Given a 4 ×4 image, find its compressed version using the...

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) $$ \left[\begin{array}{lllll} 170 & 168 & 164 & 173 \\ 179 & 167 & 167 & 167 \\ 184 & 179 & 173 & 166 \\ 183 & 179 & 184 & 173 \end{array}\right] $$ (ii) $$ \left[\begin{array}{lllll} 172 & 173 & 170 & 171 \\ 171 & 176 & 173 & 172 \\ 174 & 178 & 172 & 170 \\ 176 & 175 & 171 & 170 \end{array}\right] $$ (iii) $$ \left[\begin{array}{llll} 162 & 163 & 163 & 161 \\ 162 & 164 & 161 & 161 \\ 163 & 165 & 164 & 162 \\ 167 & 161 & 162 & 164 \end{array}\right] $$

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)
$$
\left[\begin{array}{lllll}
170 & 168 & 164 & 173 \\
179 & 167 & 167 & 167 \\
184 & 179 & 173 & 166 \\
183 & 179 & 184 & 173
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{lllll}
172 & 173 & 170 & 171 \\
171 & 176 & 173 & 172 \\
174 & 178 & 172 & 170 \\
176 & 175 & 171 & 170
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
162 & 163 & 163 & 161 \\
162 & 164 & 161 & 161 \\
163 & 165 & 164 & 162 \\
167 & 161 & 162 & 164
\end{array}\right]
$$

Key Concepts

Signal-to-Noise Ratio (SNR)

Signal-to-Noise Ratio (SNR) is a metric used to evaluate the quality of a reconstructed image after compression by comparing the level of the desired signal (the original image) to the level of background noise (errors introduced during compression and decompression). A higher SNR indicates a reconstruction closer to the original, meaning less compression-induced distortion. It is a key quantitative measure in assessing the performance of any compression algorithm.

Bits per Pixel (bpp)

Bits per pixel is a metric that quantifies the average number of bits used to represent each pixel in an image. In compression tasks, bpp serves as an indicator of the compression rate; lower bpp values imply higher compression, though this often comes with a potential loss in image quality. It is an essential measure for evaluating the efficiency and performance of compression algorithms.

Haar Discrete Wavelet Transform

The Haar Discrete Wavelet Transform (DWT) is a signal processing method that decomposes an image into different frequency sub-bands, separating the overall image content (approximation coefficients) from high-frequency details (detail coefficients). This transformation is computationally efficient and forms a basic representation of multi-resolution analysis, enabling further processing like thresholding or compression. It is particularly useful in image compression because it reduces redundancy and highlights the most significant features of the image.

Huffman Coding

Huffman Coding is an entropy coding technique used to compress data without loss by assigning shorter binary codes to more frequent symbols and longer codes to less frequent symbols. In the context of image compression, after transforming an image (for example, using the Haar DWT) and possibly quantizing the coefficients, Huffman Coding efficiently encodes the resulting symbols based on their statistical distribution, thus reducing the overall number of bits required for storage or transmission.

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