Table of Continuous-space (CS) Fourier Transform Pairs and Properties

Continuous-Space (2D) Fourier Transform (CSFT): definition and inverse transform
Forward transform $\mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }$ (info)
Inverse transform $f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv }$
2D Continuous Space Fourier Transform(CSFT) Properties
$\displaystyle f(x,y)$ $\longrightarrow$ $\displaystyle F(u,v)$
Linearity $\displaystyle af_1(x,y)+bf_2(x,y)$ $\displaystyle aF_1(u,v)+bF_2(u,v)$
Scaling $f(\frac{x}{a},\frac{y}{b})$ $\displaystyle|ab|F(au,bv)$
Shifting $\displaystyle f(x-x_o,y-y_o)$ $\displaystyle F(u,v)e^{-j2\pi(ux_o+vy_o)}$
Modulation $\displaystyle f(x,y)e^{j2\pi(xu_o+yv_o)}$ $\displaystyle F(u-u_o,v-v_o)$
Reciprocity $\displaystyle F(x,y)$ $\displaystyle f(-u,-v)$
Other Properties
Parseval’s relation $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|f(x,y)|^2dxdy }=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|F(u,v)|^2dudv }$
Initial Value $F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy }$
Symmetry Properties for Continuous Space Fourier Transform
If f(x,y) is real, the magnitued of F(u,v) is an even function; the angle of F(u,v) is an odd function.
$\displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)}$
$\displaystyle F(u,v)=F^{*}(-u,-v)$
$\displaystyle A(u,v)=A(-u,-v)$
$\displaystyle \theta(u,v)=-\theta(-u,-v)$
$f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv}$
Separability
$\displaystyle g(x)$ $\longrightarrow$ $\displaystyle G(u)$
$\displaystyle h(x)$ $\longrightarrow$ $\displaystyle H(v)$
$\displaystyle f(x,y)$ $\longrightarrow$ $\displaystyle F(u,v)$
$\displaystyle f(x,y)=g(x)h(y)$ $\longrightarrow$ $\displaystyle F(u,v)=G(u)H(v)$
For example,
$\displaystyle rect(x,y)=rect(x)rect(y)$ $\longrightarrow$ $\displaystyle sinc(u)sinc(v)=sinc(u,v)$
• Notes: If we are trying to draw rect(x,y) from a top view, it will just look like a square. In the 3D plot, keep the top view as a base, making the height as 1. The plot is a cube. Similar as sinc(u,v).
 $\displaystyle circ(x,y)$ $\longrightarrow$ $\displaystyle jinc(u,v)$
• Notes: if we are trying to draw circ(x,y) from a top view, it will look like a circle with a radius of ½. In the 3D plot, we keep the top view as a base, making the height as 1. The plot is a cylinder.
Important Continuous Space Fourier Transform Pairs
$\displaystyle x(t)$ $\longrightarrow$ $\mathcal{X}(f)$
$\displaystyle \delta (x,y)$ $\displaystyle 1$
$\displaystyle 1$ $\displaystyle \delta (u,v)$
$\displaystyle rect(x)$ $\displaystyle sinc(u) \delta (v)$
$\displaystyle \delta (x)$ $\displaystyle \delta (v)$
$e^{j2\pi(u_ox+v_oy)}$ $\displaystyle \delta (u-u_o,v-v_o)$
$\displaystyle cos[2\pi(u_ox+v_oy)]$ $\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)]$
2D Continuous Space Fourier Transform in Polar Form
$\displaystyle x=r \cos (\theta)$ $\longrightarrow$ $\displaystyle u= \rho \cos (\phi)$
$\displaystyle y=r \sin (\theta)$ $\longrightarrow$ $\displaystyle v= \rho \sin (\phi)$
 Forward transform $F(\rho,\phi)=\int_0 ^{2\pi}\int_0 ^{\infty}{f(r,\theta)e^{-j2\pi\rho r cos(\phi -\theta)}r dr d \theta }$ Inverse transform $f(r,\theta)=\int_0 ^{2\pi}\int_0 ^{\infty}{F(\rho,\phi)e^{j2\pi\rho r cos(\phi -\theta)}\rho d \rho d \phi }$

some properties related to the polar representations
Rotation $\displaystyle f(r,\theta +\alpha )$ $\displaystyle F(\rho ,\phi + \alpha)$
Circular Symmetry $\displaystyle f(r,\theta)=f_o(r)$ $\displaystyle F(\rho , \phi)=F_o(\rho)$
Circular Symmetry $\displaystyle f(r,\theta)=f_o(r)$ $\displaystyle F(\rho , \phi)=F_o(\rho)$
Convolution Theorem $\displaystyle f_1(x,y) \circledast f_2(x,y)$ $\displaystyle F_1(u,v)F_2(u,v)$
Product Theorem $\displaystyle f_1(x,y)f_2(x,y)$ $\displaystyle F_1(u,v) \circledast F_2(u,v)$

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva