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):

The inverse transform represents the spatial function as a linear combination of complex exponentials with complex weights .

• The Complex weight can be represented in polar form as

in terms of its amplitude and phase :

• The Complex exponential can be represented as

where and
• is the unit vector along the direction of in the 2D spatial frequency domain;
• is the vector along the direction of in the 2D spatial domain.
The inner product represents the projection of a spatial point onto the direction of . As the value of the complex exponential is the same for all points having the same projection , the complex exponential represents a planar sinusoid in the x-y plane along the direction (i.e. ) with frequency .

In the function on top, (2 cycles per unit distance in x) and and (3 cycles per unit distance in y), while in the function at bottom, (3 cycles per unit distance in x) and (2 cycles per unit distance in y). But along their individual directions ( and respectively), their spatial frequencies are the same .

Now the 2DFT of a signal can be written as:

which represents a signal as a linear combination (integration) of infinite 2D spatial planar sinusoids of
• ,
• ,

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

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