The main reason for conducting this experiment is to find out whether the transformation caused by a change of the context can be described by a linear or affine mapping or whether a more complex projective transformation has to be assumed. Evidence for the fact that only a projective transformation is sufficient comes from several sources:
... all clearly favor the projective model.
Before a color match is produced within the changed context B, the adjustment of the color has to be trained by producing five adjustments within the original context A. The centroid (the arithmetic mean of the colors 
-coordinates within the threedimensional chromaticity diagram) of the adjustments within context A is regarded as the color remembered by the subjects, the so-called memory-color. Considering the variability of the single adjustments around this memory color, color-discrimination-ellipses can be obtained. These ellipsoids allow the estimation of the color-discrimination depending on the targets color. There are wide and unsystematic differences in the discriminability as shown for example by Brown and MacAdam (1949; see figure 14). A simple formula predicting the size and orientation of those ellipses depending on the chromaticity point of a color has not been found yet.
Figure 14:
Color-difference-matching ellipses in the CIE 1931 chromaticity diagram (Brown & MacAdam, 1949): This figure shows 38 color-difference-matching ellipses and their axes obtained by Brown and MacAdam (1949). The ellipses are drawn ten times their actual size. Their calculation is based on adjustments (in three dimensions) of subjects viewing monocular presented stimuli of 2 
visual angle.
The mean coordinates of the colors adjusted by the subjects depart clearly from the coordinates of the color presented originally even when the adjustments take place within the original context. It is interesting to note, however, that there are only minor differences between this memory-colors and the colors produced within the changed context B. This pattern can be demonstrated for the adjustments of all subjects (see figures 15 and 16 for the adjustments of one of the subjects). The effect of memory on the color reproduced seems to be greater than the pure effect of its context. These memory-colors are not systematically related to the colors presented originally. This result can be explained by the above-mentioned tendency of memory colors to resemble the color of well known objects.
Figure 15:
Color-difference-matching ellipses in the CIE 1931 chromaticity diagram for subject EIM: The upper figure shows the color-difference-matching ellipses for the adjustments within the original context, the lower figure for the adjustments within the changed context. The ellipses are not drawn ten times their actual size.
Figure 16:
Memory- and context-effects of subject SIG: This figure characterizes the memory- and the pure context-effects on each of the ten stimuli. The left diagram (memory effect) displays the memory-effect by arrows originating in the chromaticity point of the stimulus presented within the original context and ending at the point representing the mean of the colors adjusted by the subject within the same context. The arrows in the diagram (pure memory effect) on the right start at exactly those points where the corresponding arrows in the left diagram do end and are directed towards the mean of the adjustments within the changed context.
It is especially remarkable that the colors reproduced from memory have a greater saturation then the colors presented originally (Newhall et al., 1957, report contrary findings). Highly saturated colors are matched again by highly saturated ones. Furthermore the observation, that all subjects are able to produce a match for each stimulus can only be explained by assuming a projective mapping on account of the changed context, not by assuming a linear or affine one.
Now the formal description of the transformation of the target presented within the original context to the adjustments within that context (memory color) and within the changed context (context-effect und pure context-effect) is tried: The subjects adjustments shall be predicted by that transformation.
The parameters and the likelihood of four different kinds of transformations of the colors caused by the memory- and context-effects are estimated separately for each of the subjects. These transformations shall predict the expected adjustments of the colors on the basis of the -coordinates of the target-stimuli presented within context A:
-matrix, so 6 different parameters) have to be calculated: 
).
-matrix is estimated which predicts the coordinates of the subjects' adjustments as closely as possible (69 parameters: 9 parameters of the transformation and 60 parameters for the variance-covariance-matrix).
-matrix is estimated which predicts the coordinates of the subjects' adjustments as closely as possible (72 parameters: 12 parameters of the transformation and 60 parameters for the variance-covariance-matrix).
-matrix is estimated which predicts the coordinates of the subjects' adjustments as closely as possible (75 parameters: 15 parameters of the transformation and 60 parameters for the variance-covariance-matrix; since the predictions of the projective model do not change when each of tze parameters is multiplied with a constant, only 15 of the 16 parameters of the transformation can vary independently).
The different models are fitted by maximizing their likelihood under the assumption of normality: For estimating the parameters of the transformation, it is assumed that coordinates of the single adjustments follw a (threedimensional) normal distribution with its mean at the centroid of all adjustments to one target stimulus. The parameters are estimated numerically, using Brent's (1973) PRAXIS algorithm in the implementation of Gegenfurtner (1992). Three different estimations are conducted:
The results are similar for all subjects (see table 3 for the results for one representative subject). In only one of the 36 cases (4 subjects, 3 models, and 3 kinds of effects) the unrestricted model does not describe the data significantly better than any of the restricted models. Regarding the restricted models alone, only with the pure context-effects non-significant results can be obtained: For one subject the linear and the affine model do explain the data as well as the projective model. The cross-context-matches of the remaining three subjects can be predicted significantly better by a projective model (for three out of four subjects). See table 3 for subject SIG, whose data can be predicted significantly better by the more general models having the greater number of parameters, especially by the projective one. So, in most cases the projective model explains the data significantly better than the other two regarded.
Table 3:
Results of testing the different models for one subject
There are some behavioral measures favoring the projective model besides the coordinates of the colors adjusted by the subjects: The measures discussed below indicate that the subjects can produce matches in the changed context even to highly saturated target stimuli. If the linear or affine model were valid, this would not be possible.
The number of actions taken to perform a color-match can be regarded as an indicator of the difficulties such a task poses. The average number lies between 32 and 52 action, depending on the subject. Considering the kind of action (changing hue, brightness or saturation) reveals that most of the adjustments concerned the hue of the probe and least of them its saturation (see table 4). When asked after finishing the experiment, the subjects report first trying to adjust the hue of the probe, then its brightness and at last its saturation when performing the matches. After a change in saturation, hue and brightness sometimes haad to be readjusted.
Average number of actions required for performing a single match
This behavioral measure favors the projective model, too: If a subject does not find a matching color in the changed context, a greater number of actions while performing the adjustment can be expected (since the subject would have to search for a matching color in vain). If a linear or affine model would suffice to describe the effect of changing the context, there should exist no match for highly saturated colors and a greater number of actions for adjusting matches for the highly saturated colors should result therefore. This is not the case for any of the subjects, as can be seen in table 5. Only a projective model can explain this result.
Number of actions while performing adjustment to mono- and polychromatic targets within the changed context
Finally, the average number of actions performed for one match does not decrease between the various experimental sessions (see figure 17). This result indicates that no significant practice-effects do occur while performing the experiment. The trial phase before its start seems to be long enough.
Figure 17:
Average number of actions in each experimental session and its standard deviation from the first to the 21st session of the experiment (subjects SCA and SIG).
The average time needed to perform a color match lies between 20 and 40 seconds, depending on the individual subject. This duration can be interpreted as a correlate of the difficulties encountered on a single trial. Similar to the number of actions performed, the average time needed does not decrease in the course of the experiment (see figure 18). There seems to occur no further practice-effects during the experiment.
Figure 18:
Average time to complete a match in each experimental session and its standard deviation from the first to the 21st session of the experiment (subjects SCA and SIG).
A slight training-effect can be shown within a single experimental trial: The time needed to perform the match decreases from the first to the fifth presentation of the target stimulus within the original context and sometimes increases slightly, when the target stimulus is to be produced in front of the changed context (see figure 19). But the increase in time when performing a match within the changed context is small even if compared to the time when performing a match within the original context (see table 6): The time used to perform the matches within the changed context is not significantly higher than within the original context for any of the subjects. Therefore, these matches do not seem to be especially difficult for the subjects. Again, only a projective model is compatible with this finding.
Figure 19:
Typical time used to perform matches: This figure shows the average time used to perform a color-match depending of its timing within one trial for subjects SCA and SIG: The average time for performing the adjustment and its standard error are shown for the first, second, ... fifth presentation of the target stimulus within the original context and for the sixth presentation within the changed context.
Time used to perform matches for the adjustments within the same context and within the changed context
To investigate, whether performing matches to highly saturated colors within a changed context is especially difficult, the corresponding data are compared now: One of the four subjects (SCA) does perform the adjustments for polychromatic stimuli faster than for monochromatic ones, but this pattern shows up as well for adjustments within the original context as for adjustments within the changed context. The remaining three subjects perform the adjustments for monochromatic stimuli even faster than the ones for polychromatic ones (see table 7). The analysis of the time needed to complete the adjustments does not indicate special difficulties in performing matches for highly saturated colors and thus favors the projective model, too.
Table 7:
Average time m (in seconds) used to perform matches for mono- and polychromatic stimuli and its standard error se
After performing each match (within the original context as well as within the changed context), the subjects have to indicate, how satisfatory the preceding match seems to them. As can be seen in table 8, the subjects´ ratings of the quality of their adjustments shows even better ratings for the highly saturated stimuli (except for one of the four highly saturated stimuli, for which two of the subjects regard their adjustments as significantly inferior).
Table 8:
Subjects´ ratings of the quality of their adjustments for mono-/polychromatic stimuli
This result holds as well for the adjustments within the original context as for the adjustments within the changed context (see table 9). Such ratings should definitely not occur, if the subjects cannot find matches in the changed context as predicted by the linear or affine model. Again, only the projective model matches those ratings.
Subjects´ ratings of the quality of their adjustments within the original context and within the changed context
