A linear reaction-diffusion system with interior degeneration for color image compression

This paper considers colorization-based image compression in RGB color space. In compression, we store only the compressed luminance component of the original color image and a few representative pixels extracted from the original color image. In decompression, by explicitly introducing the relation between the luminance component and the original color image into diffusion equations, a linear reaction-diffusion system with Perona‒Malik type diffusion coefficient is proposed to reconstruct R, G, and B channels simultaneously. The Perona‒Malik type diffusion coefficient is a function of the luminance component and leads to interior degenerations, in general. It yields anisotropic smoothing in the restored color image and constrains the geometry of the restored image to follow the geometry of the luminance component. The existence and uniqueness of solutions for the proposed system with a specific class of diffusion coefficients are proved in a weighted Sobolev space. The selection of representative pixels has a big impact on reconstruction results. We also propose a local-optimal strategy that splits the original color image into a series of different size subimages and searches the optimal representative pixel in each subimage. Comparisons with recent colorization-based image compression methods, as well as transform-based JPEG and JPEG2000 standards, are performed to show the potential for successful compression applications of the proposed method.