Why isn't my computation of the actual size of an object in a photograph matching the expected value?

Question

Let's say I have a full frame camera (Sony A7) with a 50 mm lens (Canon FDn 50mm f/1.4) mounted on it. At a given object (a plane) distance I should be able to compute the size of the image in real world units (meters).

To experiment this I have put an A4 sheet of paper in front of my camera and moved the camera so that the width of the sheet of paper fills the whole image.

Here are the real world results:

• Sensor size: 24 x 36 mm
• Focal length: 50 mm
• A4 sheet size: 210 x 297 mm
• Object width on the sensor: 36 mm (it covers the full width of the image)
• Object distance (from sensor to sheet of paper): 510 mm (approximately)

From this topic;

distance to object (mm) = focal length (mm) * real height of the object (mm) * image height (pixels)
                          ---------------------------------------------------------------------------
                          object height (pixels) * sensor height (mm)

In my case this can be simplified because the pixel sizes are the same:

distance to object (mm) = focal length (mm) * real height of the object (mm)
                          --------------------------------------------------
                                         sensor height (mm)

My real world experiments yelds 510 mm, not 412 mm!

50 mm * 297 mm = 412 mm
--------------
     36 mm

What is wrong with my computation?

I'm using the Fotodiox 10LA-FD-NEX adaptater, I'm able to focus the lens at infinity. The sensor/paper sheets planes are parallel, here is a picture of the setup:

Lens focus is set to the closest value (~0.45m), object distance is 525 mm

Then lens is set to infinity value, the sheet does fit the image length

Then after correcting the height, the object distance is 485 mm

485 mm is closer to the answer but still far away from the 412 mm expected!

Answer 1

I think that you have, in fact, successfully answered this inverse question: How to test actual focal length?. It's not just infinity focus, but that lens focal lengths are generally rounded to familiar even numbers. Two different nominally-50mm lenses won't always give the same field of view even when focused at infinity. (And as a side note, be aware that turning the lens as far as it will go will usually focus past infinity.)

You'll also find that some pixels from the edge of the frame aren't included in the actual image, and 36mm may not be precisely the correct value to use.

Altogether, this demonstrates another basic fact: equipment made for photography is made for photography, not for photometry. You can use a hammer to drive in screws, but you won't get ideal results.

Answer 2

The focal length of a lens is a measurement taken from the rear nodal of the lens to the focal plane when the lens is imaging an object at infinity ( ∞ ). That would be a star or an artificial star, a point of light 3000 diameters distant. In other words, the focal length measurement is only valid when the light rays arrive as parallel rays.

If the object being imaged is closer than infinity ( ∞ ), the light rays are arriving as diverging rays. Now the job of the lens is to converge the arriving light rays. This action causes the rays to take on the shape of an ice-cream cone. Now the lens has a fixed ability as to the amount of refraction (Latin “bend back”) it can perform. The result is, the apex of the cone of the image forming rays fall further from the lens when imaging nearby objects. This elongated distance, now called the image distance, is why we must cause the lens to move further away from film or chip as we focus on nearby objects.

If we are focusing close and the image is life-size -- called “unity” or 1:1 -- the lens will be racked forward one complete focal length and the image to focal plane distance will be 4x the focal length engraved on the lens. In other words, the back focus of an image at life-size is 2x the focal length. The "unity" set-up is a good way to calculate the actual focal length because at life-size (magnification 1) the focal length is the distance object to image plane divided by 4.

The bottom line is, your 50mm lens only performs as a 50mm if the object being imaged is at infinity ( ∞ ). At the subject distance you are preforming your test, the back focus is elongated, it is no longer 50mm. Your mistake is you are using the engraved focal length for your calculation when you should be using the back focus distance.

Answer 3

Simple equations of the kind you are using will only be exact for very idealised lenses as has already been pointed out. But you also have a slight error in your equation.

The simplest lens is no lens at all. A pinhole camera of 50mm focal length would consist of a pinhole at 50mm distance from the sensor (or more likely film). A quick drawing will convince you that an object 50mm from the pinhole would would lead to a life-size image on the sensor. If you compare that with your equations you will notice that the object distance should not be measured from the sensor plane, so you overestimated the distance by 50mm.