Psychophysical experiments have been previously described Weinshall, D. (1989) Nature 341, 737-739; (1991) Vision Research 31, 1731-1748 that involved the perception of many transparent layers, corresponding to multiple matching, in doubly ambiguous random-dot stereograms. Additional experiments are described in the first part of this paper. In one experiment, subjects were required to report the density of dots on each transparent layer. In another experiment, the minimal density of dots on each layer, which is required for the subjects to perceive it as a distinct transparent layer, was measured. The difficulties encountered by stereo matching algorithms, when applied to doubly ambiguous stereograms, are described in the second part of this paper. Algorithms that can be modified to perform consistently with human perception, and the constraints imposed on their parameters by human perception, are discussed.