  Visual cryptography scheme (VCS) is an encryption technique that utilizes
human visual system in recovering secret image and it does not require any
complex calculation. However, the contrast of the reconstructed image could be
quite low. A number of reversing-based VCSs (or VCSs with reversing) (RVCS)
have been proposed for binary secret images, allowing participants to perform a
reversing operation on shares (or shadows). This reversing operation can be
easily implemented by current copy machines. Some existing traditional VCS
schemes without reversing (nRVCS) can be extended to RVCS with the same pixel
expansion for binary image, and the RVCS can achieve ideal contrast,
significantly higher than that of the corresponding nRVCS. In the application
of greyscale VCS, the contrast is much lower than that of the binary cases.
Therefore, it is more desirable to improve the contrast in the greyscale image
reconstruction. However, when greyscale images are involved, one cannot take
advantage of this reversing operation so easily. Many existing greyscale nRVCS
cannot be directly extended to RVCS. In this paper, we first give a new
greyscale nRVCS with minimum pixel expansion and propose an optimal-contrast
greyscale RVCS (GRVCS) by using basis matrices of perfect black nRVCS. Also, we
propose an optimal GRVCS even though the basis matrices are not perfect black.
Finally, we design an optimal-contrast GRVCS with minimum number of shares held
by each participant. The proposed schemes can satisfy different user
requirement, previous RVCSs for binary images can be viewed as special cases in
the schemes proposed here.
