# Mathematical Formula for motion blurs (or "freezing motion")

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#### ombre

##### Senior Member
Hi guys, you may call me a freak, but I spent some time thinking and invented a mathematical formula for calculating the amount of pixel blur you will get. This is the most simplified (usable) version I have been able to come up with... (my first one had like 12 variables :S)

I'm personally not sure if anyone has done a similar math on this, I haven't found it anywhere so yea. =) (can I call it the Ombre's formula for photography blurs? haha) Please do let me know if anyone came up with something similar.

So anyway, just to share with you guys what I've found. C&C welcome. =) Just for disclaimers, I don't study mathematics. (not anymore at least)

B = fxsR / (d - f)

B = Pixels of Blur
x = object speed
s = shutter speed
R = Pixels per metre of sensor
d = distance from subject
f = lens focal length

So how much shutter speed is really needed to "freeze motion"? Say a sports man is running on a field at 4m/s, (this is about running 2.4km in 10 minutes), and you're using a 55-250 IS (handshake blurs are not taken into account). Target is 50m from you.

If you shot at 1/400s shutter speed, the pixels of blur will be 7.82289~ pixels. =) (this is calculation for 350D 8mp sensor).

Perfect accuracy of the formula is not entirely guaranteed, because the distance between your lens and sensor may vary slightly. But this should be negligible with the formula. (at this point i'm not sure).

=) Cheers.

#### Lens Fidelity

##### Member
Its great that you went through the effort to attempt in the creation of a formula for stopping motion but in my opinion, it is not practical to use it in taking photographs as by the time you finish your calculations, your subject may already be a mile away. It is faster to just set the speed through approximations.

Though impractical in my opinion in spontaneous shots, it may be useful in studio set-ups where you might need precise calculations.

##### Senior Member
This information, knowledge, how will it improve the images I produced?????

#### catchlights

##### Moderator
Staff member
This information, knowledge, how will it improve the images I produced?????
who cares, as long I have a sharp photo!

#### mcn

##### New Member
now photographie becomes a science

all hail

#### LittleWolf

##### New Member
s = shutter speed
s=Exposure time, not shutter speed. And the formula does not work for focal plane curtain shutters (i.e. SLRs).

#### blueayz

##### New Member
Interesting exercise bro. But it still does have too many variables. It will be useful for those who knows exactly what they want and have the time to plan for those one-shot-one-kill moments. To most of us who have the luxury of having several tries and the LCD preview, it won't be so critical. But not being a mathematical genius myself, I admire you being able to come up with such a formula. Maybe you should approach or is already working at DSO or ASTAR.

#### psyk1

##### New Member
Well.. I can't see how much benefit it would bring also.. but it is interesting.. I would think a formula for such a calculation is already out there somewhere.. Anyway, like others said, too much variables involved.. Like, the movement of the object being 3-dimensional.. Maybe you should have just put your initial formula with all the variables.. haha

#### ombre

##### Senior Member
Hi all, obviously it has some valid applications that I've considered. (if not I really won't do it!) Heres some.

If you think its a hassle to calculate, I've actually done everything on an excel sheet. The formula can be simpified to

B = MsxR

where M = Magnification, R won't change if you don't change cameras.

Applications include:
Answering "do you really need that F2.8 lens or F4 is enough?!" Maybe even F5.6. If your intention is just to freeze the subject under the lights that you have. You have to decide how many pixels blur is considered 'too blur' for you.. Basically, it gives you a way to decide on your threshold shutter speed or exposure time that you want for freezing subjects. After you get the shutter speed, fill in the formulas with ISO and aperture combinations, this one varies with lighting so try and error may be needed.

Yes thanks for those who pointed out that it can be for studio use! For things like sports / dance, especially jump shots. Which in fact is one of my intended use for this formula. I will be helping my friends do a ballet rehearsal photoshoot soon.

Maybe it helps you to calculate what kind of flash sync you need too? I'm not sure, never done an actual studio shoot.

Also perhaps picture quality is implied here. Do you really want to push your ISO to 6400 to get a 1/500 shutter in a pop concert, when the subject moves less than 1m/s? Use the formula to find out, maybe 1/100 is enough.

Possibly, shooting at 150mm and cropping in, results in less blur than shooting at 200mm. =) Formula should indicate this.

Aperture wise, I did make a formula last time, but aperture is much easier to approximate, still the formula is good if you want to calculate differences harder to estimate. such as F4.5 vs F4? Less practical use because its no big deal really.

S = 2log2 ( F1 / F0)

S = step gain
F = aperture (F1 is the 'smaller' aperture ie F1 = 1.8, F0 = 1.4, S = 0.7251 ) (1.653 times faster)

Personal application, besides the dance photography part, I actually planned to use this formula for candids. I'm only using a 55-250, not very high quality lens, so if can, I have to stop down abit. I figured that if people were just walking, 1/300 should be able to stop the motion, I almost wanted to get a 70-200L lens, but no money. Now i know I don't need it.... yet. 1/200 is enough to shoot walking pedestrians with 5 pixels (100%) blur, negligible. =)

All in all, trial and error can throw my formula out the window. But really, sometimes people just need a scientific confirmation (dont you agree?). Moreover I don't have alot of finances to do trial and error.

Further, but more complicated uses of this formula probably includes measuring the effectiveness of your lens IS (using the formula in reverse). basically the formula translates actual distance over time into pixel distance over time.

Reversing the formula also tells you the exact distance you were from your subject (up to the last mm!). If you want to brag about your great candid sniping... this is the way to go.

Last of all, its really fun to do, for me at least.

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#### ombre

##### Senior Member
Well.. I can't see how much benefit it would bring also.. but it is interesting.. I would think a formula for such a calculation is already out there somewhere.. Anyway, like others said, too much variables involved.. Like, the movement of the object being 3-dimensional.. Maybe you should have just put your initial formula with all the variables.. haha
I did consider that. 3 dimensional movements do not come into play or doesn't affect the calculation. Basically only the plane parallel to the camera needs to be considered. Only the X axis in this case.

Also, the subject vs background distance does not matter here. =) Only the subject distance matters.

#### ArchRival

##### New Member
Maybe you should approach or is already working at DSO or ASTAR.
Out of curiousity, what does astar do? I mean, they are taking up alot of public funds.

#### tjhan

##### New Member
The bulk of the responses to this thread is awesome, in a bad way.

I'm sure a lot of photogs are engineers and would appreciate this.

For those who aren't, let me explain that in life, almost anything can be stuffed into a formula to "approximate" a value. For example, there are equations to figure out the velocity/friction/whatever value of fluids in big pipes, small pipes, rough pipes etc. All these formulas are useful because they are a quick and easy way to figure out the value you need, without the error being too large. Of course, all engineers have a calculator in their shirt pocket lol!

I've not tested the above formula, but I noticed one crucial thing missing. WHERE ARE THE UNITS???? I notice you used m/s and metres in your explanation, but it would make more sense to add the units clearly. Also, for the sake of the CS members, do include a detailed sample calculation, with pictorial evidence. Apparently "artists" cannot do maths.

Btw the guy who said Photographie (sic) is now a science, it's based on science what.

Deadpoet gave an uncharacteristically uncle response, like "Why must retraining?" for a guy who pans others often for a lack of desire to improve.

Finally, I hate maths.

#### LittleWolf

##### New Member
I've not tested the above formula, but I noticed one crucial thing missing. WHERE ARE THE UNITS????
The units are part of the quantities that you insert when you use the formula. he formula obviously doesn't depend on a specific choice of units (if it did, there would be something very wrong with it). You can pick any unit you like as long as you calculate correctly.

#### Deming86

##### New Member
hahaa if you really have alot of spare time, you could test out the hypothesis that his formula is right! Could pump in some statistical analysis too!

#### night86mare

##### Deregistered
shoot more

understand more

sit around and calculate = waste time, because equations mean nothing irl.. unless you are hiro nakumura and can turn back time to keep reshooting

#### LittleWolf

##### New Member
understand more

sit around and calculate = waste time, because equations mean nothing irl..

Being able to derive and apply mathematical models is a sure sign of understanding. Conversely, it you aren't able to come up with the above equation (or equivalent) in less that 5 minutes, I'd take it as a sure sign that you don't understand the most basic fundamentals of geometry, perspective, and image formation by a lens.

Maybe the S'pore eduation system has failed and left you with the impression that equations are only good for memorizing to ace exams. I've met such sad cases - straight A kids - who couldn't solve the most simple real-life problems. No wonder Singapore needs to hire foreign talent.

Without such "equations that mean nothing irl", you would not have your lenses, cameras, or the computer you wrote this on.

After advocating mathematical illiteracy, what's next? Give up reading and writing? Give up speech? Will we achieve true understanding if we sit grunting in a cave around a fireplace?

#### night86mare

##### Deregistered
i never said mathematics was useless in life

just for this kind of thing, like photography.

unless you're gauss, who really has time to slowly work out formulae in their head, EVEN IF it did work? you have the powers of speed estimation? you can work out that oh, that f1 car is coming round the bend at 200km/h and i can plug it into my magic formula? dude, get real. if i do need to understand the fundamentals of geometry, you need to understand the fundamentals of reality.

stop being so worked up, it makes conversing with you a literal pain in the ass. we all know you love math and math loves you, but remember that the greatest mathematicians had undoubtedly open minds.. i'm afraid it's hard to say the same thing about you. have a nice day.

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#### shojibake

##### New Member
dimensional analysis.

#### LazerLordz

##### New Member
When you're out there shooting, getting your content is more important than pondering on the mathematical aspects of things.

Let's not forget that. View things in perspective, and not try to "shift paradigms" for the sake of doing so.

Most beginners lack a decent grounding in the basic fundamentals of photography, so get those right first.

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