How Blurry (Warning - Math)

[MarkBarbieri]

The question came up in a recent thread (Are there any PnS cameras that can do this?) as to whether a PnS camera can have a sharply focused subject and a blurry background. I thought it would be fun (knowing how much you guys love that mathematics of optics ) to explore the math on the subject.

What we want to know is how blurry a given spot that is not our subject will be in a picture and what determines that. Before we get to the formula, let's define our terms and symbols.

b - The diameter of the blurry spot. A big number means that something is very out of focus. A small number means that something is more in focus. At the plane of focus, this is 0.

f - The focal length of your lens.

m - The magnification level. This is how large something in your picture (printed or shown on the screen) is compared with the object in real life.

N - Your f-stop or aperture. I know it's a dumb letter to pick, but I'm trying match the letters used in the Wikipedia article I'll reference below.

s - The distance to your subject.

D - The distance to the thing that is not your subject. This is the thing we are trying to make blurry.

The equation is:

b = f * m / N * |s - D| / D

The |s - D| to anyone not familiar with that nomenclature is the absolute value of s - D. In other words, if s is bigger than D, then it is just s minus D. If D is bigger than s, then it is D - s.

First, let's ignore m. It's useful if you are trying to calculate the size of the actual blurry spot, but it's not useful in comparing a PnS to a DSLR. The reason is that it is the same. I am trying to compare two pictures take at the same spot zoomed in the same amount (same field of view). In both cases, the magnification (the relative size of the stuff in real life compared to it's size in the picture) is identical.

Let's look at f. The bigger it gets, the blurrier our spot gets (all other things equal). In other words. If I stand in one spot and use a longer focal length, I'll get a blurrier background. I'll also have a smaller field of view.

If you recall from earlier postings, the field of view is determined by both the focal length and the sensor size. Smaller sensors have narrower fields of view for the same focal length. Imagine that you and I are standing in the same spot and I have a PnS with a tiny sensor and you have a DSLR with a much larger sensor. If we are taking photos with the same image, I must be using a much smaller focal length. This smaller focal length is what causes me to have a greater depth of field (less blurriness).

Let's give some specific examples. Let's say that there are four of us. I'm standing there with a Canon 1Ds Mark III and an 85mm f/1.2 lens that someone has kindly donated to me for this example. GDad is standing there with his D300 (which has a sensor that has a 1.5x crop factor) and a Nikon 24-70 f/2.8 lens. Pea-N-Me is standing there with her Olympus 510 (2x crop factor) and a 35-100 f/2 lens. Bob100 is standing there with his Canon S5 (6x crop factor).

Let's say that our subject is 10 feet from us and the background (a bunch of trees) is 100 feet from us. We'll all start by shooting at f/2.8. To make our images cover the same area, I'll shoot at an 85mm focal length; GDad will shoot at 57mm (85 / 1.5); Pea-N-Me will shoot at 42mm (85 / 2); and Bob100 will shoot at 14mm (85 / 6).

A spot on the tree in the background will result in a blurry dot this big for each of us:

Mark - 27
GDad - 18
Pea-N-Me - 13
Bob100 - 4 1/2

Because we don't know the magnification (until we print or display), we don't know what unit these numbers are in. The useful thing is that it shows us the relative size of the blur spots.

Now let's play with our apertures. Bob100 will open his up as wide as possible. That makes the f-stop number lower. That makes the blur spot bigger. Alas, the widest aperture on the S5 is 2.7, so the biggest he can make his blur spot is 5. For comparison, to get the same "bluriness", Pea-N-Me would have to shoot at about f/7.5. GDad would have to shoot at f/10. I would have to shoot at f/15.

What happens if we move the trees farther away? Let's go to 1,000 feet. In that case, the bluriness doesn't increase that much. In fact, it just nudges up to 5. That's because the difference is in the |s - D| / D part of the equation. That goes from 90 / 100 to 990 / 1000. In other words, the movement of the background increased the distance factor from 0.9 to 0.99. Not much of an increase.

Now imagine that we moved the trees closer. Say go to 20 feet. Now, the distance factor goes from 0.9 to 0.5 (which is |10 - 20| / 20). So our bluriness is cut almost in half from 4 1/2 to 2 1/2.

As you can see from the formula, the bigger the difference between your subject and your background, the bigger the blur.

You can also see that the blur increases faster in front of the subject than it does behind the subject. For example, with a subject at 10 feet distant and objects 1 foot in front and behind. The one in front would have a distance blurring factor of 0.11 (which is |10 - 9| / 9). The one in front would have a distance blurring factor of 0.09 (which is |10 - 11| / 11). Switch to 5 feet in front and behind and you get a distance blur factor of 1 in front and 0.33 behind.

So how do I get the most blur with my camera? As you can see, getting the most separation between your subject and background will give you the most blur. Using your widest aperture (lowest f-stop) helps. Using a longer focal length will also give you more blur, but there is a catch. If you zoom in, you'll also need to step back to keep your subject the same size. Stepping back will then increase s, which makes it harder to blur things behind your subject.

You can also see that the larger your sensor, the easier it is to blur things. Conversely, larger sensors have a hard time getting everything in focus. That's why PnS are often so good at macro shooting.

Any questions? Any mistakes? Any still reading this?

The formula comes from the following Wikipedia article.
http://en.wikipedia.org/wiki/Depth_of_field#Derivation_of_the_DOF_formulas

 

[boBQuincy]

Any questions? Any mistakes? Any still reading this?

The formula comes from the following Wikipedia article.
http://en.wikipedia.org/wiki/Depth_of_field#Derivation_of_the_DOF_formulas

Only one thing I thought of adding, when we zoom in to get a longer focal length (and more blur) many lenses also go to a smaller aperture, somewhat negating the effect of zooming in to get more blur.

Oh yes, one more thing, many P&S have so much depth of field that a fingerprint on the lens has a much greater effect than with a SLR.

Good work, Mark!

 

[MarkBarbieri]

Oh yes, one more thing, many P&S have so much depth of field that a fingerprint on the lens has a much greater effect than with a SLR.

That reminds me of a useful trick for SLR shooters. If you need to shoot through a screen, put your camera as close as possible to the screen and shoot with a really wide open aperture (low f-stop number). The screen will be so out of focus that you probably won't see it at all.

Here is an example. You can see the look of the "screen" behind the bird but the screen in front of the bird is slow blurred that it all but disappeared.

Here are two other shots from that outing that have absolutely nothing to do with the rest of the thread. I just liked them.

Oh, and here is a shot of what must be the least functional zoo fence ever built. Yeah, that'll keep him caged in there.

 

[extreme8]

I got tired of the calculating the acceptable circle of confusion, etc a long time ago and just use a site like
http://www.dofmaster.com/dofjs.html
They even have a great downloadable DOF calculator that you can print and assemble.
Math hurts my head.

 

[Pea-n-Me]

Yeah, that'll keep him caged in there.

Gosh, what a gorgeous zebra picture!!

Thanks for taking the time to do this for us.

 

[ChrisinNJ]

but that parrot photo, is amazing, I would never know there was a screen in front of it.

I do understand the principle of what you are saying, and I how understand the idea of aperture opening affects a photo in terms of blur. And I also get the fact that how far away your background item is will affect the blur factor. But that calculation is wouldn't help me because by the time I figured it all out, the sun would have set, and I would have missed the shot completely.

I have been experimenting more with shooting on aperture priority to see what kind of effect I will get. But the battle that I seem to fight the most is trying to get my youngest dd to just stand still long enough, my older dd to stop biting her nails long enough to get a picture of her face without her hand in front of it...etc. As I am still on a learning curve, but the time I figure out what setting to use, my subject runs away!

I end up downloading 50 pictures of which only 2-3 are worth keeping...sigh...someday I will figure it out!

Oh, and I shot several pictures at a local amusement park in RAW, as well as several flower pictures in RAW, but I have yet to find the time to figure out the post processing process in order to post any of them here!

 

[wen-tom]

Wow Mark, I feel tired for you

I have been following the thread talking about DOF but I have to admit that at this stage, although I kind of understand the theory, I'm not so good in practice

I have had a few "lucky" shots though

The Zebra picture is fantastic

 

[seashoreCM]

(Math Ph.D. still needed) With all of you using the same aperture (F2/8) and shooting the same subject from the same distance, don't you all get the same depth of field as seen in the finished picture?

For example (using your figures), your blurry dot has a nominal value of 27 for a 35mm film frame. Pea-N-Me has a blurry dot value of 13. But Pea-N-Me, in order to get the same print size, say, 4x6 inches, has to enlarge her sensor image twice as much, so her blurry dot becomes the same size as yours in the finished picture. Meanwhile Bob100 with his 6x crop factor* (six of his sensor rectangular outlines fit diagonally across one 35mm film frame) needs to enlarge his sensor image six times as much as you do making his blurry dot 6 times 4-1/2 or 27, same as yours also. No?

Of course, if one of you opens his aperture, say, to f/1.2 and another of you stops down to f5/6, then the depths of field change accordingly.

Digital camera hints: http://members.aol.com/ajaynejr/digicam.htm

In another forum I mentioned the idea of making the background more out of focus by setting the actual focus in front of rather than at the desired subject. There's a little trial and error involved to end up with the subject still satisfactorily in focus and results are not guaranteed. The more mathematical discussion follows:

Now as you (and the reference article) said, b=0 (blurry dot is a point) means subject is perfectly focused. If, say, the desired subject b1 computes to zero at the same time the background you want to be blurry b2 computes to 27, next question is, is 27 blurry enough for you? Now you can focus in front of the subject causing the main subject to be not quite in focus i.e. subject's b1 to be non-zero, for example 5. The background b2 will now be blurrier, say, 32. Is 5 sharp enough for you (after enlarging to the desired print size) and is 32 blurry enough for you? I didn't do the calculations but I think you get the idea.

 

[jann1033]

so are we to assume mark got a 1ds? ( as far as the math, tomorrow is another day)

 

[MarkBarbieri]

(Math Ph.D. still needed) With all of you using the same aperture (F2/8) and shooting the same subject from the same distance, don't you all get the same depth of field as seen in the finished picture?

For example (using your figures), your blurry dot has a nominal value of 27 for a 35mm film frame. Pea-N-Me has a blurry dot value of 13. But Pea-N-Me, in order to get the same print size, say, 4x6 inches, has to enlarge her sensor image twice as much, so her blurry dot becomes the same size as yours in the finished picture. Meanwhile Bob100 with his 6x crop factor* (six of his sensor rectangular outlines fit diagonally across one 35mm film frame) needs to enlarge his sensor image six times as much as you do making his blurry dot 6 times 4-1/2 or 27, same as yours also. No?

It is my understanding that the magnification in the equation is the total net magnification. It's the ratio of how big the object appears in real life compared with how big it appears on the print. The fact is a function of field of view and print size. Using a smaller sensor doesn't change the magnification if you compensate by using a shorter focal length. Sorry for the lame explanation, but I'm a bit rushed tonight.

Now as you (and the reference article) said, b=0 (blurry dot is a point) means subject is perfectly focused. If, say, the desired subject b1 computes to zero at the same time the background you want to be blurry b2 computes to 27, next question is, is 27 blurry enough for you? Now you can focus in front of the subject causing the main subject to be not quite in focus i.e. subject's b1 to be non-zero, for example 5. The background b2 will now be blurrier, say, 32. Is 5 sharp enough for you (after enlarging to the desired print size) and is 32 blurry enough for you? I didn't do the calculations but I think you get the idea.

It's good to understand that the only place that is perfectly sharp (regardless of aperture) is the actual plane of focus. Everything else is out of focus. When people say that something is within the DOF, they mean that it is sharp enough to be considered in focus.

 

[MarkBarbieri]

so are we to assume mark got a 1ds? ( as far as the math, tomorrow is another day)

I wish. I'm still plugging along with my old 1DM2. With my companies stock dropping quickly and the overall market sagging, I'm trying to hold back on expenses. I'm afraid the boys are going to need a substantial boost to their college funds at the end of the year. I do have some money set aside for a new camera later this year or next year, but I was hoping for something like a 5D replacement coming in at $3,500. If that never materializes, I might just get a Rebel. Then again, if stocks keep getting cheaper, I may change my mind and throw it back into the market instead.

I survived Tropical Storm Eduardo today. Our offices were closed all day, so I had to work from home. The storm turned out to be a big nothing. We got 2-3 inches of rain spread out over the entire day. It never really rained hard and it never got very windy.

I'm off to Denver tomorrow night. I'm going to meet my wife, kids, and camera gear there (they've all been vacationing in Michigan). On Saturday, we're heading up to Grand Tetons and Yellowstone to go camping for a week. We'll follow that with another week near Rocky Mountain National Park in Colorado. I won't be home until late August. I doubt I'll have much connectivity while we are in Wyoming, but I should be online from time to time while we are in Colorado.

 

[fitzperry]

I'm off to Denver tomorrow night. I'm going to meet my wife, kids, and camera gear there (they've all been vacationing in Michigan). On Saturday, we're heading up to Grand Tetons and Yellowstone to go camping for a week. We'll follow that with another week near Rocky Mountain National Park in Colorado. I won't be home until late August. I doubt I'll have much connectivity while we are in Wyoming, but I should be online from time to time while we are in Colorado.

Have a great trip! We just returned from a month in Park City, Utah, and I'm missing those mountains.

 

[DisneySuiteFreak]

The question came up in a recent thread (Are there any PnS cameras that can do this?) as to whether a PnS camera can have a sharply focused subject and a blurry background. I thought it would be fun (knowing how much you guys love that mathematics of optics ) to explore the math on the subject.

What we want to know is how blurry a given spot that is not our subject will be in a picture and what determines that. Before we get to the formula, let's define our terms and symbols.

So how do I get the most blur with my camera? As you can see, getting the most separation between your subject and background will give you the most blur. Using your widest aperture (lowest f-stop) helps. Using a longer focal length will also give you more blur, but there is a catch. If you zoom in, you'll also need to step back to keep your subject the same size. Stepping back will then increase s, which makes it harder to blur things behind your subject.

So this is why when people have those expensive, heavy, 2.8 lenses, that they can get the nice creamy looking Beokeh -- because they can leave their Apetures wide open through the whole length of the lens even zoomed all the way out. Is that right?
I really do not like math...