Figure 1.
Two dramatic effects of perceptual layering and surface appearance.
(A) Adelson checkerboard image [1] adapted from http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html under the Creative Commons Attribution License: Checks labelled A and B (depicted as appearing in bright and dim illumination) have the same point-to-point luminance but check B appears light gray and check A dark gray. Checks B and D are seen through a ‘transparent shadow layer’, whereas checks A and C are seen in ‘plain view’ (without an accompanying transparent layer). Variations in illumination intensity level produce multiplicative changes in the luminance values depicted as being reflected from the checks in bright and dim illumination. (B) Anderson-Winawer effect reprinted from [12]: Chess pieces in the upper and lower rows have the same point-to-point luminance but appear white and black, respectively. The white pieces are seen through a blackish transparent ‘atmosphere’ whose transparency varies across space, while the black pieces are seen through a transparent whitish atmosphere. Variations in atmospheric transmittance levels produce additive changes in the luminance values depicted as being reflected from the black and white chess pieces. This article develops a model that aims to quantitatively predict surface lightness through transparent layers, irrespective of the physical source of the transparent layer.
Figure 2.
The representation of brightness and lightness in gamut relativity.
(A) Surface regions represented under the assumption of a single illumination level and a planar arrangement of surfaces, such as co-ordinates and
, fall on a negatively sloped ‘gamut’ line in blackness-whiteness space, where
and
denote the columns of relatively higher and lower luminance squares depicted in the insets, respectively. (B) Surface regions represented under the assumption of two different illumination intensity levels and a corrugated arrangement of surfaces, such as co-ordinates
and
, fall on two different gamut lines (termed standard and comparison, respectively). The inset figures in (A) and (B) perceptually illustrate how identical sets of luminance values can be parsed according to the assumptions of uniform or variable illumination levels, respectively. In (A), pictorial image cues indicate that the bright and dark columns of squares (sets
and
) lie in the same depth plane, favoring the assumption of uniform illumination over all squares [25], [89], [90]. Horizontal pairs of squares are thus mapped to different blackness co-ordinates,
. As blackness co-ordinates constitute the computational correlate of diffuse reflectance in gamut relativity, squares in sets
and
appear to have different diffuse reflectance. In (B), the same sets of luminance values shown in the two columns in (A) are now pictorially depicted to lie in different depth planes (the repetition of rows here enhances this depiction), favoring the assumption of variable illumination [8], [25], [85], [89], [90]. Horizontal pairs of squares in this arrangement are mapped to the same blackness co-ordinates,
, and thus appear to have the same diffuse reflectance. The horizontal vector depicts the shift of points from standard to comparison gamuts, which compensates for the presumptive illumination difference between sets
and
. Figure modified with permission from [56].
Figure 3.
Two examples of image segmentations used to guide the computation of region luminance and contrast.
(A) Adelson checkerboard image [1], modified with permission under the Creative Commons Attribution License. (B) Segmentation computed with a standard computer vision algorithm [84] (parameters: ,
). (C) The algorithm returns region labels for each image region. (D) Region labels enable the calculation of mean pixel or luminance values within each segmented region. (E-H) Same as above, except applied to a simple version of the Anderson-Winawer display (adapted from http://www.psy.ritsumei.ac.jp/~akitaoka/AIC2009.html with permission).
Figure 4.
Adelson checkerboard display parsed in the brightness and lightness modes.
The model explains the key perceptual properties implied by the Adelson checkerboard display shown in Fig. 1A. Surface gray shades are specified in a perceptual blackness-whiteness space given by the coordinates ). The free parameter
controls the balance between so-called brightness (
) and lightness (
) ‘modes’ that represent the respective assumptions of spatially uniform or variable illumination. (A) Brightness mode: According to the model, the summation of luminance and contrast vectors ensures that check B in the Adelson checkerboard display has higher whiteness than check A (
with respect to
and
) and check A has higher blackness than check B (
with respect to
and
), consistent with various data on the simultaneous contrast effect [54]. (B) Lightness mode: According to the model, an illuminant-shift process combines with the vector summation underlying simultaneous contrast to produce the Adelson checkerboard effect, i.e.
=
+
, where
is a ‘shadow vector’ with non-zero blackness and zero whiteness components that introduces the comparison luminance gamut,
. The illuminant-shift process transforms the blackness coordinates of checks B and D in relatively dim illumination towards the blackness axis, e.g.
is smaller in lightness mode than it is in the brightness mode example illustrated in subfigure (A). Checks with the same reflectance thus share the same blackness coordinates (
), and checks with different reflectance but the same luminance have very different blackness coordinates (
with respect to
and
). Due to the asymmetrical scaling of blackness coordinates relative to whiteness coordinates, blackness plays the dominant role in determining the surface gray shade [54]. The model thus explains both the independence of surface gray shades with respect to variable illumination intensity levels and the large magnitude of the Adelson checkerboard effect relative to simultaneous contrast alone. Adelson checkerboard image adapted from http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html under the Creative Commons Attribution License.
Figure 5.
Model predictions of brightness and lightness matching data relating to the Blakeslee-McCourt paint/transparency/shadow display.
(A) The model correctly predicts the influence of task instructions on perceptual matches made with surfaces seen under depicted uniform or variable illumination. The luminance of the target is shown by the dashed line, and predictions of luminance of the test target in each condition shown by the level of each bar. (B) Model luminance predictions shown in (A) were generated from minimal Euclidean distances between points representing the reference gray shades (black points, obtained from Eqn. (2) with and
) and gamut lines representing the test display (red points on gray dotted line). The test display was assumed to have background luminance values equal to
and
, and thus all grey shades in the test display fall on mixed gamut lines, since both blackness and whiteness coordinates have non-zero contrast components. (C) Data and depiction of stimuli reprinted from [58]. In total, there are 12 different test conditions: 6 of these are brightness tasks and 6 are lightness tasks. In (B), black dots indicate the blackness-whiteness coordinates,
, for 8 of these 12 conditions. As the model predictions for the ShadowL and ShadowR conditions are equally applicable to the experimental TransL and TransR (transparency) conditions, we omit the 4 transparency conditions. There are only 3 unique coordinates, since the same blackness-whiteness coordinates at approximately
are obtained for all L conditions, and the same coordinates at approximately
are obtained for both PaintR conditions and ShadowR (labelled ShadR above) in the brightness task. The final black dot at approximately
occurs uniquely for ShadowR in the lightness matching task. The red arrow indicates the minimal perceptual match between reference and test coordinates for both PaintR conditions and ShadowR in the brightness matching task.
Figure 6.
Anderson-Winawer display parsed in the brightness and lightness modes.
(A,B) The Anderson-Winawer display with blackish and whitish backgrounds, respectively. (C,D) Brightness mode: The empty gray circles ( with
) form the standard luminance gamut line for each pixel contained within each of the whitish or blackish squares shown in (A,B). The filled gray circles (
with
) form the standard increment and decrement gamut lines in (C) and (D), respectively, similar to Fig. 4A. These points, which are offset from the standard luminance gamut due to addition of the whiteness and blackness contrast vectors, would correspond to the perceived gray shades in (A,B) if the squares where rotated by
(rotation now shown here). Lightness mode: The model explains how the visual system computes separable whitish and blackish figural surface layers (
) through blackish and whitish transparent ‘ground’ layers (
) when
. The transmittance-shift process subtracts the vector
from each filled gray circle to compute each
(
with
). Surface layers are composed of the collection of every
, represented here by the empty and filled black circles falling on the whiteness and blackness axes, respectively. The vertical and horizontal rows of empty and filled black circles thus correspond to the perceptually whitish and blackish layers evident in (A,B), respectively. The labelled vector corresponds to
,
denotes the whitest pixel within the target region, and
denotes the blackest pixel in the surrounding region. Note that
.
Figure 7.
Model predictions of lightness matching data relating to the Anderson-Winawer effect.
(A,C) Model luminance predictions were generated from minimal Euclidean distances between points representing the reference gray shades (black points) and gamut lines representing the test displays (red/blue points on purple dotted lines). Each black dot represents either the highest or lowest luminance value within the target reference region associated with each Michelson contrast level, depending on the figural contrast polarity of the reference region (i.e. highest for black dots matched to blue dots, and lowest for black dots matched to red dots) (B,D) The model correctly predicts that subjects set luminance values higher than the luminance values of the target reference region appearing in plain view (black dotted lines) for both figural contrast increments (blue points/lines) and decrements (red points/lines). Each empty blue and red dot corresponds to one of the filled blue or red dots in (A,C). Higher luminance values map to higher whiteness values and lower blackness values, respectively. (E) Data reprinted from [12]. (F) Data replotted from [13].