Morgenthaler and Rosenfeld gave the first formal definition of digital surfaces. Kong and Roscoe embedded digital spaces to continuous spaces to show the intuitive meaning of Morgenthaler-Rosenfeld's surfaces. Reed and Rosenfeld discussed the recognition algorithms for digital surface. Chen and Zhang considered an intuitive and simple definition of digital manifolds in case of direct adjacency in digital spaces. This note continues the task. This note will simplify some concepts used and extended some concepts used. First, we continue to study all cases of ((alpha) , (beta) )- surfaces and eliminate some trivial and overlapping cases. Among nine types of digital surfaces, there are only two useful: (6,26)-surfaces and (18,6)-surfaces. We will develop or modify three algorithms for the three definitions. Because some visually true digital surface is not included in any of ((alpha) , (beta) )-surfaces, we will give a more general definition for digital surface in both direct and indirect adjacency. In order to develop a fast algorithm for real surface tracking, a quasi-digital-surface is defined even through sometimes the definition is weak in terms of mathematics.