In this paper, we formulate the colorization-based coding problem into an optimization problem, i.e., an L1 minimization problem. In colorization-based coding, the encoder chooses a few representative pixels (RP) for which the chrominance values and the positions are sent to the decoder, whereas in the decoder, the chrominance values for all the pixels are reconstructed by colorization methods. The main issue in colorization-based coding is how to extract the RP well therefore the compression rate and the quality of the reconstructed color image becomes good. By formulating the colorization-based coding into an L1 minimization problem, it is guaranteed that, given the colorization matrix, the chosen set of RP becomes the optimal set in the sense that it minimizes the error between the original and the reconstructed color image. In other words, for a fixed error value and a given colorization matrix, the chosen set of RP is the smallest set possible. We also propose a method to construct the colorization matrix that colorizes the image in a multiscale manner. This, combined with the proposed RP extraction method, allows us to choose a very small set of RP. It is shown experimentally that the proposed method outperforms conventional colorization-based coding methods as well as the JPEG standard and is comparable with the JPEG2000 compression standard, both in terms of the compression rate and the quality of the reconstructed color image.