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.

翻译：由多边形(相连接的脊椎)定义的面积或体积不变数量的Fourier变形的计算,往往有助于模拟波散射,或对基于实际空间矢量的量和面积的预测进行四层空间过滤。如果系统分解成一个常规阵列,快速Fourier技术可以加速由此产生的计算,但如果需要高度的空间分辨率,则最初的分解步骤可以限制性能;在另一些情况下,离散方法导致结果变异的人工制品不可接受。另一种办法是计算多边形区域的全四层整体变形,以之作为悬浮的总和,这在文献中曾使用偏差标本来得出,以减少多角表面元素周围的三维至线分解的问题,并转换成该方形的直段的一整数。我们演示了这一算法的软件实施情况,并表明它能够提供比基于块的离散化速度更快的实际形状四层变形四层的准确近似值。