Compression — practice questions

1. Analyse the data stream: First, we read the pixels as a single stream: 00000000011111100111111000000000
2. Identify the runs:
   • The first run is nine 0s.
   • The next run is six 1s.
   • The next run is two 0s.
   • The next run is six 1s.
   • The final run is nine 0s.
3. Apply RLE: We represent each run with its count and value.
   • (9, 0)
   • (6, 1)
   • (2, 0)
   • (6, 1)
   • (9, 0)
4. Final Encoded Data: The RLE representation is (9,0)(6,1)(2,0)(6,1)(9,0). The original data required 32 bits (8x4x1). The RLE representation uses 5 pairs of numbers. The exact size would depend on how many bits are used for the count and value, but the number of data items has been reduced from 32 to 10.

Part (a): Calculate uncompressed file size

1. Calculate total pixels: $Total Pixels = 4000 \times 3000 = 12,000,000 pixels$
2. Calculate size in bits: $Size (bits) = Total Pixels \times Colour Depth$ $Size (bits) = 12,000,000 \times 24 = 288,000,000 bits$
3. Convert bits to bytes: (1 byte = 8 bits) $Size (bytes) = 288,000,000 / 8 = 36,000,000 bytes$
4. Convert bytes to mebibytes (MiB): (1 MiB = 1024 KiB = 1024 × 1024 bytes) $Size (MiB) = 36,000,000 / (1024 \times 1024)$ Size (MiB) ≈ 34.33 MiB

Part (b): Calculate compression ratio

1. Ensure units are consistent: The question gives the compressed size in megabytes (MB) and we calculated in mebibytes (MiB). For exam purposes, often the distinction is ignored, but it's good practice to be consistent. Let's assume the question implies MB (1,000,000 bytes) for the compressed size and we should use the same for the uncompressed size for a fair comparison. $Uncompressed size (MB) = 36,000,000 / 1,000,000 = 36 MB$ $Compressed size (MB) = 4 MB$
2. Calculate the ratio: $Compression Ratio = Uncompressed Size / Compressed Size$ $Compression Ratio = 36 MB / 4 MB = 9$
3. State the answer: The compression ratio is 9:1.