Phenomenal transparency in achromatic checkerboards

The study explored the luminance relations that determine the occurrence of achromatic transparency in phenomenal surfaces on complex backgrounds. Let the luminances of the left and right parts of a transparent surface on a bipartite background and those of the left and right parts of the bipartite background be p and q and m and n, respectively. Metelli proposed that this surface looks transparent when the rule p < q if m < n (or p > q if m > n) is satisfied, and Masin and Fukuda that it looks transparent when the inclusion rule is satisfied, that is, when p epsilon (m, q) or q epsilon (p, n). These rules also apply to achromatic checkerboards formed by one checkerboard enclosed in another checkerboard. This study shows that only the inclusion rule correctly predicted the occurrence of transparency in these checkerboards.