Debunking Equivalence Part 2

In my previous post Debunking Equivalence I covered in detail the major flaws of this concept called “equivalence”. Mind you, not everything in equivalence is wrong. Equivalence in field of view and depth of field make total sense. What does not make sense is the equivalence in exposure. This “exposure equivalence” is what full frame fanbois sell to unsuspecting gear heads. It is supposed to prove that full frame is superior to APS-C, m43 and smaller sensor cameras. 

In this post, I will use basic math to debunk the myth. Just enough math that I learned when I was in first grade — seriously.

Recall the equivalence comparison between m43 and full frame:

m43: 25mm, f/5.6, 1/125s, ISO 100

FF: 50mm, f/11, 1/125s, ISO 400

Ignore the ISO settings for now. Let us concentrate on the f-stop and shutter speed settings. The reason, they say, that f/5.6@125 is equivalent to f/11@125 is that both gather the same amount of light by virtue of the difference in focal length. The longer 50mm lens will have the same entrance pupil diameter as the 25mm lens. The difference in focal length of course is proportional to the sensor size. 

Now let us use arbitrary units of measure and consider the ratio X/Y, where X is the total amount of light and Y is the sensor size. Supposing that for m43 we have the ratio 4/8, a full frame sensor (4x area), according to equivalence, would have a ratio 4/32. Again:

m43 = 4/8 vs FF = 4/32

So total light is constant at 4 units and the denominators are the respective sensor size units 8 and 32 for m43 and full frame respectively. Still with me? Obviously, they are not the same. Not in any known universe. This is why for the same amount of light, the full frame will come out two stops underexposed. And this is why equivalence fanbois will insist that an increase in ISO is necessary; the full frame shot is very dark! Now we know that bumping the ISO does not increase the amount of light but will make the image brighter. I’m not sure how to represent that in numbers because nothing has changed really in terms of light. ISO bump is fake. It’s a post-capture operation that does not change the exposure or captured light. Furthermore, an ISO bump introduces noise and that is why equivalence forces the comparison to be performed at the same print size. This method of cheating does miracles for the fanbois. Let’s see how it works:

If we agree to compare at the same print size of 16 units, we now have 

m43: 4/8 upsampled to 4/16

FF: 4/32 downsampled to 4/16

Magic!!! They are now the same! They are equivalent! This is true for any print size therefore equivalence is correct!!! Therefore full frame is superior because at the same f-stop it will have gathered more light! 

Well, not so fast! The amount of light did not change. It was constant at 4 units. The apparent changes in signal performance was not due to light but due to resampling. Do not equate resampling to total light. They are not the same and are completely independent of each other. Resampling is like changing your viewing distance. I can make my crappy shot look cleaner simply by viewing it farther away. Did I change total light by moving away from the image? Stupid isn’t it?

That is the very naive reasoning behind equivalence. Not only is the conclusion stupid but the assumption here is that there is absolutely NO NOISE! Noise is ever present. Noise is present and proportional to the incoming light and a property of the sensor itself. Let’s see what happens when we introduce noise. 

Supposing that noise is 1 unit. We now have:

m43: signal = 4/8, noise = 1/8

FF: signal = 4/32, noise = 4/32

Therefore the signal to noise ratio (SNR) are as follows:

m43 = 4:1

FF = 4:4 or 1:1

The full frame is obviously inferior! It makes sense because it was underexposed by two stops (4x)!!! If you boost the signal by increasing the ISO you are boosting noise as well. In low light situations where noise is more pronounced, a 4:2 SNR for m43 will be 4:8 for full frame. There is more noise than signal in the full frame image! At 4:4 SNR for m43, full frame is at 4:32. There is nothing but noise in the full frame. You just can’t underexpose indefinitely and bump the ISO! That doesn’t work in all situations. This is why images at higher ISOs look bad. There is more noise than signal in low light situations. Yet, equivalence fanbois will try to convince you that ISO 6400 on m43 is the same as two stops underexposure plus ISO 25600 in full frame. It’s not. 

So again, the equivalence fanbois could not accept this fact. At the sensor level, equivalence has made the full frame look really bad. What can they do? Cheat again! Force the comparison to use the same print size. At a print size of 16 units, noise will be increased or decreased proportional to how much you upsample or downsample. We have:

m43: signal = 4/16, noise = 2/16

FF: signal = 4/16, noise = 2/16

So now the SNR for both are equal at 4:2! Can you see how they manipulate the numbers? They are using image size (number of pixels) to circumvent noise and stupidly equate this to light gathering. The total amount of light has not changed. How could anyone possibly attribute the changes in SNR due to resampling to light? It does not make any sense at all! Look closely though because this SNR is for the entire image. Most of the signal will be concentrated on the brighter areas. In the darker areas noise will show its teeth. In instances were full frame is severely underexposed (SNR 4:32) there is no saving it. It would look crap. M43, on the other hand will happily chug along with 1:1 SNR or better. 

This is why when you start comparing two full frame cameras with different resolutions you will notice variations in SNR results at different print sizes (Megapixel Hallucinations). If the SNR changes even when the sensor size is held constant then obviously sensor size does not matter. Therefore total light, being proportional to sensor size, by itself does not tell the whole picture. What matters is the RATIO of total light to sensor size, otherwise known as EXPOSURE. For SNR to be the same, exposure must be the same for sensors with the same properties (i.e. sensel pitch, efficiency, etc…). Size does not matter. 

Equivalence debunked…again!


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