Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates

For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution-two dimensional, circular, and radial one dimensional-is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied. (C) 2009 Optical Society of America

对于用极坐标描述最佳的函数，即使该函数不具有圆对称性，二维傅里叶变换也可以用极坐标表示为汉克尔变换和傅里叶级数的组合。然而，为了像其笛卡尔坐标形式一样有用，对于平移、乘法、卷积等标准操作，需要一个极坐标形式的傅里叶运算工具集。本文推导了标准傅里叶运算所需的极坐标形式。特别是，对二维卷积、圆形卷积和一维径向卷积进行了详细讨论。结果表明，只要应用了正确的卷积定义，标准的乘法/卷积规则就适用。（C）2009美国光学学会