Digital Image Processing Basics

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

Digital Image Processing

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

Figure 1: We illustrate a "pixelized" smiley face.

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

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.

We also have discrete-space FTS:

F[u,v]=

∞

∑

m=−∞

 (

∞

∑

n=−∞

 (f[m,n]ⅇ^−(ⅈum)ⅇ^−(ⅈvm)))(2)

where F[u,v] is analogous to DTFT in 1D.

NOTE: 

"Convolution in Time" is the same as "Multiplication in Frequency"

g[m,n]=h[m,n]*f[m,n](3)

which, as stated above, is the same as:

G[u,v]=H[u,v]F[u,v](4)

EXAMPLE 2: Magnitude of FT of Cameraman Image

Figure 3

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

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