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### AuthorTopic: FOV Equivalences vs Magnification  (Read 64247 times)

#### dobson

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##### FOV Equivalences vs Magnification
« Reply #20 on: February 24, 2008, 01:52:06 PM »

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That is interesting. I didn't know that. But how can I be sure you know what you are talking about  ?  Mathematicians tell me one can prove some very absurd notions at times and that such proofs seem very logical. Paul Dirac proved mathematically that anti-matter should theoretically exist, but kept quiet about for some time fearing ridicule and loss of reputation.
[a href=\"index.php?act=findpost&pid=176866\"][{POST_SNAPBACK}][/a]

And he was right that anti-matter exists.

Back to the matter at hand.
Using the CoC equation found on wikipedia (not the best source, I know):

C= abv(s2-s1)/s2 * F^2/(N(S1-F))

C is the CoC, S1 is the focus distance, S2 is the blurred obj distance, F is focal length and N is the F-stop.

Substitute values corresponding to the different formats and an arbitrary focus distance (if you'd like):

Full Frame
C= abv(s2-10000mm)/s2 * 160mm^2/(N(10000mm-160mm))

Crop Sensor
C= abv(s2-10000mm)/s2 * 100mm^2/(N(10000mm-100mm))

Assuming identical print sizes, you have to add a 1.6x coefficient to the crop sensor equation.

C= 1.6*abv(s2-10000mm)/s2 * 100mm^2/(N(10000mm-100mm))

Set the equations equal to one another and solve for N. Or at least try to solve. They don't cancel. This proves that you cannot achieve identical CoC curves in different formats by changing aperture.
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#### dobson

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##### FOV Equivalences vs Magnification
« Reply #21 on: February 24, 2008, 02:04:06 PM »

"digital cameras with different sized sensors will produce images with identical depths-of-field"

This statement is correct, but over-simplified. Given a threshold CoC, you can indeed solve for aperture. This, however only means that only the 2 points at the threshold CoC will have identical blurring, and the object in sharp focus of course. These 3 distances do not make up the entire image, so while "DoF" is the identical using a strict definition,  the blur throughout the image will not be identical.
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #22 on: February 24, 2008, 04:07:32 PM »

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"digital cameras with different sized sensors will produce images with identical depths-of-field"

This statement is correct, but over-simplified. Given a threshold CoC, you can indeed solve for aperture. This, however only means that only the 2 points at the threshold CoC will have identical blurring, and the object in sharp focus of course. These 3 distances do not make up the entire image, so while "DoF" is the identical using a strict definition,  the blur throughout the image will not be identical.
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Some thoughts/questions (all ignoring IQ), ..

Simply 1)

-If I shrunk my hasselblad and lens in the wash why would the image change ?

-A d80 could with a 30mm could be described approximately as a shrunken blad with an 80

Mathsly 2)

The COC curves (and I have tried plotting some this weekend - anything to avoid client Post Production) are exponential or parabolic or some other curve of a cartesian nature if I remember the lingo right from shcool

The point of cartesian algebra is that given point a and point b one can extrapolate point C

(an example of the use of this algebra could be  "how much do I need to invest now in my 6% bank account to have a million in 10 years?")

Therefore there is a consistant performance as one varies the input values, (Fstop focal lenth etc)

If there is consistent factors then one can tinker until the eqivilent Fstop and Flenght are found for a different chip size

One could therefore recreaste the look with a smaller chip if a fast enough lense is avaiable

I am trying to write a bit of software theat can do the graphs - it may take some time !

SMM

« Last Edit: February 24, 2008, 04:11:53 PM by Morgan_Moore »
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #23 on: February 24, 2008, 08:55:18 PM »

Quote
"digital cameras with different sized sensors will produce images with identical depths-of-field"

This statement is correct, but over-simplified. Given a threshold CoC, you can indeed solve for aperture. This, however only means that only the 2 points at the threshold CoC will have identical blurring, and the object in sharp focus of course. These 3 distances do not make up the entire image, so while "DoF" is the identical using a strict definition,  the blur throughout the image will not be identical.
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Dobson,
Thanks for taking the trouble to find the formula. Unfortunately, I'm neither a mathematician nor a physicist so cannot dispute the accuracy of the formula you've provided. For all I know, the formula itself could be an over-simplification but perhaps to a lesser degree than the over-simplification of DoF formulas.

However, even though I'm not a physicist, if someone were to state that the distance between point A and point B is a straight line, just to be contentious, I could argue that according to Einstein there are no straight lines. However, we all know that for terrestrial purposes such errors are usually insignificant. Nevertheless, one could dogmatically stick the view that in theory there are no straight lines.

Whilst this is an extreme example, I think it demonstrates a principle that the maths used for an application does not have to be (and should not be)cumbersomely more accurate than the application requires.

As I understand, in lens design and construction, there are all sorts of compromises and trade-offs that have to be made. One cannot get all aspects of performance equally good. If a lens is optimised for best performance at f2.8 it might be too difficult for it also to have best performance at f8, although performance at f8 might still be good. (There actually are 35mm lenses that are sharper at f2.8 than at f8).

As Morgan has implied, how relevant are such inaccuracies in practice regarding uniformity of DoF with different size sensors, considering my test results comparing 250mm at f8 with 400mm at f11 and f13?

By the way, in connection with the formula you provided, you mentioned multiplying one side by 1.6 for equal size prints. You are still thinking film   . Nowadays, print size is dependent on pixel count, not sensor size.
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#### dobson

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##### FOV Equivalences vs Magnification
« Reply #24 on: February 24, 2008, 09:37:42 PM »

I agree that there may be very little real-world relevance to this exercise. I originally did the calculations to improve my understanding of the concepts of sensor size, depth of field, and hyperfocal distance. The formula I started with is indeed an over-simplification; it is true only for symmetrical lenses with identical-sized entrance and exit pupils. The calculations for asymmetrical lenses are very complicated and specific to each lens; logic shows that it would be extremely unlikely for two independently-designed lenses to show similar characteristics in two different formats. Not necessarily impossible, though (and I really don't like math enough to figure that one out).

Boy this is getting complicated for the beginner's forum. Sure is interesting, though.

Quote
By the way, in connection with the formula you provided, you mentioned multiplying one side by 1.6 for equal size prints. You are still thinking film   . Nowadays, print size is dependent on pixel count, not sensor size.
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The formula I provided is independent of recording medium. This thought experiment will show why.

Let's say I just invented a new quantum sensor/film, one that recorded perfectly every photon that hits it. I have two cameras with this sensor, one with a 35mm frame and one at APS-C. These cameras also boast perfect lenses.

I am photographing 2 points of light at different distances from the film-plane. The cameras are set up in such a way that distances to these points and angle of view are identical. The cameras focus on the primary dot while the secondary dot is blurred out of focus. We match the apertures of the two lenses so the dot is blurred identically, lets say it projects a circle 1mm in diameter (the CoCs are equal).

Now I make a 1 meter wide print of each image. This doesn't matter if the image was enlarged optically or with a printer, the result is the same; two dots in each image, one infinitesimally small and one a blurred circle. The center points of each dot lie in exactly the same place for each print.

These blurred circles are not the same size, however. Because the APS-C sensor is smaller it had to be enlarged more to achieve the desired print size. This made the blurred circle in the APS print 1.6 times the size of the circle in the 35mm print.

As a result, the APS-C image appears more out of focus. Since our goal of the first experiment was to achieve identical images, and adjustment of CoC was required to make the 2 formats appear equivalent when printed (or viewed in any manner).
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#### dobson

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##### FOV Equivalences vs Magnification
« Reply #25 on: February 24, 2008, 09:42:24 PM »

Crap - I just noticed a typo in my first proof.

The 1.6 coefficient should be added to the full frame side. I did the calculations correctly on paper, I just typed them wrong. Sorry if it caused any confusion.
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #26 on: February 24, 2008, 09:50:29 PM »

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And he was right that anti-matter exists.

[a href=\"index.php?act=findpost&pid=177093\"][{POST_SNAPBACK}][/a]

Apparently he was, but in order to find out if he was right someone had to go looking for it. I went looking for these DoF differences that you assert should exist according to the maths. I haven't found them yet. Should I continue looking   ?

Here are the rest of the crops comparing the 5D at f11, ISO 800, 400mm, and the 40D at f8, ISO 400, 250mm.

[attachment=5268:attachment]  [attachment=5269:attachment]  [attachment=5270:attachment]  [attachment=5271:attachment]  [attachment=5272:attachment]
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#### dobson

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##### FOV Equivalences vs Magnification
« Reply #27 on: February 24, 2008, 10:29:50 PM »

Now you have my curiosity. I would really like to see exactly how much difference, (if any),is visible between the two formats.

May I propose an experiment, I do not have cameras with different sized sensors.

The only issue I have with your previous tests is the difficulty I'm having measuring the blur spots.

The experiment I propose requires 3 point light sources and a dark background. Point lights could be small LED, or a hole cut in a backlit screen, or anything snall and bright. The blur circle of a point light will be very easy to measure in a print or on a monitor.

The experiment would be best performed at night (or in a dark studio) to eliminate unwanted distractions. The primary point light will be the object of focus, and closest to the camera. The secondary and tertiary points will be at different distances and out of focus.To accentuate any discrepancy, the secondary and tertiary points should be very far apart. Placing the tertiary point at infinity (such as a street light in the distance), will maximize this distance.

Then shoot the scene just as you did last time. Keeping the field of view identical and adjusting aperture to match depth-of-field. A wide aperture will make the blur spots bigger and easier/more-reliable to measure.

To measure the blur dots, resize the images to the same pixel size and measure (pixel diameters or mm). Or make identical-sized prints and measure. You will know you've done it right when either both secondary or both tertiary spots are exactly the same size (or maybe both pairs will match).

If you could perform this experiment, I would be extremely grateful. I've kicked back into engineer-mode and I am really interested in what the real-world results are. If you measure the distance to the points, we can even see how closely the calculations hold up in real-life.

Thanks,
Phillip
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #28 on: February 25, 2008, 12:48:48 AM »

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if someone were to state that the distance between point A and point B is a straight line,[a href=\"index.php?act=findpost&pid=177147\"][{POST_SNAPBACK}][/a]

The curve of focus drop off is not striaght - we are talking about the route between A and B - and where is C if one contiues taking that route !

S
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #29 on: February 25, 2008, 01:26:28 AM »

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The only issue I have with your previous tests is the difficulty I'm having measuring the blur spots.
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Phillip,
When I thought it might be worthwhile to check this matter, I chose the scene at the foot of my driveway because it contained a number of posts and readily identifiable objects at various distances receding towards infinity.

Instead of setting up an elaborate experiment, how about I just provide 800% enlargements of key features, such as the red reflectors on the white posts as they recede towards infinity? The differences you refer to appear to be so small that they might be completely obscured by other lens imperfections and quality variations. As a general principle, when trying to prove the existence of such subtle variations, as the size of blur in this case, one would have to do lots of experiments with lots of different lenses at different sets of equivalent apertures, in order to identify a trend. That task is not only too onerous for me, there just doesn't appear to be a practical issue of concern here as you can see from the following crops which, on my screen at least, represent a print size of about 30ftx20ft.

The crops from left to right are in order of their distance from the focus point. The ruler divisions at the top of each image should give you an indication of the size of the blur and any variation in the magnification of the objects I've selected.

[attachment=5277:attachment]  [attachment=5278:attachment]  [attachment=5279:attachment]  [attachment=5280:attachment]  [attachment=5281:attachment]  [attachment=5282:attachment]
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #30 on: February 25, 2008, 01:46:22 AM »

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The curve of focus drop off is not striaght - we are talking about the route between A and B - and where is C if one contiues taking that route !

S
[a href=\"index.php?act=findpost&pid=177170\"][{POST_SNAPBACK}][/a]

That might well be true. I was making a general point about theory accuracy. It could be argued that any theory that includes the concept of a straight line is inaccurate because straight lines do not exist. However, the inaccuracy is so small it can be irrelevant.

As I sit here in my studio, typing away to foreign people on the internet because I'm retired and have time to spare, I am ageing at a slightly greater rate than all those buzy people buzzing around from place to place in the city. I'm getting older than them, because I'm mostly stationary, at a faster rate of about one billionth of a second per year, but it doesn't worry me   .

You might have noticed I sometimes revert to extreme examples to get a point across   .
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #31 on: February 25, 2008, 02:10:30 AM »

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That might well be true. I was making a general point about theory accuracy. It could be argued that any theory that includes the concept of a straight line is inaccurate because straight lines do not exist. However, the inaccuracy is so small it can be irrelevant.

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Im not getting all philosopical here,

When one puts \$100 in the bank at 10% interest you have, (simply)

\$110, at the end of the first year
\$121 et the end of the second year
\$132 at the end of the third year

and \$259 at the end of the 10th year, (not \$200)

A non linear increase - your interest is not \$10 every year

while one can use a smaller principal investment with a higher intereste rate to get to the same wealth at year 10 the route will not be the same - or will it? - I dont think so

If we swap 'COC' for 'total money' and 'Duration of investment' for 'distance from focus point' a similar non linear graph is created

The question is can one create identical graphs by tinkering with chip size aperture and focal lenght

I am too stupid to be abe to answer this currently but I think with lenses the answer is it can be done

If the answer is yes then the answer is that one can recreate the same look across manys sensor sizes (excluding IQ factors)

S
« Last Edit: February 25, 2008, 02:40:18 AM by Morgan_Moore »
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #32 on: February 25, 2008, 03:04:09 AM »

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If the answer is yes then the answer is that one can recreate the same look across manys sensor sizes (excluding IQ factors)
[a href=\"index.php?act=findpost&pid=177180\"][{POST_SNAPBACK}][/a]

Wait a minute! You've just delivered an oxymoron here. The same look, excluding IQ factors? What sort of look is that?

What I could do, I suppose, is go out onto the roadside and measure the various distances between key objects in the scene I shot, then feed the results into Phillip's formula to see if they corresponded with the visual results on my monitor with respect to CoC size. If friendly neighbours happened to enquire why I appeared to be measuring their boundaries, I could try to explain that I was just checking DoF variations and they would just assume I was crazy   .
« Last Edit: February 25, 2008, 03:26:55 AM by Ray »
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #33 on: February 25, 2008, 03:46:39 AM »

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Wait a minute! You've just delivered an oxymoron here. The same look, excluding IQ factors? What sort of look is that?

What I could do, I suppose, is go out onto the roadside and measure the various distances between key objects in the scene I shot, then feed the results into Phillip's formula to see if they corresponded with the visual results on my monitor with respect to CoC size. If friendly neighbours happened to enquire why I appeared to be measuring their boundaries, I could try to explain that I was just checking DoF variations and they would just assume I was crazy   .
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-your test is too close to infinity to show significant variance (if such exists)

-BY IQ I mean, resolution, lenses distortion etc things that dont exist in a simplified theoritical world - maybe not the right definition

S
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#### Sheldon N

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##### FOV Equivalences vs Magnification
« Reply #34 on: February 25, 2008, 05:19:27 PM »

I'm also curious how the math on this shakes out, but my guess is that the theoretical curves will be the same once you equalize for aperture and format size.

I agree with Sam that there is a definite difference in how DOF and falloff appear when shooting larger sized formats. I shoot 35 FF Digital with fast primes (35 f/1.4, 85 f/1.2), MF Film, and 4x5 film so I have some experience.

I think it is rooted in a couple things.

1) For a given print size, larger formats require less enlargement. This requires less lp/mm of the lens, which means that the MTF (microcontrast) will be better, assuming equal lens quality. As a rule, the higher the lp/mm measurement the lower the MTF (contrast). This means when you enlarge less or demand less lp/mm of the lens, the contrast at the plane of focus improves. Improved contrast = increased perceived sharpness (even if resolution is equal). Increased perceived sharpness at the plane of focus contributes to the sense of DOF and subject separation, even if it is not measured by a COC calculation.

2) Larger formats typically are higher resolution (more megapixels or larger/higher resolution scans). A higher resolution file will show more fine detail when viewed at 100%. When you increase print size (which is what you are doing when viewing at 100%) then you decrease DOF. There is a point at which you cannot enlarge a print further and reduce DOF - when you run out of resolution. To me this means that a higher resolution file has the potential for shallower DOF, when printed large enough.

As an example, compare a head and shoulders portrait taken Canon 8mp digital crop with a 35mm f/1.4 L lens wide open to a 39mp MFDB shot with an 80mm f/2.8 lens wide open (or whatever the equivalent aperture is). On a 5x7 or 8x10 print the DOF may appear the same. However, on a 16x20 print the crop DSLR will be soft at the plane of focus, but the MFDB will show very fine detail on the eyelashes, with the plane of focus softening progressively as you move away from the eyes. The MFDB shot obviously has less DOF at this print size, even though the apertures were "equalized". This holds true purely on the basis of the camera's capture resolution, even if both lenses are equally sharp at their chosen apertures.

3) Tonality, highlight/shadow detail, etc. All the other image quality intangibles help things out, too.

Since photographers typically view their photos at 100% (effectively the largest possible print size), it's no wonder that all the MFDB guys insist that there is something different about MF digital.

#### Ray

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##### FOV Equivalences vs Magnification
« Reply #35 on: February 25, 2008, 08:57:32 PM »

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1) For a given print size, larger formats require less enlargement. This requires less lp/mm of the lens, which means that the MTF (microcontrast) will be better, assuming equal lens quality. As a rule, the higher the lp/mm measurement the lower the MTF (contrast). This means when you enlarge less or demand less lp/mm of the lens, the contrast at the plane of focus improves. Improved contrast = increased perceived sharpness (even if resolution is equal). Increased perceived sharpness at the plane of focus contributes to the sense of DOF and subject separation, even if it is not measured by a COC calculation.

2) Larger formats typically are higher resolution (more megapixels or larger/higher resolution scans). A higher resolution file will show more fine detail when viewed at 100%. When you increase print size (which is what you are doing when viewing at 100%) then you decrease DOF. There is a point at which you cannot enlarge a print further and reduce DOF - when you run out of resolution. To me this means that a higher resolution file has the potential for shallower DOF, when printed large enough.
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This is very true. I've made this point a number of times in discussions about DB versus high pixel count 35mm DSLRs such as the 1Ds3.

Even when the smaller format has the same pixel count, it cannot compete unless its lens is proportionally better than the larger format lens. If what's in focus is sharper in one image but what's OoF is about the same, the images will inevitably look different.

Even when comparing different formats which both use 35mm lenses, such as the 40D and 5D, results are likely to vary depending on which set of equivalent apertures are used. F8 on the 40D compared with f11 or f13 on the 5D is a fairly good match because we would expect the 35mm lens to be at least marginally sharper at f8 than at f11 or f13, and the 40D needs a sharper lens for the plane of focus to appear as sharp as it does on the 5D.

However, if we open up the apertures a bit and compare the 40D at f2.8 with the 5D at f4.5, then we could get some very different results.

This why I think it would be a very onerous exercise to try and prove the soundness or accuracy of Phillip's CoC formula by experimenting with various lenses and formats at different apertures. One would need to eliminate all the factors not related to format size that might skew the results. Pixel count would have to be the same on both formats, and lens resolution at the apertures chosen would need to be proportionally better with whatever lenses were used on the smaller format, better to the same degree that the pixel pitch is smaller, ie. system resolution, at the apertures chosen would ideally need to be the same for both formats.
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#### EricV

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##### FOV Equivalences vs Magnification
« Reply #36 on: February 26, 2008, 02:01:02 PM »

DoF calculations are ultimately based on the simple thin-lens equation:

1/s + 1/d = 1/f, where

s = distance from lens to subject,
d = distance from lens to image,
f = focal length of lens.

The relevant dimensionless quantity which affects focal length scaling is the ratio f/s, which will be small for scenes with objects far from the lens.  The rest of the calculation is ray tracing geometry which can be found on many sources on the web.

Someone already reproduced the formula for the blur diameter of an out of focus point object, derived from the lens equation.  With a little rearrangement, the formula becomes:

C = (s2-s1)/(s1s2) * (f^2/N) / (1-f/s1), where

C is the blur diameter (circle of confusion),
s1 is the (sharp) focus distance,
s2 is the (blurred) object distance,
f is the lens focal length, and N is the lens f/stop.

If this formula did not contain the pesky factor (1-f/s1) at the end, scaling the f/stop with the square of the focal length would produce identical blur across the image for objects at any distance.  The f/s1 correction makes this scaling imperfect.  In practice, this will be evident only in photographs where f/s1 is relatively large, meaning focus is set fairly close to the lens.
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #37 on: February 26, 2008, 11:14:20 PM »

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In practice, this will be evident only in photographs where f/s1 is relatively large, meaning focus is set fairly close to the lens.
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So do you think the look is 'recreatable' or not ...?

S
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#### Ray

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##### FOV Equivalences vs Magnification
« Reply #38 on: February 27, 2008, 04:24:47 AM »

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So do you think the look is 'recreatable' or not ...?

S
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What I might do tomorrow, if I have the time, is compare a close-up taken with my 10D and 17-55/2.8 zoom at 17mm and f3.5, with the same scene taken with my 5D and Sigma 15-30 zoom at 27mm and f5.6. I'm interested in comparing the sharpness and detail of these two cameras/lenses in any case.
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#### Morgan_Moore

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##### FOV Equivalences vs Magnification
« Reply #39 on: February 28, 2008, 01:48:01 AM »

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What I might do tomorrow, if I have the time, is compare a close-up taken with my 10D and 17-55/2.8 zoom at 17mm and f3.5, with the same scene taken with my 5D and Sigma 15-30 zoom at 27mm and f5.6. I'm interested in comparing the sharpness and detail of these two cameras/lenses in any case.
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I dont think youl see a lot if diffewrence wich such a wide lens

imo the differences, if there are any,  show in the 50-135 region mainly and at f4 or wider

S
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