1. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Magnification is determined by distances from the principal planes—not the planes of the pupils—so my Eq. (44) still applies to an asymmetrical lens, and Eq. (96) still holds.

Despite the title of the book, the topic of the referenced chapter is the optical image rather than a digital image that might eventually result. It seems clear that this is to what the OP is referring, so the questions are valid.

2. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Magnification at the focused point depends only on the principal plane, this is correct. However, principal point is not the center of angle of view, entrance pupil is. AOV is what determines the "scaling" of sizes in different points in space, which is what magnification at defocused points is.

When p = 1, both points are at the same spot so (44) is correct.

3. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Also, after a good night's sleep, it occurred to me that (m+p)/p is just the bellows factor so the new eq. (96) can be represented as:

k_r = m / (1 + m/p) * |x_d| / N

This is practically identical to the original, symmetric relative blur in eq. (45), the only difference being that 1+m becomes the usual 1+m/p.

4. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Originally Posted by relatively_random
AOV is what determines the "scaling" of sizes in different points in space, which is what magnification at defocused points is.
This is a new one for me. Admittedly, magnification of defocused objects isn’t a common topic; in fact, Merklinger’s discussion in The INs and OUTs of Focus is the only one of which I am aware, and he doesn’t cover asymmetrical lenses.

It seems to me that if magnification of a focused object were determined by distances from the principal planes but the magnification of defocused objects were determined by distances from the plane of the entrance pupil, magnification would be discontinuous between a focused object and an infinitesimally defocused object—which doesn’t make sense.

I am for sure no lens designer. Perhaps one of the high-powered folks like Emmanuel or Oren might be able to clarify this.

5. ## Depth of Field in Depth: Relative blur with pupil dilation issue?

Defocus magnification change is determined by the marginal ray angle in image space. That’s an easier way to derive it.

Telecentric optics do not change magnification with defocus. This is an important property for applications such as astrometry and tracking systems.

Magnification changes with field angle is distortion. You can therefore derive exact magnification calculations via Seidel aberration coefficients and/or Zernike polynomials...if and only if you want to earn extra credit in a graduate level optical aberration course.

In the real world, lens designers write a macro in Zemax or Code V to generate plots and data tables of distortion vs field angle as the image plane moves through focus while they go on their lunch break, then copy the plots into power point after they get back from lunch. Doing it this way is more accurate than calculating, since the data is generated by the software’s physics-based ray trace algorithms.

6. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

It seems to me that if magnification of a focused object were determined by distances from the principal planes but the magnification of defocused objects were determined by distances from the plane of the entrance pupil, magnification would be discontinuous between a focused object and an infinitesimally defocused object—which doesn’t make sense.
m_d = (m/p + 1) * f / (u_d - u_ep) is a continuous function for u_d > u_ep. And it's not difficult to show that m_d = m when u_d = u. Just substitute u_d=u=(1+1/m)*f and u_ep=(1-1/p)*f.

Defocus magnification change is determined by the marginal ray angle in image space. That’s an easier way to derive it.
I don't know how to do it with marginal rays in image space, but here's my derivation using chief rays in object space:

Marginal rays got me thinking that there is an easier way to derive "relative blur" in object-space:

Sorry for all the paper photos, doing all this digitally would take some learning and time.

7. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Magnification changes with field angle is distortion. You can therefore derive exact magnification calculations via Seidel aberration coefficients and/or Zernike polynomials...if and only if you want to earn extra credit in a graduate level optical aberration course.

In the real world, lens designers write a macro in Zemax or Code V to generate plots and data tables of distortion vs field angle as the image plane moves through focus while they go on their lunch break, then copy the plots into power point after they get back from lunch. Doing it this way is more accurate than calculating, since the data is generated by the software’s physics-based ray trace algorithms.
All this is waaaay above my knowledge level. I was just trying to figure out how to estimate positioning tolerance in a machine vision system, with lens data I have available.

8. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

Ah that makes more sense. If I understand the purpose correctly, Machine vision systems usually employ telecentric imaging lenses with centroid object tracking. That way the system is not sensitive to defocus. A basic blob detection scheme works perfectly fine in those cases and is simple to code (examples online). Color filters and backdrop selection are used to optimize contrast between the tracked object and background.

9. ## Re: Depth of Field in Depth: Relative blur with pupil dilation issue?

We use telecentrics when needed or when they make things simpler, yes. Often, though, they are either not feasible or simply not needed. And they don't really help with depth of field, except by making the blur symmetrical.

I had to pick some lens to order for a project with a nasty combination of properties: small objects (large magnification) which move fast (low exposure times), but relatively lots of field depth to cover. I needed to see how much depth we can cover while keeping the required details discernible enough. I had to learn how to estimate this, which is when I stumbled onto the concept of "relative blur" from this site's DOF guide. It's exactly what I needed.

10. ## Re: Using blur calculations to estimate positioning tolerance in a machine vision sys

Sorry to be a bit slow getting back here—I had a few other things to take care of the last couple of days.

On further thought, a simple diagram suggests that if magnification is to have its common meaning, the distances—for focused or defocused objects—need to be measured from the pupils. As it turns out, Applied Photographic Optics (Ray 2002, 125) gives image and object distances equivalent to what relatively_random derived; curiously, though, it mentions them only in the context of exposure compensation for lens extension (and in a later chapter, for DoF). Thanks to relatively_random for catching this.

The paper probably ought to be fixed, but unfortunately—because of changes to Word, MathType, Acrobat, and a defect in a program to convert AutoCAD images to EMFs (the only vector format Word can handle)—this is no small undertaking. And some things—like links in equation references—no longer seem to work, so that one could not click on a reference to “Eq. (44)” to jump to the equation. Suffice it to say that my thoughts on this nonsense cannot be expressed in this forum. I suppose I could post an erratum, but I’m not sure anyone would find it.

It should be mentioned that the “relative blur” concept is essentially an image-side adaptation of the “object field method” described in Harold Merklinger’s The INs and OUTs of FOCUS. Merklinger projects the blur spot into object space and compares it to the size of objects to determine whether the objects can be recognized. He gives one interesting example of an attempt to blur a distracting background in a portrait. It’s well known that a longer-focus lens gives a larger blur spot; will using a longer-focus lens solve the problem? Nope, because the magnification of the background also increases, and remains recognizable even though the blur spots are larger.

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