Image Enhancement in the Spatial Domain

Abstract

Although the transform domain processing is essential, as the images naturally occur in the spatial domain, image enhancement in the spatial domain is presented first. Point operations, histogram processing, and neighborhood operations are presented. The convolution operation, along with the Fourier analysis, is essential for any form of signal processing. Therefore, the 1-D and 2-D convolution operations are introduced. Linear and nonlinear filtering of images is described next.

Exercises

2.1

Find the complement of the $4\times 4$ 8-bit gray level image and verify that the image can be restored by complementing the complemented image.

• (i)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 112&{}148&{} 72&{}153\\ 120&{}125&{} 30&{} 99\\ 95&{}120&{} 89&{} 33\\ 170&{} 99&{}109&{} 40 \end{array}\right] $$
• (ii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 164&{}127&{}117&{} 59\\ 154&{}122&{}104&{} 83\\ 129&{}136&{}100&{} 60\\ 117&{}128&{} 80&{} 48 \end{array}\right] $$
• (iii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 46&{}48&{}46&{}45\\ 42&{}49&{}46&{}45\\ 64&{}73&{}60&{}43\\ 94&{}69&{}63&{}37 \end{array}\right] $$

2.2

Find the complement of the $4\times 4$ binary image and verify that the image can be restored by complementing the complemented image.

• (i)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 1&{}0&{}0&{}0\\ 1&{}0&{}0&{}0\\ 1&{}1&{}0&{}0\\ 0&{}1&{}0&{}0 \end{array}\right] $$
• (ii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 0&{}1&{}0&{}1\\ 0&{}0&{}0&{}1\\ 0&{}0&{}0&{}1\\ 0&{}0&{}0&{}1 \end{array}\right] $$
• (iii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 1&{}1&{}0&{}0\\ 1&{}0&{}0&{}0\\ 1&{}1&{}1&{}1\\ 1&{}1&{}0&{}0 \end{array}\right] $$

2.3

For the list of gray levels, apply gamma correction and find the corresponding new gray levels. Apply the inverse transformation to the new gray levels and verify that the given gray levels are obtained.

$$\begin{aligned} \{0, 25, 50, 100, 150, 200, 250, 255 \} \end{aligned}$$

• (i) $\gamma =0.8$.

• (ii) $\gamma =1.1$.

• (iii) $\gamma =1.8$.

2.4

Given a $4\times 4$ 4-bit image, find the histogram equalized version of it.

• *(i)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 4&{} 4&{} 3&{} 3\\ 4&{} 4&{} 4&{} 3\\ 5 &{}4&{} 4&{} 4\\ 4&{} 4&{} 4&{} 4 \end{array}\right] $$
• (ii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 1&{}1&{}0&{}1\\ 1&{}1&{}0&{}3\\ 1&{}0&{}0&{}2\\ 1&{}0&{}0&{}2 \end{array}\right] $$
• (iii)
  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 3&{}5&{}5&{}3\\ 4&{}4&{}4&{}3\\ 4&{}2&{}3&{}4\\ 4&{}2&{}2&{}3\end{array}\right] $$

2.5

Given $4\times 4$ 4-bit reference and input images, use histogram matching to restore the input image.

• *(i) The reference and input images, respectively, are

  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 4&{} 4&{} 3&{} 3\\ 4&{} 4&{} 4&{} 3\\ 5 &{}4&{} 4&{} 4\\ 4&{} 4&{} 4&{} 4 \end{array}\right] \quad \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 15&{}15&{} 0&{} 0\\ 15&{}15&{}15&{} 0\\ 15&{}15&{}15&{}15\\ 15&{}15&{}15&{}15 \end{array}\right] $$

• (ii) The reference and input images, respectively, are

  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 3&{} 3&{} 3&{} 3\\ 3&{} 3&{} 3&{} 3\\ 3&{} 2&{} 2&{} 3\\ 2&{} 2&{} 2&{} 2 \end{array}\right] \quad \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 15&{}15&{}15&{}15\\ 15&{}15&{}15&{}15\\ 15&{} 0&{} 0&{}15\\ 0&{} 0&{} 0&{} 0 \end{array}\right] $$

• (iii) The reference and input images, respectively, are

  $$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 3&{} 5&{} 5&{} 3\\ 4&{} 4&{} 4&{} 3\\ 4&{} 2&{} 3&{} 4\\ 4&{} 2&{} 2&{} 3 \end{array}\right] \quad \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 0&{}15&{}15&{} 0\\ 15&{}15&{}15&{} 0\\ 15&{} 0&{} 0&{}15\\ 15&{} 0&{} 0&{} 0 \end{array}\right] $$

2.6

Given a $8\times 8$ 8-bit image, find the binary, hard and soft thresholded versions with the threshold $T=160$.

$$ \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 255&{}255&{}255&{}117&{} 50&{} 39&{} 50&{} 56\\ 255&{}255&{}255&{}194&{} 45&{} 26&{} 48&{} 54\\ 255&{}255&{}255&{}241&{} 61&{} 25&{} 53&{} 57\\ 255&{}255&{}255&{}255&{}104&{} 32&{} 64&{} 64\\ 255&{}255&{}255&{}255&{}154&{} 37&{} 59&{} 61\\ 255&{}255&{}255&{}255&{}199&{} 54&{} 55&{} 61\\ 255&{}255&{}255&{}255&{}230&{} 71&{} 59&{} 64\\ 255&{}255&{}255&{}255&{}250&{} 95&{} 60&{} 68 \end{array}\right] $$

2.7

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ averaging filter with the borders zero-padded.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 70&{}62&{}51&{}45\\ 71&{}62&{}57&{}55\\ 73&{}65&{}56&{}60\\ 68&{}69&{}63&{}66 \end{array}\right] $$

2.8

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ averaging filter with the borders replicated.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 41&{}43&{}45&{}43\\ 40&{}41&{}42&{}41\\ 42&{}38&{}39&{}42\\ 39&{}33&{}37&{}36 \end{array}\right] $$

2.9

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ averaging filter with the borders periodically extended.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 45&{}78&{}87&{}51\\ 59&{}56&{}62&{}49\\ 59&{}39&{}44&{}57\\ 56&{}36&{}35&{}51 \end{array}\right] $$

2.10

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ Gaussian filter $(\sigma =0.5)$ with the borders symmetrically extended.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 202&{}195&{}192&{}191\\ 216&{}211&{}200&{}209\\ 224&{}212&{}215&{}227\\ 224&{}205&{}227&{}230 \end{array}\right] $$

2.11

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ Gaussian filter $(\sigma =0.5)$ with the borders periodically extended.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 202&{}195&{}192&{}191\\ 216&{}211&{}200&{}209\\ 224&{}212&{}215&{}227\\ 224&{}205&{}227&{}230 \end{array}\right] $$

2.12

Given a $4\times 4$ image, find the $4\times 4$ lowpass filtered output using the $3\times 3$ Gaussian filter $(\sigma =0.5)$ with the borders zero-padded.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 95&{}82&{}54&{}33\\ 84&{}78&{}56&{}64\\ 73&{}71&{}53&{}60\\ 73&{}73&{}54&{}36 \end{array}\right] $$

2.13

Given a $4\times 4$ image, find the $4\times 4$ highpass filtered output using the $3\times 3$ Laplacian filter

$$h(m, n)= \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }} 0&{} 1&{} 0 \\ 1 &{} -4 &{} 1\\ 0&{} 1&{} 0 \end{array} \right] $$

with the borders zero-padded.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 45&{}52&{}56&{}52\\ 49&{}60&{}55&{}55\\ 47&{}55&{}53&{}46\\ 45&{}48&{}51&{}40 \end{array}\right] $$

2.14

Given a $4\times 4$ image, find the $4\times 4$ highpass filtered output using the $3\times 3$ Laplacian filter with the borders symmetrically extended.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 64&{}62&{}62&{}68\\ 68&{}66&{}58&{}64\\ 75&{}70&{}60&{}58\\ 72&{}69&{}59&{}60 \end{array}\right] $$

2.15

Given a $4\times 4$ image, find the $4\times 4$ highpass filtered output using the $3\times 3$ Laplacian filter with the borders replicated.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 39&{}40&{}35&{}33\\ 31&{}40&{}39&{}37\\ 34&{}38&{}41&{}43\\ 37&{}39&{}42&{}43 \end{array}\right] $$

*2.16

Given a $4\times 4$ image, find the $4\times 4$ enhanced output using the $3\times 3$ Laplacian sharpening filter

$$h(m, n)= \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }} 0&{} -1&{} 0 \\ -1 &{} 5 &{} -1\\ 0&{} -1&{} 0 \end{array} \right] $$

with the borders replicated.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 190&{}206&{}228&{}238\\ 180&{}205&{}227&{}219\\ 182&{}203&{}211&{}159\\ 184&{}212&{}206&{}177 \end{array}\right] $$

2.17

Given a $4\times 4$ image, find the $4\times 4$ enhanced output using the $3\times 3$ Laplacian sharpening filter with the borders periodically extended.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 138&{}163&{}162&{}177\\ 148&{}157&{}167&{}175\\ 153&{}165&{}160&{}178\\ 157&{}162&{}164&{}188 \end{array}\right] $$

2.18

Given a $4\times 4$ image, find the $4\times 4$ enhanced output using the $3\times 3$ Laplacian sharpening filter with the borders zero-padded.

$$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 201&{}195&{}191&{}169\\ 210&{}201&{}181&{}157\\ 213&{}207&{}190&{}166\\ 204&{}204&{}197&{}159 \end{array}\right] $$

2.19

Given a $4\times 4$ image, find the $4\times 4$ median filtered output using the $3\times 3$ window with the borders zero-padded.

• (i)
  $$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 201&{}195&{}191&{}169\\ 210&{}201&{}181&{}157\\ 213&{}207&{}190&{}166\\ 204&{}204&{}197&{}159 \end{array}\right] $$
• (ii)
  $$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 138&{}163&{}162&{}177\\ 148&{}157&{}167&{}175\\ 153&{}165&{}160&{}178\\ 157&{}162&{}164&{}188 \end{array}\right] $$
• (iii)
  $$x(m, n)=\left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }} 190&{}206&{}228&{}238\\ 180&{}205&{}227&{}219\\ 182&{}203&{}211&{}159\\ 184&{}212&{}206&{}177 \end{array}\right] $$