# Thread: Depth of field and focal length

1. ## Depth of field and focal length

We all know two things about DoF: a) it gets shallower with longer lenses and b) it gets shallower as you get closer to the subject. Three things! We know three things! c) DoF gets shallower with lower f-number.

NOTE: I will NOT be talking about different sensor sizes and equivalence. You can all relax now.

One curious claim, though, is that a) and b) almost exactly cancel out if you make the *subject* the same size in each picture. Take a head shot with a 250mm lens at f/4 and you'll have the same DoF as the same headshot taken with a 120mm lens at f/4 on the same size sensor. Of course, the 120mm lens will have to be half the distance from the subject to get the same head size.

This can be seen from the formulas, or from geometric optics (ray tracing), but there's nothing like a demonstration! Between my own face and a Fuji lens, I picked the more attractive subject. Here are two shots at f/4 with the 250/4 and the 120/4, both wide open, and both cropped the same size.

250/4

120/4

Of course, the pictures are different, and the subject geometry is different, but it's about the same size in both pics, and the DoF, visible on the tape measure, is roughly equal.

Here are the same at f/8:

250/4 @f/8

120/4 @f/8

Now modern lenses are not classical thin lenses, so I wasn't sure that this would hold at all. I just always wanted to do this experiment. It looks pretty good.

Yes, there is nothing MF about this, except that I used a GFX 100 for the shots. I would have posted it to the Boring Lecture forum, but there isn't one. I should probably start it. Tired of Jim Kasson's careful measurements? Come to the Boring Lecture forum!

Thanks,

Matt

2. ## Re: Depth of field and focal length

There is a difference in background blur, but maybe I had already one Pastis too much ...

IIRC there was some discussion on how quickly the sharpness falloff is between lenses, when the new SL lenses were introduced. Leica was sort of claiming their f2 is the new f1.4. Just like Fuji claims their 44x33 is the new large format

3. ## Re: Depth of field and focal length

Originally Posted by MGrayson
One curious claim, though, is that a) and b) almost exactly cancel out if you make the *subject* the same size in each picture. Take a head shot with a 250mm lens at f/4 and you'll have the same DoF as the same headshot taken with a 120mm lens at f/4 on the same size sensor. Of course, the 120mm lens will have to be half the distance from the subject to get the same head size.
I see your lecture, and raise you a tutorial.

Or to put it another way, if you maintain the angular size of the entrance pupil as viewed from the object, DoF is maintained--more or less. Just as depth of focus (the space in front and behind the image plane) is dependent on the f-number, not the focal length, as the f-number is proportional to the angular size of the exit pupil from the image plane. (Extreme magnification and how a lens focuses can change this (lenses can focus by changing lens to image plane distance or changing focal length).)

4. ## Re: Depth of field and focal length

Originally Posted by Photon42
There is a difference in background blur, but maybe I had already one Pastis too much ...

IIRC there was some discussion on how quickly the sharpness falloff is between lenses, when the new SL lenses were introduced. Leica was sort of claiming their f2 is the new f1.4. Just like Fuji claims their 44x33 is the new large format
There IS a big difference in background blur. What's surprising is that that is unrelated to Depth of Field. A distant streetlight will make a bigger bokeh with the longer lens, because there isn't a big proportional difference between the distance to the two lenses. The effects only balance out when the 120 is half the distance to the subject as the 250. (Yes, I know that 120 isn't half of 250, but then 1.4 isn't the square root of 2.)

I'm making no claims about HOW lens sharpness falls off. It clearly differs between lenses. I'm talking about very coarse effects.

Originally Posted by Shashin
I see your lecture, and raise you a tutorial.

Or to put it another way, if you maintain the angular size of the entrance pupil as viewed from the object, DoF is maintained--more or less. Just as depth of focus (the space in front and behind the image plane) is dependent on the f-number, not the focal length, as the f-number is proportional to the angular size of the exit pupil from the image plane. (Extreme magnification and how a lens focuses can change this (lenses can focus by changing lens to image plane distance or changing focal length).)
You are, as usual, exactly right.

M

5. ## Re: Depth of field and focal length

Originally Posted by MGrayson
There IS a big difference in background blur. What's surprising is that that is unrelated to Depth of Field. A distant streetlight will make a bigger bokeh with the longer lens, because there isn't a big proportional difference between the distance to the two lenses. The effects only balance out when the 120 is half the distance to the subject as the 250. (Yes, I know that 120 isn't half of 250, but then 1.4 isn't the square root of 2.)

I'm making no claims about HOW lens sharpness falls off. It clearly differs between lenses, but I'm talking about very coarse effects.

M
I have never really studied that, but I suspect it is related to linear perspective. The ratio between the distance of the camera to subject and camera to background object means the difference in the degree of blur changes with those distances. I guess that would be hard to quantify as there is no blur at the object plane. Or this can be explained by relative magnification--but perhaps inversely proportional, which would account for less blur with the longer focal lengths.

I miss geometry...

6. ## Re: Depth of field and focal length

Originally Posted by Shashin
I have never really studied that, but I suspect it is related to linear perspective. The ratio between the distance of the camera to subject and camera to background object means the difference in the degree of blur changes with those distances. I guess that would be hard to quantify as there is no blur at the object plane. Or this can be explained by relative magnification--but perhaps inversely proportional, which would account for less blur with the longer focal lengths.

I miss geometry...
The size of the point spread is the intersection of the cone based on the aperture and the focal plane. When the two apertures "look" the same from the focal plane, these cone intersections are fairly similar for nearby points. A distant vertex (the streetlight) will make a cone that looks like a cylinder based on the apertures, and will thus be twice as large intersecting the focal plane (which is still at the subject).

Here are the two (coincident) cones on the larger and smaller apertures, both the same f-stop. An OOF point makes cones that intersect the focal plane in small circles. Remember that the two lenses are capturing the same amount of the focal plane, so circles on the focal plane will look the same size (head shots) in both captures.

If we zoom way in, we see that the two circles are almost the same size.

But for a distant object, the cones become cylinders, and the circles are twice as big for the larger opening.

M

7. ## Re: Depth of field and focal length

While DoF is similar, the 120 makes your lens look fat. Just sayin...

8. ## Re: Depth of field and focal length

Originally Posted by jack
while dof is similar, the 120 makes your lens look fat. Just sayin...

Thanks, Jack. It takes a lot to get a laugh these days.

Matt

9. ## Re: Depth of field and focal length

Perhaps of interest to your discussion, a somewhat weird but older recognized phenomena will be noticed when shooting larger formats; and that is lens extension required to focus on the subject, and then along with the associated increase in focal length to accomplish it, you in turn further reduce DoF. For example, a "normal" lens for a 16x20 large format view camera is 600mm. However, to make a "head and shoulders" portrait, you need to focus nearly to 1:1 with that camera, and in so doing, your 600mm lens has become effectively 1200mm in length. So that one lens can give you both a normal view at infinity focus, and the roughly ideal doubling of focal length to make the portrait...

10. ## Re: Depth of field and focal length

Originally Posted by MGrayson
The size of the point spread is the intersection of the cone based on the aperture and the focal plane. When the two apertures "look" the same from the focal plane, these cone intersections are fairly similar for nearby points. A distant vertex (the streetlight) will make a cone that looks like a cylinder based on the apertures, and will thus be twice as large intersecting the focal plane (which is still at the subject).
Matt, thanks for all that. I certainly agree with your analysis. I am uncertain it explains some of the difference in the Bokeh/CoC. While I think the geometric representation you have is right, for me, there is still the issue of the appearance of the image. What your diagrams don't address is the rendering of the scale of the objects behind the subject--the magnification in both setups are not the same. What does the image of the CoC look like in the image?

(I am just thinking through this so bare with me.)

The ratio of image size is equal to the ratio of the camera to the objects in an image. In this case, we are using two focal lengths to create the same magnification at the object plane. The distance in that case, just to keep the math simple, is proportional to the focal length--so the focal length and (lens) object distance between the two cases could be x and 2x for a 125mm lens and a 250mm lens. However, the far object in both cases is 2ft. For simplicity, lets assume the 125mm lens is also 2ft from the lens so x = 2ft. So the ratio in size between the lens and ruler width is the ratio between the camera and lens distance and the camera and ruler distance or x:2x (the 125mm case) and 2x:3x (the 250mm case)(forgive my math notation, I am not a mathematician, just in case you have not worked that out).

So here is the question: does the ratio of object sizes based in camera to object distances apply equally to the image of the CoC?

11. ## Re: Depth of field and focal length

Originally Posted by Shashin
Matt, thanks for all that. I certainly agree with your analysis. I am uncertain it explains some of the difference in the Bokeh/CoC. While I think the geometric representation you have is right, for me, there is still the issue of the appearance of the image. What your diagrams don't address is the rendering of the scale of the objects behind the subject--the magnification in both setups are not the same. What does the image of the CoC look like in the image?

(I am just thinking through this so bare with me.)

The ratio of image size is equal to the ratio of the camera to the objects in an image. In this case, we are using two focal lengths to create the same magnification at the object plane. The distance in that case, just to keep the math simple, is proportional to the focal length--so the focal length and (lens) object distance between the two cases could be x and 2x for a 125mm lens and a 250mm lens. However, the far object in both cases is 2ft. For simplicity, lets assume the 125mm lens is also 2ft from the lens so x = 2ft. So the ratio in size between the lens and ruler width is the ratio between the camera and lens distance and the camera and ruler distance or x:2x (the 125mm case) and 2x:3x (the 250mm case)(forgive my math notation, I am not a mathematician, just in case you have not worked that out).

So here is the question: does the ratio of object sizes based in camera to object distances apply equally to the image of the CoC?
If I understand your question correctly, the answer is "sorta". Here's a picture of a more distant object with CoC of its two endpoints. Then I'll work out the formulas...

The Blue intervals on the focal plane are the Bokeh of the ends of the arrows in the 120/4 capture. The Red intervals (almost completely covering the Blue ones) are for the 250/4. Both the magnifications (distances between the intervals of the same color) and the CoC's are different.

OK.. to be exact. Lenses are distance 2 and 4 from the focal plane. their physical apertures are .5 and 1, respectively. I'll let x be distance from the focal plane. Then the bokeh for the 120/4 satisfies (similar triangles) .5/(2+x)=b/x, so b = .5 x/(2+x) = x/(4+2x). For the 250/4, it is x/(4+x). When x is near zero, these are close because the 4 in the denominator overwhelms the x vs. 2x. When x is large, the disk sizes converge to 1/2 and 1 respectively. For magnification, let's go to a pinhole camera. Then an arrow (it's always an arrow) of length 1 at position x on the central axis... well, it's tail goes to the center of each image. It's head goes to ... similar triangles again... for the 120mm, t/2=1/(2+x), or t=2/(2+x). For the 250mm, t/4=1/(4+x) or t=4/(4+x). Again, when x is zero, we have the same magnification (wrt the final image). As x grows, t decreases for both lenses (as it should), but for large x, the t for the 120 goes as 2/x, and for the 250, it goes as 4/x... Exactly what you'd expect from their focal lengths!

Here are plots as functions of distance from the focal plane.

Note! While the bokeh increases the same to first order with both lenses (in this particular setup), the magnifications agree only to zeroth order - they start changing linearly with x. This is why the geometry (appearance) of your subject changes a lot, but the DoF doesn't.

M

12. ## Re: Depth of field and focal length

Matt, thank you. I appreciate you taking so much time and effort! (I hope the COVID isolation is not getting to you.)

Geometric optics is fairly straightforward, but the visual implications are more complex. The last plot is particularly interesting in this regards.

13. ## Re: Depth of field and focal length

Weirdly, the Magnification and Bokeh curves cross at the same distance (x=4) for both lenses. The object looks smaller with the 120, but it is also blurred less, so at THAT distance from the focal plane, the objects will look exactly the same, but with a magnification difference! I'll have to check that one with an actual camera and see...

The 120/4 can focus at 2 feet, but the 250/4 can't quite do 4 feet. I'll have to change the numbers a bit. I can do 6 feet and 3 feet, but then the critical object distance will be 6 feet further out, or 12 feet from the 250, 9 feet from the 120.

To be continued...

Matt

Oh, another prediction of geometric optics is that if you focus at infinity, and take pictures of Bob at 10 feet and 100 feet, and blow up the 100 foot image so that the two Bob's are the same size, then they will look identically blurry. Have to test that one, too.

And in all the stuff above, I am, of course, ignoring the effect that Jack mentioned above about lens "focal length" changing as you extend the bellows.

14. ## Re: Depth of field and focal length

Originally Posted by MGrayson
And in all the stuff above, I am, of course, ignoring the effect that Jack mentioned above about lens "focal length" changing as you extend the bellows.
You know more about the Fuji optics, but the Pentax the 120mm macro works by changing the lens to image plane distance. But the longer focal length lenses are internal focal (IF), which changes focal length to focus--the Fuji 250mm looks like it is IF. That difference will influence results, especially at shorter object distances. Not sure how much those matter in this case, but I imagine the mathematical modelling will be close to actual use.

And thanks again for doing this work. I am enjoying this thread.

15. ## Re: Depth of field and focal length

Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.

16. ## Re: Depth of field and focal length

Originally Posted by Abstraction
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
Yes, that's what the first set of images was supposed to demonstrate. The size of the OOF disk at the focal plane is determined by the angle at the vertex of the "light cone" and distance from the focal plane. Magnification tells you how big THAT disk will be in the final image. For nearby points, the angle won't change much.

But looking at the pictures brought up the question about how that relationship changes when you move further from the plane of focus. Different focal length lenses will change both the magnification and the angular width of the cone differently with distance. Formulas are great, and necessary, but I always want pictures, either diagrams or photographs, to go with them.

Matt

17. ## Re: Depth of field and focal length

Very interesting discussion! I follow ....

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics...

G

18. ## Re: Depth of field and focal length

Originally Posted by Godfrey
Very interesting discussion! I follow ....

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics...

G
I’m ignoring everything that happens once the light hits the entrance pupil ( where the aperture appears to be from outside ), and I’m assuming that it doesn’t move when focusing. These assumptions are wildly false, especially when either a) the subject to lens distance is comparable to the lens to sensor distance or b) the lens is complex with many and/or floating elements. The experiments are to see how good the simplest approximations work in non-macro photography, and to illustrate some of the less obvious relationships. Jack’s example counts as macro. A head shot on an 16x20 camera is definitely around 1:1, and, as Jack said, the distance from lens to film can double, throwing the simple arguments off. So yes, it is more like a pinhole with the longer focal length. If you stop down to f/256, or f/32,768 it would BE a pinhole camera of that longer focal length.

I’m not mistaking this for photography! The best photographer I know is unaware of any optics theory. He knows what camera settings work and he concentrates on his lighting and subject. Since I’m 90% mathematician and 10% photographer, I find this stuff fun.

Matt

19. ## Re: Depth of field and focal length

Originally Posted by Abstraction
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
Actually, that would not be true (I assume you are using the term magnification as linear magnification at the image plane, which is the usual meaning of the term). DoF is a perceptual/subjective quality of an image (which is why it needs to define a permissible circle of confusion based on human perception). Format will impact DoF because format effects viewing conditions, which is why DoF scales change with format--you could also display images based on the the size of their relative format, where DoF would appear the same, but no one does that.

And by aperture, that is not the same as f-number. If you mean the angular size of the entrance pupil, then that would be more accurate, but format is still going to influence the results across formats.

And this is were the conversation become complex--how do we perceive DoF?

20. ## Re: Depth of field and focal length

Originally Posted by Godfrey
Very interesting discussion! I follow ....

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics...

G
I doubt you would ever get to such small apertures that a lens would have f-number as high as those of a pinhole camera--one definition of a pinhole is that it has so much DoF it does not need optics. But you are right that the effective aperture can be significantly different from the marked aperture. Basically, if you draw out a lens at twice its focal length, for example, having a 300mm lens 600mm from the image plane, the effective aperture will be two stops less--it will go from f/11 to f/22, for example, explaining why exposures can be significantly impacted in macro photography. Qualities such as diffraction and magnification will change by the same degree as well.

21. ## Re: Depth of field and focal length

Originally Posted by MGrayson
The best photographer I know is unaware of any optics theory.
The great thing about photography is seeing is believing. Hard to argue something does not look the way it does.

This is what makes photography as a science extremely difficult. It isn't just describing a physical process, but a perceptual/cognitive one as well. DoF is a perceptual quality--it actually does not exist in any absolute sense--if you have a picture and no one sees it, does it have DoF? Photographic systems ultimately are trying to imitate the human visual system. The fact they work so well is pretty amazing, but the limitations are obvious too.

22. ## Re: Depth of field and focal length

Originally Posted by Abstraction
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
This is a consequence of a deeper statement than I first thought. (Or equivalent to it, if by DoF one really means "knowing the size of every OOF spot in the final image"). DoF is a choice we make from that information.

Theorem: Adjusted for magnification, bokeh disks will be identical in two images of the same f-number and same focus plane. Magnification will vary from point to point, but the relationship holds point-wise.

Here we go... We take two random lenses of the same f-stop, physical apertures a1 and a2, and place them anywhere, but focus them on the same plane:

b1, the bokeh spot as seen by lens 1 on the focal plane, has size b1/x=a1/(x+y1), so b1 = a1*x/(x+y1). Similarly (pun intended) b2 = a2*x/(x+y2). Now a1 and a2 are apertures at the same f-stop, so the focal lengths f1 and f2 are f*a1 and f*a2. So magnification m1 = f*a1/(y1+x) and m2 = f*a2/(y2+x).

We calculate: b1 adjusted for magnification is ...

b1*m2/m1 = a1*x/(x+y1)*f*a2/(y2+x)/(f*a1/(y1+x)) = a2*x/(y2+x) = b2.

Hot damn, it's true EVERYWHERE.

Matt

23. ## Re: Depth of field and focal length

But are we confusing depth of focus, the distance on each side of the image plane that will result in a perception of sharpness, which is only dependent on f-number (and the permissible circle of confusion), or depth of field, the distance on each side on the object plane, which is dependent on focal length and f-number (and permissible circle of confusion)? The description of object space and image space is not the same. Any lens at the same f-number will have the same depth of focus, but not the same depth of field.

Or to put it another way: what is the relationship of the Numeric Aperture (the angular size of the entrance pupil from the object plane) to the f-number (the angular size of the exit pupil from the image plane)?

24. ## Re: Depth of field and focal length

Originally Posted by Shashin
But are we confusing depth of focus, the distance on each side of the image plane that will result in a perception of sharpness, which is only dependent on f-number (and the permissible circle of confusion), or depth of field, the distance on each side on the object plane, which is dependent on focal length and f-number (and permissible circle of confusion)? The description of object space and image space is not the same. Any lens at the same f-number will have the same depth of focus, but not the same depth of field.

Or to put it another way: what is the relationship of the Numeric Aperture (the angular size of the entrance pupil from the object plane) to the f-number (the angular size of the exit pupil from the image plane)?
Isn't that ratio magnification?

My difficulty here is that I never learned the proper names for things. I learned (and taught) optics as a problem in linear algebra - lenses and air gaps are just different 2x2 matrices. A telescope, periscope, or microscope is a matrix with a desired form. How do you combine lens and spacing matrices to get you your telescope? Real optics design uses different names - not always, but often enough to leave me confused.

25. ## Re: Depth of field and focal length

Originally Posted by MGrayson
Isn't that ratio magnification?

My difficulty here is that I never learned the proper names for things. I learned (and taught) optics as a problem in linear algebra - lenses and air gaps are just different 2x2 matrices. A telescope, periscope, or microscope is a matrix with a desired form. How do you combine lens and spacing matrices to get you your telescope? Real optics design uses different names - not always, but often enough to leave me confused.
Sorry, ratio magnification is a new term for me.

And I have appreciated your approach on this topic. I have never seen it defined in the terms you have shown and I find it interesting. I have a bachelors of science from RIT in imaging and photographic technology and so look at this issue in applied photo science terms. DoF is one of those tricky subjects as you are trying to predict human perception of an image displayed under certain criteria using optical choices of a photographer (format, focal length, f-number, distance) with a subjective constant to join them--permissible circle of confusion. And while we can only perceive the resulting image in an image space, DoF is trying to make predictions in the object space when the image is taken--the photographer wants to be able to predict what will appear sharp in the scene in front of him/her. And we have wandered into a more complex idea of how the OFF area will be percieved.

Personally, this is what I find interesting, how do these factors influence what we perceive in an image. (Beyond an upward pointing arrow in object space will point down in image space. )

26. ## Re: Depth of field and focal length

Sorry - I meant "Isn't the ratio of angles you just mentioned the same thing as magnification?"

That's why I like to talk about bokeh disks as things like "imagine that your subject has a disk on her shoulder the size of the physical aperture (or entrance pupil, or whatever). That's what a streetlight will look like. That is concrete to the photographer, and they can draw whatever conclusions they want about what is or is not in focus. If the subject is a mountain climber on a distant slope, the disk is insignificant. If they are 20 feet from you and you have a 400mm f/2.8 monster, then it will be a very big disk indeed. To get the sensor size and final print size in the mix is unnecessary confusion, although very important if you're making that print!

I'm also fond of taking the convolution of Bob with a dime to express an image of a person with a dime sized point blur.

27. ## Re: Depth of field and focal length

Originally Posted by Shashin
But are we confusing depth of focus, the distance on each side of the image plane that will result in a perception of sharpness, which is only dependent on f-number (and the permissible circle of confusion), or depth of field, the distance on each side on the object plane, which is dependent on focal length and f-number (and permissible circle of confusion)? The description of object space and image space is not the same. Any lens at the same f-number will have the same depth of focus, but not the same depth of field.

Or to put it another way: what is the relationship of the Numeric Aperture (the angular size of the entrance pupil from the object plane) to the f-number (the angular size of the exit pupil from the image plane)?
I'm confused again. Something in my definitions must be wrong. I'm trying to put everything in the object plane, which is easier to visualize. I think I don't understand something, and I think I'm confusing two meanings of magnification.

I blame it on the pandemic.

M

28. ## Re: Depth of field and focal length

Originally Posted by MGrayson
I'm confused again. Something in my definitions must be wrong. I'm trying to put everything in the object plane, which is easier to visualize. I think I don't understand something, and I think I'm confusing two meanings of magnification.

I blame it on the pandemic.

M
Right. So everything I said in the last post with diagrams was wrong. Not that the math was wrong, but what it MEANT was not what I meant it to mean . I'm one of those "don't look it up, work it out from scratch" kind of annoying mathematicians, but that way you can find new ways of looking at things. More often, you just rediscover the wheel.

I'm going to go back to the simpler case of uniform magnification on the focus plane and see what we get from there.

M

29. ## Re: Depth of field and focal length

Originally Posted by MGrayson
Right. So everything I said in the last post with diagrams was wrong. Not that the math was wrong, but what it MEANT was not what I meant it to mean . I'm one of those "don't look it up, work it out from scratch" kind of annoying mathematicians, but that way you can find new ways of looking at things. More often, you just rediscover the wheel.

I'm going to go back to the simpler case of uniform magnification on the focus plane and see what we get from there.

M
In imaging, magnification is usually linear magnification, the difference between object size and image size. This should not be confused with visual instruments like telescopes and microscopes, with describes angular magnification (also note, that definition is different between telescopes and microscopes (a telescope working at 1x is not the same as a microscope working at 1x). But the difference is practical. Linear magnification allows us to predict the image size of an object in an imaging system, where angular magnification allow us to predict how something will appear when we look at it.

And this is where linear magnification can give two perceptually different results. If you shoot at 1:1 (or at 1x) on a medium format camera and APS-C camera, the resulting image will appear that the APS-C has more "magnification" simply because it has a smaller sensor. A 1cm object will appear larger on a smaller sensor simply because 1cm is proportionally larger on the small sensor.

I am not sure this will be helpful, but although image and object planes are conjugate, they are not the same appearance and are not directly related to point spread. So an f/2 lens will have the same depth of focus, the focused light cone will intersect the image plane at the same angle regardless of focal length, but the displacement from the image plane will represent different object distances because of focal length--think how far long focal length on a view camera has to be moved to focus from infinity to 1m compared to wide angle illustrating the image-side point spread is not the same as the object-side point spread (although those can coincide as you are doing). DoF is describing the appearance of the object-side point spread.

So how does this impact the visual appearance of the OFF image? So as you know, the difficultly with equivalence is it is like Home Depot--it almost has what you want. So in equivalence you can get many, but not all variables to coincide. And the same thing seems to be happening here. So if you look at the specular highlight on the ruler two feet behind the lens, the out of focus image does not appear the same, even though the DoF is equal. I think it is related to relative size in relation to perspective, as I posted above--the difference in the sizes of two object in an image is proportional to their distances from the camera. So in this case, the 250 is further from the lens, but the ruler end is fixed at 2ft for both. So while the DoF is identical in each image, the linear perspective (at least in terms of identifiable points in the image) is different.

In short, you can make DoF coincide, but not linear perspective and hence magnification, at least for objects within the frame. This may be the reason we can perceive the same DoF, but not Bokeh.

Just a thought.

30. ## Re: Depth of field and focal length

Actually, scratch that. I think point spread at the image plane can still work, but the distance from the image plane along the cone does not represent the same distance in object space--it might also be proportional to the linear perspective thing above.

Or something...

31. ## Re: Depth of field and focal length

My goal is to move everything to the object plane or, even better, to the entire 3D scene. The latter means "what do I do to reality so that a pinhole camera image agrees with the 50 f/2 image taken from here." I understand the truth of what you're saying, but I'm after an "interpretation" that will help a photographer visualize the effect of focal length, aperture, focus distance and OOF objects OUTSIDE the camera.

For instance, a distant streetlight looks like my lens's entrance pupil placed next to the subject in focus. This is universally true.

Or "If I focus on the mountains with a 2cm entrance pupil, then EVERY object in the scene is smeared by a 2cm disk." 2cm at a distance will be smaller than the CoC, and so that part of the image will be "in focus". Every enlargement will have objects that appear smeared by "real" 2cm. Enlarge the the object, enlarge the 2cm disk. If I focus at infinity, then even the mountains have a 2cm blur, but 2cm looks very tiny at that distance.

I'm hoping for other simple universal relationship between these variables. I think there may still be a few.

Matt

32. ## Re: Depth of field and focal length

Matt, you absolutely can describe an image from object space--conjugate planes after all. I am more thinking out loud. Most of the formula used for things like DoF are very simple and are only really interested in what looks sharp for one particular condition. The OOF question is interesting. There is a strange idea the sharp and OOF are just binary conditions and so sharp is sharp and OOF and OOF, yet looking at images, it is not that simple.

Looking forward to seeing what you find.

33. ## Re: Depth of field and focal length

Just a little thinking...

Images are simply a single plane sample from a projected volume. Using a simple lens, the volume is the same regardless of focal length, just different sizes--magnification. But you should be able to intersect any volume with any size plane (format) and get the same image of the object plane--the in focus image. So, how does the entrance/exit pupil and size of the image plane effect the appearance of the resulting image? And how does the point spread change with those variables to create the OOF area?

Boy, am I glad you are the mathematician. I will stick with pub quizes on 70's pop music...

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