Lempel-Ziv Coding

• dictionary encoding
• near-optimal

1. find a sequence s of bits (length l > 0) not in the dictionary
2. subsequence of length l - 1 must be in dictionary
3. encode s as the position of the left substring, plus the new bit
4. number of bits to encode position is log2 (|dictionary|)
5. decoding consists of following pointer

Lempel-Ziv Example

• 1
• 1, pointer to 1 (a) followed by bit 1
• 1, a/1, 0
• 1, a/1, 0, pointer to 11 (b) /0
• 1, a/1, 0, b/0, c/1
• 1, a/1, 0, b/0, c/1, c/0
• 1, a/1, 0, b/0, c/1, c/0, e/1
• ...

Differential Pulse Code Modulation

• DPCM
• start with a reference symbol
• then only list the differences between successive symbols.
• Example:
  • data (24 bits) is 0000 0001 0001 0010 1001 1000
  • compressed to 0 (symbol), +1, +0, +1, 9 (new symbol), -1
  • encoded by 2 bits for difference
  • code 10 marks a new symbol
  • 10.0000 01 00 01 10.1001 11
  • Compression ratio 20 / 24 = 5/6.

DCPM Characteristics

• lossless.
• like RLE, but allows small variations in the data (e.g. real-world images).
• good for reducing the dynamic range of pixel values (adjacent pixels are usually similar).

GIF image compression

• Graphical Interchange Format
• lossy compression for 24-bit color images.
  1. identify the colors used in the picture.
  2. select 256 colors, replace 24-byte pixels with 8-byte indices into table (lossy).
  3. Run LZ over result (sequences of pixels are the "strings").
• Up to 10-to-1 compression ratio (10%).

JPEG image compression

• Joint Photographic Experts Group
• lossy compression for color images
  1. Color (RGB or YUV) separation (steps following are for each color).
  2. Divide into 8 * 8 blocks.
  3. Discrete Cosine Transform (DCT) on each block.
  4. Quantization (lossy).
  5. RLE/DPCM Encoding
• JPEG standard lets the compression quality be traded against image quality and viceversa.

JPEG DCT

• Discrete Cosine Transform is similar to Fast Fourier Transform (FFT).
• DCT computes spatial frequencies for a data set (for JPEG, for an 8 * 8 block).
• Except for quantization errors, the spatial frequencies can be converted (using inverse DCT) back into the data set, so DCT is not lossy.
• spatial frequency (0, 0) is the "DC", "constant", or "offset" for the entire block (how light or dark the entire block is).
• higher frequencies correspond to variations in the image: highest frequencies are the sharpest edges.

JPEG Quantization

• the lowest spatial frequencies are most important for picture quality (for humans)
• JPEG uses more bits for the low spatial frequency components.
• Only the more significant bits of higher spatial frequency components are retained
• achieved by dividing each coefficient by a value stored in a table (larger values for larger frequencies)
• example: value for (0,0) might be 4, so only the 2 least significant bits are lost.
• example: value for (7,7) might be 32, so the 5 least significant bits are lost
• JPEG specifies a set of quantization tables.

JPEG Encoding

• JPEG uses a variant of run-length encoding in a diagonal pattern over the data produced by quantization.
• variant: only the length of runs of zero values are encoded
• high-frequency components have been divided by larger values, so they tend to be zero
• codewords for Huffman coding are length of zero-run and following non-zero value
• DC coefficient is likely to be the same as in the previous block, and is encoded using DPCM.