This thesis examins the problem of efficient compression using graph-morphology operations and answers the question, 'can a lossless image compressor be developed using the information contained in the nodes of the sieve tree?' It shows how humans are sensible to the small scale contained in images and a new way of measuring image quality is introduced, showing that this method can be normalised for comparing the results of different image coders. It provides a review of the scale-space decomposition algorithm denominated sieve and the hierarchical structure that is obtained from applying this filter to an image, and presents the descriptive statistics intended for getting an understanding of the distributions stored and for better exploiting the correlations found in the nodes of the sieve tree. The thesis also includes a thorough review of different lossless region coding algorithms, namely skeletons and chain codes, and evaluates their performance for coding the regions contained in the nodes of the sieve tree. Redundancy found between the parent-child relations of this tree is also addressed, for increasing the compressing ratios. Comparisons against the best lossless image compressors are performed, for evaluating the performance of the sieve-based compressor.