Binary morphological transformations based on the residues (ultimate erosion, skeleton by openings, etc.) and their associated functions which are based on the analysis of the residue evolution in every point of the image are extended to functions. In this approach, the associated function indicates the value of the residue index for which the evolution is the most important. The approach also has the advantage of supplying effective tools for shape analysis and of allowing the definition of new residual transforms together with their associated functions. Two of these numerical residues will be introduced, called, respectively, ultimate opening and quasi-distance and, through some applications, the interest and efficiency of these operators will be illustrated. Finally, this residual approach will be extended to more complex operators.