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Lossy image compression is generally formulated as a joint rate-distortion optimization problem to learn encoder, quantizer, and decoder. Due to the non-differentiable quantizer and discrete entropy estimation, it is very challenging to develop a convolutional network (CNN)-based image compression system. In this paper, motivated by that the local information content is spatially variant in an image, we suggest that:(i) the bit rate of the different parts of the image is adapted to local content, and (ii) the content-aware bit rate is allocated under the guidance of a content-weighted importance map. The sum of the importance map can thus serve as a continuous alternative of discrete entropy estimation to control compression rate. The binarizer is adopted to quantize the output of encoder and a proxy function is introduced for approximating binary operation in backward propagation to make it differentiable. The encoder, decoder, binarizer and importance map can be jointly optimized in an end-to-end manner. And a convolutional entropy encoder is further presented for lossless compression of importance map and binary codes. In low bit rate image compression, experiments show that our system significantly outperforms JPEG and JPEG 2000 by structural similarity (SSIM) index, and can produce the much better visual result with sharp edges, rich textures, and fewer artifacts.