A Computational Introduction to Digital Image Processing, 2nd Edition

154 A Computational Introduction to Digital Image Processing, Second Edition

1. There is no scaling factor 1/N

2. The sign inside the exponential function has been changed to positive

As with the forward transform, we can express this as a matrix product:

f = F

−1

F

with

F

−1

=























1 1 1 1 1 ··· 1

1

ω

1

ω

2

ω

3

ω

4

··· ω

N−1

1 ω

2

ω

4

ω

6

ω

8

··· ω

2(N−1)

1 ω

3

ω

6

ω

9

ω

12

··· ω

3(N−1)

1 ω

4

ω

8

ω

12

ω

16

··· ω

4(N−1)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

ω

N−1

ω

2(N−1)

ω

3(N−1)

ω

4(N−1)

··· ω

(N−1)

2























where

ω =

1

ω

= exp



2πi

N



.

In MATLAB or Octave, we can calculate the forward and inverse transforms with

fft and

ifft. Here fft stands for Fast Fourier Transform, which is a fast and efficient method of

performing ...