Unified approach to exact geometric computation over integer points

我们重点介绍了仅使用整数计算的网格点上的2D旋转。我们研究了给定数字图像被旋转成相同数字图像时的任意旋转角度。然后，我们提出了一种从给定的欧几里德角度和从一对数字图像中获得这些角度的上下界的方法。此外，我们对该方法进行了扩展，使其能够处理3D旋转。即研究了三维半网格与旋转平面的交点，提出了三维铰链角度。然后，我们开发了一种仅通过整数计算就可以对所有3D铰链角度进行排序的方法。这一研究扩展到刚性变换。此外，我们还讨论了在存在离群点的情况下将一条离散直线或多项式曲线拟合到给定的整点的问题。然后提出了一种通过局部搜索有效地获得局部最优解的方法

We focused on 2D rotations on grid points computed by using only integers. We investigated rotation angles with any of which a given digital image is rotated to the same digital image. We then proposed a method for obtaining the upper and lower bounds of such angles from a given Euclidean angle and from a pair of digital images. Furthermore, we extended this method so that it can deal with 3D rotations. Namely, we studied the intersection between the 3D half-grid and the rotation plane and proposed 3D hinge angles. Then, we developed a method to sort all 3D hinge angles with integer computations alone. This investigation is extended to rigid transformations. In addition, we addressed the problem of fitting a discrete line or polynomial curve to given integer points in the presence of outliers. We then proposed a method that effectively achieves a locally optimal solution by using a local search