Image Compression

Abstract

Image compression is as essential as its processing, since images require large amounts of memory and, in their original form, it is difficult to transmit and store them. There are two types of image compression: (i) lossless and (ii) lossy. In lossless compression, the original image can be reconstructed exactly from its compressed version. Lossy compression is based on the fact that the magnitude of the frequency components of typical images decreases with increasing frequency. Emphasis is given to using the DWT, since it is a part of the current image compression standard.

Exercises

13.1  Given a $4\times 4$ image, compute the entropy. Find the Huffman code representation of its unique symbols and the bpp.

* (i)

$$ \left[ \begin{array}{rrrr} 144&{} 113&{} 121&{} 107\\ 144&{} 110&{} 121&{} 103\\ 129&{} 109&{} 120&{} 99\\ 116&{} 108&{} 121&{} 103 \end{array}\right] $$

(ii)

$$ \left[ \begin{array}{rrrr} 209&{} 190&{} 179&{} 179\\ 143&{} 136&{} 132&{} 129\\ 131&{} 130&{} 125&{} 117\\ 113&{} 109&{} 118&{} 143 \end{array}\right] $$

(iii)

$$ \left[ \begin{array}{rrrr} 85&{} 91&{} 91&{} 89\\ 79&{} 83&{} 88&{} 87\\ 90&{} 86&{} 86&{} 90\\ 97&{} 93&{} 88&{} 90 \end{array}\right] $$

13.2  Given a $4\times 4$ image, decompose it into 4 bit planes and represent each of them by run-length coding. Use both the methods.

(i)

$$ \left[ \begin{array}{rrrr} 6&{} 6&{} 15&{} 14\\ 8&{} 8&{} 4&{} 11\\ 9&{} 9&{} 10&{} 12\\ 10&{} 14&{} 13&{} 1 \end{array}\right] $$

* (ii)

$$ \left[ \begin{array}{rrrr} 11&{} 9 &{} 15 &{} 14\\ 13&{} 7&{} 6&{} 7\\ 6&{} 5&{} 5&{} 5\\ 11&{} 4&{} 4&{} 2 \end{array}\right] $$

(iii)

$$ \left[ \begin{array}{rrrr} 13&{} 9&{} 12&{} 10\\ 13 &{} 13&{} 12&{} 9\\ 13&{} 13&{} 12&{} 9\\ 11&{} 13&{} 12 &{} 10 \end{array}\right] $$

13.3  Given a $4\times 4$ image, find the linear predictive code. Find the entropies of the input and the code.

(i)

$$ \left[ \begin{array}{rrrr} 15&{} 16&{} 20&{} 20\\ 15&{} 15&{} 19&{} 22\\ 15&{} 16&{} 19&{} 20\\ 15&{} 17&{} 19&{} 16 \end{array}\right] $$

(ii)

$$ \left[ \begin{array}{rrrr} 158&{} 157&{} 154&{} 149\\ 168&{} 153&{} 157&{} 149\\ 170&{} 152&{} 157&{} 149\\ 166&{} 153&{} 157&{} 142 \end{array}\right] $$

* (iii)

$$ \left[ \begin{array}{rrrr} 106&{} 103&{} 98&{} 99\\ 121&{} 122&{} 108&{} 93\\ 102&{} 102&{} 100&{} 99\\ 100&{} 101&{} 102&{} 96 \end{array}\right] $$

13.4  Given a 1-digit sequence, find its arithmetic code. Reconstruct the sequence from the code. Let there be three symbols $\{1,2,3\}$ and number of occurrences of the symbols, respectively, be $\{2,1,1\}$ in a sequence of length 4.

(i) $\{1\}$

(ii) $\{2\}$

13.5  Given a sequence x(n), find the 1-level DWT coefficients using the 9 / 7 filter. Assume whole-point symmetry at the borders. Verify that the reconstructed signal is the same as the input.

* (i)

$$ \{1, -4, 1, 3, 3, 1, 3, 0, 2, 2, 3, 1, -5, 2, 0, 3 \}$$

(ii)

$$ \{1, 2, 1, 1, 3 , 4 , 0, 3 , 1, 3 , 2, -1, 0 , 1, 4, -3 \}$$

(iii)

$$ \{ -2, 0, 2, -2, 1, 0, 3, 1, -2, 1, 2, -1, 2, 0, -2, 1 \}$$

13.6  Given a $4\times 4$ image, find its compressed version using the 1-level Haar DWT and the Huffman code. What is the bpp and SNR.

(i)

$$ \left[ \begin{array}{rrrr} 170&{} 168&{} 164&{} 173\\ 179&{} 167&{} 167&{} 167\\ 184&{} 179&{} 173&{} 166\\ 183&{} 179&{} 184&{} 173 \end{array}\right] $$

* (ii)

$$ \left[ \begin{array}{rrrr} 172&{} 173&{} 170&{} 171\\ 171&{} 176&{} 173&{} 172\\ 174&{} 178&{} 172&{} 170\\ 176&{} 175&{} 171&{} 170 \end{array}\right] $$

(iii)

$$ \left[ \begin{array}{rrrr} 162&{} 163&{} 163&{} 161\\ 162&{} 164&{} 161&{} 161\\ 163&{} 165&{} 164&{} 162\\ 167&{} 161&{} 162&{} 164 \end{array}\right] $$