We assume a 2D function f(\bm{r}) can be decompose into angular Fourier series: \begin{equation} f(\bm{r}) = \sum_m a_m(r) e^{\mathrm{i} m \theta}, \end{equation} where (r, \theta) is 2D polar coordinates, and the integer m is the spin number, and a_m is radial dependency function. To have a better understanding of this decomposition, we look into a specific radius at r_0, and f(\bm{r}) at the radius is a 1D function of \theta. The expansion is an angular Fourier series of the signal at the radius: \begin{equation} f(r_0, \theta) = \sum_m a_m(r_0) e^{\mathrm{i} m \theta}, \end{equation} and a_m(r_0) can be calculated as \begin{equation} a_m(r_0) = \frac{1}{2\pi}\int f(r_0, \theta) e^{-\mathrm{i} m \theta} \mathrm{d}\theta\,. \end{equation}
In practice, one can divide the 2D signal into different radial bins labeled with i and the mean radius of the is denoted as r_i. Subsequently, a_m(r_i) can be estimated for each radial bin.
In the last section, we show how to decompose a 2D image into different spin component and determine the radial dependency function for each spin component. If we have an ensemble of galaxy images, and each image has a radial dependency function for each spin number. Here we focus on m=0 components. The galaxies in the ensemble is labeled with j, and we have a number of a_0^{(j)}(r) from this ensemble. We can perform principal component decomposition to \{a_0^{(j)}\} to find a minimal set of 1D orthogonal function to represent the images in the ensemble.