In the inverse-square law derived perceived size equation, the measure-point (Mp) is the sole human term inserted, whereas it is unnecessary to have this term for all the other radiations to which the law applies when nobody is around to make the measurements. Although the measure-point is likened to an intensity detector earlier, it is still not satisfactory as to why it can change our size perception so effectively. Under certain circumstances, the measure-point (or focal distance) plays a role that is quite contradictory to our common sense, such as the images in the second mirror in Case C. It is essential for us to investigate the issue further here.
The main standpoint for the function of Mp is that Mp works the way it does because all the objects we perceive and measure are mentally projected images. The first clue comes with our new understanding of mirror image formation. Our mind figures that the reflected light is from the direction in which the light source has originated. So the mind intends to project brain image Bi or perceived size Ps (they are interchangeable as far as the object’s size is concerned) in the direction of the reflected light rays back to its original location and size in the mirror. Similarly, our mind could project the object’s Bi or Ps back to its original location and size along the direction of the incoming light, which has of course reflected directly off the object.
From the philosophical point of view, we have enough reasons to believe that the ordinary or veridical objects have basically the same qualities as the mirror images. First of all, ordinary objects and mirror images are mathematically identical. The perceived size equation can apply to both ordinary objects and mirror images equally well, which has been confirmed by the experiments in Case A. The distance term in the equation is the total distance traveled by light from the object to the eye. The light travels directly from the object to our eye in a straight line when we perceive an ordinary object, whereas the light travels in a zigzag fashion by reflecting off mirrors when we perceive the images of objects in the mirrors. Once we understand this distance equality, then it is quite clear that the perceived size equation for ordinary objects and that for mirror images are by and large the same.
Secondly, ordinary objects and mirror images are experimentally indistinguishable as demonstrated by the cases A, B, and C above. The measured image size, which is the perceived image size for mirror images and the perceived size for ordinary objects, is exactly the same for both ordinary objects and mirror images as long as the values of H, Mp, and d are the same. This means that the mirror images, like the ordinary objects, can be measured, and also the measurements are highly precise. One demarcation of the veridical and non-veridical objects is the measurability; the veridical objects can be measured truthfully and the non-veridical ones cannot. For instance, the illusory objects cannot be measured precisely; and the hallucinatory objects cannot be measured at all. Thus, the ordinary objects and the mirror images are both veridical; and they are qualitatively not distinguishable.
Thirdly, ordinary objects and mirror images are phenomenologically indiscernible. Once all the clues about the existence of a mirror are removed, such as in a photo or video, it is impossible for us to tell a mirror image apart from an ordinary object. In sum, since ordinary objects and mirror images are largely the same in many respects, like the mirror images the ordinary objects we see could be also the projected images.
Notwithstanding, it is the measure-point term in the equation that provides the most compelling evidence to support the position, that is, the ordinary objects we see in our daily lives are the mental projections. Let’s assume that, or you are actually watching a TV. A straight line from the top of the TV screen is imagined to extend to your eye, enter the pupil and land on the retina. We call this line the top line and the landing spot on the retina the point a. In the same manner, a bottom line from the bottom of the TV screen projects onto the retina and lands on the point b. The top line and the bottom line intersect at the entrance point and form the visual angle θ. These two lines cross each other as they pass the entrance and land on the opposite positions on the retina, meaning that the point b is on the top and the point a is at the bottom. The angle inside the eye has the same degree as the visual angle θ so that θ determines the retinal image size according to the formula: R=tan θ · n, where the nodal distance n is a constant. This is the conventional description and explanation of our size perception.
However, this geometrical explanation has some unrelenting quandaries. The reflected light ray from any point on the top line or the bottom line can project onto the same retinal cell. As such, on the basis of the retinal image alone, we are unable to know whether the point a or b came from a nearer position or farther position in the environment. For instance, the combination of a nearer position on the top line and a farther position on the bottom line would produce the same sized image on the retina. Our brain would not be able to perceive the proper size of any image located along these two converging lines.
The conventional causal chain of vision states that the reflected light from an object in the world enters the eye, and the pattern of points of the light on the retina forms an image, where it is transduced to a neural signal in the optic nerves, from which it is transformed into a pattern of activation in the visual cortex. In other words, a physical process starts from a visible object (as a cause) and produces an effect on the retina and then in the brain. But, this causal chain poses the so-called inverse problem (Zygmunt, 2001).
The inverse problem is the determination of the “cause” of a phenomenon from measurements of the “effect” (i.e., the phenomenon itself). As for our size perception, there is a fundamentally ambiguous mapping between sources of retinal stimulation and the retinal images that are caused by those sources. For example, the size of an object and its distance from the observer are conflated in the retinal image. For any given projection on the retina, there are infinite number of pairings of object size and distance so that the image on the retina does not specify which pairing did in fact cause the image, as we have discussed earlier about the combination of nearer and farther positions on the top and bottom lines. Therefore, we are precluded from any possibility to know correctly an object’s size and distance.
Now let’s return to watching TV. This time, you will make an adjustable frame through which to watch the TV. You will soon find out that to match the TV frame you have to enlarge your adjustable frame when moving it closer to the TV set, and vice versa. According to the discussions above, the adjustable frame should have zero effect on the retinal image size no matter where the frame is placed because, on the basis of our conventional wisdom, our brain should not be able to perceive any size changes of the TV screen which casts a same image size on our eye. But, why do we have to change the sizes of the frame to fit around the TV screen when it is moved back and forth between the TV set and our eye?
To solve the puzzle, we have to completely reverse the causal chain of vision. Instead of regarding the object as the “projector” and the retina as the intercepting screen, we take the mind as the “projector” with the eye as its lens and the end of the projection as the location of the final intercepting screen to capture the image which is the original size of the object. We can also consider the adjustable frame as a transparent movable screen, which can essentially intercept the projecting image anywhere between the TV set and the eye. The size of the intercepted images by the frame, which is the perceived size, is proportional to the distance of the frame from the eye, as stipulated by the perceived size equation, because this adjustable frame is the measure-point itself.
As a matter of fact, if we put measuring marks on the frame, we can use the adjustable frame as a ruler to measure the perceived size of objects. The frame can definitely be conceived as being equivalent to the ruler. In addition, mirrors and windows, as noted before, can act in exactly the same way as the frame. As a result, rulers, frames, mirrors, and windows are all the measure-points as long as they are focused upon. All of them can effectively intercept the projecting images mid-way and produce the perceived image size proportional to the distances of these measure-points. When we look at a mirror image or an image through a window (or even an ordinary object), we are actually looking at a framed-in image that has been captured on its way of projecting onto the end screen. This new approach to the visual processing has essentially turned the inverse problem in vision into a forward problem where the size perception is concerned. As such, we know exactly how much each of the terms, i.e., perceived size, distance, and measure-point, has contributed to the final projected image. The external objects we see are the projected images; and the projected image we perceive are the external objects.
Besides, when we measure an object with the ruler touching the object, we have no doubt that we are measuring an external object. Yet, when we move the ruler some distance away from the object and then measure the same object with the ruler, now we do not think that we are measuring the object anymore, but the image of the same external object. We also believe that the image measured is located in the external world. Furthermore, the measurement of this external image must be exact because the object’s size and its distance are exact.
However, it is impossible for us to measure the incoming reflected light from the objects. It is because images have not been formed yet before the reflected light enters the eye and reaches the sensory surface, i.e., the retina. We simply cannot perceive the reflected light itself. In addition, it is impossible to measure precisely the representational images inside the brain (if there is a way to measure the brain image at all). According to the formula S=R·D, the subjective distance D is changeable from individual to individual and from situation to situation so that the perceived size S is not exact. This is not in agreement with the end image such as a mirror image which is precisely the same size as the original object by calculations and measurements. Accordingly, the image we are measuring is neither the incoming light nor the internal brain image. The only possibility left is that the image measured is the projected image out of the mind and in the external world. When we measure the image, we are actually measuring the original object. The original object and its projected image are inseparable so that the original object is the projected image and vice versa.
In fact, the thesis that what we see are the projected images has more supporting evidence. As mentioned earlier, our mind probably has the motivation to restore the object’s size and location. All that the mind needs to so is to reverse the process while projecting. In the case of the size restoration, the brain image or the perceived size is acquired by the process of dividing the distance, as shown in the perceived size equation. To restore the original size, the mind simply does a reverse operation by multiplying the distance, which is also indicated in the perceived size equation and has been confirmed by the projected images in the plane mirrors. What's more, the mind is also motivated probably to restore something else, i.e., the orientation of objects. It is generally believed that the retinal image or the pattern of light points on the retina is upside down. To restore the orientation, the mind simply reverses the brain image and projects it out as a right-side up image since the projection is a reverse process. As a matter of fact, the actual movie projector does exactly the same thing, that is, to project the upside down films out as the right-side up images onto the screen. The fact that the ordinary objects we see are all right-side up may prove that reversing the orientation of the brain images indeed occurs thanks to the mental projection.
Using a black paper to cover your eye and cutting a hole as big as your pupil, you will find out that more than half of the usual visual field has disappeared when looking through the hole at the world. Our eye is always likened to a camera which has an aperture to let in images. The aperture as small as a pinhole is all needed to capture the full scene. If our eye is like a camera, the pupil is all needed to let in the whole visual field for the brain to “see”. This experiment demonstrates that the pupil may merely let in the patterns of reflected light which the mind cannot perceive yet until they are transduced into the brain images being projected out. The plausible reason for this phenomenon is that the whole eyeball is acting as the lens for the “projector”, the mind. We can only perceive the projected images, not the images still in the brain.
Looking straight into a movie projector, you cannot see any image. When a screen or window is put in between, now you can see the images on the window or screen. As understood conventionally, the retinal image should be formed with or without the transparent window. The window should have no effect on the retinal image. The probable reason is that the window serves as a surface upon which the brain image can be projected.
If you wear glasses (or get a pair of glasses if you do not), take them off and hold them in front of you, letting the front side facing you. Look at objects through the two lenses. Now you will find that for each single object there are two images, one for each lens.
Get nine of such lenses and glue them together in three rows with three lenses in each row. Now you will see nine images of each single object on the multiple lenses. But, if you project a picture on a transparent sheet through the multiple lenses onto a wall or screen, only is one image shown on the screen. This means that a single image would land on the retina. Therefore, we should see just one image rather than nine if our conventional conception of visual perception were correct. This simple experiment provides an important evidence to support the mental projection thesis. Just like many images of an object seen in the multiple mirrors, which are projected by the mind individually and separately, the plausible explanation for the multiple images on the glass lenses we see is that our mind projects those images separately on each lens.All the above observations, experiments and analyses offer the favorable evidence for the position that the ordinary objects we normally see in the world are the projected images.
Gregory, R.L. (1998). Eye and brain (5th ed.). Oxford: Oxford University Press.
Hollins, M. (1976). Does accommodative micropsia exist? American Journal of Physiology, 89, 443-454.
Kaneko, H, & Uchikawa, K. (1997). Perceived angular size and linear size: The role of binocular disparity and visual surround. Perception 26 (1), 17–27.
Kosslyn, S.M. (1975). Information representation in visual images. Cognitive Psychology, 7, 341-370.
Zygmunt, P. (2001). Perception viewed as an inverse problem. Vision Research, 41(24), 3145-3161.
Related Information on the Web: