Perception, 2015, volume 44, pages 243 – 268

doi:10.1068/p7912

The achromatic object-colour manifold is three-dimensional

Alexander D Logvinenko

Department of Life Sciences, Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK; e-mail: [email protected] Received 12 November 2014, in revised form 3 March 2015 Abstract. When in shadow, the achromatic object colours appear different from when they are in light. This immediate observation was quantitatively confirmed by Logvinenko and Maloney (2006, Perception & Psychophysics, 68, 76–83) who, using multi­dimensional scaling (MDS), showed the two-dimensionality of achromatic object colours. As their experiments included only cast shadows, a question arises: is this also the case for attached shadows? Recently, Madigan and Brainard (2014) argued in favour of the negative answer. However, they also failed to confirm the two-dimensionality for cast shadows. To resolve this issue, an experiment was conducted in which observers rated the dissimilarity between achromatic Munsell chips presented in light and in shadows of both types. Specifically, the chips were presented in four conditions: in front in light; at slant in light; in front in shadow; and at slant in shadow. MDS analysis of the obtained dissimilarities confirmed the twodimensionality of achromatic colours for both types of shadow. Furthermore, the dimension induced by the cast shadow (shadowedness) was found to be different from that induced by the attached shadow (shading). In the three-dimensional MDS output configuration these were represented by clearly different dimensions. This quantitatively supports a fact, well-known to artists, that attached and cast shadows are experienced as different phenomenological entities. It is argued that a shading gradient is perceptually experienced as shape (ie spatial relief ). Keywords: lightness, brightness, slant, multidimensional scaling

1 Introduction Achromatic object colours are traditionally assumed to form a one-dimensional continuum from white through various shades of grey to black (Fairchild, 2005; Gilchrist, 2006; Wyszecki, 1986). This belief has been supported by many experiments with homogeneously illuminated reflecting objects (Wyszecki & Stiles, 1982). Yet, as noted by Katz (1911/1999) more than a century ago, the same objects under more (or less) intense illumination seem to form a different continuum. In Katz’s words, the achromatic object colours become more (respectively, less) pronounced. This can be easily verified by observing a series of grey cards in light and in shadow. A well-known fact is that, given a grey card in light, it is impossible to find a card in shadow that will make an exact match to it. It follows that the achromatic continuum in light is different from that in shadow.(1) It is for this reason that Katz claimed the achromatic object colours to be two-dimensional. More recently, the two-dimensionality of the achromatic object-colour palette has been confirmed more objectively by using multidimensional scaling (MDS) analysis (Logvinenko & Maloney, 2006). Specifically, two attributes of achromatic object colours were revealed: one is mainly correlated with the reflecting surface, and the other (1)

 This difference becomes literally obvious when an apparent inversion of depth relief (eg by using a pseudoscope) occurs, and the shadowed area is then perceived as a pigmented spot (Logvinenko & Menshikova, 1994). In this case all the achromatic colours in the hitherto shadowed area alter their appearance. All the achromatic colours now seem to belong to the same one-dimensional continuum. Interestingly, the apparent contrast which the apparent pigmented spot makes with the surround appears stronger than that between the shadow and the surround, despite the fact that the luminance contrast remains the same (Logvinenko, 2005b).

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with surface illumination (2) (referred to as surface brightness). So, we experience a shadow as the achromatic object-colour variation along the surface brightness attribute. The surface brightness is not the same as the subjective intensity of the reflected light. It has been shown that objects reflecting lights of approximately equal intensity might have very different surface brightness (Logvinenko, 2005a; Logvinenko & Maloney, 2006). Also, surface brightness does not seem to be the same as apparent illumination, if the latter is understood to be the subjective intensity of the light source or the ambient illumination. Immediate observation shows that the surface brightness is experienced as a perceptual attribute of the surface rather than the light source or the ambient illumination.(3) However, since the existence of surface brightness in Logvinenko and Maloney’s experiment has been shown for cast shadows (that are produced by the illuminant intensity variation), these two perceptual attributes—apparent illumination and surface brightness—were confounded. Yet, they can be disentangled if attached shadows (produced by, say, surface slant variation) are used instead of cast shadows. Besides this, it is an important and interesting question in itself whether attached shadows are also experienced as the achromatic object-colour variation along some specific perceptual dimension which cannot be reduced to luminance (or brightness). In particular, will the achromatic object-colour continuum change for the stimulus surfaces observed at a slant? And if it does, will it be the same surface brightness dimension or another one? So far, the only attempt to address this issue failed to reveal an additional dimension due to slant (Madigan & Brainard, 2014). However, this study also failed to confirm the two-dimensionality of achromatic object colours under varying illumination intensity. More specifically, these authors failed to replicate Logvinenko and Maloney’s results despite using a very similar approach: MDS of observers’ dissimilarity judgments. It should be pointed out that there were a few important methodological differences between these two studies (see section 4). As these differences might have been crucial, we decided to follow up the previous study (Logvinenko & Maloney, 2006) varying not only stimulus illumination but also stimulus slant. Our objectives were: first, to find out whether we can replicate the previous finding concerning the special dimension of achromatic object colours induced by cast shadows; second, to check whether a similar dimension also exists for attached shadows; and, third, to ascertain whether these dimensions are the same for attached and cast shadows. 2 Methods 2.1  Apparatus Four identical sets of eight matt Munsell chips (4 × 5 cm each) served as stimuli (table 1). They were mounted on the vertical rectangular stimulus display covered by white paper with a random-dot design (figure 1). One (left) half of the display was perpendicular to the observer’s view; the other was bent at 70°. A digital projector illuminated the stimulus display so that the upper part (of both halves) was in light, and the bottom part was in shadow. So, there were four quadrants (24 × 40 cm each) filled with the same Munsell chips: (i) the left one in light, ie bright in frontal view (denoted as BF); (ii) the right one in light, ie bright at slant (denoted as BS); (iii) the left one in shadow, ie dim in frontal view (denoted as (2)

 It should be borne in mind that the two-dimensionality in question exists only in real (natural) perception—that is, for real scenes comprising real objects illuminated by real lights. Achromatic pictorial perception has been found to be one-dimensional (Logvinenko, Petrini, & Maloney, 2008). (3)  When the apparent inversion of depth relief results in the shadowed area being perceived as a pigmented spot, its surface brightness increases, becoming equal to that of the surround (Logvinenko & Menshikova, 1994). This happens despite the fact that the physical illumination of the scene remains unchanged.

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Table 1. Munsell chips employed in the experiment.

1 2 3 4 5 6 7 8

Munsell Value

Reflectance/%

N 2.75/ N 3.25/ N 3.75 N 4.5/ N 5.5/ N 6.5/ N 7.75/ N 9.0/

5.5 7.7 10.4 15.6 24.6 36.2 54.8 78.7

Figure 1. Experimental setup (see text). The four-part stimulus display was presented against a wall covered with black cloth. The right half of the display made an angle of 110° with the left half. Quadrants BF and BS were illuminated by bright light; DF and DS were illuminated by dim light produced by the digital projector next to the observer. The location of the Munsell chips (table 1) is marked with eight squares per quadrant. The Munsell chip array was random in each quadrant. The location of the pieces of white and black paper providing anchors for the maximum of dissimilarity is designated by numbers 1, 2 (white), 3, and 4 (black).

DF); and (iv) the right one in shadow, ie dim at slant (denoted as DS). They determine four different stimulus conditions (BF, BS, DF, and DS), respectively. The luminances of the light reflected by each chip under all the four stimulus conditions are presented in figure 2. They vary from 2161.3 cd m−2 (white, N 9.0/, in frontal view in light) to 5.9 cd m−2 (black, N 2.75/, at slant in shadow). Notably, the decrease in the reflected light intensity produced by the decrease in the illumination (in the bottom part of the display) is approximately equal to that induced by the obliquity (the green and red lines in figure 2 are close to each other). The average (across Munsell chips) luminance ratio for BF vs DF stimulus conditions is 10.1; that for BF vs BS stimulus conditions is 11.6. The average luminance ratio for BF vs DS stimulus conditions is 55.8; that for DF vs BS stimulus conditions is 1.17. Hence, stimulus conditions DF and BS were practically equiluminant.

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BF DF BS DS

Figure 2. [In colour online, see http:// dx.doi.org/10.1068/p7912] The lumi­ nances of the light reflected from the stimulus Munsell chips. The mean of four measurements of each stimulus made at the various stages of the experiment is presented.

102

101  101

Reflectance/%

The illumination was arranged so as to keep the average amount of light constant throughout the experiment. For this purpose two vertical flank strips (ie the extensions of the panel to the right and left of the stimulus grids) were used (figure 1). It is their illumination that varied depending on illumination of the four quadrants so as to counterbalance the overall illumination over conditions. Specifically, their illumination was the same as that of the stimulus grids in the main panel when two lights were engaged. However, as implied in subsection 2.4, in some experimental sessions (eg BF vs BF or DS vs DS) the whole panel was illuminated with only one light. In this case the average illumination differed by a factor of 2. To counterbalance such an increase (or decrease, depending on which light— bright or dim—was used), the extensions (the area of each of which was two thirds of a single panel)(4) were illuminated with the missing light. For example, when the stimulus grids were lit by the bright light (as in BF vs BF), the extensions were illuminated by the dim light, and vice versa. Next to each Munsell chip, an LED was built into the stimulus display board. All the LEDs were driven by a PC that turned them on and off to indicate a pair of Munsell chips in different quadrants (under different or same stimulus conditions(5)). 2.2  Observers Four undergraduate optometry students with normal vision served as observers. All had no prior experience in psychophysical studies. Two of them were partly aware of the purpose of the experiment, the other two being completely naive (with respect to the purpose of the experiment). 2.3  Experimental task and procedure Observers were asked to rate with a number (0 to 100) the dissimilarity in colour appearance between the pair of Munsell chips indicated by the LEDs. They were asked to assign a 0 when completely identical appearance was observed. A rating of 100 was suggested for a pair made of a piece of white paper attached in the bright quadrant on the left (ie under BF stimulus condition) and a piece of black paper in the dim quadrant on the right (ie under DS stimulus condition). This pair was always present in the observer’s view, and served as a standard of dissimilarity. Realising that it might be difficult for observers to choose between numbers when rating, we advised them to consider ratings as ordinal numbers. (4)

 The extensions were made smaller than the main panels because the former were made of white paper whereas the latter were covered by random-dot paper and with the Munsell chips, the reflectance of which was considerably lower. (5)  Recall that in 40% of experimental sessions the stimulus conditions were identical.

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Specifically, they were instructed to keep the order between given numbers in line with the order between the corresponding dissimilarities. In other words, if the dissimilarity between stimuli x and y is judged as larger than the dissimilarity between stimuli x l and y l , then the number assigned to (x, y) should be larger than that assigned to (x l , y l  ). The observer used a keypad to enter his or her ratings in response to the indicated pair of Munsell chips. The next trial did not start until the observer’s response to the current trial was entered. The observer sat 1.3 m from the stimulus display in a semidarkened room. Although there was no particular restriction on the condition of viewing, the observers were advised to refrain from head movements when making similarity judgments. Viewing time was unlimited. 2.4  Experimental design All ten possible pairs of the four stimulus conditions were tested five times each: BF vs BF; BF vs BS; BF vs DF; BF vs DS; BS vs BS; BS vs DF; BS vs DS; DF vs DF; DF vs DS; and DS vs DS. In one experimental session one pair of stimulus conditions was tested once. Consider, as an example, stimulus condition pair BF vs DF. In an experimental session testing these stimulus conditions each of the eight Munsell chips in the bright left quadrant (BF) was successively indicated in a random order with each of the eight Munsell chips in the dim left quadrant (DF). An outcome of a single experimental session was an 8 × 8 dissimilarity response matrix, the entries of which were the ratings assigned to the pairs of Munsell chips. 2.5  Ranking the stimulus conditions In the beginning of the first three experimental sessions the observer was asked to find the maximally dissimilar pair in each of the ten pairs of stimulus conditions when all were presented simultaneously. (Mostly the choice was white, N 9.0/, versus black, N 2.75/, when the two quadrants compared were identical. When the quadrants were different, it was white, N 9.0/, in the brighter quadrant versus black, N 2.75/, in the darker quadrant.) Then, they were asked for which of the ten pairs of stimulus conditions the maximally dissimilar pair in it was smallest as compared with the other pairs. Subsequently, they were asked to find the smallest one among the other 9 stimulus condition pairs, and so on. As a result, all the 10 maximally dissimilar pairs were rank ordered. They were then asked to rate them according to the standard of 100. This ranking of and rating maximally dissimilar pairs was intended to help the observers maintain the same standard of dissimilarity across experimental sessions. 3 Results All five dissimilarity response matrices for a single pair of stimulus conditions were averaged for each observer and combined into the whole dissimilarity response matrix. The average (across the four observers) dissimilarity response matrix is presented in figure 3. 3.1  Least-dissimilarity analysis As one can see in figure 3, the minimum dissimilarity for symmetrical stimulus condition pairs (eg BF vs BF )—that is, in the four 8 × 8 submatrices on the main diagonal—is mainly achieved for the diagonal entries. However, this is not the case for asymmetrical stimulus condition pairs (eg BF vs DF). Specifically, the minimum dissimilarity is achieved mostly from the main diagonals in the other 8 × 8 submatrices. For each chip under each stimulus condition, the least-dissimilar match was evaluated. More specifically, for a given chip under a given stimulus condition (referred to as the test chip and the test stimulus condition, respectively) we found the chip under another stimulus condition (referred to as the match stimulus condition) that was least dissimilar (from the test chip under the test stimulus condition). Following the terminology used by Logvinenko

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100 90

DS

80 70 DF

60 50 40

BS

30 20 BF

10 BF

BS

DF

DS

Figure 3. [In colour online.] Average dissimilarity response matrix. It comprises sixteen 8 × 8 submatrices, each of which corresponds to a stimulus condition pair (eg BF vs BS). Within each such submatrix the rows and columns are ordered with respect to Munsell Value (increasing from left to right and from bottom to top). Note that submatrices below and above the main diagonal are identical simply because they represent the same data.

and Tokunaga (2011a), this chip will be referred to as the least-dissimilar match. The results are presented in figures 4–7. Figure 4 shows the least-dissimilar matches for the chips under the symmetrical stimulus condition pairs. Figure 5 shows the least-dissimilar matches for the chips under the asymmetrical stimulus condition pairs that differ in illumination with the orientation remaining the same. Figure 6 presents the same for the asymmetrical stimulus condition pairs differing only in orientation. Figure 7 displays the least-dissimilar matches for the chips under the asymmetrical stimulus condition pairs when both illumination and orientation are different. As can be seen in figure 4, for the symmetrical stimulus condition pairs the leastdissimilar match differs (by one step)(6) from the physical match for not more than one

8

Munsell chip number

7

BS BF DS DF

6 5 4 3 2 1 1

(6)

2

3 4 5 6 Munsell chip number

7

8

Figure 4. [In colour online.] Leastdissimilar match for the symmetrical pairs of stimulus conditions. The test chip is plotted along the horizontal axis. The least-dissimilar match is on the vertical axis.

 Here a step means the distance between adjacent chips in the sample used in the experiment.

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(of eight) chip. Specifically, under the symmetrical stimulus condition pairs, BF vs BF and DF vs DF, all the least-dissimilar matches coincide with the physical match. For the other two symmetrical stimulus condition pairs the least-dissimilar match deviates from the physical match by one eighth of a step on average. DS vs BS BS vs DS BF vs DF DF vs BF

8

Munsell chip number

7 6 5

Figure 5. [In colour online.] Leastdissimilar match for the asymmetrical pairs of stimulus conditions differing in illumination, the orientation being the same. The test chip is plotted along the horizontal axis. The least-dissimilar match is on the vertical axis.

4 3 2 1 1

3 4 5 6 Munsell chip number

7

8

BF vs BS BS vs BF DS vs DF DF vs DS

8 7 Munsell chip number

2

6 5

Figure 6. [In colour online.] Leastdissimilar match for the asymmetrical pairs of stimulus conditions differing in orientation, the illumination being the same. The test chip is plotted along the horizontal axis. The least-dissimilar match is on the vertical axis.

4 3 2 1 1

9

3 4 5 6 Munsell chip number

7

8

BS vs DF DF vs BS BF vs DS DS vs BF

8 7 Munsell chip number

2

6 5

Figure 7. [In colour online.] Leastdissimilar match for the asymmetrical stimulus conditions differing in both illumination and orientation. The test chip is plotted along the horizontal axis. The least-dissimilar match is on the vertical axis.

4 3 2 1 0

0

1

2

3 4 5 6 Munsell chip number

7

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9

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The average least-dissimilar mismatch (in step units) for all the stimulus condition pairs is presented in figure 8 and table 2. Note the asymmetry in this table. For example, as shown in figure 7, Munsell chip #6 under the dim illumination at slant (stimulus condition DS) is least dissimilar from Munsell chip #4 under the bright illumination oriented perpendicular to the observer’s line of sight (stimulus condition BF). However, it does not follow that the latter (ie Munsell chip #4 under the bright illumination oriented perpendicular to the observer’s line of sight) will be least dissimilar from Munsell chip #6 under the dim illumination at slant. As can be seen in figure 7, it is Munsell chip #3 under stimulus condition BF that makes the least-dissimilar match when Munsell chip #6 under stimulus condition DS is a test.

2.0 1.5 1.0 0.5 0.0  DS

h

atc

M

DF

rn

tte

pa

BS BF

DF

BF

DS

BS ttern Test pa

Figure 8. [In colour online.] Average least-dissimilar mismatch (table 2) presented as a bar graph.

Table 2. Average least-dissimilar mismatch. Test

BF BS DF DS

Match BF

BS

DF

DS

0.000 0.375 0.875 1.875

0.375 0.125 0.625 1.125

0.750 0.500 0.000 1.125

1.875 1.000 1.125 0.125

The least-dissimilar mismatch averaged across chips and over the four stimulus condition pairs represented in figure 5 is 0.94 steps, whereas that in figure 6 is 0.75 steps. Hence, if least-dissimilar mismatching is taken as an indicator of achromatic object-colour inconstancy, then one can conclude that achromatic object-colour constancy with respect to illumination was found to be poorer than achromatic object-colour constancy with respect to orientation. When a stimulus condition pair differs in both illumination and orientation, the leastdissimilar mismatch is different for different pairs. For stimulus condition pair BS vs DF (as well as for DF vs BS) the average least-dissimilar match is much less than that for stimulus condition pairs DS vs BF and BF vs DS, which take the maximal value: 1.875. In order to give an idea as to how inconstant the achromatic object-colour perception was in our experiment in terms of more conventional measures, the Thouless (1931) ratio—that is, a logarithmic version of the Brunswick ratio—was evaluated for each

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least-dissimilar match:

ln ^ LMMh - ln ^ LTTh , (1) ln ^ LMTh - ln ^ LTTh where LTT is the luminance of the light reflected by the test chip in the test stimulus condition; LMT is the luminance of the light reflected by the test chip in the match stimulus condition; and LMM is the luminance of the light reflected by the match chip in the match stimulus condition. Average values of ratio (1) are presented in table 3. Being rather high, Thouless ratios indicate that, in the traditional terminology, quite high achromatic object-colour constancy was observed in our experiment. Table 3. Average Thouless ratios. The test stimulus condition is the same for every row; the match stimulus condition is the same for every column. NA indicates when the denominator is too close to zero for the ratio to be stable. Test

BF BS DF DS

Match BF

BS

DF

DS

NA 0.93 0.86 0.73

0.96 NA NA 0.74

0.89 NA NA 0.82

0.90 0.85 0.87 NA

3.2  MDS analysis The maximal dissimilarity for each of the ten stimulus condition pairs was evaluated from the average dissimilarity response matrix (figure 3) and is displayed in table 4. Although the individual rankings and ratings obtained in the beginning of the experiment (see section 2.5) varied somewhat from time to time as well as between observers, they are, by and large, in line with those in table 4. Yet, there are two distinctions. Firstly, the range of maximal dissimilarities in the middle column in table 4 is slightly narrower than that of individual ratings. Specifically, the maximal dissimilarity for stimulus condition pair DS vs DS was on average 70% that for stimulus condition pair BF vs DS. Secondly, all the observers yielded more or less equispaced ratings, whereas the differences between some values in the middle column in table 4 are very small (eg between BS vs DF and BS vs BF ). The likely reason for Table 4. Maximal dissimilarity values for various stimulus condition pairs as derived from the observer response matrix (middle column), and after rescaling (right column).

1 2 3 4 5 6 7 8 9 10

Stimulus condition pair

Maximal dissimilarity

Rescaled maximal dissimilarity

DS vs DS DF vs DF DS vs DF BS vs BS BF vs BF BS vs DF BS vs BF BF vs DF BS vs DS BF vs DS

78.2 79.5 80.2 81.6 84.4 87.3 87.4 89.8 90.2 98.6

70.0 73.3 76.7 80.0 83.3 86.7 90.0 93.3 96.7 100.0

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this discrepancy might be observers’ inability to keep the standard of dissimilarity constant across experimental sessions. So, the dissimilarities underwent rescaling so as to expand the range and to make them equidistant (the rightmost column in table 4). More specifically, the dissimilarities in each 8 × 8 submatrix corresponding to a particular stimulus condition pair were multiplied by a number so as to make the maximal (rescaled) dissimilarity equal to the corresponding value in the rightmost column in table 4. For example, for the submatrix corresponding to DS vs DS this multiplier was 70.0/78.2. 3.2.1  Dissimilarity shift induced by illumination. In order to compare our results with previous experiments where only illumination varied (Logvinenko & Maloney, 2006; Madigan & Brainard, 2014), the submatrix (of the dissimilarity response matrix in figure 3) made by stimulus condition pairs BF vs BF, BF vs DF, DF vs BF, and DF vs DF was used as an input to the nonmetric MDS routine (namely, the Matlab code ‘mdscale’). The rationale was that such a submatrix can be considered as the result of an experiment in which, along with reflectance, only illumination varied (at two levels). The output configuration is shown in figure 9. 15 10 BF DF

Dimension 2

5 0 −5 −10 −15 −50

0 Dimension 1

50

Figure 9. The output configuration resulting from nonmetric multidimen­ sional scaling. Black dots stand for the bright illuminant; grey dots indicate the dim illuminant. Note the different scales on the axes in this and the following figures (10–17).

Black markers stand for the eight chips lit by the bright light, and grey markers indicate the same chips lit by the dim light. The leftmost dots indicate the blackest chips (N 2.75/ ), and the rightmost indicate the whitest ones (N 9.0/ ), with the intermediate dots being ordered by Munsell Value. As one can see, the curve made up of the black markers is clearly shifted downwards with respect to the curve made up of the grey markers. This apparent vertical shift between the curves representing the two illumination conditions in figure 9 can be quantified in terms of average distance between corresponding chips in the curves, which has been evaluated as 16.2. It equals 18.3% of the maximal distance between the black and white chips under the same (bright) illumination. Although the distances in figure 9 are, by and large, in a good correspondence with the dissimilarities, the shift directly evaluated in terms of dissimilarities is somewhat different from that evaluated in terms of distances. Specifically, the mean dissimilarity between the same chips illuminated by different lights (in table 5 this is referred to as mean diagonal dissimilarity) is 11.1—that is, 13% of the maximal dissimilarity between equally illuminated black and white chips (table 5). As the minimal dissimilarity between the chips under different illumination is not always achieved for the pair of identical chips (see figures 5–7), the mean minimal dissimilarity between the chips under different illumination is slightly less: 9.1 (10.8%).

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Table 5. Minimal dissimilarities. The second column presents the minimal dissimilarity between the chips under two stimulus conditions (indicated in the first column) averaged across repetition and observer. The third column presents the minimal dissimilarity between the chips of the same Munsell Value under the two stimulus conditions averaged across repetition and observer. The fourth column presents the smallest dissimilarity between the same chips under the two stimulus conditions. The last column is the dissimilarity from column 3 normalised by the dissimilarity between equally illuminated black and white chips. Stimulus condition pair

Mean minimal dissimilarity

Mean diagonal dissimilarity

Minimal diagonal dissimilarity

Relative shift/%

BF vs DF BS vs DS BF vs BS DF vs DS BS vs DF BF vs DS

9.1 9.7 9.9 10.4 10.7 15.4

11.1 18.9 10.0 12.9 12.6 23.5

6.7 5.7 5.8 7.1 5.3 9.5

13.2 23.4 13.1 16.6 15.6 29.1

The smallest dissimilarity between the chips of the same Munsell Value was found to be 6.7 (7.6%). Of all these three measures, the mean relative (in percentage) dissimilarity between the chips of the same Munsell Value seems to be the most appropriate index of the dissimilarity shift induced by illumination. It is referred to in table 5 as relative shift. Figure 10 represents the output configuration resulting from MDS analysis of the submatrix made by stimulus condition pairs BS vs BS, BS vs DS, DS vs BS, and DS vs DS. It emulates a similar experiment in which illumination varied at the same two levels except that all the chips were slanted rather than being perpendicular to the observer’s view. A similar vertical displacement of the curves relative to each other can be observed in figure 10. The mean minimal dissimilarity between the differently illuminated chips is 9.7 (13.0% of the dissimilarity between equally illuminated black and white chips), the mean dissimilarity between identical (but differently lit) chips being 18.9 (23.4%). 15

BS DS

Dimension 2

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−10 −60

  −40

 −20

0 Dimension 1

20

40

60

Figure 10. The output configuration resulting from the nonmetric multi­ dimen­sional scaling. Black squares stand for the bright illuminant; grey squares indicate the dim illuminant.

3.2.2  Dissimilarity shift induced by slant. Figures 11 and 12 display the output configurations for the submatrices made by stimulus condition pairs BF vs BF, BF vs BS, BS vs BF, and BS vs BS; and DF vs DF, DF vs DS, DS vs DF, and DS vs DS, respectively. They emulate two experiments in which the illumination was fixed, and slant varied. There is an obvious dissimilarity shift between the curves representing equioriented chips (see also table 5).

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BF BS

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10

Figure 11. The output configuration resulting from nonmetric multi­dimen­ sional scaling. Dots stand for the chips in front; squares indicate the chips at slant.

DF DS

8 6

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4 2 0 −10 −20 −30 −40 −40

  −30

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Figure 12. The output configuration resulting from nonmetric multidimen­ sional scaling. Dots stand for the chips in front; squares indicate the chips at slant.

Note that the leftmost black dot (N 2.75/ in stimulus condition BF) and the rightmost grey dot (N 9.0/ in stimulus condition DF) in figure 9 are very far apart despite the fact that these chips (N 2.75/ and N 9.0/) in the corresponding stimulus conditions reflect lights of very close intensity (see figure 2). At the same time the separation of these markers (representing the stimuli with close luminance) is a few times greater than the separation between the points representing the chips of the same Munsell Value in this figure that reflect lights, the intensity of which differ by a factor of 10 (see table 5). That is also the case for the leftmost black square (N 2.75/ in stimulus condition BS) and the rightmost grey square (N 9.0/ in stimulus condition DS) in figure 10, and the case for the leftmost black dot (N 2.75/ in stimulus condition BF) and the rightmost black square (N 9.0/ in stimulus condition BS) in figure 11, as well as the case for the leftmost grey dot (N 2.75/ in stimulus condition DF) and the rightmost grey square (N 9.0/ in stimulus condition DS) in figure 12. Thus, the stimuli, the luminances of which were very similar, were judged as nearly maximally dissimilar, whereas the stimuli with tenfold luminance ratio were judged as a great deal less dissimilar. All this shows that the dissimilarity judgments in this experiment (as in the previous experiment by Logvinenko and Maloney, 2006) are based on anything but the luminance (or luminance ratio) of the reflected light.

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3.2.3  Dissimilarity shift induced by both illumination and slant. Figures 13 and 14 present the output configurations produced for submatrices made by stimulus condition pairs DF vs DF, DF vs BS, BS vs DF, BS vs BS; and BF vs BF, BF vs DS, DS vs BF, DS vs DS, respectively. In these, the stimulus conditions differ in both illumination and orientation. However, while stimulus conditions DF and BS were approximately equiluminant (see figure 2)—the mean luminance ratio for the chips with equal Munsell Value under stimulus conditions DF and BS was 1.17—the difference in luminance in the stimulus condition pair BF and DS was maximal, the mean luminance ratio for the chips with equal Munsell Value under stimulus conditions BF and DS being 55.8. Yet, the dissimilarity shift in figure 13 is as pronounced as in figure 14. True, the dissimilarity shift in the stimulus condition pair BF and DS is nearly twice as much as that in the stimulus condition pair DF and BS (see table 5). However, it is highly unlikely that it could be accounted for by the difference in luminance ratio between these stimulus condition pairs. 15 10

BS DF

Dimension 2

5 0 −5 −10 −15 −50

0 Dimension 1

 50

15

Figure 13. The output configuration resulting from nonmetric multidimen­ sional scaling. Notation is same as in figures 9 and 12.

BF DS

10

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It is worth pointing out that the maximal luminance ratio produced by Munsell chips (N 9.0/ vs N 2.75/) is 14.3, whereas the average luminance ratio produced by the shadows in the BF vs DS stimulus condition pair is 55.8. However, as can be seen in table 5, the dissimilarity invoked by the shadows does not exceed even one third of that invoked by the reflectance difference. Interestingly, the luminance ratio produced by Munsell chips N 9.0/ and N 3.25/ is

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10.2—that is, practically equal to that produced by the cast shadow in the BF vs DF stimulus condition pair (10.1). Yet, the dissimilarity between N 9.0/ and N 3.25/ is more than 7 times larger than that between the identical chips in this stimulus condition pair (see table 5). It follows that the luminance-contrast contribution to the observer’s judgment depends on whether it is experienced as lightness contrast or shadow contrast. This is in line with the previous finding, mentioned above, that a shadow of the same luminance as the pigmented spot appears much less pronounced (ie it seems to be of lower apparent contrast) than this pigmented spot (Logvinenko, 2005b). 3.2.4  3‑D output configuration. Figure 15 presents the output configuration resulting from nonmetric MDS analysis of the full dissimilarity response matrix (figure 3) when the number of dimensions is restricted to three. It is clearly three-dimensional (3‑D). The stress value for this 3‑D solution is found to be 0.0500, the corresponding stress values for two-dimensional and four-dimensional solutions being 0.0653 and 0.0446, respectively. 15

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There is a view (figure 16) where the markers of the same colour (representing the chips of the same illumination) largely overlap, the curves with black markers being distinctively shifted with respect to the curves with grey markers. The direction along which this shift takes place lends itself to being tentatively named as the illumination direction. This seems to be the dissimilarity shift revealed in figures 9 and 10. BS BF DS DF

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On the other hand, there is a view (figure 17) in which there appears to be a common dissimilarity shift for the markers of the same shape (representing the chips of the same orientation). The direction of this shift can be tentatively named as the orientation direction. The displacement along this direction is likely to be just another manifestation of the same dissimilarity shift as in figures 11 and 12.

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Figure 17. One more view for the output configuration presented in figure 15.

4 Discussion MDS analysis of the dissimilarity judgments of Munsell chips differently illuminated but equally oriented reveals their two-dimensional structure (figures 9 and 10) in line with the results of Logvinenko and Maloney (2006), and contrary to Madigan and Brainard (2014). Furthermore, in contrast to Madigan and Brainard’s study, the two-dimensional structure of the dissimilarity judgments was also revealed when using equally illuminated Munsell chips of different orientation (figures 11 and 12). There are two main important distinctions between the present study and Logvinenko and Maloney’s (2006), on the one hand, and Madigan and Brainard’s (2014), on the other. Firstly, in the present study (as well as in Logvinenko and Maloney’s) all the stimuli were presented in every trial, keeping a standard (the dissimilarity maximum) permanently in the observers’ view, whereas Madigan and Brainard (2014) presented one stimulus pair at a time without displaying a standard. Keeping the standard of dissimilarity in view provides an opportunity to make a judgment with respect to this standard that, in turn, makes the judgments more consistent. Perhaps Madigan and Brainard’s failure to reveal the twodimensionality in question can be partly accounted for by the absence of the standard of dissimilarity permanently in the observers’ view. Secondly, we suggested that the observers should evaluate the dissimilarity between the achromatic object colours, whereas Madigan and Brainard (2014) asked their observers to provide a dissimilarity rating based on either “the apparent reflectance of the two test cards” or “the apparent amount of light reflected from the two cards” (page 57). Such a difference in instruction alone might have predetermined the one-dimensionality of the output configuration obtained by them. Indeed, while it is not quite clear how the observers understood the notion of ‘apparent reflectance’ (see section 4.2 on the controversy concerning the notion of reflectance, not to mention ‘apparent reflectance’), if they interpreted it as something like ‘lightness’, or any other subjective correlate of a one-dimensional physical variable, then it is hardly surprising that the output configuration was one-dimensional. At any rate, the onedimensional output configurations yielded by Madigan and Brainard’s observers instructed this way do not seem to contradict the two-dimensionality of achromatic colour claimed by

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Logvinenko and Maloney (2006) and in the present study. Suppose the achromatic object colour has two dimensions: say, lightness and surface brightness (or pronouncedness). When instructing observers to judge the dissimilarities between achromatic object colours, one might expect both the dimensions to contribute to dissimilarity judgments, producing two-dimensional output configurations.(7) However, if one instructs observers to base the dissimilarity judgments on only one of the two dimensions, the most likely result would be a one-dimensional output configuration.(8) MDS analysis of the full dissimilarity matrix results in the three-dimensionality of the output configuration (figure 15). One of the three dimensions in figure 15 is obviously related to the surface reflectance. From the other two, one is clearly associated with surface illumination, and the other with surface orientation. One might argue that these two dimensions are cognitive rather sensory ones. For example, one could suggest that the cognitive (inferred) difference in illumination might have contributed to observers’ dissimilarity judgments. However, this is hardly the case since Logvinenko, Petrini, and Maloney (2008) showed that this second dimension does not appear when a picture with a strong impression of difference in illumination between the different areas is observed. If not cognitive, could then these two dimensions have been subjective correlates of the reflected light—a sort of derivative of the brightness of the light reflected by the surface? Indeed, one of these is clearly associated with cast shadow, and the other with attached shadow. Physically, shadows of both types produce the same effect: a decrease in light reaching the eye. Yet, figure 13 testifies that either of these two dimensions cannot be interpreted as a derivative of brightness of the reflected light. Moreover, it shows that these two dimensions are perceptually different. Indeed, stimulus condition DS can be considered, on the one hand, as a (cast) shadowed version of stimulus condition BS and, on the other hand, as an (attached) shadowed version of stimulus condition DF. Now, given DF, it is first shadowed (in terms of attached shadow) (DF → DS), and then brightened (in terms of cast shadow) (DS → BS). Therefore, quadrant BS is a slanted version of quadrant DF illuminated so as to make up for the attached shadow induced by the slant. If attached and cast shadows represent the same perceptual dimension, they should cancel each other. As a result, the appearance of the same chips under stimulus conditions BS and DF should be identical (remember that the luminance of the light reflected from such chips under stimulus conditions BS and DF is practically equal). Still, it is not the case. Admittedly, there are some residual differences in luminance between the corresponding chips. However, they can hardly explain the dissimilarity shift in figure 13. Such a large dissimilarity shift could have emerged only if attached and cast shadows do not interact, being independent perceptual dimensions. In other words, attached and cast shadows are experienced as different perceptual dimensions. 4.1  Achromatic object-colour manifold A traditional view on achromatic object colour as a one-dimensional continuum of various shades of grey from black to white holds true for only flat homogeneously illuminated and equally oriented (with respect to observer) surfaces. A series of neutral Munsell chips on a desktop lit by a single overhead light gives a sample of this continuum. Note that, having two natural end points (black and white), it can be thought of as topologically equivalent to a closed segment. (7)

 Logvinenko and Maloney (2006, page 79) tested statistically the hypothesis concerning the form of a pooling function in terms of which the two dimensions might have combined into the similarity judgment. (8)  All this also applies to the instruction on apparent amount of reflected light. The most plausible inter­ pretation of it would be ‘brightness of the reflected light’, which is by all standards a one-dimensional entity.

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The perceptual dimension in terms of which the achromatic object colours vary is usually referred to as lightness (Gilchrist, 2006; Wyszecki, 1986). It should be borne in mind, however, that lightness is also believed to be one of the three dimensions constituting the manifold of all (not only achromatic) object colours. Specifically, the variety of object colours is thought of as a cylinder made by the Newtonian hue disc (with grey in the centre and pure hues on the circumference) multiplied by the lightness dimension (Fairchild, 2005; Kuehni, 2001; Wyszecki & Stiles, 1982). Yet, such a cylindrical model of object colour runs into a serious problem. It implies that lightness achieves the same maximum for every hue. However, when Munsell (1915) tried to arrange a sample of object colours in a 3‑D space, one of dimensions of which was lightness (‘Value’ in Munsell’s terminology), he found that, first, it is hard to estimate lightness (Value) for different hues and, second, the lightness (Value) maximum was different for different hues. As a result, his sample of object colours resembled more a sphere flattened at the poles rather than a cylinder (Munsell, 1915; Wyszecki & Stiles, 1982). The spherical representation is more in line with that the volume of cone excitations produced by all possible objects (ie their spectral reflectance functions) has the shape of a somewhat deformed sphere, the poles of which are made by the perfect reflector (ie ideal white) and the perfect absorber (ie ideal black) (Schrödinger, 1920; Wyszecki & Stiles, 1982). Recently, using a new (partial hue-matching) technique (Logvinenko, 2012), in line with Hering’s (1920/1964) conjecture, black and white have been confirmed to be component colours on a par with yellow, blue, red, and green (Logvinenko & Beattie, 2011; Logvinenko & Geithner, in press). Still, contrary to Hering’s view, grey was found to be no component colour at all (Logvinenko & Beattie, 2011). That is, grey does not serve as a component for any other object colour. More specifically, the partial hue-matching technique implies that observers should evaluate whether two colours contain any shade in common. From observers’ responses, the chromaticity classes (ie the classes of colours sharing at least one common shade) are then derived. Munsell chips appearing grey make up a class on their own whereas the chromaticity classes including yellow, blue, red, green, black, and white colours comprise the few chips sharing the common colour in question. These six colours lend themselves to be called component colours or component hues (including achromatic hues: black and white). The observers were found to be able to line up the colours in the same chromaticity class with respect to pronouncedness (strength) of the component hue constituting this class (Logvinenko & Beattie, 2006). This makes it possible to describe an object colour in terms of only two variables: hue and purity (Logvinenko, 2015). Hue is characterised in terms of component hues. Purity is a quantitative characteristic reflecting the fact that the component hues can vary in strength (pronouncedness). All the object colours can be then represented as a sphere on the boundary of which the colours of maximal purity lie, grey being in the centre, and purity gradually decreasing along the radii (Logvinenko, 2009a, 2015). The partial hue-matching technique allows objective determination of those colours which consist of the only component (unique or pure colours in the traditional terminology) (Logvinenko & Geithner, in press). It has been confirmed with this technique that there are colours which contain as the only component either black or white. Along with grey (which contain no component at all), these one-component colours (of various purity) are usually referred to as achromatic object colours. As purity varies gradually from its maximum (the ideal black or the ideal white) to zero (grey), one can consider the continuum of white colours (of various purity) and the continuum of black colours (of various purity). These combined with grey will be referred to as the blackness and whiteness continua. In the object-colour solid (as well as in the object-colour sphere) they are represented by two radii connecting the centre (grey) with the corresponding pole (that, actually, make a diameter). Therefore, the set of achromatic object colours can be thought of as the union of the blackness and whiteness continua through the grey point.

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Importantly, the partial hue-matching technique reveals that the black and white component colours are not contained at the same time in any colour (as conjectured by Hering). Therefore, combining them into a single continuum, while formally possible, does not make a single (homogeneous) dimension. If it were, the notion of lightness would be justified. However, it is not. So, being a union of three heterogeneous entities (namely, the blackness and whiteness continua, and grey), lightness seems to be an artificial notion, like warmness (and/or coldness) of colour, which can characterise colour from a specific (aesthetic) point of view but which is not necessary from the scientific standpoint. At any rate, I will avoid the term ‘lightness’, using the term ‘blackness–whiteness’ to refer to the achromatic object-colour continuum. The difference between the traditional term of ‘lightness’ and that of blackness–whiteness is that lightness is believed to be an attribute of any object colour, whereas this is not the case for blackness–whiteness. In other words, according to the traditional view, any object colour (except for black) has some nonzero value of lightness, whereas there are many colours that contain neither black nor white (nor grey) (Logvinenko & Beattie, 2011). Consider now two identical sets of neutral Munsell chips on the same desktop lit by the same light—only one in direct light and the other in shadow (say, produced by some obscuring screen above half the desktop). The sets will now appear different. In particular, being physically identical, the white chips will appear different. A traditional way to describe this difference is to say that the whites look of the same lightness but different apparent illumination. It implies that achromatic object colours can be described by using a one-dimensional continuum of lightness and a one-dimensional continuum of apparent illumination. More formally, the achromatic object colours are believed to be represented as a Cartesian product of two independent continua: lightness and apparent illumination. This view is, in fact, a particular case of a more general view according to which each object colour can be represented by two triplets of numbers: one triplet characterises the reflectance and the second triplet the illumination (MacLeod, 2003). However, it has been proved that such a representation is impossible because of metamer mismatching (Logvinenko, 2013). Such a description suggests that introducing shadow, leaving lightness unchanged, just adds a new dimension: apparent illumination. For example, we see the white chip as the same white but of different apparent illumination. However, if this were the case, it would be natural to expect the least dissimilar match to be the physical match. Indeed, in this case any other pair of differently lit chips (ie which is not the physical match) would differ not only in apparent illumination but also in lightness. Even more importantly, the claim that achromatic object colour can be factored into two independent variables—lightness and apparent illumination—contradicts our immediate introspective experience. Indeed, a shadowy white does not look like a compound percept: white plus something else. To the contrary, achromatic object colour in multi-illuminant scenes is experienced as unitary a percept as in single-illuminant scenes. An alternative view is that each achromatic object colour has its qualitative uniqueness that does not allow it to reduce to just two factors (eg lightness and apparent illumination). For instance, there is the whole continuum of different whites as well as there being the whole continuum of blacks. Different illuminants might invoke different whites as well as different blacks. Moreover, one can perceive a particular black and a particular white as belonging to the same continuum (ie as having the same apparent illumination). Still, one can hardly see any common perceptual component in these blacks and whites. Consider a number of identical series of neutral Munsell chips arranged as a matrix so its reflectance varies horizontally but remaining constant along the vertical dimension. Under neutral homogeneous illumination, each row of such a matrix represents the same continuum of achromatic object colours from black to white. Let us cover this surface with a neutral

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light, the intensity of which gradually varies along the vertical dimension (remaining constant along the horizontal one) and vanishing at the bottom row of chips.(9) Each row of such a display will now represent different blackness–whiteness continua. All these continua differ from each other in what will be referred as shadowedness. It must be said that in the present context this term refers to what was previously called ‘surface brightness’ by Logvinenko and Maloney (2006). Surface brightness will be retained as a generic term. A change in surface brightness can be induced by a highlight as well as a shadow (either a cast or attached shadow). As surface brightness varies in shadows of both types, one has to terminologically distinguish between them. So, the term ‘shadowedness’ will be used to refer to the perceptual attribute of objects induced by cast shadows as such, and ‘shadowedness– brightness’ to refer to the quantitative dimension along which shadowedness varies. Likewise, the term ‘shading’ will be used to refer to the perceptual attribute of objects induced by attached shadows, and ‘shading brightness’ to refer to the quantitative dimension along which shading varies. [The terms ‘shadowedness’ (‘shading’) and ‘shadowedness–brightness’ (‘shading brightness’) are related to each other in the same way as width (height) and length. The horizontal side of a rectangle is usually called width, and the vertical one height.] Thus, a column in this display represents a blackness–whiteness value of various shadowedness–brightness, and a row represents a shadowedness–brightness value of various blackness–whiteness. As shown in table 5, the dissimilarity between black and white in bright illumination is greater than that in a dim one. This is in line with Logvinenko and Maloney (2006), who also found that the blackness–whiteness continuum shrank when the illumination intensity decreased. Furthermore, it seems natural to suppose that at the limit, when the light intensity approaches zero, the blackness–whiteness continuum collapses to a point.(10) Because of this, Logvinenko and Maloney (2006) used a curvilinear triangle to represent the achromatic object colours of various blackness–whiteness and shadowedness. The base of the triangle will represent the upper row in our display, the bottom row mapping to its apex. The lateral sides of the triangle each represent the shadowedness continua of black and white. Analogous experiments with chromatic Munsell chips (specifically, yellow, grey, and blue) showed that the shadowedness dimension came up for these chips as well (Tokunaga, Logvinenko, & Maloney, 2008). Furthermore, MDS analysis of the dissimilarities between chromatic Munsell chips lit by chromatic lights showed that, like achromatic object colour splits into blackness–whiteness and shadowedness, chromatic object colour splits into material colour (the blackness–whiteness being the achromatic part of it) and lighting colour (the shadowedness being the achromatic part of it) (Tokunaga & Logvinenko, 2010a, 2010b, 2010c). In other words, in variegated scenes lit by chromatically inhomogeneous illumination, colour varies in the two domains: the 3‑D material colour and the 3‑D lighting colour, the blackness–whiteness and the shadowedness being their achromatic dimensions (Logvinenko, 2013). Consider now a vertical half-cylinder made of white paper under the same illumination (ie with a similar vertical gradient of light intensity). The illumination gradient will generate a vertical wave of shadowedness on the half-cylinder surface. At the same time along each horizontal direction there will be a gradual change of apparent white induced by the curvature of the half-cylinder surface. The perceptual dimension along which this apparent gradient occurs will be referred as shading according to the terminology put forward above.(11) In other words, the apparent white varies vertically in shadowedness, and horizontally in shading. (9)

 This can be implemented by using a neutral film sheet, the transmittance of which varies, say, sinusoidally along the vertical dimension. (10)  In the dark all cats are grey. (11)  One could use the terms ‘attached shadowedness’ versus ‘cast shadowedness’; however, for the reason which will be clear below I prefer ‘shading’ to ‘attached shadowedness’.

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As in the flat display described above, the illumination vanishing at the bottom side of the half-cylinder makes shadowedness ultimately thick. As a result, the white points along the bottom side perceptually merge into the single point—the apparent full dark. The same happens at the occluding (right-hand side) edge of the half-cylinder. Because of occlusion, the white points along the right side also perceptually merge into the apparent dark. Gluing together the points of the two sides of the rectangle representing the half-cylinder surface yields a two-dimensional geometrical figure shaped as a petal with two sharp apexes: one representing the dark, and the opposite one representing the white with no shadowedness and no shading (ie the white observed in direct light with head‑on orientation). Such a petal will be referred to as the shadowedness–shading continuum of white. Using a surface with an arbitrary albedo to make a half-cylinder, one can see that not only white but any shade of blackness–whiteness can be represented as a petal-like shadowedness– shading continuum. Now, replacing a radius in the curvilinear triangle representing the blackness–whiteness and shadowedness with the corresponding petal-like shadowedness– shading continuum, one can get a 3‑D manifold representing all the achromatic object colours. More formally, the tridimensional set of achromatic object colours is the one‑dimensional bundle of the two-dimensional petal-like shadowedness–shading continua. 4.2  Relevance to the colour constancy problem The heart of the present approach to achromatic object colours is that they are defined as a subset of the entire set of object colours—namely, those which contain either just one of the two (black or white) components or none. The partial hue-matching technique provides the operational means to distinguish achromatic object colours from the others. Accordingly, the theory of achromatic object colour should be just the general theory of object colours applied to only the achromatic subset. There is no need for a special theory of achromatic object colours, just as there is no need for a special theory of, say, red or green colours. The colour constancy problem is a good opportunity to illustrate what sort of problems one runs into when trying to develop a special theory of lightness (as a special dimension) and to consider lightness constancy as a problem on its own. The major problem with the notion of lightness constancy is that neither lightness nor its constancy are properly defined. Maybe the most common definition of lightness is perceived reflectance (Gilchrist, 2006), or perceived albedo. Albedo is usually defined as the ratio of the energy of reflected radiation from the surface to that of incident radiation upon it. However, so defined, albedo will depend on the spectral power distribution of the incident radiation. In other words, even being illuminated by lights of equal energy, a surface will reflect light of different energy depending on the spectral power distribution of incident light (Logvinenko, Funt, & Godau, submitted). Hence, it is impossible to define surface albedo so that it is constant for the same surface irrespective of the illumination. In other words, albedo itself is not constant, so it is difficult to expect perceived albedo (or perceived reflectance), whatever it might mean, to be constant. The situation cannot be rescued by ‘improving’ the definition of albedo. The entire goal of defining some sort of ‘colour descriptor’ that is independent of the illuminant, and determined by only the object, is doomed to failure (Logvinenko, 2013) because of so‑called metamer mismatching (Wyszecki & Stiles, 1982). Owing to metamer mismatching, objects appearing achromatic under one illumination can look coloured under other lights (Logvinenko, Funt, & Godau, 2014). So, generally, object colour cannot be a constant (intrinsic) characteristic of an object in principle (Logvinenko, 2013). Yet, inability to properly conceptualise reflectance with a single colour descriptor like albedo (thus, lightness) does not set aside the perceptual phenomenon that while a white cat at night reflects as much light as a black cat in daylight, we recognise the former as white and

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the latter as black.(12) Textbooks will describe a phenomenon like this as an example of lightness constancy. Replacing a controversial term ‘lightness’ with ‘achromatic object colour’, does this phenomenon really tell us a story of achromatic object-colour constancy with illumination? Quite unlikely. The achromatic object colour of the cat in daylight is different from the achromatic object colour of the same cat in moonlight. These differ in surface brightness. Therefore, strictly speaking, there is no achromatic object-colour constancy with respect to illumination. Nor is there such with respect to orientation. It follows that, as argued before (Logvinenko & Maloney, 2006), asymmetric achromatic colour matching is, strictly speaking, impossible. Does this mean that recognising a cat as white irrespective of illumination has no sensory basis—that it is a purely cognitive phenomenon? By no means. Such a basis is belongingness to the same shadowedness–shading petal. In other words, two achromatic object colours tend to be categorised as the same if they differ in only shadowedness and/or shading. Although asymmetric achromatic colour matching is impossible, the asymmetric colourmatching technique has been used in a number of experiments on lightness. Logvinenko and Maloney (2006) suggested that, being unable to accomplish the asymmetric–achromaticcolour-matching task, observers in these experiments were most likely to perform a match based on the least-dissimilarity criterion. Specifically, the conjecture was that, being unable to find a pair of achromatic chips illuminated by different lights that completely match each other, observers chose the pair of chips that were least dissimilar from each other. In the present experiment observers were explicitly asked to evaluate dissimilarity between Munsell chips. Then, for each chip, the chip least dissimilar from it was evaluated from the dissimilarity judgments obtained for each stimulus condition. It was referred to as the least-dissimilar match. It is interesting to compare the least-dissimilar matches derived with the asymmetric achromatic colour matches obtained in the previous experiments on lightness constancy. First of all, it should be pointed out that the least-dissimilar matches were found to be rather close to the exact match, yielding a higher Thouless ratio than in the previous experiments. Notably, Thouless ratios derived for the stimulus condition pairs in which orientation varied were generally higher than for those in which illumination varied. This is in contrast to previous experiments which demonstrated higher lightness constancy with respect to illumination as compared with that with respect to orientation (eg Madigan & Brainard, 2014). Still, there were systematic deviations of the least-dissimilar matches from exact match. Interestingly, the direction of these deviations was similar to those that occurred in the previous lightness constancy experiments. Specifically, when the test chip reflected brighter (respectively, dimmer) light in the test stimulus condition as compared with the match stimulus condition, then the least-dissimilar chip had higher (respectively, lower) Munsell Value. In Thouless’s (1931) words, the “phenomenal regression to the object” was not complete. The ‘phenomenal regression’ is supposed to occur from the equiluminant stimulus (resulting in a zero Thouless ratio) toward the physically equal chip reflecting dimmer (respectively, brighter) light (resulting in a Thouless ratio of 1). The closer the Thouless ratio is to 1, the more complete the phenomenal regression. Inconsistency in Thouless’s metaphor and its inadequacy in the present context become obvious in the view of the least-dissimilar matches derived for the pairs BS vs DF and DF vs BS made up from approximately equiluminant stimulus conditions. In the case of these two stimulus condition pairs, chips making an exact match turned out to be equiluminant. (12)

 As can be seen in figure 2, Munsell chip N 9.0/ in stimulus conditions BS and DF reflects nearly as much light (202.6 cd m−2 and 204.7 cd m−2, respectively) as chip N 2.75/ in stimulus conditions BF (167.4 cd m−2 ). Nevertheless, N 9.0/ is perceived as white and N 2.75/ as black. Furthermore, chip N 2.75/ in stimulus conditions BF reflects 6.5 times as much light as chip N 9.0/ in stimulus condition DS (25.8 cd m−2 ). Still, N 2.75/ is perceived as black and N 9.0/as white.

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There is no ‘room’ and no need for phenomenal regression. Nevertheless, for these two stimulus condition pairs the deviations from exact match do take place as well and in the same direction as for the other stimulus condition pairs. Logvinenko and Tokunaga (2011b) suggested that such a deviation of the least-dissimilar match from the exact match can be accounted for by illumination discounting instigated by low articulation of the scene. It is widely recognised that neither luminance nor luminance contrast can uniquely determine achromatic object colour (Gilchrist, 2006). Indeed, Munsell chips under stimulus conditions BS and DF not only have practically the same luminance (figure 2) but also make the same luminance contrast with their immediate surround. Yet, their appearance is very different, which is reflected in the high average dissimilarity between the corresponding chips—higher, in particular, than in stimulus conditions BF and DF (table 5), where the difference is produced by the tenfold luminance ratio. It has been hypothesised that a particular value of luminance contrast determines the whole set of admissible achromatic object colours, the dimensions of which (eg blackness–whiteness and shadowedness; or lightness and surface brightness in the terminology used by Logvinenko and Maloney, 2006) are bound by a particular relationship (Logvinenko, 1997; Logvinenko et al., 2008). As an early version of such a relationship, one can point out the so‑called albedo hypothesis (Beck, 1972; Koffka, 1935), or its more recent modifications (Logvinenko, 1997; Logvinenko & Menshikova, 1994). When the scene is low-articulated there are several possible achromatic object colours which are compatible with the luminance contrasts in the retinal image (Logvinenko, 2002). For example, the well-known Gelb (1929) effect shows that a single reflectance lit by a single light can be perceived either as a black surface lit by a bright light or as a white surface lit by a dim light. However, the more articulated the scene is, the less ambiguous it is. As a result, in well-articulated scenes high lightness constancy has usually been recorded (Gilchrist, 2006; Maloney & Schirillo, 2002). The experimental display in our experiment was well articulated with respect to reflectance but not with respect to illumination, as light intensity varied at only two levels. Therefore, a partial Gelb effect had possibly happened and led to the observed deviation of the least-dissimilar matches as described by Logvinenko and Tokunaga (2011b). 4.3  Relevance to illumination perception It would be incorrect to consider shadowedness as apparent illumination, since the latter notion implies an apparent attribute of illuminant, whereas shadowedness is an attribute of object. To be more exact, it is not an independent perceptual attribute at all. Admittedly, shadowedness renders illumination variation. However, it does this in a special way. Specifically, for each level of shadowedness there is its own palette of blackness–whiteness. Therefore, to represent an illumination border in a scene, the visual system chooses different palettes to represent achromatic object colour on different sides of the border. 4.4  Relevance to shape perception As pointed out by many authors, attached shadows remain almost unnoticed (Gilchrist, 2006; Kardos, 1934). “The attached shadow is an integral part of the object, so much so that in practical experience it is generally not noted but simply serves to define volume” (Arnheim, 1954/2004, page 315). It is a commonplace idea that attached shadows contribute to 3‑D shape perception. A number of computational algorithms have been put forward to derive ‘shape from shading’ in computer vision (Horn, 1989; Zhang, Tsai, Cryer, & Shah, 1999). It should be borne in mind, however, that in these algorithms shading is thought of as luminance, or illuminance (eg on the retina), image; shape is understood as a physical parameter of the surface producing this image (eg surface normal, surface slant, and the like). As the general belief is that the visual system does derive somehow apparent 3‑D shape from the retinal image, it is important to understand how the apparent shape is represented in our mind.

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Do we perceive shape, or do we only think of it? What do we actually experience when we see a slanted surface? Is there any sensory correlate of slant? In other words, what is the sensory fabric of apparent 3‑D shape? A tentative answer might be that it is the dimension of achromatic object colour discovered in the reported experiment and called ‘shading’ above that represents shape in our perception. Gibson (1979) made a strong case, arguing that spatial relief can be perceived directly as a gradient of texture. However, we know that one can readily see the spatial relief of even a textureless surface. I believe that in this case it is the gradient of shading (as defined above) that renders the surface shape. Figuratively speaking, shape is a shading gradient. Of course, this does not undermine the role of texture gradient in shape perception. However, shading (as a special dimension of achromatic object colour) is the sensory fabric of apparent shape. All this explains why attached shadows are so inconspicuous. As their gradient renders ecologically the much more important impression of spatial relief, they will get overridden in our consciousness by that. However, if conscious attention is drawn to shadows, they are immediately recognised and understood as such. Yet, they are often (and erroneously) understood as a perceptual attribute of the reflected light rather than as a perceptual colour attribute on its own. It should be said that our ability to experience reflected light has been considerably overestimated. I would rather defend the stance that we do not realise what light is reflected by a particular object at all. For example, it has been shown that, when a yellow card is lit by blue light, and a blue card is lit by yellow light such that both reflect the lights of the same chromaticity as a grey card lit by daylight, the observer would not believe that the reflected lights in all these three cases are metameric (Tokunaga & Logvinenko, 2010b). 4.5  Implications for possible mechanisms of object-colour perception The long history of studying achromatic colours (for a review see Gilchrist, 2006) can be summarised as a search for the elusive stimulus for lightness. The hope was that either luminance or luminance contrast (or luminance ratio) could be normalised (‘anchored’) so as to predict lightness. Some vision scientists have succeeded to some extent (they could be called ‘doers’), though at a cost of restrictive conditions under which their models or algorithms apply. Yet, as a rule, the others (they could be called ‘destroyers’) have come up with some demonstrations serving as a counterexample of the proposed model or algorithm. Such a race has been going on and on for the last few decades. It should be noted that a stimulus for lightness is implicitly assumed to be one-dimensional. However, as the achromatic object-colour manifold has been found to be 3‑D, the stimulus cannot be one-dimensional. Indeed, if it were one-dimensional, then it should ‘trifurcate’ into blackness–whiteness, shadowedness, and shading, which cannot be done in a unique way. The uniqueness of the relationship between the stimulus and the percept is the cornerstone of the psychophysics inherited from behaviourism. If there is no unique relationship between achromatic object colour and the physical variable, then this variable cannot be a stimulus for the achromatic object colour. There are plenty of demonstrations testifying against this uniqueness (Logvinenko, 2009b). An alternative approach suggests that luminance contrast (or luminance ratio) serves as a constraint rather than stimulus for achromatic object colour (Logvinenko, 2002, 2009b). For example, the border dividing quadrants BF and DF is perceived as a shadowed edge. However, as is well known, a cast shadow can convert into a pigmented area (Logvinenko, 2009b; Logvinenko & Menshikova, 1994). That is, the same luminance edge can invoke either a shadowedness edge or a blackness–whiteness edge. Therefore, a luminance edge does not determine a unique achromatic colour edge but rather constrains object-colour perception at the edge (Logvinenko, 2009b).

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5 Summary Achromatic object colours are defined as those which contain either one Hering’s component (unique) colour (black or white) or none (grey). Operationally, the partial hue-matching technique (Logvinenko, 2012) can be used to single out the achromatic from the rest of object colours without resorting to colour naming. Thus, a Mondrian-like sample of flat objects (eg Munsell chips) appearing achromatic falls into, generally, three disjunctive subsets: whites, greys, and blacks. When illuminated homogeneously and presented at the same orientation, the whites can be linearly ordered with respect to the amount of white in them. Such a linearly ordered series of shades of white is referred to as the whiteness continuum, the difference between various shades in it being referred to as the difference in purity (of white). Likewise, the series of shades of blacks linearly ordered with respect to purity (of black) is referred to as the blackness continuum. MDS analysis of the dissimilarities between such chips produces a one-dimensional configuration, which can be interpreted as a union of the blackness and whiteness continua through the grey. It is referred to as the blackness–whiteness continuum. It is believed to be what is traditionally called ‘lightness’. When two identical Mondrian-like samples, as described above, are presented under different illuminations—one in light, and one in (cast) shadow—the multidimensionalanalysis output configuration for such chips becomes two-dimensional. Specifically, in the proximity space the blackness–whiteness continua get shifted approximately parallel to each other in a transversal direction. This shift results from the fact that the white objects in a multilight scene can be ordered not only in purity but also in shadowedness (the dimmer the illuminant, the thicker the shadowedness). Although all the white (respectively, black) colours under multiple illumination appear different (so an exact asymmetrical match is impossible), observers are willing to establish asymmetrical matches between the whiteness (respectively, blackness) continua. The hypothetical basis for this is that they put in correspondence the whites (respectively, blacks) of the same purity. Such an ability is usually referred to as achromatic object-colour (‘lightness’) constancy. Admittedly, the terminology is rather misleading since, as a matter of fact, the asymmetric match rarely coincides with the physical match. Varying slant along with the illumination results in attached as well as cast shadows, and leads to the fact that the proximity (dissimilarity) space gains one more dimension corresponding to the attached shadow. Introspectively, this is experienced as that the whites (respectively, blacks) can be linearly ordered with respect to not only whiteness–blackness purity and shadowedness, but also in what is called ‘shading’ in this paper. MDS analysis shows that shading does not interact with shadowedness. Contrary to the terminology in computer vision where shading is understood as luminance distribution, here shading is thought of as a perceptual attribute. Shading usually eludes attention because the shading gradient is experienced as shape (ie spatial relief ). Acknowledgments. I would like to thank GCU optometry students Christopher Connaghan and Zaeem Ali for collecting data and help in conducting experiments. I also wish to thank Brian Funt and Adam Reeves for reading the manuscript and valuable comments. References Arnheim, R. (2004). Art and visual perception: Psychology of the creative eye. Berkeley, CA: University of California Press. (Original work published 1954) Beck, J. (1972). Surface color perception. Ithaca, NY: Cornell University Press. Fairchild, M. D. (2005). Color appearance models (2nd ed.). Chichester, Sussex: Wiley. Gelb, A. (1929). Die Farbenkonstanz der Sehdinge. In A. Bethe (Ed.), Handbuch der normalen und pathologischen Physiologie (pp. 594–678). Berlin: Springer.

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