Fourier Analysis

Abstract

Transforms provide an alternate representation of images, which usually facilitates easier interpretation of operations and fast processing. The most important of all the transforms, the Fourier transform, decomposes an image in terms of sinusoidal surfaces. This transform is of fundamental importance to image processing, as is the case in almost all areas of science and engineering. As in the case of the convolution operation, both the 1-D and 2-D versions are described. Although the image is a 2-D signal, some of the important operations are decomposable and can be carried out in one dimension with reduced execution time. Another advantage is that understanding of the 1-D version is simpler. Definition, properties, and examples of the transforms are presented.

Exercises

3.1

The discrete periodic waveform x(n) is periodic with period 4 samples. Express the waveform in terms of complex exponentials and, thereby, find its DFT coefficients X(k). Find the 4 samples from both the expressions and check that they are the same. Find the least-squares errors, if x(n) is represented by its DC component alone with the values X(0), 0.9X(0), and 1.1X(0).

• *(i)
  $$\begin{aligned} x(n)= 1+ 3\cos \left( \frac{2\pi }{4}n + \frac{\pi }{3}\right) + 2\cos \left( 2\frac{2\pi }{4}n\right) \end{aligned}$$
• (ii)
  $$\begin{aligned} x(n)= -2+ \cos \left( \frac{2\pi }{4}n - \frac{\pi }{3}\right) + \cos \left( 2\frac{2\pi }{4}n\right) \end{aligned}$$
• (iii)
  $$\begin{aligned} x(n)= 2+ \cos \left( \frac{2\pi }{4}n + \frac{\pi }{6}\right) + \cos \left( 2\frac{2\pi }{4}n\right) \end{aligned}$$

3.2

Find the DFT of the 4 samples using the matrix form of the DFT definition. Reconstruct the input from the DFT coefficients using the IDFT and verify that they are the same as the input. Verify Parseval’s theorem.

• (i)
  $$\begin{aligned} \{x(0)=2,x(1)=1,x(2)=3,x(3)=2\} \end{aligned}$$
• (ii)
  $$\begin{aligned} \{x(0)=1,x(1)=1,x(2)=2,x(3)=-3\} \end{aligned}$$
• (iii)
  $$\begin{aligned} \{x(0)=-1,x(1)=0,x(2)=-3,x(3)=2\} \end{aligned}$$

3.3

The discrete periodic image x(m, n) is periodic with period 4 samples in both the directions. Express the image in terms of complex exponentials and, thereby, find its DFT coefficients X(k, l). Find the $4\times 4$ samples from both the expressions and check that they are the same. Find the least-squares errors, if x(m, n) is represented by its DC component alone with the values X(0, 0), 0.9X(0, 0), and 1.1X(0, 0).

• *(i)
  $$\begin{aligned} x(m, n)= 1+ 2\cos \left( \frac{2\pi }{4}(m+n) - \frac{\pi }{3}\right) + \cos \left( 2\frac{2\pi }{4}(m+n)\right) \end{aligned}$$
• (ii)
  $$\begin{aligned} x(m, n)= 2+ 2\cos \left( \frac{2\pi }{4}(m+2n) + \frac{\pi }{3}\right) - \cos \left( 2\frac{2\pi }{4}(m+n)\right) \end{aligned}$$
• (iii)
  $$\begin{aligned} x(m, n)= -1+ 2\cos \left( \frac{2\pi }{4}(2m+n) - \frac{\pi }{6}\right) + 2\cos \left( 2\frac{2\pi }{4}(m+n)\right) \end{aligned}$$

3.4

Find the DFT of the image x(m, n) using the row–column method. Reconstruct the input from the DFT coefficients using the IDFT and verify that they are the same as the input. Verify Parseval’s theorem. Express the magnitude of the DFT coefficients in the center-zero format using the log scale, $\log _{10}(1+|X(k, l)|)$. The origin is at the top-left corner.

• *(i)
  $$x(m, n)= \left[ \begin{array}{rrrr} 112&{}148&{} 72&{}153\\ 120&{}125&{} 30&{} 99\\ 95&{}120&{} 89&{} 33\\ 170&{} 99&{}109&{} 40 \end{array}\right] $$
• (ii)
  $$x(m, n)= \left[ \begin{array}{rrrr} 143&{}107&{}183&{}102\\ 135&{}130&{}225&{}156\\ 160&{}135&{}166&{}222\\ 85&{}156&{}146&{}215 \end{array}\right] $$
• (iii)
  $$x(m, n)= \left[ \begin{array}{rrrr} 164&{}127&{}117&{} 59\\ 154&{}122&{}104&{} 83\\ 129&{}136&{}100&{} 60\\ 117&{}128&{} 80&{} 48 \end{array}\right] $$

3.5

Find the DFT of the $4\times 4$ impulse image x(m, n) using (i) the row–column method and (ii) using the shift theorem.

• (i)
  $$ x(m, n)= \left[ \begin{array}{rrrr} 0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0 \end{array}\right] $$
• (ii)
  $$ x(m, n)= \left[ \begin{array}{rrrr} 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0 \end{array}\right] $$
• (iii)
  $$ x(m, n)= \left[ \begin{array}{rrrr} 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1 \end{array}\right] $$

3.6

Using the DFT and IDFT, find: (a) the periodic convolution of x(m, n) and h(m, n), (b) the periodic correlation of x(m, n) and h(m, n), and h(m, n) and x(m, n), (c) the autocorrelation of x(m, n).

• *(i)
  $$x(m, n)= \left[ \begin{array}{rrrr} 2 &{} 1 &{} 3 &{} 3 \\ 1 &{} 0 &{} 1 &{} 2 \\ 4 &{} 1 &{} 0 &{} 1 \\ 2 &{} 0 &{} 1 &{} 2 \end{array}\right] \quad h(m, n)= \left[ \begin{array}{rrrr} -2 &{} 1 &{} 3 &{} 2 \\ 1 &{} 1 &{} -1 &{} -2 \\ 4 &{} 0 &{} 0 &{} -1 \\ 1 &{} 0 &{} 2 &{} 2 \end{array}\right] $$
• (ii)
  $$ x(m, n)=\left[ \begin{array}{rrrr} 1&{} 2&{} -2&{} 1\\ -2&{} 0&{} 1&{} 4\\ 1&{} 1&{} -1&{} 2\\ 0&{} 1&{} 2&{} 4 \end{array} \right] \quad h(m, n)=\left[ \begin{array}{rrrr} 0&{} -1&{} 2&{} 2\\ -3&{} 1&{} 1&{} -1\\ 1&{} 1&{}-3&{} 0\\ 0&{} 1&{} 1&{} 2 \end{array} \right] $$
• (iii)
  $$ x(m, n)=\left[ \begin{array}{rrrr} 2&{} 1&{} 3&{} 4\\ -2&{} 0&{} 1&{} 4\\ 1&{} 1&{} 3&{} 2\\ 3&{} 1&{} 0&{} 4 \end{array} \right] \quad h(m, n)=\left[ \begin{array}{rrrr} 3&{} 1&{} 2&{} 4\\ 0&{} 1&{} -1&{} 2\\ 1&{} 1&{} -2&{} 2\\ 0&{} 1&{} 2&{} 1 \end{array} \right] $$

3.7

Compute the DFT of the column vector $x(m)=\{1,1,-1,-1\}$ and the row vector $x(n)=\{1,1,-1,-1\}$. Using the separability theorem, verify that the product of the vectors in the time domain and the 2-D IDFT of the product of their individual DFTs are the same.

3.8

Compute the DFT of the column vector $x(m)=\{ 0.2741, 0.4519, 0.2741\}$ and the row vector $x(n)=\{ 0.2741, 0.4519, 0.2741\}$. Using the separability theorem, verify that the product of the vectors in the time domain and the 2-D IDFT of the product of their individual DFTs are the same.

3.9

Compute the DFT of the column vector $x(m)=\{0, 1, 0, -1\}$ and the row vector $x(n)=\{1, 0, -1, 0\}$. Using the separability theorem, verify that the product of the vectors in the time domain and the 2-D IDFT of the product of their individual DFTs are the same.