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DOCUMENTATION CENTER SEARCH
User's Guide
Image Transforms
8.2 Discrete Fourier Transform
The discrete Fourier transform (DFT) is a frequency domain representation of finite-extent sequences. The DFT is a decomposition of a finite-extent sequence by a family of complex exponential sequences. The DFT of a discrete-time sequence of length N is itself a sequence of length
N
.
The DFT plays a central role in the implementation of many signal and image processing algorithms. Particularly important has been the discovery of a fast algorithm, the fast Fourier transform (FFT) [
Coo65
], that reduces the number of complex multiplications needed to compute an
N
-point DFT by a factor of
.
Transform
Short
DiscreteFourierTransform
[
img
]
DFT[
img
]
discrete Fourier transform of
img
InverseDiscreteFourierTransform
[
coef
]
IDFT[
coef
]
inverse discrete Fourier transform of
coef
FourierMatrix
[
n
]
returns an
n
×
n
matrix of
n-
Fourier basis sequences
Image transforms.
The 1D discrete Fourier transform (1D DFT) is a unique decomposition of a discrete-time, finite-length signal onto a finite family (basis) of discrete, periodic, complex exponential sequences of the form
, where k,n,N
(integers). The ratio
is the fundamental frequency. The symmetric DFT formulation is given by the following pair of equations:
X[k] is known as the
N
-point DFT of the sequence x[n]. Equations (8.2.1) and (8.2.2) are the analysis and synthesis equations, respectively.
This loads the
ImageProcessing
package.
In[1]:=
We conclude this section with a couple of examples of computing the DFT. We first calculate the DFT of the following sequence:
Here is the sequence x[n] of length 32 and its DFT.
In[2]:=
Out[2]=
The 2D discrete Fourier transform (2D DFT) is a straightforward extension of the 1D formulation. The DFT X[k
_{1}
,k
_{2}
] of a 2D sequence
x[n
_{1}
,n
_{2}
] , with
, is defined as
for 0≤k
_{1}
≤N
_{1}
-1, 0≤k
_{2}
≤N
_{2}
-1 and
for 0≤ n
_{1}
≤N
_{1}
-1, 0≤n
_{2}
≤N
_{2}
-1.
Next we obtain the DFT of a 2D sequence defined by the following formula:
Here is the N×N matrix representation of this sequence (N=8).
In[3]:=
Out[3]//MatrixForm=
Here is the 32-point DFT of the sequence x[n
_{1}
,n
_{2}
].
In[4]:=
Here we plot the magnitude and phase of the 2D example sequence.
In[5]:=
Out[5]=
Finally, we compute the 2D DFT of one of the example images.
In[6]:=
Here is the original image.
In[7]:=
Out[7]=
Here we display the magnitude (left) and phase (right) spectra of the image.
In[8]:=
Out[8]=
In visualizing the frequency response of a sequence, it is common to use a centered DFT domain.
In[9]:=
Out[9]=
The energy compaction property of the DFT is clearly visible. Most of the signal's energy is concentrated at the four corners of the DFT magnitude matrix, which are the low-frequency zones of the transform domain. Note that a large majority of the Fourier coefficients have relatively small values and therefore contribute little to the signal's total energy.
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.