Delving into Arithmetic Coding
Arithmetic coding is a sophisticated form of entropy encoding used in lossless data compression. Unlike algorithms like Huffman coding which assign a specific integer number of bits to each symbol, arithmetic coding can assign fractional bits, often leading to greater compression efficiency, especially for sources with a small alphabet or highly skewed probabilities.
How Arithmetic Coding Works
The core idea of arithmetic coding is to represent an entire message (or sequence of symbols) as a single fraction, a number in the interval [0, 1). As the message becomes longer, the interval representing it becomes smaller, and the number of bits needed to specify that interval grows.
Here's a simplified view of the process:
- Model Probabilities: First, you need a model that provides the probabilities of the symbols in the input data. This could be a fixed model or an adaptive one that updates probabilities as it processes the data.
- Interval Division: Start with the range [0, 1). For each symbol in the input sequence, this range is narrowed based on the symbol's probability. The current range is subdivided into sub-intervals, with the size of each sub-interval proportional to the probability of the corresponding symbol.
- Select Sub-interval: The sub-interval corresponding to the next symbol in the input sequence becomes the new current range.
- Repeat: This process is repeated for all symbols in the message.
- Final Code: The final compressed message is a number that uniquely identifies the final, very small, interval. In practice, only enough bits to distinguish this interval from all other possible final intervals are transmitted.
Advantages of Arithmetic Coding
- Higher Compression Ratios: Often achieves compression ratios closer to the theoretical limit (entropy) than Huffman coding, particularly when symbol probabilities are not close to powers of 1/2.
- Adaptability: It can be easily combined with adaptive probability models, where symbol probabilities change as data is processed, leading to better compression for varying data sources.
- Separation of Model and Coding: The probability model and the encoding process are distinct, allowing for easier experimentation with different models.
Key Differences from Huffman Coding
While both are entropy coders, Huffman coding assigns a fixed-length prefix code (e.g., 0, 10, 110) to each symbol. Arithmetic coding, on the other hand, effectively allocates a "fractional" number of bits to each symbol, based on its probability, leading to a single floating-point number representing the entire input sequence.
Disadvantages of Arithmetic Coding
- Computational Complexity: Generally slower than Huffman coding due to the more complex mathematical operations involved in maintaining and subdividing intervals.
- Patents: Historically, its use was complicated by patents, though many have now expired.
- Implementation Sensitivity: Requires careful implementation, especially regarding floating-point precision and scaling to avoid underflow/overflow.
Prominent Use Cases
Despite its complexities, arithmetic coding is used in several standards due to its efficiency:
- JPEG 2000: The image compression standard uses arithmetic coding (specifically, the MQ coder, a binary arithmetic coder).
- H.264/AVC and H.265/HEVC (CABAC): Context-Adaptive Binary Arithmetic Coding (CABAC) is a significant part of these modern video compression standards.
- It has also been used as a component in other compression utilities like bzip2 (though bzip2 primarily uses Burrows-Wheeler Transform and Huffman coding, it can use arithmetic coding as an optional final stage).
For a more in-depth mathematical treatment, you can refer to resources like the Wikipedia page on Arithmetic Coding.