Solution

Although this was not a competition problem, two correct solutions have been submitted: by Dinis Cheian and Ivan Ivashkovskiy.

The Ivan's solution is very thorough – with error analysis (as should be for such a semi-experimental problem) and is presented below on two images. Note that he defines n as the ratio of two lengths, h_im and l, which he can measure from the photo and thus calculate n. On the other hand, he finds expressions for h_im and l, and using these expressions shows that n =H/D. Knowing the object lenth H, he can find now D.

In addition to that solution, I have two comments. First, the usage of the Newton formula would have simplified slightly the mathematics: x₁x₂ = F², where x₁ = d – F and x₂ = f – F are the distances of the object and of the image from the respective focal planes (using Ivan's notations, d and f are the respective distances from the lens). So, the diameter of the circle of confusion l = Dx₂ / F = FD / x₁ = FD / (d – F). Second, pay attention to the main result: the diameter of the lens equals to the diameter of the circle of confusion created by a far-away dot-source, if measured by the image of an in-focus ruler. So, if you have a subject for your photo (eg. friends face for a portrait) and you want to blur the (far-away) background, the degree of the background blur is defined purely by the diameter of the lens: a tele-lens 600mm/4 creates circles of confusion of the size of the face, leaving no visible details at the background; a point-and-shoot camera with a small sensor and 8mm/4 lens creates almost no background blur: the diameter of the circles of confusion are smaller than the pupils of the portrait.

Finally, let us analyse the benefits of a large camera: you have a full-frame DSLR, and your friend has a small point-and-shoot camera of a 4 times smaller sensor (in linear size).  Your friend takes a photo of something, zooming to the focal length of 13 mm and is using the full aperture of F/4 (ie. the focal-length-to-diameter ratio equals to 4). You want to obtain exactly the same result: in order to have the same angle of view (and perspective), you need to take the focal length equal to 13*4 = 52 (your 50mm/1.4 standard prime lens works well). In order to have the same blur, you need to shut down the lens — down to the aperture F/16 (using the diaphragm you decrease the effective diameter). On the other hand, if you take a photo at the full aperture of F/1.4, your friend would need the aperture of F/0.35, which is theoretically impossible (would violate the second law of thermodynamics). Meanwhile, if you want to have both sharp foreground and sharp background, the point-and-shoot is better: you can use F/22, which would correspond to F/88 for the DSLR. While theoretically this is possible, the smallest aperture is typically only F/32 (starting from ca F/16, diffraction starts degrading the image).

– Jaan Kalda – Academic Committe –