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#1




The danger of "equivalent fnumber"
****** We have certainly gotten used to the concept of "equivalent focal length". We say that a lens with a focal length of 50 mm, on "our" camera, with a sensor whose linear dimensions are 0.625 times the dimension of a " fullframe 35mm"size sensor, will give the same field of view as would a lens with a focal length of 80 mm (1.6 × 50) on a camera with a fullframe 35mmsize sensor (and from now on I will for conciseness refer to that as a "ff35" camera). And thus we say that "our" sensor has a "1.6 equivalent focal length factor". So it at first does not seem to be a surprise that there has emerged (especially on a certain other forum) the notion of an "equivalent fnumber". How that works is this, considering "our" camera as discussed above. A lens on our camera with an aperture of f/2.0 is said to have an "equivalent fnumber" of f/3,2 (2.0 × 1.6). (The "1.6" is in fact the equivalent focal length factor for our camera's sensor size.) Now, historically, our first concern with the fnumber has been its effect on exposure. So does the notion above mean that a lens on our camera with an aperture of f/2.0 will have the same effect on exposure that a lens with an aperture of f/3.2 would have on an ff35 camera? No. A lens with an aperture of f/2.0 will have on our camera the same effect on exposure as a lens with an aperture f/2.0 would have on an ff35 camera. No, the significance of the "equivalent tnumber" has to do with our second concern with aperture: its effect on depth of field (and its counterpart, outoffocus blur performance). Now, if we consider two cases, involving two cameras with different sensor sizes, and all these factors are the same between the two cases: • The lenses have focal, lengths such that the field of view of the two cameras are the same.then, with an f/2.0 lens on "our" camera, we will get the same depth of field as we would get with an f/3.2 lens on an ff35 camera. Now, the good people on that other forum often are so kind as to provide us with graphs that show the variation with focal length in maximum available aperture for a certain camera (with a certain zoom lens aboard), over the focal length range of the lens. And that's very handy. But this is not given in terms of fnumber, but rather in terms of effective fnumber, as described above. Hmm. Now often two or more different cameralens rigs have this plotted on the same graph. Is this to tell us how the "speed" of the lenses on the two cameras differ at different focal lengths (often a matter of great interest)? No, it it to show how the effect of the lenses (wide open) on depth of field. Now, if all the cameras plotted on the chart have the same sensor size, then the plot indeed also shows the difference between the "speed" of the two lenses at different focal lengths. But if, as is often the case, the cameras being compared have different sensor sizes, then we cannot determine from such a chart the difference in lens "speed" between the two cameras at some focal length. I leave it to the reader to determine how to feel about this practice. I do not have a positive feeling about it. A second supposed utility of the concept of "equivalent focal length" has to do with the effect of fnumber on diffraction. The premise of that notion is complicated (and questionable), but I will not go into that here. Best regards, Doug 
#2




We see the risk in this scheme in a discussion on that forum of the differences between the Panasonic ZS100 and its predecessor, the ZS60.
The ZS60 lens has (at its smallest focal length) a maximum aperture of f/3.3. The ZS100 has (at its smallest focal length, the same as for the ZS60) a maximum aperture of f/2.8. So (at the lowest focal length) the ZS100 lens is about 0.5 stop "faster". But the "equivalent focal length factor" for the ZS60 is about 5.7, while for the ZS100 (which has a substantially larger sensor) it is about 2.7. So the infamous "equivalent aperture chart" shows, at the lowest focal length for both lenses, an equivalent fnumber for the ZS60 of f/18.8, and for the ZS100, f/7.6. If those were real apertures, that difference would be about 2.6 stops. And based on that, the discussion says that "the ZS60 is effectively 2 stops slower than the ZS 100." But of course the ZS60 is not in any way "2 stops slower" than the ZS100. Yes, its "effective aperture" (whatever that's worth) is actually 2.6 stops "less" than the effective aperture of the ZS100. But to use the term "slower" for this difference is wholly inappropriate, and could easily mislead the reader. For example, the reader might think that, for the same ISO sensitivity and the same scene, we would need to use a 2stop greater shutter time (2.6, really) to get the same exposure. But in fact, the shutter time needed would be only 0.5 stop greater. Best regards, Doug 
#3




Quote:

#4




Hi, Jerome,
Quote:
• For depth of field purposes, we define "the limit of acceptable blurring" in terms of the blurring that significantly compromises the camera's resolution (rather than the traditional outlook in which "acceptable blurring" is defined in terms of what would be perceptible to the viewer viewing the image at an arbitrary size and arbitrary distance). • The two cameras have a different pixel pitch (and thus a different pixel count). Best regards, Doug 
#5




It is sometimes said that the "diffraction limit" occurs for the same "equivalent fnumber" on cameras with different sensor sizes. This is at least closely true if:
• We consider the "limit" to occur when diffraction significantly compromises the resolution of the camera.(We usually do that, but there is an alternative outlook that may be appropriate in certain cases.) • The resolutions of the cameras bear the same relationship to their respective pixel pitches. (Maybe, in terms of line pairs per mm, 0.75/(2p), where p is the pixel pitch in mm, a common situation.) • The pixel pitches of the cameras are the same same as a fraction of the sensor size (that is, all the cameras have the same pixel count). Best regards, Doug 
#6




Quote:
Actually, if you try to lenses with the same aperture and the same focal length on two cameras with either the same sensor or the same sensor size and resolution, you will also often notice that the depth of field is not visually the same, wherein I define depth of field as what appears similarly sharp or similarly unsharp to a panel of viewers. 
#7




Hi, Jerome,
Quote:
To what can we possibly attribute that difference between cameras in which all pertinent parameters are seemingly the same (which is assume is included in your case)? Best regards, Doug 
#8




Quote:
How did you determine that the depth of field is the same? 
#9




Well, I've had breakfast, and my serum glucose level is up.
Quote:
There are two widely used approaches to defining what degree of blurring from imperfect focus is "just barely acceptable" (a pivotal aspect of reckoning the expected depth of field of some "camera setup"). Outlook 1 ("traditional"). This says that the limit of acceptable blurring is when the blurring is enough to be noted by an "average" person viewing of the image at some arbitrarilychosen size (perhaps 8" × 12") from some arbitrarilychosen distance (perhaps 17"). This is quantified by saying that the "limiting diameter" of the blur figure is some arbitrary small multiple of what is generally considered to be the resolution of the human eye, as applies to viewing of the image under the arbitrary conditions I mentioned above. That is then normalized so it can be spoken of in terms of the diameter of the blur circle on the sensor, with the adopted "limit" often coming out to be 1/1400 of the sensor diagonal dimension. Outlook 2 (adopted by many modern workers). This says that the "limiting diameter of the blur figure" is where it degrades the resolution of the camera proper. That is usually quantified as the diameter of the blur circle being some (arbitrary) small multiple of the pixel pitch. If we embrace this outlook, then for cameras whose pixel count is substantially greater than is "needed" for viewing under the arbitrary conditions I mentioned above, this outlook will lead to a reckoning of the expected depth of field that may be much smaller than if we proceed under Outlook 1. My discussion in an earlier post here was, I think, predicated on Outlook 2. Best regards, Doug 
#10




Nope. Rather "How do you determine depth of field". I bet is has something to do with the 10 Deutsche Mark banknote.

#11




Hi, Jerome,
Quote:
If so, I would certainly agree. In "practice", we rarely see that analytical approach followed! Rather, under "outlook 2", as I mentioned, workers often use the criterion, "when the expected 'diameter' of the blur figure exceeds n times the pixel pitch", where n has been chosen based on empirical observation of actual results. Thanks for that nice link to history! Best regards, Doug 
#12




Quote:
The handsome subject of the banknote did considerable work in optics. Great guy BTW, a true genius. 
#13




Gaussian optics is what I had in mind. The formulas often used to compute depth of field are derived from the formalism of Carl Friedrich Gauß.

#14




Hi, Jerome,
Quote:
Neat! Best regards, Doug 
#15




Good to have you back on board. Was breakfast good? Did you find out why, if you try to lenses with the same aperture and the same focal length on two cameras with either the same sensor or the same sensor size and resolution, you will also often notice that the depth of field is not visually the same, wherein I define depth of field as what appears similarly sharp or similarly unsharp to a panel of viewers?

#16




Hi, Jerome,
Oh, very. Quote:
My guess is that it may have to do with the radial illuminance distribution on the blur figure (circle of confusion, in the proper use of that term). Best regards, Doug 
#17




Yes, it does. But the formulas which are used to determine "equivalent fnumbers" are derived from the principles of Gaussian optics. Do the principle of Gaussian optics allow to determine that illuminance distribution?

#18




Hi, Jerome,
Quote:
If we wish to predict the actual distribution of illumination across the figure, we must take into effect the recognized aberrations of the lens as well as diffraction. This analysis can be done within the paradigm of Gaussian optics. But we have to recognize that this paradigm gives us only an approximation of the actual paths of the rays. Of course, the analyses actually used today in lend design are based on ray tracing that is "precise" and is not premised on the paraxial approximation (that is, goes outside the paradigm of Gaussian optics. I think. As to the determination of "effective fnumbers": Indeed the fnumber of a lens may be approximately calculated within the paradigm of Gaussian optics. But having in hand the fnumber of a lens at, say, a certain focal length (in the case of zoom lenses), the socalled equivalent fnumber is obtained merely by multiplying that supposed actual fnumber by the ratio of (a) 43.3 mm (the diagonal dimension of the "fullframe 35mm" format) to (b) the diagonal dimension of the format of the camera on which the lens is assumed to be mounted. There is no actual "optical calculation" involved in that stage. In any case, I note that when the concept of "effective fnumber" is actually used to compare the "expected" depth of field (or out of focus blur performance) of a lens (in terms of what lens would be expected to give the same performance in one of those two ways on a fullframe 35mm size format camera), the result even theoretically is not exact, but will be quite close for most realistic cases. For example, if we consider the size of the blur figure for these two actual setups; a. Lens focal length 50 mm, f/4, camera focused at 5 m, "background object" at a distance of 20 m. b. Lens focal length 25 mm, f/2, camera focused at 5 m, "background object" at a distance of 20 m. We find that the reckoned diameter of the circle of confusion (on the sensor) for case (a) is 0.0947 mm and for case (b) it is 0.0471. So if in fact the format in case (a) is twice the linear size of the form in case (b), and if we consider the diameter of the blur circle as it would be seen on equalsized printed, then the two blur circles would be very nearly the same diameter. Best regards, Doug 
#19




Well said, but the problem with this approach, I think, is that blur is also a subjective feeling and that feeling also depends on the distribution of illumination across the figure. A theory like Gaussian optics which neglects the shape of the blur figure is thus likely to predict that some shapes are blurred, while they are visually recognisable or that some shapes are recognisable while the distribution of illumination across the figure increases the blur to a point where the viewer does not see a sharp image any more.
I actual usage of depth of field (that is with reasonably fast lenses), the effect is quite noticeable and expects why the image given by two lenses which should otherwise be similar can loom surprisingly different. 
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