Wednesday, July 20, 2011

Two-Dimensional Fourier Transform notes


Two-Dimensional Fourier Transform

Fourier transform can be generalized to higher dimensions. For example, many signals $f(x,y)$ are functions of 2D space defined over an x-y plane. Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal is periodic and discrete.

  • Aperiodic, continuous signal, continuous, aperiodic spectrum

    \begin{displaymath}F(u,v)=\int \int_{-\infty}^{\infty} f(x,y) e^{-j2\pi(ux+vy)} dx dy \end{displaymath}




    \begin{displaymath}f(x,y)=\int \int_{-\infty}^{\infty} F(u,v) e^{j2\pi(ux+vy)} du dv \end{displaymath}


    where $u$ and $v$ are spatial frequencies in $x$ and $y$ directions, respectively, and $F(u,v)$ is the 2D spectrum of $f(x,y)$.

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