2-D Fourier变换的定义

定义 2-D Fourier变换
$$F(\xi ,\eta )=\mathcal{F}[f(x,y)]={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy$$

2-D Fourier逆变换
$$f(x,y)={\mathcal{F}}^{-1}[F(\xi ,\eta )]={\int }_{-\infty }^{+\infty }F(\xi ,\eta )e^{j2\pi (\xi x+\eta y)}d\xi d\eta$$

这个定义要推广到$N$维也是很容易的，
$$\begin{gathered} F(\vec{\xi })=\mathcal{F}[f(\vec{x})]={\int }_{-\infty }^{+\infty }f(\vec{x})e^{-j2\pi \vec{\xi }\cdot \vec{x}}d\vec{x} \\ f(\vec{x})={\mathcal{F}}^{-1}[F(\vec{\xi })]={\int }_{-\infty }^{+\infty }F(\vec{\xi })e^{j2\pi \vec{\xi }\cdot \vec{x}}d\vec{\xi } \end{gathered}$$

例1 可分函数(separable function)：
$$f(x,y)=g(x)h(y)$$

它的Fourier变换为
$$\begin{array}{rl}F(\xi ,\eta )& ={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy\\ & ={\int }_{-\infty }^{+\infty }g(x)h(y)e^{-j2\pi (\xi x+\eta y)}dxdy\\ & =\left({\int }_{-\infty }^{+\infty }g(x)e^{-j2\pi \xi x}dx\right)\left({\int }_{-\infty }^{+\infty }h(y)e^{-j2\pi \eta y}dy\right)\\ & =\mathcal{F}[g(x)]\mathcal{F}[h(y)]\end{array}$$

对于一般函数，
$$\begin{array}{rl}F(\xi ,\eta )& ={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy\\ & ={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy\\ & ={\int }_{-\infty }^{+\infty }\left({\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi \xi x}dx\right)e^{-j2\pi \eta y}dy\\ & ={\mathcal{F}}_y[{\mathcal{F}}_x[f(x,y)]]\end{array}$$

也就是说多元函数的Fourier变换是可以序贯地对每个变量做Fourier变换的。

2-D Fourier变换的性质

性质	原函数	Fourier变换
Scaling	$f(ax,by)$	$\frac{1}{ab}F(\xi /a,\eta /b),ab>0;-\frac{1}{ab}F(\xi /a,\eta /b),ab<0$
Shifting	$f(x-x_0,y-y_0)$	$e^{-j2\pi (\xi x_0+\eta y_0)}F(\xi ,\eta )$
卷积定理	$f(x,y)\ast h(x,y)$	$F(\xi ,\eta )H(\xi ,\eta )$
卷积定理(频域)	$F(\xi ,\eta )\ast H(\xi ,\eta )$	$f(x,y)h(x,y)$

其中2-D卷积的定义是
$$f(x,y)\ast h(x,y)={\int }_{-\infty }^{+\infty }f(\alpha ,\beta )h(x-\alpha ,y-\beta )d\alpha d\beta$$

极坐标系中的2-D Fourier变换

从定义出发，用二重积分的换元公式得到极坐标系中的2-D Fourier变换，
$$F(\xi ,\eta )=\mathcal{F}[f(x,y)]={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy$$

将$(x,y)$变换到极坐标$(r,\theta )$中，将$(\xi ,\eta )$变换到极坐标$(\rho ,\varphi )$中，即
$$\{\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array},\{\begin{array}{l}\xi =\rho \cos \varphi \\ \eta =\rho \sin \varphi \end{array}$$

所以
$$x\xi +y\eta =r\rho (\cos \theta \cos \varphi +\sin \theta \sin \varphi )=r\rho \cos (\theta -\varphi )$$

Jacobian为
$$\left|\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right|=\left|\begin{array}{cc}\cos \theta & r\sin \theta \\ \sin \theta & -r\cos \theta \end{array}\right|=-r$$

因此，对$f(x,y)=f(x(r,\theta ),y(r,\theta ))$根据积分换元公式
$$\begin{array}{rl}F(\rho ,\varphi )& ={\int }_{-\infty }^{+\infty }f(x,y)e^{-j2\pi (\xi x+\eta y)}dxdy\\ & ={\int }_0^{2\pi }{\int }_0^{+\infty }rf(r,\theta )e^{-j2\pi r\rho \cos (\theta -\varphi )}drd\theta \end{array}$$

这就是极坐标中的函数$f(r,\theta )$的Fourier变换公式。

Hankel变换

对于任意极坐标系中的函数$f(r,\theta )$，它可以展开为
$$f(r,\theta )=\sum _{k=-\infty }^{+\infty }{f}_k(r)e^{jk\theta }$$

它的Fourier变换为
$$\begin{array}{rl}F(\rho ,\varphi )& ={\int }_0^{2\pi }{\int }_0^{+\infty }rf(r,\theta )e^{-j2\pi r\rho \cos (\theta -\varphi )}drd\theta \\ & =2\pi \sum _{k=-\infty }^{+\infty }(-j{)}^k{e}^{jk\varphi }{\mathcal{H}}_k[f_k(r)]\end{array}$$

假设$f(r,\theta )$是极坐标系中的可分函数，即
$$f(r,\theta )=f_R(r)f_{\Theta }(\theta )$$

则它的展开式中$f_k=f_R$，Fourier变换为
$$\begin{array}{rl}F(\rho ,\varphi )& =\sum _{k=-\infty }^{+\infty }{c}_k(-j{)}^k{e}^{jk\varphi }{\mathcal{H}}_k[f_R(r)]\end{array}$$

这里$c_k$表示Fourier级数展开的系数，
$$c_k=\frac{1}{2\pi }{\int }_0^{2\pi }{f}_{\Theta }(\theta )e^{-jk\theta }d\theta$$

${\mathcal{H}}_k$表示$k$阶Hankel变换，
$${\mathcal{H}}_k[f_R(r)]=2\pi {\int }_0^{+\infty }r{f}_R(r)J_k(2\pi r\rho )dr$$

其中$J_k$表示$k$阶第一类Bessel函数。

例 Radially Symmetric Function
假设极坐标系中的可分函数满足$f_{\Theta }=1$，就称其为Radially Symmetric Function，它的Fourier变换为
$$\begin{array}{rl}F(\rho ,\varphi )& ={\int }_0^{2\pi }{\int }_0^{+\infty }rf(r,\theta )e^{-j2\pi r\rho \cos (\theta -\varphi )}drd\theta \\ & ={\int }_0^{2\pi }{\int }_0^{+\infty }r{f}_R(r)e^{-j2\pi r\rho \cos (\theta -\varphi )}drd\theta \end{array}$$

因为
$$J_0(2\pi r\rho )=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-j2\pi r\rho \cos (\theta -\varphi )}d\theta$$

用Fubini定理，
$$F(\rho ,\varphi )=F(\rho )=2\pi {\int }_0^{+\infty }r{f}_R(r)J_0(2\pi r\rho )dr$$

这个变换为0阶Hankel变换，也被称为Fourier-Bessel变换，它的逆变换为
$$f_R(r)=2\pi {\int }_0^{+\infty }\rho F(\rho )J_0(2\pi r\rho )d\rho$$