Threshold visual cryptography schemes with specified whiteness levels of reconstructed pixels (Q5959833)

Consider an image that is a collection of black and white pixels. Each pixel is divided into \(m\) subpixels. In a \((k,n)\)-threshold visual cryptography system (VCS) each of \(n\) participants is given a share (or transparency) consisting of a collection of subpixels such that if \(k\) of the \(n\) participants stack their transparencies together the image is revealed but if fewer than \(k\) participants do so, the image will be indistinguishable from random noise. More formally define two sets of binary \(n \times m\) basis matrices \({\mathcal C}_0\) and \({\mathcal C}_1\). To encode a white pixel, a random \(C \in {\mathcal C}_0\) is chosen and on share \(i\) is placed \(m\) subpixels indicated by the nonzero elements of row \(i\) with a similar operation for black pixels by using a matrix of \({\mathcal C}_1\). The construction of these sets is crucial to the VCS. It should be noted that reconstruction of the image is not perfect in the sense that white pixels, under reconstruction, will not be all white and black pixels will not be all black. If the exact image is obtained on reconstruction with \(k\) or more shares, it is called a strong VCS. In section 2 the model of a VCS is given more formally including the use of the basis matrices. A new measurement is then given for contrast measurement for use in the reconstruction process. In section 6 new \((2,n)\) schemes are given with the new contrast parameters and is shown to be optimal with respect to pixel expansion. The main result of the work is given in section 6, which gives a new bound on \(m\) for \((2,n)\) schemes for given values of the contrast parameters. Section 7 considers the effect of forcing the basis matrices for white pixels to have constant weight and similarly for basis matrices for black pixels. Section 9 shows how to pose the minimum pixel expansion for \((2,n)\) schemes for specified contrast parameters as an integer linear programming problem. It is also shown how well the bounds derived here predict the actual pixel expansion. Some open problems in this area are also considered.