多边形面积和体积的高效直接傅里叶变换的实现
An implementation of an efficient direct Fourier transform of polygonal areas and volumes
摘要：
在由多边形（连接的顶点）定义的区域或体积上的常量的傅里叶变换的计算通常在建模波散射或在实空间基于矢量的体积和区域投影的傅里叶空间滤波中是有用的。如果系统被离散化到一个规则的阵列上，快速傅立叶技术可以加快计算速度，但是如果需要高空间分辨率，离散化的初始步骤可能会限制性能;在其他时候，离散化方法会导致结果变换中不可接受的伪影。另一种方法是将多边形区域的全傅里叶积分变换计算为顶点上的和，其先前已在文献中使用发散定理导出，以将问题从三维简化为多边形表面元素的周界上的线积分，并转换为该轮廓的直段上的和。我们演示了该算法的软件实现，并表明它可以提供准确的近似的真实形状的傅里叶变换的收敛速度比基于块（体素）的离散化。
英文摘要：
Calculations of the Fourier transform of a constant quantity over an area or volume defined by polygons (connected vertices) are often useful in modeling wave scattering, or in fourier-space filtering of real-space vector-based volumes and area projections. If the system is discretized onto a regular array, Fast Fourier techniques can speed up the resulting calculations but if high spatial resolution is required the initial step of discretization can limit performance; at other times the discretized methods result in unacceptable artifacts in the resulting transform. An alternative approach is to calculate the full Fourier integral transform of a polygonal area as a sum over the vertices, which has previously been derived in the literature using the divergence theorem to reduce the problem from a 3-dimensional to line integrals over the perimeter of the polygon surface elements, and converted to a sum over the straight segments of that contour. We demonstrate a software implementation of this algorithm and show that it can provide accurate approximations of the Fourier transform of real shapes with faster convergence than a block-based (voxel) discretization.
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