Rethinking the Functionality of Latent Representation: A Logarithmic Rate-Distortion Model for Learned Image Compression

End-to-end optimized Learned Image Compression (LIC) has demonstrated remarkable performance in terms of Rate-Distortion (R-D) efficiency. However, the R-D characteristics of LIC codecs remain underexplored. Previous research has attempted to investigate the R-D behavior through numerical and statistical approaches, but these methods often provide only empirical results, lacking theoretical insights. In this work, we introduce a novel methodology for studying the R-D characteristics of LIC. By rethinking the LIC paradigm from a fresh perspective, we propose a plug-and-play module, the Latent-domain Auto-Encoder (LAE). This innovative approach not only naturally leads to Variable Bit-Rate (VBR) compression, but also allows for a theoretical modeling of the R-D behavior of LIC codecs. Our findings reveal that the bit-rate is the logarithmic sum of the neurons <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sub> in our designed network’s last layer, plus a constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> introduced by image content, formally expressed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sub> = Σ log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i>. This insight is pivotal, as it underscores how the bit-rate can be systematically derived from the latent representations. Further analysis demonstrates that our proposed R-λ model enables effective rate control for learned image codecs, enhancing their adaptability and accuracy. Experimental results validate that our VBR method surpasses fixed-rate coding by 2.9% in terms of BD-rate. Additionally, the proposed R-λ model exhibits superior rate control performance, suggesting that it not only elucidates the underlying R-D characteristics of LIC but also significantly enhances its practical deployment in real-world applications.