# The "reach" of a lens

#### Doug Kerr

##### Well-known member
People often want to speak of the "reach" of a lens, perhaps to recognize the impact of a certain focal length lens as used on a camera with a certain sensor size. Intuitively, the number should increase with focal length, and for a given focal length, should decrease with sensor size. In fact, if that was all we needed to do, the "full-frame 35-mm focal length" would be a credible metric for "reach".

But why do we want a lens of "longer" focal length on a camera for certain work?

Well, the simplistic reason (recognized in the above concept of "reach") is to make the field of view smaller. But is that really an advantage that warrants a longer focal length lens? After all, we can do that by cropping the image.

The usual actual reason (although we rarely articulate it) is to have a given subject covered by as many lines of resolution as we can, so long as the entire desired scene is included in the field of view. Increase in focal length increases that, proportionately to focal length if the longer focal lens does not significant degrade the overall camera resolution (as expressed, say, in lines per picture height).

So perhaps our metric for "reach" should reflect our success in achieving that objective (not just in making the field of view smaller).

To quantify this, we must think in terms of the number of lines of resolution embraced by a certain distance across a subject at a certain distance from the camera.

But, to generalize it, that turns out to be equivalent to a certain number of lines of resolution per unit of angle - the angular resolution of the camera with the lens of interest in place (and that is with resolution defined in a way that the number is larger for "finer" resolution - often angular resolution is defined the other way up).

Even though "line pairs" would be better, from a theoretical standpoint, than "lines" (it can be related more directly, for example, to the matter of MTF) I have suggested a metric based on "lines" of resolution. The reason is that there is a (very) approximate equivalence between lines of resolution (per picture height, for example) and pixels.

Now what unit should we use for our metric? We could use lines per radian. But the number would turn out to be very large for most cases we will work with. For example, if we assume a sensor 24 mm high, a resolution of 2000 lines per picture height, and a focal length of 200 mm, the "reach" would be about 16,700 lines per radian.

We could also do it in terms of lines per degree. For the example above, the "reach" would be about 290 lines per degree. That's easier to write. So why not use lines per degree.

Now, if we instead mounted a 400 mm lens, and the resolution at the sensor did not decline because of it, our reach would be about 580 lines per degree.

But if the optical performance of our 400 mm lens was so poor that the camera resolution with it in place was only 1500 lines per picture height, now our reach would be only about 435 lines per degree (only an increase of 1.5 times over the 200 mm lens).

Best regards,

Doug

#### Cem_Usakligil

##### Well-known member
Hi Doug,

A very well formulated article, as usual. Thanks for that.

...But why do we want a lens of "longer" focal length on a camera for certain work?

Well, the simplistic reason (recognized in the above concept of "reach") is to make the field of view smaller. But is that really an advantage that warrants a longer focal length lens? After all, we can do that by cropping the image.

The usual actual reason (although we rarely articulate it) is to have a given subject covered by as many lines of resolution as we can, so long as the entire desired scene is included in the field of view. Increase in focal length increases that, proportionately to focal length if the longer focal lens does not significant degrade the overall camera resolution (as expressed, say, in lines per picture height)...
Indeed. Before somebody else jumps in to indicate that there may be other reasons for using a different focal length, such as wanting to have a different Depth of Field characteristics, it is imperative to read your sentence I have marked in bold above. Within the context of having more reach (i.e. creating a larger image of the subject on your film/sensor) those other reasons would not be relevant.

...The usual actual reason (although we rarely articulate it) is to have a given subject covered by as many lines of resolution as we can, so long as the entire desired scene is included in the field of view. Increase in focal length increases that, proportionately to focal length if the longer focal lens does not significant degrade the overall camera resolution (as expressed, say, in lines per picture height).

.......

But if the optical performance of our 400 mm lens was so poor that the camera resolution with it in place was only 1500 lines per picture height, now our reach would be only about 435 lines per degree (only an increase of 1.5 times over the 200 mm lens).
Very astute demonstration of the fact that the effective increase of the reach is not always linearly related to the lens' focal length. For example, if one would use a poorly designed teleconvertor (aka extender) to extend the focal length of a given lens and the resulting lines per degree is actually lower compared to the lines per degree without the use of the convertor, one would be better off shooting without the TC and by cropping instead. Of course, this is not taking into account the possibility of intelligent uprezzing the picture in the post, which might tip the balance for shooting without the TC even if the absolute value of the lines per degree with TC is higher. OTOH, the possibility of applying some clever deconvolution sharpening to the TC image would tip the balance (the break even point) in the other direction. So there are many variables to reckon with. Your recent demonstration of the Canon 2x extender is an interesting read for those who haven't done so already.

PS: There is one aspect of this measure of 290 lines per degree which we should remember. That is, it is sensor specific; among others. If the 24mm high sensor would have 3000 lines instead of the 2000, and if the lens would be capable of resolving to that level, then the lines per degree would be 435. So the total chain consisting of the lens resolution, focal length, sensor size and number of pixels and the raw conversion program used (or the built in camera software to create jpg files) would dictate the resulting lines per degree. Bart's resolution test target can help us to calculate exactly that.

#### Doug Kerr

##### Well-known member
Hi, Cem,

All well said. Thanks for adding to this topic.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
Cem aptly reminded us that another reason to choose a "longer" focal length lens has to do control of depth of field.

As he said, this really doesn't interfere with the point of my note. But it is an important matter, so I thought I would expand a little on it.

As so often when we deal with depth of field issues, we have to be careful about the conditions under which we make a comparison. I'll choose what I think is a "useful" premise.

Let's consider two cases:

Case A

• Camera with 36 mm x 24 mm sensor.

• 200 mm lens, f/3.5

• Distance to subject: 10 m

• Desired scene: that which fills the full frame with the 200 mm lens

• Delivered image: from full taken image (entire scene)

Case B

• Camera with 36 mm x 24 mm sensor.

• 400 mm lens, f/3.5

• Distance to subject: 10 m

• Desired scene: same scene as in case A

• Delivered image: from 18 x 24 mm crop of taken image (entire scene)

Now, lets do some depth of field calculations. We must first adopt a circle of confusion diameter limit (COCDL). For case A, let's choose 1/1400 the diagonal size of the delivered image (as measured on the focal plane). That is 0.031 mm.

Why did I say "1/1400 the diagonal size of the delivered image (as measured on the focal plane)" rather than "1/1400 the diagonal size of the sensor"? In this case, these are the same. But I used the language I did to set the stage for something to come.

So on that basis, for case A, the total depth of field calculates as 0.53 meters.

Now for case B. The premise of a COCDL based on image size has to do with the perceived blurring. In case A, when we chose a COCDL of 1/1400 the diagonal dimension of the delivered image (on the focal plane), that means that the maximum "allowed" blur circle diameter, on the print, is 1/1400 the diagonal size of the print.

In this case, we have a "delivered" image of the very same scene, and lets say on the very same size print. There is no reason why we should consider "just acceptable" any different blur circle diameter on the print than we did for for case A. That means that again, the maximum "allowed" blur circle diameter, on the print, is 1/1400 the diagonal size of the print.

But of course this print is printed with twice the print magnification as for case A.

Thus, the appropriate COCDL for Case B would be 1/1400 of the region (on the sensor) that contains the scene (18 x 12 mm in size), or 0.0155 mm.

So on that basis, for case B, the total depth of field calculates as 0.07 meters, substantially less than in case A.

Now for those who don't agree with my outlook on a COCDL for case B, you may want to calculate the depth of field using the same COCDL as for case A.

In that case, the total depth of field calculates as 0.13 meters.

Does that mean that the scene is "more in focus" than before? Of course not - it is the same print. It will look the same, regardless of the COCDL you adopt, or even if you do not calculate the depth of field at all or never heard of depth of field.

Recall that depth of field is not a physical or optical result - it is just a manmade way to "score" what happens, based on an arbitrary criterion.

Interesting.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
Well, zut alors! Cem has been kind enough to point out (privately) that the demonstration above is all backwards.

It is so hard to get good help anymore!

So I will do the whole thing over.

************

Cem aptly reminded us that another reason to choose a "longer" focal length lens has to do control of depth of field.

As he said, this really doesn't interfere with the point of my note. But it is an important matter, so I thought I would expand a little on it.

As so often when we deal with depth of field issues, we have to be careful about the conditions under which we make a comparison. I'll choose what I think is a "useful" premise.

Let's consider two cases:

Case A

• Camera with 36 mm x 24 mm sensor.

• 400 mm lens, f/3.5

• Distance to subject: 10 m

• Desired scene: that which fills the full frame with the 400 mm lens

• Delivered image: from full taken image (the entire "desired scene")

Case B

• Camera with 36 mm x 24 mm sensor.

• 200 mm lens, f/3.5

• Distance to subject: 10 m

• Desired scene: same scene as in case A

• Delivered image: from 18 x 24 mm crop of taken image (the entire "desired scene")

Depth of field calculations

Case A

Now, lets do some depth of field calculations. We must first adopt a circle of confusion diameter limit (COCDL). For case A, let's choose 1/1400 the diagonal size of the delivered image (as measured on the focal plane). That is 0.031 mm.

Why did I say "1/1400 the diagonal size of the delivered image (as measured on the focal plane)" rather than "1/1400 the diagonal size of the sensor"? In this case, these are the same. But I used the language I did to set the stage for something to come.

So on that basis, for case A, the total depth of field calculates as 0.13 meters.

Case B

Now for case B. The concept of depth of field has to do with the perceived blurring. In case A, when we chose a COCDL of 1/1400 the diagonal dimension of the delivered image (on the focal plane), that means that the maximum "allowed" blur circle diameter, on the focal plane, is 1/1400 the diagonal size of the image; thus on the "full frame" print (where everything is magnified by some consistent ratio from the situation on the focal plane), the maximum "allowed" blur circle diameter is 1/1400 the diagonal size of the print.

In case B, we have a "delivered" image embracing the very same scene, let's say on the same size print. There is no reason why we should consider "just acceptable" any different blur circle diameter on this print than we did for for case A. That means that again, the maximum "allowed" blur circle diameter, on the print, is 1/1400 the diagonal size of the print.

But of course this print is printed with twice the print magnification as for case A.

Going back from this to the focal plane, the appropriate COCDL for Case B would be 1/1400 of the region (on the sensor) that contains the entire scene (18 x 12 mm in size), or 0.0155 mm.

So on that basis, for case B, the total depth of field calculates as 0.27 meters, about twice that of case A.

In summary, if we take Case A (the 400 mm lens case) as our starting point, we find that by using a 200 mm focal length instead, for a delivered image spanning the same scene, we have a greater depth of field.

Now for those who don't agree with my outlook on a COCDL for case B, you may want to calculate the depth of field using the same COCDL as for case A (0.031 mm).

In that case, the total depth of field calculates as 0.53 meters, twice what we had under my suggested COCDL.

Does that mean that the scene appears "less blurred overall" than before? Of course not - it is the same print! It will look the same, regardless of the COCDL you adopt, or even if you do not calculate the depth of field at all, or if you never heard of depth of field.

Recall that depth of field is not a physical or optical phenomenon or result - it is just a manmade way to "score" the image, based on an arbitrary outlook and an arbitrary criterion.

************

My apologies for the error the first time. And thanks, Cem, for catching the error.

Best regards,

Doug

#### StuartRae

##### New member
Hi Cem (and Doug),

I don't know if this is the proper place for this comment, but..........

Cem_Usakligil said:
..........this is not taking into account the possibility of intelligent uprezzing the picture in the post, which might tip the balance for shooting without the TC.........

Did you have any particular software in mind? Although software upsampling is useful and sometimes unavoidable, I've yet to see an example that doesn't look somewhat artificial.

For me at least, what Doug's '2x extender' article proved is that optical zoom is preferable to digital zoom. There were too many potential variables involved to draw any other deeper conclusion. (Apologies Doug, I should maybe have said this as a reply to your original post)

Regards,

Stuart

#### Cem_Usakligil

##### Well-known member
....Did you have any particular software in mind? ...
I think that Bart knows the answers to this and I expect he shall chime in. What I normally use are the QImage Studio from Mike Chaney (for uprezzing to the print resolution using the Hybrid SE algorithm) and the ImageMagick. If I understand correctly, the ImageMagick uses the best possible algorithms for uprezzing and also downrezzing. Besides, it is free. There are some other commercial apps out there but I am not experienced with them.

#### StuartRae

##### New member
I think that Bart knows the answers to this and I expect he shall chime in. What I normally use are the QImage Studio from Mike Chaney (for uprezzing to the print resolution using the Hybrid SE algorithm) and the ImageMagick. If I understand correctly, the ImageMagick uses the best possible algorithms for uprezzing and also downrezzing. Besides, it is free. There are some other commercial apps out there but I am not experienced with them.

Thanks Cem. Yes, I use QI for printing, and the up-sampled images look fine in print. I've yet to find anything that (as Doug did in his test) gives natural looking results when used for a simple 2x up-sampling for the screen.
I hadn't realised that ImageMagick was free; I'll try it for the very few times when I need to re-size.

Regards,

Stuart

#### Cem_Usakligil

##### Well-known member
Thanks Cem. Yes, I use QI for printing, and the up-sampled images look fine in print. I've yet to find anything that (as Doug did in his test) gives natural looking results when used for a simple 2x up-sampling for the screen.
I hadn't realised that ImageMagick was free; I'll try it for the very few times when I need to re-size.t
Take a look at this webpage. ImageMagick uses the Mitchell algorithm as a default for enlarging images. Lanczos filter is used for downrezzing. For important images I post on the web, I use ImageMagick. For regular ones, LR3 is very good too.

Thanks Cem.

#### Cem_Usakligil

##### Well-known member
Thanks Cem.
No thanks needed. I have been educated by Bart so we should thank him, lol.

Try the following and you will be amazed. Install the ImageMagick. Create a new batch file (with the extension .bat) on your desktop. Edit it and add the following text:
Code:
``convert %1 -depth 16 -filter Mitchell -gamma 0.4545455 -resize 200%% -gamma 2.2 "%~n1_IM_200_pct.tif"``
Drag and drop any tif file you want to resize onto this icon. The batchfile will then resize your image to 200% and output a new 16-bit tif file with the addition of _IM_200_pct to the original file name. It will create this new file in the same directory as the original image which is dragged and dropped. The resizing works in the 16 bit domain making use of a gamma of 1. The output may need a slight sharpening which you can do in your favorite program.

PS: The idea of using batchfiles with ImageMagick like this is of course courtesy of Bart. You can also create this batch file in the Send To folder so that you can right-click on any file and execute the action.

PS2: You can also use QImage to uprezz even if you are not printing. Just choose the output type "file" and create a upsized document using an imaginary printer size with an imaginary ppi (together resulting in the required final image dimensions in pixels).

#### Cem_Usakligil

##### Well-known member
Apologies to Doug for getting a bit off-topic. We shall continue this elsewhere if necessary.

#### Doug Kerr

##### Well-known member
Hi, Cem,

Apologies to Doug for getting a bit off-topic. We shall continue this elsewhere if necessary.
No problem, this is all relevant to the same overall concern. The various suggestions are quite helpful.

Indeed I sometimes use Qimage to uprez to a new file.

It's also so nice to hear of new batch file work - I have lately slightly ignored that tool (used to use it a lot). I also keep forgetting about Send To.

Best regards,

Doug

#### Cem_Usakligil

##### Well-known member
Hi, Cem,

No problem, this is all relevant to the same overall concern. The various suggestions are quite helpful.

Indeed I sometimes use Qimage to uprez to a new file.

It's also so nice to hear of new batch file work - I have lately slightly ignored that tool (used to use it a lot). I also keep forgetting about Send To.

Best regards,

Doug

I have done some very interesting testing today. Took an image of mine and I have resized it to 200% using LR3, QImage Studio (Hybrid SE method) and ImageMagick (Mitchell-Netravali filter). LR3 results were OK-ish but not as good as the other two. QI and IM were almost on par with each other, with a slight advantage to QI. Could we perhaps uprezz your 200mm shot (from the TC comparison) using this method and add deconvolution sharpening with FocusMagic as the last step? It could then show us whether the advantage of using the 2x TC to get to 400m is still there. Of course it will be, I am just musing...

Last edited:

#### Bart_van_der_Wolf

##### pro member
It's also so nice to hear of new batch file work - I have lately slightly ignored that tool (used to use it a lot). I also keep forgetting about Send To.

Hi Doug,

I hope you don't mind my adding another bit of batch file text that might be useful (it assumes ImageMagick is installed).
Code:
``[B][COLOR="Red"]For %I in (*.tif) Do [/COLOR][COLOR="SeaGreen"]convert [COLOR="Red"]%I[/COLOR] -filter lanczos -resize 100x100[/COLOR][COLOR="red"] %~nI[/COLOR][COLOR="seagreen"]_T.jpg[/COLOR][/B]``

The part in red is MS DOS code, the part in green is basic ImageMagick code. The ImageMagick part can be enhanced by wrapping it in a temporary gamma linearization step (as shown in Cem's code snippet) and the JPEG compression quality and chroma subsampling can also be forced to other settings than the reasonable defaults. One can also have it create the thumbnails in a subdirectory by adjusting the path for the output filename.

This processes all files with a TIF extension in a subdirectory, and produces (in this example) a JPEG thumbnail of 100 pixels maximum with a "-T"suffix to the filename.

And now to get back on topic, yes, it is better to have an optical magnification rather than a digital interpolation. However, it depends on how well the extender/converter plays together with the lens in question.

In the past I have used Kenko converters because they gave better results in the image corners than the Canon exenders, but that was with specific lenses. With my some of my current telelenses they would produce (or rather introduce(!)) horrible Chromatic Aberrations that were virtually impossible to correct and thus destroyed the overall look of the image, even though the center indeed had gained optical resolution. Therefore I prefered to interpolate, and sold the Kenkos.

I've tried the new EF 70-200mm f/4L IS with the EF 2x II Extender, and was pleasantly surprised with how well it performed on my 24x36mm full frame sensor. The corners, although not as good as center frame, had an adjustable type of lateral Chromatic Aberration that is tolerable.

So this specific combination does produce a workable solution, that is if one uses it on a camera that allows to still Auto Focus at f/8. Working from a tripod is often also mandatory with such a widest aperture solution. An EF 1.4x would produce a more workable solution for different camera models.

The experience has made me careful not to generalize the usefulness of one solution over another to easily. Apparently, it depends on the particular combination.

Nevertheless, it won't hurt to have a quality interpolation solution in one's toolchest. Qimage is a formidable solution for output, and its Hybrid SE interpolation does an excellent job with almost artifact free upsampling for display as well (while avoiding jaggies on high contrast slanted edges). ImageMagick has a price that cannot be beaten, but does require some moderately advanced computer skills. The output results are of what I'd call industry proven quality.

And then there is a dedicated software called Photozoom Pro by Benvista, which can interpolate files up to a million pixels per dimension, while converting sharp edges to a vector format. I do recommend using it's spline related interpolation with restraint though, because it can produce a mental disconnect between edge sharpness and surface detail / material structure that looks artificial.

It's nice to have options to choose from.

Cheers,
Bart

#### Doug Kerr

##### Well-known member
Hi, Cem,

Could we perhaps uprezz your 200mm shot (from the TC comparison) using this method and add deconvolution sharpening with FocusMagic as the last step? It could then show us whether the advantage of using the 2x TC to get to 400m is still there. Of course it will be, I am just musing...
Let me try that as soon as I can. I didn't have Focus Magic until about ten minutes ago, so it will take me a little while to learn how to run it!

Best regards,

Doug

#### Doug Kerr

##### Well-known member
Hi, Cem,

No thanks needed. I have been educated by Bart so we should thank him, lol.

Try the following and you will be amazed. Install the ImageMagick. Create a new batch file (with the extension .bat) on your desktop. Edit it and add the following text:
Code:
``convert %1 -depth 16 -filter Mitchell -gamma 0.4545455 -resize 200%% -gamma 2.2 "%~n1_IM_200_pct.tif"``
How does the O/S know which convert.exe/convert.com to call? There are often several executable by that name in place (including a standard Windows utility for converting a volume from one file system to another - very dangerous).

Best regards,

Doug

#### Bart_van_der_Wolf

##### pro member
Hi, Cem,

Let me try that as soon as I can. I didn't have Focus Magic until about ten minutes ago, so it will take me a little while to learn how to run it!

Hi Doug,

Forget the standalone application which only processes JPEGs, just try the Photoshop plug-in. Active support (or rather the lack thereof) for FocusMagic doesn't bode well for the (64-bit Windows OS) future, so use your 10 trial conversions with care. You can try as much as you like as long as you cancel before converting a full image.

I usually recommend to start with a large percentage, say 300%, and gradually increase the radius from 0 to whatever works best. There comes a point where increasing resolution flips to 'fatter' contours, then back off to a lower radius, and drop the percentage to taste.

A more future proof (64-bit Windows and Mac compatible) deconvolution sharpener is offered as a plug-in by Topaz Labs, but they need to do a bit of work on it to become better than the current version 1, or as good as FocusMagic (only 32-bit).

Cheers,
Bart

#### Bart_van_der_Wolf

##### pro member
Hi, Cem,

How does the O/S know which convert.exe/convert.com to call? There are often several executable by that name in place (including a standard Windows utility for converting a volume from one file system to another - very dangerous).

Hi Doug,

The setup file adjusts you system's Path statement by adding the ImageMagick program directory to the search string. If you get into a conflict with another convert application or command, then you can prefix the batch version with the full path to it.

Cheers,
Bart

#### Doug Kerr

##### Well-known member
Hi, Bart,

Hi Doug,

The setup file adjusts you system's Path statement by adding the ImageMagick program directory to the search string.

Sure.

If you get into a conflict with another convert application or command, then you can prefix the batch version with the full path to it.

Sure.

Thanks.

Best regards,

Doug

#### Asher Kelman

##### OPF Owner/Editor-in-Chief
Outstanding work, Doug, Cem and Bart!

This deserves to be stuck!

Asher