Contributors: Robert T. Arrigo: Programming Gordon Redding: Author
 Additional Credits: FundingThis module was supported by National Science Foundation Grants #9981217 and #0127561.

Illusions

A basic assumption that seems to be built into our visual system is that the objects that we see are from a three-dimensional world. So, if we are presented with a two-dimensional drawing, our visual system tends to interpret the image as if it were caused by three-dimensional objects. Look at the picture to the left. It appears that the three men in the picture are roughly the same height, right? We make that judgment even though the drawing of the man at the back of the picture is much smaller than the drawing of the man at the front of the picture. The drawing was made to conform to the physical laws of projective geometry (i.e., the size of the objects in the picture vary inversely with the distance portrayed in the scene).

Consider a picture that violates the laws of projective geometry (that is, the size of the objects in the picture do not change with the portrayed distance). When we look at that picture, the three men pictured do not appear to be the same size. But, if you were to use a ruler, you could confirm that the three figures are exactly the same size.

So this is an illusion. Things appear differently than they really are. Of course it is good that we are susceptable to this illusion. It results because our visual system has been adapted for use in a three-dimensional (not a two-dimensional) world. This assumption -- that images projected onto our retina are reflected off of a three-dimensional world (and thus subject to the laws of projective geometry) -- is built into our visual system.

Illusions, then, are especially important in the study of vision, because they provide dramatic clues concerning the underlying assumptions that are built into our visual systems.

The Müller-Lyer illusion

The Müller-Lyer illusion is one of the oldest geometric illusions (Müller-Lyer, 1889/1981). Lines with arrow junctions attached to the ends are perceived as shorter and lines with attached fork junctions are perceived as longer, compared to plain lines without junctions.

One hypothesis (Redding, 2000; Redding & Hawley, 1993) is that the size illusion is the result of our interpreting the line drawings as representations of three-dimensional corners. This is called the linear perspective hypothesis because it postulates that our visual system interprets the lines as a linear perspective drawing, which is a drawing that produces the impression of depth by projecting a three-dimensional scene onto a two-dimensional picture plane. When we view such a drawing our visual system attempts to solve the inverse problem by recovering the three-dimensional scene that would typically produce two-dimensional images of this kind (Kubovy, 1986).

The Linear-Perspective Hypothesis The linear-perspective hypothesis assumes that the Müller-Lyer arrow and fork junction stimuli are interpreted by the visual system as linear perspective drawings depicting right-angled corners either projecting toward the viewer and in front of the picture plane (a convex corner) or projecting away from the viewer and behind the picture plane (a concave corner). We call a three-dimensional corner that is represented by a two-dimensional drawing, a "virtual" corner (because it isn't a real corner, it only looks like one).

The image above shows the corner of a building projecting towards the viewer with the walls on either side receding away from the viewer. If the corner of the building is perceived to be projecting in front of the picture plane, then the line representing the corner will be perceived as being closer to the viewer and thus shorter than a simple straight line located in the picture plane (remember the law of projective geometry).

 Conversely, the image on the left shows the corner of a building that projects to the back of the picture plane with the walls on either side coming toward the viewer. If the corner of the building is perceived to be projecting behind the picture plane, then the line representing the corner will be perceived as being further from the viewer and thus longer than a simple straight line located in the picture plane.

In other words, if the corner of the building (i.e., the virtual corner) is receding into the distance, then that tells the visual system that it is further away and thus that it is really taller than the image it projects.

If, on the other hand, the corner of the building is projecting out towards the viewer, then that tells the visual system that the corner line is very close to the viewer and thus it is judged to be shorter than a single line without the arrow junctions.

Of course, the size of the two-dimensional drawing (what is called "the picture plane size") also influences perception and the illusion is never as large as the difference in virtual corner sizes would predict.

Hypothesis testing

The linear perspective hypothesis makes sense, but is it a good explanation of the illusion? One way of testing the hypothesis is to derive from it predictions about how the illusion should change in other conditions not yet observed.

For example, if we mathematically rotate the virtual corners about the vertical axis in the picture plane, the corner size must also be changed to keep the same size in the drawing because the corners move closer to the picture plane. The rotated convex corner must be made larger and the rotated concave corner must be made smaller. This means that the illusions should decrease. Just this result has been found when drawings of rotated corners are presented to people for their judgment of size. The effect is small, but detectable (Redding, 2000).

Our vision experiment used four experimental stimuli. There were two arrow junction stimuli, one of which depicted a non rotated convex virtual corner and the other a virtual convex corner rotated 20 degrees.

There were also two fork junction stimuli, one of which depicted a non rotated concave virtual corner and the other a virtual concave corner rotated 20 degrees. Therefore, the four experimental stimuli were different in terms of kind of virtual corner (convex or concave) and amount of rotation (0 or 20 degrees). Kind of virtual corner and amount of rotation were the manipulated variables. Manipulated variables are call "independent variables".

## What behaviour are we studying?

The behavior studied was participants' report of perceived size. This was measured by having participants adjust a response line to match the perceived size of the target line. Matching size was the measured behavioral variable. A measured variable is called a dependent variable. So, during the experiment, when you changed the length of the line on the right, trying to match the length of the simuli on the left, you produced behavior that we can measure and analyze (i.e. a dependent variable).

Differences in matching length between each of the experimental stimuli and the control stimulus provided a measure of the illusion. That is, the experiment recorded your response to both the experimental stimuli (the arrow and fork junctions) and to the control stimuli (the sideways "H" shape). The greater the difference in length between the line you matched to the experimental simuli and the line you matched to the control stimuli -- the greater the illusion was for you.

This illusion score enabled us to test predictions from the hypothesis. But there is more to the experimental design. Why?

First, why include a control stimulus? What did it control for?

You might think that we could have just compared the matching size of each experimental stimulus with the physical length of the line. This procedure would assume that participants would have been perfectly accurate in matching the size of the experimental stimuli, except for the distortion produced by the arrow and fork junctions. But participants are not usually perfectly accurate in their perceptual reports.

Arrow and fork junctions are not the only reasons people make errors in their perceptual reports. For example, people generally tend to underestimate size, especially for larger objects. And, from trial to trial, participants are more or less attentive, more or less careful in their matching response. The control stimulus controls for such inaccuracies in perceptual report. Because these sources of inaccuracy are arguably present for both experimental and control stimuli, when we take the difference between these stimuli we subtract out such inaccuracies and don't confuse them with the illusion.

(1) The experiment controls the KINDS of corners that subjects see.: Each subject sees equal quantities of both convex virtual corners (arrow-shaped)

and concave virtual corners (fork-shaped).

So the shape of the virtual corners (concave vs. convex) is an independent variable (i.e., manipulated by the experimenter) and it is within-subjects (i.e., everyone tested gets the same percentage of both concave and convex corners).

(2) The experiment controls the Degree of Rotation of the Corners: Each subject sees equal quantities of unrotated corners (i.e., where the perspective is straight onto the corner and not at an angle).

So the degree of rotation of the virtual corners (0 degrees or 20 degrees) is also an independent variable and it is also within-subjects.

(3) The experiment controls the Direction of Rotation of the Corners: Each subject did not see corners rotated in the same direction.

Of the rotated corners that subjects saw, some subjects saw more / all rotations in a clockwise direction, while others saw more / all rotations in a counter-clockwise direction. So the direction of rotation is a control variable and it is between-subjects.

## CONTROL VARIABLE: REPEATED TRIALS = Repeated Blocks (4 times) of Experimental and Control Stimuli. WHY?

We presented first a block of practice stimuli followed by four repeated blocks of the experimental and control stimuli. Why? The practice trials familiarized the participant with the task so that when the experimental trials occurred we could be reasonably sure participants were doing what we asked them to do. In particular, the practice stimuli were all plain lines, without junctions, so that participants would become used to matching lines and not include the junctions on the experimental stimuli in their matching response. The repeated blocks of the experimental stimuli were intended to produce more reliable data. No one is ever perfectly accurate all of the time. By averaging our data over repeated trials we can be more confident that we have accurately measured our participants' behavior.

## CONTROL ("filler") STIMULI: Plain Lines (without junctions) Mixed in with Experiment and Control Stimuli WHY?

But there were plain line stimuli without attached junctions mixed in with the experimental and control stimuli. Why? They were included to keep participants making a range of matching responses so that the illusion effect isn't reduced through the habit of making similar line judgments. This is especially important because all of the experimental and control stimuli were actually the same length! Even with the illusion, participants might have begun to make a narrow range of response and attenuate the illusion. The filler stimuli also kept the participants in the habit of matching only the length of lines, not including the junctions in their judgments.

## CONTROL VARIABLE: Randomized Order of Stimuli WHY?

Behavior is influenced by previous behavior. For example, after making a long matching response a participant may feel like making a short response, regardless of what the stimulus is. In such cases, response to an arrow junction stimulus would possibly be different depending upon whether it was preceded by another arrow junction stimulus or a fork junction stimulus. Previous experience "carries over" to later experience and influences behavior. Hence, such influences are called "carryover effects". When we randomized the order of stimuli within blocks we know that, in the long run at least, each stimulus will precede and follow every other stimulus an equal number of times. Thus, any carryover effects are averaged out when we combine our data over participants.

## EXPERIMENT RESULTS

Here is a graph that shows the results of trials done with one group of subjects.

The illusion was calculated by finding the difference between matching size for experimental stimuli (arrow and fork junctions) and control stimuli (T junctions). According to the linear perspective hypothesis, the lines with arrow junctions should produce a negative illusion score (less than the control), while the lines with fork junctions should produce a positive score (greater than the control). And the prediction was that lines with asymmetrical junctions (rotated corners) should produce less illusion than lines with symmetrical junctions (unrotated corners).

Does the data provide evidence that is consistent with the truth of the linear perspective hypothesis? Yes. It is what the theory predicts (within a reasonable degree of accuracy). Since the evidence is consistent with the theory, does that mean that the linear perspective hypothesis has been proven to be true? No. That is too strong a claim. The author of this experiment (Dr. Gordon Redding) claims that any difference in the illusion for stimuli depicting rotated and non-rotated virtual corners is due to the difference in virtual corners. Do you think he is right? Maybe he is. But we can't draw this conclusion with absolute certainty because there are other possible explanations for the data, as Dr. Redding himself is willing to admit. The experimental stimuli have been divided into two groups: those with rotated virtual corners and those with unrotated virtual corners. The two groups do vary with respect to the "degree of the rotated corner". But, they may also differ with respect to any number of other properties. And if they do, the explanation for the experimental results may turn out to depend primarily on this other property, rather than on the property identified by the linear perspective hypothesis (i.e., the degree of the rotated corner).

## COMPETING EXPLANATIONS

THEORY #1: Linear Perspective Hypothesis

THEORY #2: The "Total Height of Figure" Distortion Hypothesis

THEORY #3: The Stimulus Averaging Theory