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Digital Image Processing Basics

Module by: Robert Nowak. E-mail the author

Summary: The module provides an introduction to the concepts of digital imaging processing through basic equations and examples.

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Digital Image Processing

A sampled image gives us our usual 2D array of pixels f⁢mn f m n (Figure 1):

Figure 1: We illustrate a "pixelized" smiley face.
Figure 1 (smiley.png)

We can filter f⁢mn f m n by applying a 2D discrete-space convolution as shown below (where h⁢mn h m n is our PSF):

g⁢mn=h⁢mn*f⁢mn=∑k=−∞∞∑l=−∞∞h⁢m−kn−l⁢f⁢kl g m n h m n f m n k l h m k n l f k l
(1)

Example 1: Sampled Image

Figure 2: Illustrate the "pixelized" nature of all digital images.
Figure 2 (sec2_eg1.png)

We also have discrete-space FTS:

F⁢uv=∑m=−∞∞∑n=−∞∞f⁢mn⁢e−(i⁢u⁢m)⁢e−(i⁢v⁢m) F u v m n f m n u m v m
(2)
where F⁢uv F u v is analogous to DTFT in 1D.

note:

"Convolution in Time" is the same as "Multiplication in Frequency"
g⁢mn=h⁢mn*f⁢mn g m n h m n f m n
(3)
which, as stated above, is the same as:
G⁢uv=H⁢uv⁢F⁢uv G u v H u v F u v
(4)

Example 2: Magnitude of FT of Cameraman Image

Figure 3
Figure 3 (cam_mag.png)

To get a better image, we can use the fftshift command in Matlab to center the Fourier Transform. The resulting image is shown in Figure 4:

Figure 4
Figure 4 (cam_mag_center.png)

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