如何将点映射到扭曲的网格上 [英] How to map a point onto a warped grid

问题描述

假设您有一个坐标在笛卡尔坐标系上的点的集合.

Say you have a collection of points with coordinates on a Cartesian coordinate system.

您想绘制另一个点，并且您知道它在同一个笛卡尔坐标系中的坐标.

You want to plot another point, and you know its coordinates in the same Cartesian coordinate system.

但是，您绘制的情节与原始情节有所不同.想象一下，将原始平面打印在橡胶纸上，并以不对称的方式(没有重叠或任何复杂的方式)在某些地方拉伸它并在其他地方捏合它.

However, the plot you're drawing on is distorted from the original. Imagine taking the original plane, printing it on a rubber sheet, and stretching it in some places and pinching it in others, in an asymmetrical way (no overlapping or anything complex).

(来源)

您知道每个点集的拉伸和未拉伸坐标，但不知道底层的拉伸函数.你知道一个新点的未拉伸坐标.

You know the stretched and unstretched coordinates of each of your set of points, but not the underlying stretch function. You know the unstretched coordinates of a new point.

如何根据附近点的拉伸位置估计在拉伸坐标中绘制新点的位置?它不需要精确，因为除非您有更多信息，否则您无法从一组重新映射的点中确定实际的拉伸函数.

How can you estimate where to plot the new point in the stretched coordinates based on the stretched positions of nearby points? It doesn't need to be exact, since you can't determine the actual stretch function from a set of remapped points unless you have more information.

其他可能的关键字:warped 扭曲网格 网格平面坐标 unwarp

other possible keywords: warped distorted grid mesh plane coordinate unwarp

推荐答案

好的，这听起来像是图像变形.这是你应该做的:

Ok, so this sounds like image warping. This is what you should do:

1. 创建一个未扭曲网格的 Delaunay 三角剖分，并利用您对扭曲和非扭曲网格之间的对应关系，以创建扭曲网格的三角剖分.现在您知道了每个图像中对应的三角形，并且由于没有重叠，您应该可以毫不费力地执行下一步.

1. Create a Delaunay triangulation of your unwarped grid and use your knowledge of the correspondences between the warped and unwarped grid to create the triangulation for the warped grid. Now you know the corresponding triangles in each image and since there is no overlapping, you should be able to perform the next step without much difficulty.

现在，在变形图像中找到对应的点A:

Now, to find the corresponding point A, in the warped image:

1. 找到三角形A所在的位置，利用未扭曲网格中的三角形与扭曲网格之间的变换来找出新的位置.

1. Find the triangle A lies in and use the transformation between the triangle in the unwarped grid and the warped grid to figure out the new position.

这有详细解释 这里.

另一种(更复杂的)方法是 薄板样条(也解释了在上面的幻灯片中).

Another (much more complicated) method is the Thin Plate Spline (which is also explained in the slides above).

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