  We propose a novel joint lossy image and residual compression framework for
learning $\ell_\infty$-constrained near-lossless image compression.
Specifically, we obtain a lossy reconstruction of the raw image through lossy
image compression and uniformly quantize the corresponding residual to satisfy
a given tight $\ell_\infty$ error bound. Suppose that the error bound is zero,
i.e., lossless image compression, we formulate the joint optimization problem
of compressing both the lossy image and the original residual in terms of
variational auto-encoders and solve it with end-to-end training. To achieve
scalable compression with the error bound larger than zero, we derive the
probability model of the quantized residual by quantizing the learned
probability model of the original residual, instead of training multiple
networks. We further correct the bias of the derived probability model caused
by the context mismatch between training and inference. Finally, the quantized
residual is encoded according to the bias-corrected probability model and is
concatenated with the bitstream of the compressed lossy image. Experimental
results demonstrate that our near-lossless codec achieves the state-of-the-art
performance for lossless and near-lossless image compression, and achieves
competitive PSNR while much smaller $\ell_\infty$ error compared with lossy
image codecs at high bit rates.
