On the Composition of Digitally Continuous Multivalued Functions

Abstract

In Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin, pp. 81---92, 2008), Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810. Springer, Berlin, pp. 275---287, 2009) and Escribano et al. (J Math Imaging Vis 42:76---91, 2012), a notion of continuity in digital spaces, which extends the usual notion of digital continuity, was introduced. In Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin, pp. 81---92, 2008), it was claimed that the composition of two digitally continuous multivalued functions is a digitally continuous multivalued function. However, this result is only correct in the following situation: If $$X\subset {{\mathrm{{\mathbb {Z}}}}}^m , Y\subset {{\mathrm{{\mathbb {Z}}}}}^n , Z\subset {{\mathrm{{\mathbb {Z}}}}}^p $$X?Zm,Y?Zn,Z?Zp, and $$F:X\leadsto Y$$F:X?Y and $$G:Y\leadsto Z$$G:Y?Z are, respectively, a $$(k,k')$$(k,k?)-continuous multivalued function and a $$(k',k'')$$(k?,k??)-continuous multivalued function, with $$k=3^m-1$$k=3m-1, then $$GF:X\leadsto Z$$GF:X?Z is a $$(k,k'')$$(k,k??)-continuous multivalued function. In this paper, we give a proof of this result. On the other hand, in Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810. Springer, Berlin, pp. 275---287, 2009) and Escribano et al. (J Math Imaging Vis 42:76-91, 2012), the continuity of the composition of continuous multivalued functions is used in some proofs. Therefore, those proofs are only valid for $$k=8$$k=8. We give in this paper new proofs for the case $$k=4$$k=4 not requiring this property, with the exception of a result on the sequential deletion of 4-simple points which we state here as Conjecture 1.