Transparency Interpretation of X Junctions

Eric Saund
Xerox Palo Alto Research Center

Psychophysicists have proposed an "atmospheric" model for surface lightness perception. The lightness of a surface patch is interpreted as a combination of the lightness of some underlying surface, and a contribution from an occluding, semitransparent surface. Metelli's law states:

r = b * alpha + t * (1 - alpha)

where

• r is the measured lightness (0 = black, 1 = white)
• b is the lightness of the background surface
• t is the inherent lightness of the occluding atmosphere
• alpha is the opacity of the atmosphere (0 = opaque, 1 = transparent)

This model can be applied to X-junctions under the assumption that some underlying contrast edge (p q) is partially occluded by a semitransparent surface resulting in lightnesses (r s).

Adelson and Anandan noted that depending on the behavior of contrast signs as a lightness boundary is traversed through an X-junction, one of three conditions holds: a semitransparent overlap interpretation is possible in one, both, or neither direction. In other words, under some combinations of lightnesses, the pair (q s), or others, could be interpreted as an occluding semitransparent surface.

The applet demonstrates this atmospheric model for the interpretation of transparency at an X-junction.

• Use the sliders on the right to adjust lightnesses of patches a, b, c, and d.
• Drag the circle to provide a motion cue for a moving circle/hole object.
• The two-dimensional graphs illustrate the solutions to the Metelli equation across the vertical and horizontal overlap directions. The left and right graphs manipulate alternative transparent interpretations.

  Moving points (a c) or (b d), in the left graph, adjusts

  • background consisting of a field with a circle painted on it
  • occluding surface splitting the image in two

  Moving points (a b) or (c d), in the right graph, adjusts

  • background consisting of a two rectangular patches
  • occluding surface which is either a circle or a complete surface with a hole cut in the middle

• Icons activate to indicate which transparent interpretations are predicted by the atmospheric model. Compare with your own perception.
• A possible transparent interpretation is predicted when both points are on the same side of the t-diagonal.
• The blue line shows how the lightness point pairs imply the parameters of the atmosphere.
  • The point furthest from the diagonal is the diagonal is the background; the point nearer is the occluder.
  • The intersection of the blue line with the t-diagonal indicates the occluding surface's inherent lightness t.
  • The position of the lightness point along the blue line indicates the occluder's opacity alpha.
• As you adjust surface lightnesses, opacity alpha and occluding surface lightness t are updated in the sliders at the bottom. Use these sliders to adjust alpha and t directly. (You can also drag t directly on the 2d graphs.)
• Points can be arranged to generate lightness combinations permitting
  • no transparent interpretations (double-reversing junctions)
  • one transparent interpretations (single-reversing junctions)
  • two perpendicular transparent interpretations (non-reversing junctions)

Notes and Observations

• Some lightness combinations look like a spotlight shining on a dimly-lit background.
• Some lightness combinations result in t < 0 (smoke blacker than black) or t > 1 (fog whiter than white). These look like regular smoke and fog to me.
• When the points are nearly parallel to the t-diagonal, it becomes ambiguous whether the atmosphere is a region of smoke or the complementary region filled with fog. I tend to see smoke more readily than fog.
• I tend to see the strongest transparency effects when t is in the middle range, neither black smoke nor white fog, but intermediate grey.
• Whenever two transparent interpretations are possible, they both imply the same underlying lightnesses t for their respective occluding surfaces. Their opacities alpha may differ. This arises directly from the Metelli equations. If this prediction can be shown to be violated psychophysically it would indicate a failure of the purely multiplicative atmospheric model.
• One way to help yourself perceive the transparency is to drag the circle around and imagine it being either in front of both of the rectangular panels, or sandwiched between them. Transparency interpretations are intimately related to relative surface depth layering.

This kind of analysis will be required to extend computational models of the perceptual organization of occluding surfaces to images containing X-junctions, which provide cues for surface transparency.

References

• Ted Adelson and P. Anandan, "Ordinal Characteristics of Transparency," AAAI-90 Workshop on Qualitative Vision, Boston, MA. 1990.
• Bart Anderson , "A Theory of Illusory Lightness and Transparency in Monocular and Binocular Images: The role of contour junctions", Perception, 26(4), 1997.
• Metelli, F., "The Perception of Transparency," Scientific American, Vol 230, pp. 90-98, 1974.
• Saund, E., "Perceptual Organization of Occluding Contours of Opaque Surfaces", Computer Vision and Image Understanding: Special Issue on Perceptual Organization", Vol. 76, No. 1, Oct 1999, pp. 70-82.
• Saund, E., "Toward Richer Labels for Visual Structure," in K. L. Boyer and S. Sarkar, eds, Perceptual Organization for Artificial Vision Systems, Kluwer, 2000.