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

1. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Hi guys, the bulk of response is quite a bit more than I expected, I can't entirely respond to all but lets pick out the crucial ones

Originally Posted by Deming86
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!
Well I don't! I worked out this formula on a train ride on the way to psychology class. (I'm no mathematician!). After that I just wrote it out on paper / microsoft excel, just to confirm my results.

Someone mentioned that this is an estimate. The formula for Magnification, M = f / (d-f), f=focal length, d= subject distance. Is taken from elsewhere, its a known formula and should be accurate. Assuming that formula is accurate, my formula should be pin sharp. (unless your sensor is less than ideal and doesn't picks up the image) Or maybe shutter lag. But seriously, do you need to be that sharp?

Originally Posted by tjhan
The bulk of the responses to this thread is awesome, in a bad way.

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.

Finally, I hate maths.
Units aren't part of the forumla. Of course if you did math/physics in school, the units are always prescribed. But at this level, I assume people who are really interested in the formula know how to standardize units. If you decide your distance should be in meters, just make sure you convert your focal length, subject distance, displacement/second, all into meters.

Essentially the formula is more of a ratio, thus no units really either. The ratio goes like
distance travelled (in real) : distance travelled on image (in pixels) (=blur)

Originally Posted by night86mare
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.
This is precisely why you need my formula! I never said you calculate the formula on the spot during the F1 race!!! I don't mean to offend you, but thats being a bit short-sighted. You would have done the calculations the night before as preparation. I shall build on your example, and apply the formula.

Say you're on the stand, about 50 meters from the "turn". Car is turning at 200km/h, we will use 55 m/s here (actual is 55.555556~ m/s). And you brought your 200mm F2.8 for a night race. Lets assume after you expose properly, you discover that you're at 1/640 seconds of exposure time. This results in ~54 pixels of blur! Assume you're at max ISO and aperture. 54 pixels is quite abit obviously.

What will you do?
Some won't shoot,
some would bring a shorter lens (135 F2 L?) and crop...
Some will bring a 300mm F2.8 and sit further away, this actually reduce the blur due to the mathematics, you wouldn't have otherwise realize this would you? =)
I personally will shoot underexposed and post process. Noise is better than blur for me.

Essentially given the above scenario, using the formula, you should have in mind that at this event, you don't want to shoot below 1/2500 at 50 meters. This gives 14 px of blur, live with it, you cant remove it completely.

How to improve? Sit further back, at 100m away, with the 300mm, 1/2000 will give you 13 px of blur. If you print A4, maybe the blur is only "zoomed out" to 2-3 px, assuming its 72px per cm, not observable.

I'm not sure how many times the F1 car will come around the bend for you to review or go home and find out that all your pics got blurred, and then you learn by experience. Sometimes, mathematics changes your pictures for the year or a few. Worse yet if the car crashes at the turn. =) In this case, pre-calculating and remembering your minimum exposure time you want to prepare for the shoot is a good option, agree? I'm someone who prefers using Manual mode in such instances anyway.

Note: I used a 8mp calculation for all the above, figures changes according to camera pixel density.

2. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by ombre
Hi all, obviously it has some valid applications that I've considered [...]
Exactly that's the key phrase here: valid applications. I can see where you are coming from in deriving and developing your formula and I will give it a try to get some indications for my next shoots. Thank you so much for this little trip into mathematics
I have seen t-shirts from Nikon with Hyperfocal Distance scales - maybe soon somebody will have your calculations at his/her chest?
Of course: if the subject is not moving (portrait, landscape) then the formula and the whole considerations are simply not necessary. No need to get worked up. The right tool for the right purpose.

3. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

mathematical formula for motion BLUR

after reading, i still blur. hahahah!

but seriously, good effort in translating it into a formula that some people might be able to put to use.

4. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

hi, freak !! no la, just joking.

although i'm not sure if any smoke is let off here, i salute u for the effort u've put in. well done !!

5. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Thanks for all the encouraging replies!

Thanks for the critics too, indirectly helped me explore some more distinct applications that I wouldn't have thought of (like the F1 car).

And yes, definitely a tool with limited valid applications. Can be thrown out the window if you have enough time and resources to experiment on your own. The formula simply overcomes the learning curve, and nothing more. Its pretty similar to the DOF and focal length calculators you have online. Oh and by the way, I simply do not understand the DOF calculator. Haha. I personally couldn't be bothered to use the DOF calculator despite preferring calculations, I thought that was one aspect that impromptu judgment (not even trial and error) is more appropriate. That said, it probably applies more to candids, and sports especially when your shutter speed is greatly compromised and your subject is moving quickly.

Having that said, I shall just put a quick estimated chart of the shutter you need to remove the blurs. I will use 85mm and subject at 12m (a candid distance that fills your subject in a horizontal frame) for the example. These are about the minimum I will use.
Strolling person - ~1m/s - shutter : 1/200
Brisk walking person - 1.5m/s - shutter: 1/320
Runner / normal sports - 4m/s - 1/1000
motor sports / roller coaster - 20m/s - about 1/4000

Of course these are the harshest conditions with your subject in the whole frame. As you increase your distance from the subject, the shutter speed drops somewhat exponentially. To put it on the safe side, if you fill half the frame, you can half the amounts prescribed above.

6. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by ombre
motor sports / roller coaster - 20m/s - about 1/4000
Just to illustrate the problem when you're using a traveling-curtain focal plane shutter:

1) Moving car really looks like this:

2) Moving car taken with a horizontally traveling shutter curtain:

or

3) Moving car taken with a vertically traveling shutter curtain:
or

The motion blur gets similarly sheared, i.e. it can be more or less than what you calculated, depending on the direction in which the object is traveling. You can estimate how much the image gets sheared across the frame by using your formula by using the X synchronisation time of your camera as the exposure time.

7. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by ombre

This is precisely why you need my formula! I never said you calculate the formula on the spot during the F1 race!!! I don't mean to offend you, but thats being a bit short-sighted. You would have done the calculations the night before as preparation. I shall build on your example, and apply the formula.
and how many photographers shoot your f1 car properly without a formula? the empirical evidence is against you.

besides, if car come around bend, and whee, it's going at 205 km/h a slight shift from 200 km/h, do you realise that your example doesn't apply any more? you will need to plug it in again. and in the first place, this is my point - how the HELL are you going to be able to tell the speed of the car?

do you also realise that your formula does not seem to take into account angles when objects are moving?

8. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by tjhan
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.
hmm.. about the unit it does make sense... in the engineering point of view... u will need the unit to get better accuracy of the result... even if without unit, its not wrong too.. u will get a close estimation of the result
this formular will become an empirical formular... if u have study physic or engineering, some formular are actually empirical... meaning, the formular are obtain through experiment that give very close estimation... now i guess wat he need is to do experiment to prove it right and repeatable...

9. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Once again, there needs to be some application of dimensional analysis first.

10. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

The free, unhampered exchange of ideas and scientific conclusions is necessary for the sound development of science, as it is in all spheres of cultural life. (Albert Einstein, 1952)

11. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by night86mare
and how many photographers shoot your f1 car properly without a formula? the empirical evidence is against you.

besides, if car come around bend, and whee, it's going at 205 km/h a slight shift from 200 km/h, do you realise that your example doesn't apply any more? you will need to plug it in again. and in the first place, this is my point - how the HELL are you going to be able to tell the speed of the car?

do you also realise that your formula does not seem to take into account angles when objects are moving?
Mathematically its not possible to calculate the speed of the car visually.

In psychology, Howard Gardner calls this Visuo-spatial Intelligence. (Visual + Space) The ability to make accurate judgment of space, distance, speed, or other geometrical estimations using vision alone. Every human has this in varying degrees and it can be trained.

Being a relatively trained photographer yourself now, I believe you can roughly judge how far is your subject from you, by looking with your naked eyes, without looking through the viewfinder. Calculating speed builds on this, just factoring a new dimension, time.

My example was never designed to give you a pin sharp calculation of how much blur you will get. Its just so prepare you for the shoot. For instance, going into the shoot with the mentality "I must shoot at least at 1/2500 at the F1 race", or "I must shoot 1/640 for a badminton match". If for some reason you feel that you cannot get a good exposure of minimum 1/2500 at the race, try one of the contingency plan, sit further back with a longer lens.

Yes, many professional and serious amateurs have been successful in capturing F1 race cars. But you have not considered the rest who went home disappointed. Furthermore, this brings out the need for the formula once more:

1) How many can afford F1.2 lenses? Or even the so called bang for bucks 135 F2 L? There are serious photographers who are unwilling to spend more than 400 for a lens. Not everyone has the luxury or so called 'passion' of lavishing their hard earned money on lenses. (me included). But that does not mean they do not throughly enjoy photography. Neither does it imply they do not deserve the shot. As mentioned in my first opening and subsequent post, this formula is those who don't have time and resources to try all the settings and lenses.

2) Some may go with the notion that "1/500s" or other loosely quoted figures freezes motion. Does it really? Not everyone has prior experience of an F1 race.

Notice I posted this on the newbie forum too. I am not out to wow the professionals, if anything, I am objectively and mathematically trying to answer the question, "what min. shutter speed do I need to freeze motion?".

12. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by LittleWolf

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?
YOU MUST BE JOKING.

Coming up with this formula is a sure sign of having absolutely no idea what photography is.

Understanding this formula, if this formula is in fact valid, is a sure sign of analysis paralysis and guaranty of being a lousy photographer.

The problem here is, an art form had been hijacked by engineers. Engineers like to have definitive answers and solutions. An art is anything but definitive.

Photography is an art, for most of us, and it's something for us to create, not to follow formulas.

13. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by LittleWolf
Just to illustrate the problem when you're using a traveling-curtain focal plane shutter:

1) Moving car really looks like this:

2) Moving car taken with a horizontally traveling shutter curtain:

or

3) Moving car taken with a vertically traveling shutter curtain:
or

The motion blur gets similarly sheared, i.e. it can be more or less than what you calculated, depending on the direction in which the object is traveling. You can estimate how much the image gets sheared across the frame by using your formula by using the X synchronisation time of your camera as the exposure time.
Thanks for pointing this out! Although this complicate matters way beyond I would calculate. I assume in reality, this is taking into consideration your ability to pan, which can significantly reduce the exposure time required. And since my formula calculates the threshold, any reduction would be most welcome, I wouldn't be picky about the calculation then. (Or else i'd be a true math freak!)

Still I think the formula will work fine. Just that when calculating the subject speed, it is now a relative calculation factoring camera movement and only the vectors parallel to your image sensor. Most probably this will be somewhat a circular movement calculation I assume.

So in any case, the formula still is ultimately interested only in the plane parallel to the sensor. This multi-dimensional calculation must be done independently before plugging into the equation. Its like Newton's law of F = ma, doesn't consider the angles of movement nor units either, in application, the physicist would rule in all these on their own.

14. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by ombre

Being a relatively trained photographer yourself now, I believe you can roughly judge how far is your subject from you, by looking with your naked eyes, without looking through the viewfinder. Calculating speed builds on this, just factoring a new dimension, time.
it is one thing to judge manual focus roughly by eye, and be spot on because you are using f/8..

another thing to judge speeds of cars, etc. it's all relative. let's say you ARE 50m away from the car you wish to shoot. let's say that you are REALLY REALLY good and you can estimate the speed of the car is 200km/h or 55.55 m/s.

do you really think you have the time to a) estimate, and realise it, b) plug it into your magical formula, c) react and shoot a proper picture?

i thought c was already more than enough for any human to handle.

i don't question the use of your formula. i question its practicality, it's like cracking your brains to come up with a formula that allows me to calculate the optimum way i should breathe to maximise aerobic capacity when i run. is it useful? perhaps. are there uses for it? probably. the use to a common person, the value? a big fat zero.

there are rough guides to shooting various degrees of frozen motion; i suggest that you go out to shoot, internalise it, and realise that there are many, many more factors in reality than your simple shutter speed etc. what of light? what of external factors? learning to take on photography as a whole, not just the technical dimension (which is important nonetheless).. that gives you much more valuable lessons than discussing war strategies on a paper.

15. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by night86mare
it is one thing to judge manual focus roughly by eye, and be spot on because you are using f/8..

another thing to judge speeds of cars, etc. it's all relative. let's say you ARE 50m away from the car you wish to shoot. let's say that you are REALLY REALLY good and you can estimate the speed of the car is 200km/h or 55.55 m/s.

do you really think you have the time to a) estimate, and realise it, b) plug it into your magical formula, c) react and shoot a proper picture?

i thought c was already more than enough for any human to handle.

i don't question the use of your formula. i question its practicality, it's like cracking your brains to come up with a formula that allows me to calculate the optimum way i should breathe to maximise aerobic capacity when i run. is it useful? perhaps. are there uses for it? probably. the use to a common person, the value? a big fat zero.

there are rough guides to shooting various degrees of frozen motion; i suggest that you go out to shoot, internalise it, and realise that there are many, many more factors in reality than your simple shutter speed etc. what of light? what of external factors? learning to take on photography as a whole, not just the technical dimension (which is important nonetheless).. that gives you much more valuable lessons than discussing war strategies on a paper.
Hi Nightmare, I'm saying this for the 3rd time now. I don't mean any offense, but I feel that after highlighting it twice and you still don't get it, may simply imply bias and that you're not objectively reading my arguments.

One who chooses to use the formula should PREPARE a figure in his head, before he steps into the field.

YOU MUST BE JOKING.

Coming up with this formula is a sure sign of having absolutely no idea what photography is.

Understanding this formula, if this formula is in fact valid, is a sure sign of analysis paralysis and guaranty of being a lousy photographer.

The problem here is, an art form had been hijacked by engineers. Engineers like to have definitive answers and solutions. An art is anything but definitive.

Photography is an art, for most of us, and it's something for us to create, not to follow formulas.
A really blunt reply, but I shall take it with a pinch of salt. I leave you with this thought. I agree photography is some form of artistry, in fact that is why I came into it. But I don't deny that there are other elements in photography that thrills me equally, such as the technicalities.

But if you are shooting a high speed sports car, do you consider it artistry? Or is it now a sport? Sure, you can say you did it perfectly, got a dead sharp picture with excellent panning blur, WITHOUT a formula! All these high precision skill set are still indicative of a sportsman, rather than an artist.

Really the camera is just a tool. Holding a camera doesn't make you an artist. Neither is photography necessarily art. What you shoot, is what you are. However, that said, I will agree that its a mixture of both, with inclination to one field. If you shoot sports, you're in a fashion more inclined to sports, if you shoot stills and abstract, you may like to call yourself an artist, but ultimately, focusing, exposure, etc etc, is still indispensible.

Just to run a little off topic at this opportune moment. Forgive me here if I am a little blunt, or offensive, nothing personal. I sometimes disagree with the way photographers uphold themselves as "artists". I feel that many of them are at best 'commercial artists', or 'pop artists'. Coming from another art field myself, I've seen my fair share of who I will call true artists. Be it dancers/painter/musician or any art, these people are those who dedicate their entire life since childhood to the mastery and perfection of their art form, we are talking about from 4 years old to death, literally. These people sometimes also earn less than \$1200 SGD equivalent, just to stay in the field of their passion, while the pop/commercial artists presenting much more untrained work, earn 4 times more. How many photographers do that? I've seen enough who buy top of the line equipment, shoot a few popular looking pictures and 'self-console' themselves that they are artists or artists in practice.

All in all, I don't want to get into a firefight, all I hope to do is spark a little self-reflection / introspection. The practical realities of life goes on, while the debate of art vs sports vs science will go on forever.

16. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by ombre
Hi Nightmare, I'm saying this for the 3rd time now. I don't mean any offense, but I feel that after highlighting it twice and you still don't get it, may simply imply bias and that you're not objectively reading my arguments.

One who chooses to use the formula should PREPARE a figure in his head, before he steps into the field.
there's no bias here, i like to be prepared as anybody else, but i am amazed by your abilities to say, forecast what speed a f1 car is going to be doing when it turns around a bend. you have not offered any new points on this, which leads me to think that you have been smoking the wrong wine. then again, what sort of refutation can one provide to that question, i have no idea.

no matter, please go forth and bring glory with your mathematical formula.

17. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

The problem here is, an art form had been hijacked by engineers.
Firstly, doing photography always was first and foremost an engineering problem. Thanks to the skill of engineers, the technology was refined to become eventually so idiot-proof that even intellectually challenged individuals can take a photo nowadays.

Secondly, science and engineering are traditional arts, in the traditional sense of artistic=skillful (and as still evidenced by terms such as "state of the art"). The wishy-washy touchy-feely interpretation of the term "art' is a rather new phenomenon.

Thirldly, even if you have no intention to improve the state of the art, photography still is a craft performed with a set of tools and materials. Without an understanding of your tools and materials, you won't have the craftsmanship to be a competent photographer.

Photography is an art, for most of us, and it's something for us to create, not to follow formulas.
Well, it may not have occured to you, but engineering is a highly creative discipline. A capable engineer doesn't blindly follow formulas; they are tools for a purpose, and you need to understand your tools. On the other hand, when you take a photo with an electronic camera, or you use photoshop, you are using canned formulas - and if you do all this without understanding what's going on, you follow formulas that you don't even understand blindly, which is what you seem to criticize in your statement.

18. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by night86mare
there's no bias here, i like to be prepared as anybody else, but i am amazed by your abilities to say, forecast what speed a f1 car is going to be doing when it turns around a bend. you have not offered any new points on this, which leads me to think that you have been smoking the wrong wine. then again, what sort of refutation can one provide to that question, i have no idea.

no matter, please go forth and bring glory with your mathematical formula.
Ok at least a little more neutral now.

You don't have to forecast every single detail I guess, application is ultimately up to the shooter. Say you prepare 1/2500s for a 55m/s car (honestly thats damn fast!). On that day, the car turns slower than that, your preparation would have sufficed. You shoot at 1/2500 and get a happy shot with less blur than anticipated. No problem right? If you were very much into the sport, I don't think its very hard to get statistical figures on how fast these cars travel / turn. Its quite well documented. Other sports such as soccer, basketball or whatever, its not very hard to find out what a maximum human ability is, just calculate the worse scenario and you'll be safe shooting all the lesser sportsman.

Keyword is to determine the 'Threshold' 'desirable (implies subjectivity)' minimum shutter speed, not 'exact spot on' shutter speed that everyone must use.

19. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Does having all the right mathematical and physical formulas enable us to take good photos? - NO.
Inversely, if a person takes great photos, with flawless technical details, does it mean he understands all the maths and physics behind photography? - NO.
Are there people who actually find that maths and physics gives them better understanding of what they're doing in photography? - YES.
Does it actually help them in photography? - It depends.

We are different people enjoying the same hobby, so it's only natural that we approach it from different angles. I think that it's good that TS is thinking about photography. Does it really matter if it's a science or an art? I'm an engineer, and I actually make it a point to understand the physics behind photography because I like to do so, even though it's hardly helpful to me when I'm actually shooting.

Similarly, does a painter need to study the chemistry of paints? Or, for those familiar with the programme "Food Science", does a cook need to understand the science behind cooking? It's not necessary at all! But why discourage the healthy curiosity of those looking to pursue such an approach?

20. ## Re: Mathematical Formula for motion blurs (or "freezing motion")

Originally Posted by Gengh
Does having all the right mathematical and physical formulas enable us to take good photos? - NO.
Inversely, if a person takes great photos, with flawless technical details, does it mean he understands all the maths and physics behind photography? - NO.
Are there people who actually find that maths and physics gives them better understanding of what they're doing in photography? - YES.
Does it actually help them in photography? - It depends.
TS didn't entirely think about photography actually. It was \$\$ issues.

I used the formula to help me decide that 55-250 IS was sufficient for me, instead of a 70-200 F4 L. The 1 step would do little to help me "freeze" the motion of my subjects in the situations where I needed it. And that its okay to use 50mm F1.8. =) Or at least, in each scenario, the cost vs utility was not sufficient to convince me.

Of course after coming up with it, there were a lot more applications so why waste it. =)

In some sense, the application of this formula is in reverse. Formulas are usually designed to advance sciences to greater limits. This one simply points out what you need, what you don't. In my applications, it reduced me to using the bare minimums. Of course it still advances you in theory.

Lastly after looking at this for awhile, I'm starting to wonder why I got so many critiques... I never forced anyone to use it and acknowledged each limitation where necessary.

All in all, I'm pretty happy to have come up with something few realized / tried to do. Even if its not necessarily a revolutionary, PhD level math. Like I said before, I'm not a math person, from a point of view, going into math is a creative process for me.

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