Abstract
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.
Highlights
Entropy-constrained scalar quantization (ECSQ) is a well-known compression scheme where a scalar quantizer q(·) is followed by a block lossless entropy-constrained encoder [1,2]
Several works have addressed the design of scalar quantizers for noisy channels (e.g., [10,11,12]). All these works present conditions and algorithms to optimize the scalar quantizer given that it is followed by a noisy channel. This is similar to the lossy-bit entropy-constrained scalar quantization (Lb-ECSQ) setup, where the lossy binary encoder behaves like a “noisy channel”, with an important and critical difference: in our problem, the distortion introduced by the lossy encoder is a parameter to be optimized, and acts as an additional degree of freedom
While this assumption is somewhat unrealistic, our main goal in this paper is to analyze the fundamental limits of the proposed scheme, as one would do in ECSQ when assuming that the scalar quantizer’s output is compressed at a rate equal to its entropy
Summary
Entropy-constrained scalar quantization (ECSQ) is a well-known compression scheme where a scalar quantizer q(·) is followed by a block lossless entropy-constrained encoder [1,2]. The compression scheme is straightforward, as we only need to expand ECSQ with an additional bit that encodes if the source symbol was in the left half or the right half of the quantization region that contained the source symbol In other words, this scheme codes the least significant quantization bit lossily, allowing a certain Hamming distortion D H. All these works present conditions and algorithms to optimize the scalar quantizer given that it is followed by a noisy channel This is similar to the Lb-ECSQ setup, where the lossy binary encoder behaves like a “noisy channel”, with an important and critical difference: in our problem, the distortion introduced by the lossy encoder (the“error probability” of the channel) is a parameter to be optimized, and acts as an additional degree of freedom.
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