Fourier Transform Theorems

$$\newcommand{\FT}[1] {\mathcal{F}\left(#1\right)}
\newcommand{\iFT}[1] {\mathcal{F}^{-1}\left(#1\right)}$$

The Transform

The Fourier Transform (FT) is represented here by the operator \(\FT{.}\).

The transform is

$$\FT{f(x)}=F(u)=\int_{-\infty}^{\infty} f(x)e^{-2 \pi i u x} dx$$

And the inverse is

$$\iFT{F(u)}=f(x)=\int_{-\infty}^{\infty}F(u)e^{2 \pi i u x} dx$$

In two dimensions the transform is

$$\FT{f(x,y)}=F(u,v)=\int \int_{-\infty}^{\infty} f(x,y)e^{-2 \pi i (ux+vy)} dxdy$$

and the inverse is

$$\iFT{F(x,y)}=f(x,y)=\int \int_{-\infty}^{\infty} F(u,v)e^{2 \pi i (ux+vy)} dxdy$$

Fourier Transform Properties

We will show the two dimensional case here, but the theroems apply to any
number of dimensions.

Scaling

If \(a\) and \(b\) are a real and non-zero constant, then

$$\FT{f(ax,by)}=\frac{1}{|ab|}F(\frac{u}{a},\frac{v}{b})$$

Shifting

If \(a\) and \(b\) are real constants, then

$$\FT{f(x-a,y-b)}=F(u,v)e^{-i2 \pi (au+bv)}$$

Convolution

If

$$g(x,y)=f(x,y) \otimes h(x,y)$$

then in Fourier space

$$G(u,v)=F(u,v)*H(u,v)$$

$$G(u,v)=\FT{ \int \int_{-\infty}^{\infty}f(\xi,\eta)h(x-\xi,y-\eta)d \xi d \eta}$$

Parseval’s Theorem

$$\int \int_{\infty}^{\infty}|f(x,y)|^{2}dx dy=\int \int_{-\infty}^{\infty}|F(u,v)|^{2}du dv$$

Linearity Theorem

$$\FT{af(x,y)+bg(x,y)}=a \FT{f(x,y)}+b \FT{g(x,y)}$$

Autocorrelation Theorem

$$\FT{f(x,y) \odot f(x,y)} = \FT{ \int \int_{-\infty}^{\infty}f(\xi,\eta)f^{\ast}(\xi-x,\eta-y) d \xi d \eta}=|F(u,v)|^{2}$$

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