Perception is three-dimensional

The brain is optimized for three dimensional perception. This makes sense, because it is very important to perceive three dimensions in order to establish how fast you have to run to escape being eaten. For instance, in the picture shown, no one has any doubt that the corners of my house are about the same length, even though the size of the vertical lines is extremely different

Perception is three-dimensional

Graphics are two-dimensional

This illusion is the Muller-Lyer illusion: The vertical line in “A” appears to be longer than the line in “B”, even though the segments are the same length. This is because we see image “A” in perspective as an interior corner of a three-dimensional object (which implies it would be further away than “B”). Since they are the same size in the image, “A” must be bigger than “B” taking perspective into account.

Perception is three-dimensional

The lines drawn onto the image above show the real-world context for this effect; the relative size of the two corners of the house make sense in context, but when the three-dimensional context is removed in an artificial two-dimensional setting, the illusion appears. There are many situations where three-dimensional perception conflicts with artifical two-dimensional graphics; our brain is not entirely trustworthy when we are dealing with synthetic images.

Ambiguous Figures

Necker cubes are another situation where there is some ambiguity in mapping the three-dimensional world back into two dimensions. A wire-frame cube as shown in the left-most image can be interpreted in two different ways, shown in the other panels. When looking at the necker cube, it will often “flip” back and forth between the two interpretations; the brain interprets the ambiguous image but cannot decide what three-dimensional object would give rise to the image on the retina.

Artistic Trickery

These images show other ways the brain mis-interprets two-dimensional stimuli as three-dimensional objects. The elephant appears normal (but look at the legs!), and the triangle seems perfectly fine until you look closer and realize that it is an “impossible object” that has no obvious three-dimensional interpretation. Next, we will examine one example of the disconnect between two and three-dimensional perception, looking at an illusion that occurs frequently in statistical graphics.

The Sine Illusion

The Sine Illusion

The image shown on the previous page is the sine illusion. The lines at the top and bottom of the image seem longer than the lines in the middle of the curve (at the inflection point of the underlying function). In reality, the lines are actually the same length; they appear uneven because the brain interprets this image as three-dimensional. Interestingly enough, it seems that at least one person who does not have binocular depth perception due to a congenital condition is immune to this illusion. This suggests that we are correct in believing that like the Muller-Lyer illusion, the sine illusion is the result of misapplied perceptual heuristics which add a three-dimensional context to the illusion. This context is shown on the next slide

Three Dimensional Context

Three Dimensional Context

The previous slide shows the three-dimensional context for the illusion; when the vanishing point is close, the illusion slightly resembles the three-dimensional object, but when we move the vanishing point out to infinity, the illusion and the three dimensional context map almost exactly. The way we interpret the “length” of the lines in the illusion makes sense given the 3d context; we would measure the width of a ribbon in three dimensions by looking at the orthogonal distance across the curve, but for the purposes of the statistical graphic, we should be considering the vertical length. While the sine illusion is relatively artificial, we run into this problem in scatterplots and time-series graphs as well.

Example: US Gas Prices

Example: US Gas Prices

Due to the change in the average price of gas over time, we miss the gradual increase in variance - the standard deviation almost doubles over the 19 year period. The dips in the mean don’t actually affect the variance in the price of gas, but we might miss important information about the variance structure by plotting the data, even though such plots are very important to build and evaluate models.

Understanding the Illusion

We can correct the illusion by increasing the blue line’s length until the red line is equal to the actual distance. More Information about the correction

Tufte’s Lie Factor

\[ \text{Lie Factor} = \frac{\text{Effect Size}_{graphic}}{\text{Effect Size}_{data}} \]

Psychological Lie Factor

\[ \text{Lie Factor (Brain)} = \frac{\text{Effect Size}_{perception}}{\text{Effect Size}_{graphic}} \]

Experiment Goals

• Is the sine illusion a significant factor in reading simple plots?
  Do participants always choose the uncorrected plot with uniform line length?
  
  
• How large is the effect of the sine illusion? Does it depend on the underlying function?
  
  
• Would two individuals have similar psychological lie factors?

Experimental Design

• Show participants sets of 6 plots with the sine illusion corrected to different degrees (including some overcorrection)
  • 3 underlying functions: sin(x), exp(x), and 1/x.
    
    
• Participants choose the plot of the 6 shown which has lines that are the most uniform in length
  
  
• Participants had to complete at least 10 trials to be included in the study

Experimental Design

Experimental Design

• Compute the “Psychological Lie factor”, \(D^\ast\), for the chosen plot
  
  
• Normalize the chosen lie factor so that the lowest possible selection corresponds to 1.
  
  
  
  106 participants completed 1542 trials
  (411 exponential, 316 inverse, and 815 sine)

Model Details

Hierarchical Bayesian model for \(\theta\), the overall lie factor

- Uniform priors on \(1\leq \mu\leq 4\), \(.1\leq\sigma\leq 2\)

- \(f(\theta | P, \mu, \sigma)\sim \text{Truncated Normal}(\mu, \sigma)\), \(\theta>1\)

Results

4 individuals who completed the most trials

Results

Overall Results

Results

Individual and Overall Credible Intervals

Conclusions

• The sine illusion has a significant effect on our perception of simple plots
  
  
• The effect does depend on the function
  
  
• The effect seems to be fairly consistent across individuals, but there is some variance