New Methods for Multiplication-Free Arithmetic Coding

Arithmetic coding is a well-known technique for lossless coding or data compression. It is based on splitting a current interval into sub-intervals, each sub-interval corresponding to a possible symbol to be encoded and the size of each sub-interval corresponding to the probability of the associated symbol. This involves multiplying the current interval size A by one or more probabilities, which is a bottleneck for a high-speed implementation. Fortunately, it turns out that the multiplications can be performed with low accuracy, at only a small loss of performance. Generally, low-accuracy multiplication methods are based on approximating the interval size A before multiplying it with the symbol probabilities, with the purpose of reducing the complexity of the multiplication. For example, one method is to simply truncate A (which is kept normalized between 1/2 and 1) to its x most significant bits. Typically, x is small compared to the number of bits used for representing A (even x=2 is practically useful). An implementation of arithmetic coding that uses such low-complexity approximations of the full-accuracy multiplication is called “multiplication free”. We have developed two new multiplication-free methods. Our first new method is to round A to x bits instead of truncating it. Rounding is equivalent to truncating A to its x most significant bits if the (x+1)th most significant bit of A is a 0 and adding 1 to the truncated representation if the (x+1)th most significant bit is a 1. The rounding that is applied in our new method increases the complexity (compared to truncation), since, in about half of the cases, 1 has to be added to the truncated representation. As an alternative, we therefore developed a second new method, which we call “partial rounding”. By partial rounding we mean that 1 is only added to the truncated representation of A in case the (x+1)th most significant bit is a 1 and the xth most significant bit is a 0. In the implementation this means that the xth bit of the approximation of A equals the logical OR of the xth and (x+1)th most significant bits of the original A. The partial rounding of this second new method results in the same approximation as the “full rounding” of the first method in about 75% of the cases, but its complexity is as low as that of truncation (since the complexity of the OR is negligible). Applying the various multiplication-free methods in the arithmetic coder in the scheme described in [1–3] has demonstrated that our new rounding-based method outperforms the previously published multiplication-free methods. The “partial rounding” method outperforms the previously published truncation-based methods.