Update: The original website with its digitally-altered ‘miniworld’ pics seems to have been taken down. I’ve constructed a few of my own ‘miniworld’ pics to give the reader an idea of what I’m talking about:

From now on, I save an archive of all photos I use from the internet!

While stumbling through the ‘tubes the other day, I came across a very cute series of photos, an example of which is pictured below:

Is it a picture of a real place, or a picture of a miniature model? All the pictures are apparently of real places, digitally modified via Photoshop to make them look tiny! If you are skeptical, you can see the original picture of Stonehenge in the snow here, and what looks to be the original of Tokyo Tower here.

I’m not a digital photo expert, but it looks like a number of ‘tricks’ were used to make the pictures look miniature, including the simple psychological effect of always having views from above. The most significant modification, though, is of the focus: in the modified pictures, the foregrounds and backgrounds are out of focus, and this alone can fool the eye. Why is this?

This effect is a clever exploitation of the concept of depth of field, and its related ‘cousin’, depth of focus. I thought I’d do a short post about the idea (my friend Personal Demon suggested the topic months ago, actually) and the related mathematics. Along the way we’ll introduce a few simple lens topics from geometrical optics.

Without going into too much detail, according to the theory of geometrical optics, in free space light travels along straight line paths called rays. One can see how this idea came about: light passing through gaps in a cloud or through gaps in the foliage of trees appears as a continuous line of light.

A lens can be used to focus light through the process of refraction. Light rays passing through the glass of the lens are deflected at different angles depending on the angle at which the ray approaches and the thickness of the glass at the point of entry. For instance, the image below shows the effect of a lens upon a collection of parallel rays:

For our purposes we only need two numbers to characterize the properties of a lens: its focal length and its -number. The focal length is the distance between the center of the (presumably thin) lens and its focal point, as illustrated above. The -number is the diameter of the usable part of the lens, usually expressed in units of focal lengths. In other words, a lens whose diameter is half of a focal length has an -number of , while a lens whose diameter is an eighth of a focal length has an -number of .

The distance at which a lens forms an image is characterized entirely by the aptly-named thin lens equation, given by

,

where is the distance of the object to be imaged from the front of the lens and is the distance at which the image appears behind the lens. This is illustrated below:

For this example, the object is placed at , which by the thin lens equation results in an image at . For the picture, the -number is , though it is important to note that the -number has no effect on the axial position of the image. It will, however, play an important role in the discussion of depth of field and depth of focus.

The thin lens model can be used to understand the operation of simple cameras and the human eye. In a camera, a lens (or, usually, a system of lenses) projects an image onto photographic film or a CCD sensor. In the human eye, a lens projects an image onto the retina in the back of the eye, where the image information is converted into neuron signals:

It is to be noted, however, that the thin lens equation suggests that every object distance is imaged to a different image distance. In other words, the image of an object which is 15 feet away will appear at a different location behind the lens than the image of an object which is 10 feet away. If a camera (or an eye) is to be useful, it must be able to image objects at different distances. In a camera, one can adjust focus by changing the distance between the lens and the film; in the human eye, muscles in the eye change the focal length of the lens by distorting its shape.

What happens if our film (or retina) is not placed at the ideal image distance ? The image will be increasingly blurry the further we move from the ideal image distance. There will be, however, a range of distances around the ideal distance where the blur will not be noticeable. This range of distances is known as the depth of focus.

We may also ask what happens to the images of objects which are not located at the ideal object distance . Again, there will be a range of distances around the ideal distance where the blur will not be noticeable. This range of distances is known as the depth of field.

We can summarize both this ideas as follows:

1. Depth of focus. For a given object distance , the depth of focus is the range of image distances for which the object is in focus.

2. Depth of field. For a given image distance , the depth of field is the range of object distances for which the object is in focus.

Both of these concepts are illustrated below.

The ideas of depth of focus and depth of field both concern the ability of an image-forming device to make sharp images, but under different circumstances. When we are focusing on a single object and want to know the range of positions within the camera which are effectively in focus, we are referring to depth of focus. When we are imaging a collection of objects at different distances and want to know how many of these objects will be in focus at a given image plane, we are referring to depth of field.

Neither of these concepts are ‘absolute’ properties of an imaging system: they both depend on the resolution requirements at the end of the system. For instance, a high-resolution digital camera requires its images to be in much sharper focus than my 20/200 near-sighted eyes do. A point object which is being improperly imaged will produce a circular spot, rather than a point image, as illustrated below. It can be seen that the object on axis at A will produce a point image at the point I; the object on axis at B, however, will produce a point image at an earlier location. At I, object B produces a circular spot. It is important to note, and one can see from the diagram below, that the size of the circular spot depends upon the size of the lens aperture ( -number).

We define the circle of confusion as the largest diameter circular spot we are willing to tolerate in our imaging system.

Now we can do some math to quantify these concepts! We’ll restrict ourselves to studying depth of field, as it is directly relevant to the ‘miniature’ pictures listed earlier. Let’s start by looking at the behavior on the imaging side of the lens, as shown below:

The lens is taken to have an aperture , and we assume for future sanity that all objects lie at distances greater than the focal length . We choose an image plane which lies at distance ; images which focus at distances (red lines) and (blue lines) both produce a circular spot at the image plane ; it is assumed that this spot is of diameter , equal to the circle of confusion. A little bit of geometry gives us the following formulas for and ,

,

.

Now we can transform this to the object side of the lens; we have the corresponding figure:

It is to be noted that lies closer to the lens than , because lies closer to the lens than . We can transform our equations for the image locations into their corresponding object locations using the thin lens formula. After a bit of work, one finds that

,

,

where is the -number of the lens. The total depth of field is simply the difference of these values, .

What can we learn from these formulas? The formulas for and are identical except for a minus sign which appears in the denominator of . This sign makes all the difference, however, because that denominator can therefore go to zero, which means that becomes infinite. This happens when the lens is focused onto an object at a distance such that

.

This distance is known as the hyperfocal distance. When a lens is focused at or beyond the hyperfocal distance, everything beyond that distance is always in focus. Herein lies the key to understanding the ‘miniature’ pictures mentioned at the beginning of this post. When we focus our eyes on an object that is extremely close to us, we have a relatively narrow depth of field: objects immediately in front of and behind the object we’re looking at are out of focus. When we focus our eyes on a distant subject, we are looking past the hyperfocal distance and see almost everything in focus. Incidentally, one of the things that happens when you turn a digital camera onto ‘landscape’ mode is that the focus is set at or beyond the hyperfocal distance.

Though you probably don’t consciously realize it, your brain is taking some of its distance cues from this depth of focus behavior. The ‘miniature’ pictures help make big objects appear small by simulating a narrow depth of field.

Can we ‘fake it’ the other way, and make miniature objects appear larger? In principle this could be done by taking pictures of miniatures with a pinhole camera. I’ve talked about the image-forming abilities of the camera obscura in a previous post; a figure of this is shown below:

The image of a camera obscura (or, with film, the pinhole camera) is created by simple geometric projection. A perfect image is formed regardless of object or image location relative to the pinhole, which means that the pinhole camera has infinite depth of focus.

Of course, if you try to use a pinhole camera to make a miniature scenery look real, you had better have some extremely well-detailed miniatures!