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# Two-Dimensional Fourier Transform

Fourier transform can be generalized to higher dimensions. For example, many signals 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

where and are spatial frequencies in and directions, respectively, and is the 2D spectrum of .

• Aperiodic, discrete signal, continuous, periodic spectrum

where and are the spatial intervals between consecutive signal samples in the and directions, respectively, and and are sampling rates in the two directions, and they are also the periods of the spectrum .

• Periodic, continuous signal, discrete, aperiodic spectrum

where and are periods of the signal in and directions, respectively, and and are the intervals between consecutive samples in the spectrum .

• Periodic, discrete signal, discrete and periodic spectrum

where and are the numbers of samples in and directions in both spatial and spatial frequency domains, respectively, and is the 2D discrete spectrum of . Both and can be considered as elements of two by matrices and , respectively.

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 direction ,
• is a vector along the direction in the 2D spatial domain.
The inner product represents the projection of a spatial point onto the direction of . As all points on a straight line perpendicular to the direction of have the same projection, 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:

Matrix Form of 2D DFT

Consider the 2D DFT:

where, as defined before

We further define

and rewrite the 2D transform as

The above two equations are the two steps for a 2D transform:
1. Column Transform:

First consider the expression for . As the summation is with respect to the row index of , the column index can be treated as a parameter, and the expression is the 1D Fourier transform of the nth column vector of , which can be written in column vector (vertical) form for the nth column:

or more concisely

i.e., the nth column of is the 1D FT of the nth column of . Putting all columns together, we have

or more concisely

where is a by Fourier transform matrix.

2. Row Transform:

Now we reconsider the 2DFT expression above

As the summation is with respective to the column index n of , the row index can be treated as a parameter, and the expression is the 1D Fourier transform of the kth row vector of , which can be written in row vector (horizontal) form for the kth row:

or more concisely

i.e., the kth row of is the 1D FT of the kth row of . Putting all rows together, we have

or more concisely

But as , we finally have

This expression indicates that 2D DFT can be carried out by 1D transforming all the rows of the 2D signal and then 1D transforming all the columns of the resulting matrix. The order of the steps is not important. The transform can also be carried out by the column transform followed by the row transform.

Similarly, the inverse 2D DFT can be written as

It is obvious that the complexity of 2D DFT is (assuming ), which can be reduced to if FFT is used.

A 2D DFT Example

Consider a real 2D signal:

The imaginary part . The 2D Fourier spectrum of this signal can be found by 2D DFT. The real part of the spectrum is:

and the imaginary part of the spectrum is:

Pay close attention to the even and odd symmetry of the spectrum.

Next: Two-dimensional Fourier Filtering Up: Image_Processing Previous: Fast Fourier Transform
Ruye Wang 2007-11-15