Fourier Transform
31
Since the DFT of a real signal is symmetric across the middle time point
(see Problem 2.8), it is often more desirable to see only half of the signal as the
remaining half is repetitious. The last two lines are used to label the axes.
2.5 TWO-DIMENSIONAL DISCRETE FOURIER TRANSFORM
Just like in continuous signals, 2-D FT is a rather straightforward extension of the
1-D transform. Mathematically, the 2-D DFT is defined as follows:
N −1 N −1
p ux vy)
2 ( +
G u v = ∑∑ g x y e
, )
− j
N
(2.29)
( , )
(
x=0 y=0
where u and v are frequency axes in which G(u, v) is described. The inverse transformation, i.e., 2-D IDFT is then defined as follows:
1 N −1
+ )
N −
2p (ux vy
g x y
( , ) =
G u v e
N
( , )
(2.30)
∑∑
j
x=0 y=0
Some of the major conceptual properties of 2-D DFT are essentially the extension
of the same conceptual properties in the 1-D DFT. For instance, the high-frequency
components of the 2-D DFT represent fast variations in the gray level of the neighboring points in the signal. Such fast variations (i.e., high frequencies) occur at the
edges (borders) of the objects or in the texture of an object. Edges by definition are
places where the intensity of pixels changes rapidly across a boundary. This rapid
change is what we refer to as high frequency. Similarly, texture is defined as the
rapid and semirandom changes of intensity in a region. This also identifies high
frequencies, i.e., frequencies far from the origin in the 2-D frequency space. Edge
detection and texture analysis play important roles in the analysis of biomedical
images. For example, the intensity of the points (pixels) in a tumor is often different from their surrounding normal tissues. This difference in intensity identifies an
edge between the tumor and its surrounding normal tissues. Tumor detection and
segmentation is often performed based on Fourier and similar types of analysis as
will be discussed later.
As in 1-D DFT, we will apply MATLAB to calculate 2-D DFT. The following
example explores using MATLAB for 2-D DFT images.
Example 2.6
Consider the image g(x, y) shown in Figure 2.12. This is a tomographic image of
pulmonary veins in atrial fibrillation. As shown in the following code, the main
command in MATLAB for calculating the 2-D DFT of images is “fft2”. The magnitude of the 2-D DFT of the image g is shown in Figure 2.13. The command
“image” in MATLAB is used to display an image and is often accompanied by
the command “colormap”, which defines the type and range of gray level or
colors to be used for presenting images. Another command, “imshow”, can also
31
Since the DFT of a real signal is symmetric across the middle time point
(see Problem 2.8), it is often more desirable to see only half of the signal as the
remaining half is repetitious. The last two lines are used to label the axes.
2.5 TWO-DIMENSIONAL DISCRETE FOURIER TRANSFORM
Just like in continuous signals, 2-D FT is a rather straightforward extension of the
1-D transform. Mathematically, the 2-D DFT is defined as follows:
N −1 N −1
p ux vy)
2 ( +
G u v = ∑∑ g x y e
, )
− j
N
(2.29)
( , )
(
x=0 y=0
where u and v are frequency axes in which G(u, v) is described. The inverse transformation, i.e., 2-D IDFT is then defined as follows:
1 N −1
+ )
N −
2p (ux vy
g x y
( , ) =
G u v e
N
( , )
(2.30)
∑∑
j
x=0 y=0
Some of the major conceptual properties of 2-D DFT are essentially the extension
of the same conceptual properties in the 1-D DFT. For instance, the high-frequency
components of the 2-D DFT represent fast variations in the gray level of the neighboring points in the signal. Such fast variations (i.e., high frequencies) occur at the
edges (borders) of the objects or in the texture of an object. Edges by definition are
places where the intensity of pixels changes rapidly across a boundary. This rapid
change is what we refer to as high frequency. Similarly, texture is defined as the
rapid and semirandom changes of intensity in a region. This also identifies high
frequencies, i.e., frequencies far from the origin in the 2-D frequency space. Edge
detection and texture analysis play important roles in the analysis of biomedical
images. For example, the intensity of the points (pixels) in a tumor is often different from their surrounding normal tissues. This difference in intensity identifies an
edge between the tumor and its surrounding normal tissues. Tumor detection and
segmentation is often performed based on Fourier and similar types of analysis as
will be discussed later.
As in 1-D DFT, we will apply MATLAB to calculate 2-D DFT. The following
example explores using MATLAB for 2-D DFT images.
Example 2.6
Consider the image g(x, y) shown in Figure 2.12. This is a tomographic image of
pulmonary veins in atrial fibrillation. As shown in the following code, the main
command in MATLAB for calculating the 2-D DFT of images is “fft2”. The magnitude of the 2-D DFT of the image g is shown in Figure 2.13. The command
“image” in MATLAB is used to display an image and is often accompanied by
the command “colormap”, which defines the type and range of gray level or
colors to be used for presenting images. Another command, “imshow”, can also