next up previous
Next: Matrix Form of 2D Up: Fourier_Analysis Previous: Two-Dimensional Fourier Transform

Physical Meaning of 2DFT

Consider the Fourier transform of continuous, aperiodic signal (the result is easily generalized to other cases):

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


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

The inverse transform represents the spatial function $f(x,y)$ as a linear combination of complex exponentials $e^{ j2\pi(xu+yv)}$ with complex weights $F(u,v)$.

Now the 2DFT of a signal $f(x,y)$ can be written as:

\begin{displaymath}
f(x,y)=\int\int_{-\infty}^{\infty} \vert F(u,v)\vert e^{j\an...
...t e^{j(\angle{F(u,v)}+2\pi w(\vec{r} \cdot \vec{n}))}
\;du\;dv
\end{displaymath}

which represents a signal $f(x,y)$ as a linear combination (integration) of infinite 2D spatial planar sinusoids $F(u,v)$ of

The 2D function shown below contains three frequency components (2D sinusoidal waves) of different frequencies and directions:

sin_wave.gif


next up previous
Next: Matrix Form of 2D Up: Fourier_Analysis Previous: Two-Dimensional Fourier Transform
Ruye Wang 2003-11-17