A Computational Introduction to Digital Image Processing, 2nd Edition

The Fourier Transform 179

where in each case the parameter n is called the order of the filter. The size of n dictates

the sharpness of the cutoff. These functions are illustrated in figures 7.20 and 7.21.

Note that on a two-dimensional grid, where x in the above formulas may be replaced

with



x

2

+ y

2

as distance from the grid’s center, assuming the grid to be centered at (0, 0),

then the formulas for the low pass and high pass filters may be written

f(x, y) =

1

1 +



x

2

+ y

2

D

2



n

and

f(x, y) =

1

1 +



D

2

x

2

+ y

2



n

respectively.

f(x)

x

1

f(x)

x

1

(a) Low pass (b) High pass

FIGURE 7.20: Butterworth filter functions with n = 2

f(x)

x

1

f(x)

x

1

(a) Low pass (b) High pass

FIGURE 7.21: Butterw ...