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Scale-space representation has been extensively studied in the computer vision community for analyzing image structures at different scales. This paper borrows and develops useful mathematical tools from scale-space theory to facilitate the task of image compression. Instead of compressing the original image directly, we propose to compress its scale-space representation obtained by the forward diffusion with a Gaussian kernel at the chosen scale. The major contribution of this work is a novel solution to the ill-posed inverse diffusion problem. We analytically derive a nonlinear filter to deblur Gaussian blurring for 1D ideal step edges. The generalized 2D edge enhancing filter only requires the knowledge of local minimum/maximum and preserves the geometric constraint of edges. When combined with a standard wavelet-based image coder, the forward and inverse diffusion can be viewed as a pair of pre-processing and post-processing stages used to select and preserve important image features at the given bit rate. Experiment results have shown that the proposed diffusion-based techniques can dramatically improve the visual quality of reconstructed images at low bit rate (below 0.25bpp).