Variable Length Lossy Coding Using an LDPC Code

Junya Honda, Hirosuke Yamamoto

Abstract: In this paper, a new variable length coding scheme using a low density parity check (LDPC) code is proposed for the lossy compression of general i.i.d. finite sources. It is proved that the proposed scheme achieves the rate-distortion function asymptotically for an LDPC ensemble. For our setting, Miyake-Muramatsu already proposed an asymptotically optimal LDPC coding scheme. In their scheme, a source sequence is first vector-quantized using an LDPC matrix and then it is compressed losslessly by fixed length coding with another LDPC matrix. However, it is not shown whether their scheme can attain good performance practically. This is mainly because of the difficulty of lossless coding by a fixed length code. In the proposed scheme, the lossless compression is performed by arithmetic coding instead of the fixed length code. Combined with vector-quantization using the reinforced belief propagation, the proposed scheme attains performance near the rate-distortion function practically with time complexity roughly equal to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\bf O}(n\log n)$</tex></formula> for length <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> source sequences.