilon ,\alpha )\) - \({{\mathrm{IoU}}}\) 的估计器，到三维边界框。最后，我们在\({\mathbb {R}}^2\)和\({\mathbb {R}}^3\)中进行了许多数值实验，以测试计算\({ {\mathrm{IoU}}}\)，并比较\((\epsilon ,\alpha )\) -estimator 对\({{\mathrm{IoU}}}\) 的启发式估计的效率. 数值研究表明，\((\epsilon ,\alpha )\) -estimator 的特点是精度和实现的简单性，而精确计算方法的特点是精度和速度。