Constantdifferenceseries experiments have proved very useful in investigating both the spatial nonlinearity (see 2.0 Complex channels ) and the intensive nonlinearity in texture segregation (now thought to be normalization rather than an earlylocal nonlinearity, see 3.5 and 4).
What's on this page?
 The matrix of stimuli in a full constantdifferenceseries experiment
 Diagram of constantdifferences series
 Comparing simplechannels model predictions to results of constantdifferenceseries experiments
 A bit about complex channels' predictions for constantdifferenceseries experiments
 General interpretation of constantdifferenceseries experiments
 Some results using three different kinds of squareelement patterns
 Some results from gratingelement experiments in three different contrast ranges
The figure above shows an example of an elementarrangement pattern containing two different textures. These textures are made from the same two kinds of elements but in different arrangements. Elements of one type are dark squares. Elements of the other type are light squares. The central region is filled with a checkerboardarrangement texture. The side regions with a striped arrangement.
The full set of stimuli includes all combinations of equallyspaced positive and negative contrasts. (Since the two elements have identical spatial characteristics, the other half of the matrix below can be ignored.)
The difference between the contrasts of the two element types is constant along positivelysloped obliques.
The ratio of contrasts of the two element types is constant along the rays coming from the origin labeled in terms of angular degrees away from the ray of oppositebutequal contrasts. This angle  called the contrastratioangle below  is a convenient way to keep track of different types of stimuli.
The stimuli along a positive diagonal will be called a constantdifferenceseries and are of particular interest.
Two such constantdifference series are diagramed in the next figure. Each little drawing in the figure represents the luminance profiles of the two element types in one pattern.
Examples of samesignofcontrast, oneelementonly, and oppositesignofcontrast patterns from two constantdifference series, one of squareelement patterns and the other of gratingelement patterns are on another page.
The results of one constantdifferenceseries experiment are plotted below in the top panel and the predictions of the simplechannels model (see description of SimpleChannels model on the ComplexChannels page) are given in the panel below. The segregation rated by the observer or predicted by the model is plotted on the vertical axis. The contrastratio angle characterizing the pattern is plotted on the horizontal axis. Each curve connects the patterns in a constantdifference series. (Other examples of results are given at the bottom of this page.
Results of an experiment using squareelement patterns (results from Graham, Sutter, Beck, 1992).


Predictions from the simplechannels model (described on Complexchannels page). Note that the predicted value plotted here is assumed to be monotonic with the ratings given by the observer (but not necessarily strictly proportional) 


Notice that the simplechannel model predicts that all members of a constantdifferenceseries should be approximately equally segregatable. We attempt to give the intuition behind this prediction in Graham, 1991, and Graham, Beck and Sutter, 1992.
The experimental results above differ from the simplechannel model predictions quite dramatically. These differences turn out to be explained by a combination of two nonlinearities:
(i) a spatial nonlinearity like that in complex channels, and
(ii) a compressive intensive nonlinearity.
The signatures of these two kinds of nonlinearity in results from constantdifferenceseries experiments are illustrated in the next two figures. For more information about the spatial nonlinearities go to ComplexChannels page. For more information about possible intensive nonlinearities go to the Normalization page and/or t
If complex channels were substituted for the simple channels (still assuming no intensive nonlinearity), what would happen? This complexchannel model would NOT predict that all members of a constantdifferenceseries are equally detectable. Instead it predicts a dip (illustrated below) in the middle of the series, for oppositesignofcontrast patterns . But such a model does predict that all samesignofcontrast members of a constantdifferenceseries should be equally segregatable.
As shown in the first figure below, the results for oppositesignofcontrast patterns are particularly informative about whether given patterns are detected by simple or complex channels (or a combination of both, as in the squareelement experimental results above).
Both simple and complex channels (in the absence of any intensive nonlinearity) are equally sensitive to all samesignofcontrast patterns in a particular constantdifference series.
The next figure shows that results for samesignofcontrast patterns are particularly informative about the intensive nonlinearities. For further explanations see the published papers using these experiments (Graham, 1991; Graham, Beck, and Sutter, 1992; Graham and Sutter, 1996; Graham and Sutter, 2000).
Why this downturn at the ends of the curves is naturally called "compressiveness" might be clearer after reading the explanation given on the EarlyLocalNonlinearity page of how a model with a compressive earlylocalnonlinearity predicts this downturn. An expansive earlylocalnonlinearity would work the opposite way.
An explanation of how normalization predicts this downturn can be found on the Normalization page. See the Normalization page also for a possible explanation of this "expansiveness" within the context of a normalization model.
Some further examples of constantdifferenceseries experimental results are given next. For published examples and interpretations see Graham, 1991; Graham, Beck, and Sutter, 1992; Graham and Sutter, 1996; Graham and Sutter, 2000,and Wolfson and Graham, in press 2004. (The first many papers used subjective ratings of perceived texture segregation. The 2004 paper replicated these results using several objective forcedchoice tasks.)
Oppositesignofcontrast examples of the three kinds of patterns are shown here. Regularlyspaced large squares are shown on the left (in the experiment each square subtended 0.33 deg), regularlyspaced small squares are shown in the middle, and sparselyspaced small squares are shown on the right.
A series of three members of a ConstantDifferenceSeries of regularlyspaced squares can be seen on the ConstantDifferentSeries patterns page.
The results below were all collected for stimuli within a contrast range from 12% to 60% (the size of the contrast step in the matrix of stimuli shown above) was 12%.The horizontal axis shows contrastratioangle
In the results above, look particularly at the results for contrastratioangles between 45 and +45, which is the oppositesignofcontrast range. These provide information about the complex channels. (See General interpretation of ConstantDifferenceSeries experiments above.) The results here suggest that the sparselyspaced patterns were segregated by complex channels, the regularlyspaced smallsquare patterns by simple channels, and the regularlyspaced largesquare patterns by a combination of simple with some complex channels (Graham and Sutter, 2000).
The derived relativelyearlylocal functions corresponding to these results are on the EarlyLocalNonlinearity page.
(Another set of results for regularlyspaced largesquareelement patterns was given in the section on the SimpleChannels Model above.)
These three experiments were all done using elementarrangement patterns with elements that were Gabor patches at a spatial frequency of 12 c/deg. The three experiments differed in their contrast ranges as indicated above each panel. Three patterns from a gratingelement Constant Difference Series are shown on another page (although the patches in this example contain only 1/4th as many cycles as did the 12 c/deg patches used for the data below).
Look at the ends of the curves for contrastratioangles less than 45 and greater than 45 (samesignofcontrast patterns) in all six examples above. Note that in 5 of 6 panels there is "compressiveness" shown for samesignofcontrast patterns.
For the lowcontrast gratingelement case (lower left), however, there is "expansiveness".
The derived relativelyearlylocal functions corresponding to these results are on the EarlyLocalNonlinearity page.