Entropy-Constrained Scalar Quantization with a Lossy-Compressed Bit.

We consider the compression of a continuous real-valued source X using scalar quantizers and average squared error distortion D. Using lossless compression of the quantizer's output, Gish and Pierce showed that uniform quantizing yields the smallest output entropy in the limit D -> 0, resulting in a rate penalty of 0.255 bits/sample above the Shannon Lower Bound (SLB). We present a scalar quantization scheme named lossy-bit entropy-constrained scalar quantization (Lb-ECSQ) that is able to reduce the D -> 0 gap to SLB to 0.251 bits/sample by combining both lossless and binary lossy compression of the quantizer's output. We also study the low-resolution regime and show that Lb-ECSQ significantly outperforms ECSQ in the case of 1-bit quantization.