Full sets of pictures to encode pictures

A picture, or two-dimensional (2D) string, is a rectangular array of symbols over a finite alphabet. In this paper, we introduce the notion of fullness for sets of strings and sets of pictures. Fullness is a local counterpart of completeness. While in 1D fullness coincides with completeness, in 2D complete sets of pictures are a subset of full ones. This new notion allows introducing the encoding of a picture. The definition of encoding is based on the one of cutting decomposition. If a set of pictures X is full then any picture has an encoding over X; furthermore, the encoding is unique if X is a univocally full set. Univocally full sets coincide with the maximal strong prefix codes of pictures that were recently introduced. At last, we show an encoding algorithm for pictures, which relies on a new tree data structure to represent univocally full sets.