1. A preference for global convexity in local shape perception
Michael S. Langer Heinrich H. Bülthoff
Max-Planck-Institute for Biological Cybernetics Tübingen, Germany
[Presented at European Conference on Visual Perception, Oxford U.K. Aug. 2000]

2. Shape from Shading
The human visual system has a remarkable ability to interpret shading patterns on a surface in terms of the underlying 3-D surface shape. Computational analysis has shown that shape from shading problem is strictly speaking impossible. For a given shading pattern, there are infinitely many shapes, lighting conditions and surface reflectances that can produce that pattern.
To solve the ill-posed problem of shape from shading, the visual system relies on prior assumptions about the scene. In this talk, we are going to examine several of these prior assumptions.

3. Depth-reversal ambiguity in shape-from- shading
Our study makes use of a classical ambiguity in shape-from-shading, which is known as the depth reversal ambiguity. An example of the depth reversal ambiguity is shown here. A valley illuminated from one direction looks the same as a hill illuminated from the opposite direction.
The depth reversal ambiguity applies to general surfaces as well. By reversing the surface in depth, and moving the light source direction in the appropriate way, one obtains the same retinal image.
In this task, we will present an experiment that addresses how the visual system resolves this two-fold ambiguity.
valley
hill

4. Hollow Mask Illusion (Luckiesh, 1916)
One well known method for resolving the ambiguity is to use prior knowledge about the objects. For example, if we are shown an image of a face, then our visual system will assume a face is indeed present even though there is an equally valid interpretation which is a hollow mask. In fact, the object shown here is actually a how mask, rather than a face. This hollow mask illusion is quite strong. It can often overwhelm texture cues, stereo cues, and motion cues.
SHOW VIDEO.
I think you will agree that it is very hard to see the object correctly as a rotating mask.

5. Hollow mask illusion is the sum of two factors (Johnston et a
Hollow mask illusion is the sum of two factors (Johnston et a. ’92, Hill & Bruce ’94)
face
familiarity global convexity
It was observed by Alan Johnston and colleagues and later by Hill and Bruce that the hollow face mask illusion could be the combination of two factors. The first factor is familiarity. The visual system resolves the depth reversal ambiguity by perceiving the surface it is familiar with, namely a face. The second factor is global convexity. The visual system makes a prior assumption that a surface seen in isolation is globally convex, rather than globally concave. To demonstrate the prior for global convexity, the authors used a hollow mould of an unfamiliar object, namely an arbitrary potato. They showed that the hollow mask illusion holds for a hollow potato as well, although the illusion is not as strong because the familiarity factor has been removed.

6. Global shape discrimination is easy
convex concave
“face” “mask”
In this talk, I will report on experiments we recently carried out which are related to the hollow potato illusion but in which we examined local shape perception, rather than global shape perception.
Our experiments used randomly corrugated surfaces which were rendered using computer graphics. Examples are shown here. The surface on the left is globally convex and the one on the right is globally concave. We can think of the surface on the left as the face condition, and the surface on the right as the mask condition . Both surfaces are rendered seperately under perspective projection.
When the surfaces are viewed under the correct perspective, it is trivial to decide which of these surfaces is globally convex and which is globally concave. The surface on the left is globally convex since it bulges in the middl and the surface on the right is globally concave since it is narrow in the middle. The bulging and narrowing is due to the perspective projection.
Thus these surfaces are fundamentally different from those used in the hollow potato illusion, where observers were heavily biased to perceive the global shape as convex.
The question we addressed in our experiments was how well can observers judge the local shape of these surfaces.

7. Procedure
The experiment goes as follows. On each “trial”, the observer is shown a cylindrical surface with no random corrugations.

8. Fixation Mark (1 sec.)
Then a single point is marked on the surface and the observer makes an eye movement toward that marked point. One second later, the cylindrical surface is replaced by a randomly corrugated surface having the same global shape and the same lighting condition as the cylindrical surface.

9. Task: hill or valley ?
The observer’s task is to judge whether the marked point is on a local hill or in a local valley. In the experiment, the size of the black probe point was much smaller than what is shown here so that the marked probe does not interfere significantly with the surrounding shading pattern.
The response time for each trial was restricted to a maximum of 3 seconds. Observers were typically responded “hill” or “valley” in less than one second. They responded by pressing one of two keys on the keyboard.
I will say more about the design in a few minutes.

10. Three prior assumptions were tested
light source direction (Rittenhouse 1786,…..)
viewpoint direction (Reichel & Todd 1990, Mamassian & Landy 1998)
global shape (Johnston et. al 1992, Hill & Bruce )
We tested three different prior assumptions that observers could use to solve perform this local shape discrimination task. We explain these in more detail in the following two slides.
The first prior is the classical prior on light source direction. Observers prefer the light source to be from above the line of sight rather than from below. The second prior is for viewpoint direction. Observers prefer the viewpoint to be from above rather than from below. The third prior is for global shape. Observers prefer globally convex objects over globally concave objects.

11. Example in which all three priors assumptions are met
1. light from above viewpoint from above
3. shape is convex
Let me clarify the three priors using a cartoon example.
Here we show a situation in which all three priors assumptions are met. The person looking at an object, namely a cube on the ground. The object is illuminated from above. The viewpoint is from above. And the object is convex.

12. Example in which all three prior assumptions fail
shape is concave viewpoint from below
light from below
Here we have a situation in which all three prior assumptions fail. We have a person who is looking at a concave object that is overhead. The object is illuminated by a light source that is below the line of sight.
In our experiment, we looked at possible combinations the three priors assumptions. Let me show you some examples.

13. light source direction (collimated source)
from
above
below
On half of the trials in our experiment, the light was from above and on the other half of the trials the light was from below.
In the image on the left, the light is 10 degrees above the line of sight. In the image on the right, the light is 10 degrees below the line of sight. Notice that the two images are quite different even though the surfaces are identical and the lighting directions are separated only by 20 degrees.
Because the visual system makes a prior assumption that the light comes from above, observers are better at discriminating the hills from the valleys in the situation on the left. We have shown this is previous studies using surfaces similar to those shown here.

14. viewing direction (Reichel and Todd 1990)
view from above
view from below
The second prior assumption we consider is on viewing direction. For a globally convex surface shown here, points in the upper part of the image have a surface normal pointing upwards. This corresponds to a viewpoint from above. Points in the lower part of the image have a surface normal pointing downwards. This corresponds to a viewpoint from below.
Another way to think of these two conditions is a floor orientation and a ceiling orientation. The viewpoint from above is a floor orientation and the viewpoint from below is a ceiling orientation.
If observers make a prior assumption that the viewpoint is from above, then they will be better at discriminating the local shape in the upper half of this image, than in the lower half.

15. viewing direction (globally concave surface)
view from below
view from above
For the case of a globally concave surface, the upper half of the surface is now viewed from below and the lower half of the surface is viewed from above. In this case, if observers used a prior assumption that the viewpoint is from above, then they would be better at discriminating the local shape in the lower half of the surface.
We also note that the previous demonstrations of the prior for viewpoint from above (such as the study by Reichel and Todd and later by Mamassian and Landy), only considered surfaces that were globally flat. Because the surfaces we used in our study are globally curved and contained both floor and ceiling regions, it was not clear whether the prior would be used here as well.

16. Design three factors : - light direction - viewpoint - global shape
2 x 2 x 2 within observer
512 trials (64 per condition)
The experiment we present here used a three factor, within-observer design. The three factors were 1. Light direction, 2. Viewpoint direction and 3. Global shape.
For each factor, there were two levels. Light from above vs. below, viewpoint from above vs. below, globally convex vs. concave
Observers ran 512 trials each, consisting of 64 trials for each of the eight conditions. The trials were randomly ordered for each observer. Let us have a look at the data.

17. ANOVA Results (12 naïve observers)
Main effects:
light direction F(1,11) = 6.8, p = .025
viewpoint F(1,11) = 9.6, p = .01
global shape F(1,11) = 46.1, p < .001
An ANOVA revealed a main effect for all three factors. The largest effect was for global shape, in the sense that the p value was the lowest for this factor. A closer examination of the data showed that the low p value for global shape was due to lower variability between observers.
* A significant interaction was also found between global shape and viewpoint ( F(1,11) = 11.6, p = .006) Maybe it is best not to mention this since we have limited time.

18. Linear Regression percent correct = 51 + 10 * light source direction
* viewing direction
* global shape
(Each factor had value of –1 or 1)
We also carried out a linear regression to get a sense of which of the factor was strongest.
We see that each of three of the factors added or subtracted about 10 percent from the score, depending on whether the factor took the preferred value or not.
Also notice that the overall percent correct score was Thus, observers were at chance.

19. Examples: 87% 15% (best) (worst)
Here are the best and worst cases. The upper left shows the best case. The light is coming from above. The surface region is viewed from above. And the surface is convex. In this condition, observers were 87 percent correct in discriminating hills from valleys.
The upper right shows the worse case. The light is from below, the surface region is viewed from below, and the surface is concave. In this case, observers were 15 percent correct. That is, observers were systematically fooled under this condition. .

20. Conclusion
The prior for global convexity is used in local shape from shading.
The global convexity prior had roughly the same strength as the light-from-above and viewpoint-from-above priors.
We draw two main conclusions from this study. First, the visual system uses a prior assumption for global convex surface when discriminating local surface shape. Even though information about the global shape was readily available to the observers, for example from perspective cues, observers did not use these cues. Instead observers relied on a prior assumption about the global surface shape.
The second conclusion is that the prior assumption of global convexity had roughly the same strength as a priors for light from above and for viewpoint from above. Both of these priors had been demonstrated in previous studies. Our study today shows that all three priors play a significant role.