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Author Topic: How to derive a focal lenght according to an image-crop?  (Read 9052 times)

cmi

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How to derive a focal lenght according to an image-crop?
« on: July 03, 2009, 05:32:10 pm »

I have an image here shot at (for example) 180mm APC-C, and I have it cropped. I want to know wich lens to use to cover the cropped area horizontally. So, wich formula can I use to determine FOV and focal lenght, regardless of camera (sensor size) from the pixel size of the crop? Furthermore, is it correct, that if I double my focal lenght, my FOV angle halfes? My tests and also the fov-data of an ef200, 400 and 800mm I checked indicates that.

Christian
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frugal

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How to derive a focal lenght according to an image-crop?
« Reply #1 on: July 03, 2009, 10:42:56 pm »

I believe you are right that doubling the focal length halves the field of view (or vice versa).

Assuming you're keeping the same aspect ratio in the crop, I think that you can take the percentage change in horizontal measurements and the necessary change in focal length will be the same change. For example, if you have a 2000 pixel wide image and crop it to 1000 pixels that's a 50% decrease in width, so I think you'd then need a 50% increase in focal length to match that field of view. I haven't checked this math though.
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01af

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How to derive a focal lenght according to an image-crop?
« Reply #2 on: July 04, 2009, 05:43:55 am »

Quote from: Christian Miersch
I want to know wich lens to use to cover the cropped area horizontally. So, wich formula can I use to determine FOV and focal length, regardless of camera (sensor size) from the pixel size of the crop?
That's easy:

fe = (pa / pc) * fa

where

fe = equivalent focal length---equivalent with regard to the uncropped image, not to 35-mm format (except when the uncropped image is 35-mm format, of course).
fa = actual focal length of the lens used
pa = actual pixel count of the uncropped image in one dimension, e. g. width.
pc = pixel count after cropping, in the same dimension pa refers to.


Quote from: Christian Miersch
Furthermore, is it correct, that if I double my focal lenght, my FOV angle halfes? My tests and also the fov-data of an ef200, 400 and 800mm I checked indicates that.
No, that's not correct.

Only when dealing with small angles of view (less than, say, 20°, i. e. telephoto lenses) then your assumption is a fairly good approximation for all practical intents and purposes. However the wider the angles of view, the less accurate the approximation. That's because the relation between focal length and angle of view (on a given image format) is given by the tangent function which is almost linear at small angles but highly nonlinear at wide angles.

-- Olaf
« Last Edit: July 04, 2009, 10:49:47 am by 01af »
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Dick Roadnight

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How to derive a focal lenght according to an image-crop?
« Reply #3 on: July 04, 2009, 07:15:24 am »

Quote from: 01af
Only when dealing with small angles of view (less than, say, 20°, i. e. telephoto lenses) then your assumption is a fairly good approximation for all practical intents and purposes. However the wider the angles of view, the less accurate the approximation. That's because the relation between focal length and angle of view (on a given image format) is given by the tangent function which is almost linear at small angles but highly nonlinear at wide angles.
-- Olaf
This (above) is correct, but your formula (below) is also only approximately correct for small angles
Quote
fe = (pa / pc) * fa
Can anyone come up with a accurate formula before I do?
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01af

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How to derive a focal lenght according to an image-crop?
« Reply #4 on: July 04, 2009, 08:17:19 am »

Quote from: Dick Roadnight
Can anyone come up with a accurate formula before I do?
Good luck finding a 'better' formula!  

While my formula isn't perfect, you are looking into the wrong direction for a better one. At long focusing distances, my formula holds for all angles of view (except when fish-eye lenses are involved). After all, it doesn't deal with different angles at all but with equivalent focal lengths, i. e. same angle on both sides of the equation. So angle of view gets cancelled out of the equation.

Still, it is inaccurate indeed at large magnifications, so you cannot use it for macro photographs. Here, you'd have to calculate equivalent magnification rather than equivalent focal length. And that's easy enough to do: simply replace the focal lengths with the respective magnifications in the formula given above.

-- Olaf
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Dick Roadnight

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How to derive a focal lenght according to an image-crop?
« Reply #5 on: July 04, 2009, 08:29:29 am »

Quote from: 01af
Good luck finding a 'better' formula!  

While my formula isn't perfect, you are looking into the wrong direction for a better one. At long focusing distances, my formula holds for all angles of view (except when fish-eye lenses are involved). After all, it doesn't deal with different angles at all but with equivalent focal lengths, i. e. same angle on both sides of the equation. So angle of view gets cancelled out of the equation.

Still, it is inaccurate indeed at large magnifications, so you cannot use it for macro photographs. Here, you'd have to calculate equivalent magnification rather than equivalent focal length. And that's easy enough to do: simply replace the focal lengths with the respective magnifications in the formula given above.

-- Olaf
I have the geometric construction sorted, and, as you say, the angles are irrelevant, so I thinking of using Pythagorus rather than trigonometry.
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cmi

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How to derive a focal lenght according to an image-crop?
« Reply #6 on: July 04, 2009, 08:55:28 am »

Thanks Olaf, since it was indeed about long focal lenghts as I suggestet, your formula works perfectly, thats all I needed, thank you very much. According to this I need 2350mm at crop 1.6 - DOH! (I had a quite small horizontal snippet at 179mm...) Had I come to something like 1000mm I would consider getting some mirror tele, but now it definitely looks like I have to collect similar views elsewhere and then do some compositing to recreate the scene.

Dick, I dont need a more correct formula, but if you want to do it out of curiosity, feel free to go ahead.

Christian
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01af

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How to derive a focal lenght according to an image-crop?
« Reply #7 on: July 04, 2009, 09:38:52 am »

Quote from: Christian Miersch
... since it was indeed about long focal lenghts as I suggested, your formula works perfectly ...
Umm ... now I am under the impression you didn't fully understand the formula, and when it's accurate and when not. As a matter of fact, it's accurate for all focal lengths, not just for long ones. However it's not accurate at short focus distances, i. e. in the close-up or macro range. You know the difference between focal length (= Brennweite in German language) and focus distance (= Einstellentfernung), don't you?


EDIT: In case someone is technically inclined enough to be interested in the accurate formula that holds for all focal lengths and all focus distances up to a magnification of 1:1 (but not beyond)---here it is:

fe  =  q * q * fa  -  q * (q - 1) * (d/2 - sqrt(d * (d/4 - fa)))

where

q  = pa / pc
d  = focus distance
fe = equivalent focal length---equivalent with regard to the uncropped image
fa = actual focal length of the lens used
pa = actual pixel count of the uncropped image in one dimension, e. g. width
pc = pixel count after cropping, in the same dimension pa refers to

Note how this formula becomes fe = q * fa when d approaches infinity---which is the same as the formula given above.

-- Olaf
« Last Edit: July 04, 2009, 10:49:21 am by 01af »
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cmi

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How to derive a focal lenght according to an image-crop?
« Reply #8 on: July 04, 2009, 11:02:05 am »

Olaf, now Im confused.

You said it is wrong that if the focal lenght doubles, the FOV halfes, because that is only an aproximation for long focal lenghts, right? I understood your simple formula as a proportion wich I assumed to be linear. But since you said the underlying relationship is not linear but depends on some circle functions, I assumed that the first formula would not hold true for short focal lenghts (wide FOV's). Where is my error?

But in any case, my problem is solved, even if my understanding of the whole thing is not correct. Yes thats a bit ignorant, but my math skills are not that great

Christian


Quote from: 01af
Umm ... now I am under the impression you didn't fully understand the formula, and when it's accurate and when not. As a matter of fact, it's accurate for all focal lengths, not just for long ones. However it's not accurate at short focus distances, i. e. in the close-up or macro range. You know the difference between focal length (= Brennweite in German language) and focus distance (= Einstellentfernung), don't you?


EDIT: In case someone is technically inclined enough to be interested in the accurate formula that holds for all focal lengths and all focus distances up to a magnification of 1:1 (but not beyond)---here it is:

fe  =  q * q * fa  -  q * (q - 1) * (d/2 - sqrt(d * (d/4 - fa)))

where

q  = pa / pc
d  = focus distance
fe = equivalent focal length---equivalent with regard to the uncropped image
fa = actual focal length of the lens used to take the uncropped image
pa = actual pixel count of the uncropped image in one dimension, e. g. width
pc = pixel count after cropping, in the same dimension pa refers to

Note how this formula becomes fe = q * fa when d approaches infinity---which is the same as the formula given above.

-- Olaf
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Dick Roadnight

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How to derive a focal lenght according to an image-crop?
« Reply #9 on: July 04, 2009, 11:42:18 am »

Quote from: 01af
Umm ... now I am under the impression you didn't fully understand the formula, and when it's accurate and when not. As a matter of fact, it's accurate for all focal lengths, not just for long ones. However it's not accurate at short focus distances, i. e. in the close-up or macro range. You know the difference between focal length (= Brennweite in German language) and focus distance (= Einstellentfernung), don't you?
I am an engineer, I enjoy deriving formulae.

I have read and understood Merklinger's Books "The ins and out of Focus" and "Focusing the view camera"
Focal length was defined as the extension (film to lens distance) at which an object at infinity would be in focus, but this only applies to normal, rather than tele-photo retro-focus lenses.

In the English language, as far as I am aware, there is no technical phrase "focus distance", but I imagine that you are thinking of the lens to object (subject) distance "D"... or do you mean the back-focus distance "B" which is the lens to sharp image distance?

The geometric construction I was working on assumed that lenses were elaboration of pinholes, and that B = f, which is only the case when focused at infinity.

I am now going away for the weekend.

You have a 90 degree lens, and you want double the field of view... what is the focal length of a 180 degree lens?

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01af

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How to derive a focal lenght according to an image-crop?
« Reply #10 on: July 04, 2009, 12:48:54 pm »

Quote from: Christian Miersch
Olaf, now I'm confused.
I'd say that's only because you don't read my postings carefully enough.  


Quote from: Christian Miersch
You said it is wrong that if the focal lenght doubles, the FOV halfes, because that is only an aproximation for long focal lenghts, right?
Yeah, right.


Quote from: Christian Miersch
I understood your simple formula as a proportion wich I assumed to be linear.
Yes, it is linear ... at long focus distance, that is.


Quote from: Christian Miersch
But since you said the underlying relationship is not linear but depends on some circle functions, I assumed that the first formula would not hold true for short focal lenghts (wide FOV's). Where is my error?
Your error is that you're confusing two different things.

You asked two questions, remember? The first question was: How do I compute the equivalent focal length for a cropped area in my photograph? The second question essentially was: Is the relationship between focal length and angle of view a linear one?

And the equivalent focal length is linearly proportional to the width of the crop (question #1), but focal length is not linearly proportional to the angle of view (question #2). Field of view is not the same as angle of view.


Quote from: Dick Roadnight
I am an engineer, I enjoy deriving formulae.
 


Quote from: Dick Roadnight
I have read and understood Merklinger's books "The ins and out of Focus" and "Focusing the view camera".
So have I.


Quote from: Dick Roadnight
Focal length was defined as the extension (film to lens distance) at which an object at infinity would be in focus ...
Actually, it's the lens-to-film distance (or more generally speaking, the lens-to-image distance) at infinity focus. As an engineer, you'll understand the (subtle) difference---in optical imaging systems, distances are always measured in subject-to-image orientation.


Quote from: Dick Roadnight
... but this only applies to normal, rather than tele-photo retro-focus lenses.
Actually, it applies to all kinds of lenses, be them regular, telephoto, or retro-focus. The difference between the three is where exactly 'the lens' formally is located, in relation to the mass of the glass.


Quote from: Dick Roadnight
In the English language, as far as I am aware, there is no technical phrase "focus distance" ...
There isn't!? Oops! I thought so. In fact I often saw that term ... but maybe that's just layman's talk, as opposed to a technical term.

So please tell me: what is the technical term for the distance between subject and image? That's the sum of the subject-to-lens distance plus the lens-to-image distance (plus, if we are scrupulous, the distance between the lens' principal planes). It's the distance engraved on the lenses' distance scales. I always thought the technical term for this was 'focus distance' ... no?


Quote from: Dick Roadnight
... what is the focal length of a 180° lens?
It's 0 mm. Except when it's a fish-eye lens.

-- Olaf
« Last Edit: July 04, 2009, 12:58:22 pm by 01af »
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cmi

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How to derive a focal lenght according to an image-crop?
« Reply #11 on: July 04, 2009, 02:12:13 pm »

Olaf,

I dont want to be rubbed at that I dont understand the topic fully. I dont see myself as dumb, I try my best reading you carefully and I sure remember that I asked two questions. Please spare me the rethorical part, it comes off a bit high-horsed.

But back to topic:

Yes, I used the term FOV-Angle and FOV when I instead meant the angle of view. Fuzzyness on my part, sorry!

So youre saying your simple formula holds true also to extremely wide angle of views?

Christian
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01af

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How to derive a focal lenght according to an image-crop?
« Reply #12 on: July 04, 2009, 02:34:15 pm »

Quote from: Christian Miersch
So you're saying your simple formula holds true also to extremely wide angle of views?
Yes, exactly---provided the, umm, 'focus distance' is long in relation the focal lengths involved ... and not for fish-eye lenses. For the latter, the respective formula would be pretty complex as it depends on the fish-eye projection function, and it's not linear to how much you crop.

So for regular (i. e. non-fish-eye) telephoto, standard, and wide-angle lenses at long 'focus distance', focal length is proportional to field of view, field of view is not proportional to angle of view, and thus focal length is not proportional to angle of view.

-- Olaf
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cmi

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How to derive a focal lenght according to an image-crop?
« Reply #13 on: July 04, 2009, 02:59:12 pm »

//disregard
« Last Edit: July 21, 2009, 08:00:55 pm by Christian Miersch »
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