A discussion on the relationship between probabilistic visual cryptography and random grid

In a (k,n) visual cryptography scheme (VCS), the dealer encodes the secret into n shadow images. Each pixel of the secret image is “expanded” into m sub-pixels in each share. In a (k,n)-VCS, the secret can be visually reconstructed when k or more shares are available. The reconstruction process employs the human visual system and no computations are required. To solve the pixel expansion problem of VCSs, probabilistic VCSs are directly converted from VCSs with no pixel expansion. Another well-known secret image sharing scheme, random grid (RG), is also provided with this novel stacking-to-see property. Compared with VCS, the most appealing benefit of RG is that there is no pixel expansion. In this paper, we use the probabilistic VCS without pixel expansion, to study the construction and performance of RG. This paper has two main results: (i) We show that each step of the share generation process in all existing (2,2)-RG, (2,n)-RG, (n, n)-RG, (k,n)-RG, incremental RG can be mapped to a corresponding step in PVCSs, and their shadow images between PVCS and RG are perfectly indistinguishable and the reconstructed images are the same, also include our (2,n) RG. (ii) From the quality of reconstructed image, pixel expansion, recognized region size, and image types to be considered for evaluating PVCSs and RGs, we point out that RG and PVCS have no difference other than the terminology. Furthermore, RGs is a subset of PVCSs.