The other day, psychologist Tom Hartley tweeted "Your reflection is always half the size of the real thing - no matter how far from mirror. Hard to believe but true." and linked to this post in which someone demonstrates this effect. I had never quite thought about it, but realised it was of course always true: the mirror is at half the distance specified in the reflection. Then I read this post linked from the original, which reviewed an article by Lawson et al (2007) describing how people misjudge the size of objects on mirrors - specifically, they think the projection is larger than it actually is. This got me thinking about some work by Gibson on slant perception (Gibson, 1950) and then I realised that this really is an interesting topic. So I'm taking a break from the brain this week to blog about some optics instead.
People's understanding of reflection
In the world, we typically exhibit size constancy: we perceive a given object to be the same size regardless of it's distance from us, even though it's angular size and projection on the retina shrink with distance. Critically, you need information about the distance to the object for this to work reliably, and in traditional cognitive approaches you must combine the two cues to size and distance to achieve your percept of the object. Ecologically, of course, there is typically extensive motion based information for distance in the behaviour of the object in question (e.g. this demo based on Jokisch & Troje, 2003 - an excellent paper, by the way) so there's no cue combination required.
People also show size constancy for objects reflected in mirrors when they are treated as objects at the specified distance - this is useful, because it allows for cheap virtual reality (e.g. Coats et al, 2008). Lawson & Bertamini (2006), however, showed that people lose this size constancy when asked about the size of the projections in mirrors; the 2D structure at the surface of the mirror. People robustly over-estimate the size of the projection - in other words, people don't see that the projection is, in fact, only half the size of the real object. Lawson et al (2007) claim that this suggests there is no percept for the projection: it isn't a distal object and thus isn't perceived as such.
Lawson et al (2007) conducted some experiments to compare people's judgements of the size of a physical object (bamboo sticks) and the size of the projection of the object, and compared performance using mirrors and windows (which has a faint image on it's surface in the right lighting). The stick was held over the observer's head in the mirror condition, and held by an experimenter at the matching distance (2m) in the window case. People were asked to judge the physical size of the stick, and the size of the image of the stick in the mirror or window. The results showed that physical size was judged quite accurately, but that the projection size was widely overestimated in both the mirror and window cases. People rated the projection to be about 80-85% of the physical size (remember the correct answer was 50%).
They next varied the distance; They had people stand at 1.5m or 6m from the mirror or window. This had no effect on the over-estimation of the size of the projection or the essentially accurate judgements of physical size. Next, they made the distance of the observer to the mirror/window different to that of the bamboo stick (to make the actual ratio of projection size:physical size closer to .86). People's judgements of physical size remained accurate, and their judgements of projection size did not change, and were thus 'accurate' in the varied distance condition. This last actually means that participants aren't perceiving the projected size at all, because otherwise their judgements would have changed in response to the size of the projection.
Lawson et al (2007) conclude that people cannot perceive the projection on the surface of the mirror or window, and instead only perceive the distal object that is at some distance 'into' the mirror. Roughly, people aren't overestimating the size of the projection: they are only perceiving the size of the physical object and then combining this with some other knowledge about the presence of the mirror, dragging the estimate down. People only perceive objects, and not mere projections.
Gibson and slant perception
The basic result here is that people correctly perceive the 3D structure of the world 'in' the mirror, and don't perceive the 2D structure 'on' the mirror. When asked about the latter, they systematically misjudge the size of that 2D structure, in the direction of the 3D structure. This pattern reminded me of a study by Gibson (1950), who was studying the perception of visual surfaces (this is early work on his proposal that we don't perceive space, we perceive surfaces, and then, in his later work, what those surfaces afford). One of the experiments involved judgements of optical slant, which is an angular measure of a surface's deviation from the perpendicular to the line of sight (0°). A surface which is slanted relative to the line of sight extends in distance, and presents a texture gradient to the observer - the optical size of the texture elements at the close end are larger than those at the far end, but people exhibit size constancy and, instead of perceiving the projected size, instead perceive the correct physical size and that these elements are varying in distance (as in Lawson et al, 2007).
Gibson presented observers with vertical surfaces (i.e., 0° slant) painted with optical texture gradients specifying a non-0° slant. The displays were viewed in a headrest and monocularly, to reduce the motion and binocular information specifying that the surface was actually vertical. Participants then used a palm board to reproduce the visually specified slant. While their judgements covaried with the visually specified slant, they still systematically underestimated the slant. Essentially, they responded to the texture gradient information but were also sensitive to the fact that this gradient was actually on a vertical surface.
I think this kind of analysis helps firm up the otherwise weak conclusions drawn by Lawson et al (2007). They are basically correct in their explanation, but they haven't done an analysis of the optics and I think this would help make the whole thing less vague. It ties into my big gripe about perception work, not doing the information because it's hard.
The optics of reflection
Reflection is a surprisingly complicated effect; it's explained in clear fashion on this Ask A Mathematician/Ask a Physicist post (a great blog, by the way). One of the interesting things is that, for analysis purposes, reflection is best explained by pretending the surface isn't there and describe the reflection event in terms of an identical (but negative) wave travelling in the opposite direction. This figure demonstrates the analysis, and the figure caption is especially useful:
"Mathematically, as long as you restrict your attention to the left side [your side of the mirror], the following cases are exactly the same: 1) a wave reflecting off of a non-permeable surface and 2) no surface, but two waves, one positive, one negative, passing each other such that they exactly cancel."
So when you are viewing yourself 'in' the mirror, it is as if the light has, indeed, travelled twice the distance from you to the surface of the mirror. The optical structure, therefore, reflects (no pun intended) having travelled that distance. The speed at which optical flow elements move varies inversely and linearly with distance - the further the object, the slower it appears to move (motion parallex). When you move in the world, your reflection 'in' the mirror moves as well, and optically, moves in a manner appropriate to the distance 'in' the mirror, rather than the distance 'to' the mirror itself.
Optically, then, the setup is this: you are at, say, 2m from the surface of the mirror. Your reflection is light that has been structured as if it is 4m away from you (2m to the mirror, 2m 'into' the mirror') and it flows in a manner that is entirely correct for this longer distance. Your projection is the projection of your reflection onto the surface of the mirror; the reflection is 2m 'into' the mirror, and the mirror is only 2m from you, so this projection is always half the size of the real thing (you) and of the reflection. The reflection moves and changes in a manner that is perfectly consistent with an object 4m away from you, and you are thus able to show size constancy about the physical size of the reflections (motion and parallex are very informative about distance and thus support the constancy without the need for you to 'know' the size of the object; eg Jokisch & Troje, 2003). So the physics of reflection mean that the world in the mirror behaves, optically, exactly as it should for the distance it appears to be. Lawson et al (2007) then demonstrate that it is this flow structure is what we perceive, and not the projection structure.
That said, you can ask people to make judgements about the projection structure. The current results suggest, however, that people are staggeringly insensitive to this projection structure, and while they can generate a response, that response does not change with changes in perspective structure. So people aren't overestimating projection size, regardless of the article title; they are actually generating responses based on the reflection structure and producing underestimates.Why underestimates? Because while the reflection structure specifies an object at 4m, there is also information that the mirror is actually at 2m, and while the 3D reflection structure dominates, the 2D surface information also seems to matter.
What is this surface information that tells people about the distance to the mirror? The mirror is not simply hanging in empty space, nor is it taking up your entire field of view. The mirror has an edge, and edges are highly salient for vision because they are where the flow of optical texture exhibits a discontinuity. As I mentioned, motion parallex is the fact that optic flow changes speed smoothly with distance. Edges occur at the end of surfaces, and if this surface is in front of another surface the texture on the nearer surface flows faster than the texture on the far surface; at the edge, this shows up as an abrupt discontinuity in the speed of the optic flow.
In the mirror, the world appears to continue on into the mirror and the optical structure is appropriate for that; within the bounds of the mirror, optical distance increases. In contrast, the wall the mirror is mounted on remains at a constant distance (2m) from you. And, importantly, the edge of the mirror is 2m away and the optic flow discontinuity specifies this fact. You now have conflicting information, that the object being judged is simultaneously 4m and 2m away. It's apparent size is correct for 4m, but if that same apparent size was at 2m, it would appear to be a smaller object (size constancy).
Cue-combination - what might the ecological equivalent look like?
The real question is why does the final judgement look like some kind of weighted average of these two distance measurements (weighted in favour of the stronger, more robust reflection structure)? Traditional cognitive views would see this as a post-perceptual process, similar to how you get size constancy, where you perceive these conflicting cues and resolve the conflict by some process of cue-combination (using the latest in sexy statistics to make the result useful). Ecologically, this cue combination should really be happening in the world; the structure of the light should instantiate the combination. But given the geometry I've laid out, its not clear to me how or why this could work. Perhaps it's the answer you get when you ask a system calibrated using the correct optic flow and motion parallax a question about something which violates the form of that calibration: not quite random nonsense. I'd be interested in experiments which recalibrated the rate at which optic flow speed varies with distance and looked for consequences. But to be honest, I don't quite have an explanation for cue-combination, other than it's a hint we've asked the wrong question.
This foray into mirrors makes clear (I hope) the idea that we do not perceive light, or structure in light. We don't perceive the light as on the surface of a mirror (where it really is), but instead we perceive what the structure in light specifies. The reflected world in the mirror doesn't really exist; it is a purely optical structure. But we happily perceive it as a normal world of the specified size, because it behaves perfectly appropriately. The only transformation it does impose (the left-right reversal) has no effect on the structure and behaviour of the light, and thus no effect on our perception of the world specified by that light. There is, to the visual system, no difference between looking in a mirror and looking out a window. The only weird part is what happens when you ask about the projection on the mirror itself, and the data clearly support the conclusion that our perception of the reflected world is not based in any way on this projected image - variations in projected size simply aren't perceived and asking about the projection simply produces noisy results derived from the perception of the extended mirror world and the actual distance of the mirror. Cue combination? Maybe; the judgements aren't random. But the answer, as ever, lies in careful analysis of the optical structure in the task, and this must form a critical part of future work on this topic.
Coats, R., Bingham, G.P. & Mon-Williams, M. (2008). Calibrating grasp size and reach distance: Interactions reveal integral organization in reaching-to-grasp movements. Experimental Brain Research, 189, 211-220. Download
Gibson, J. (1950). The Perception of Visual Surfaces The American Journal of Psychology, 63 (3) DOI: 10.2307/1418003
Jokisch, D. and Troje, N. F. (2003) Biological motion as a cue for the perception of size. Journal of Vision, 3, 252-264. Download
Lawson, R., Bertamini, M., & Liu, D. (2007). Overestimation of the projected size of objects on the surface of mirrors and windows. Journal of Experimental Psychology: Human Perception and Performance, 33 (5), 1027-1044 DOI: 10.1037/0096-15126.96.36.1997