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Rate-distortion (RD) theory is a fundamental theory for lossy image compression that treats compressing the original images to a specified bitrate with minimal signal distortion, which is an essential metric in practical application. Moreover, with the development of visual analysis applications (such as classification, detection, segmentation, etc.), the semantic distortion in compressed images are also an important dimension in the theoretical analysis of lossy image compression.

In this paper, we propose a neural implementation of a companded quantization scheme allowing to train and implement optimal scalar quantization in data compression systems based on neural networks. The advantage of companded quantization lies in the fact that it allows to implement optimal non-linear quantization in a simpler form based on uniform quantization. In our work, we consider two different models of uniform quantization. Further on, in order to verify the effectiveness of the proposed approach, we made a series of experiments on natural grayscale images.