Usable Magnification
17th Mar 2017 00:14 UTCRonald J. Pellar Expert
Maximum Usable Magnification
When using a microscope objective or a separate reversed lens there is a maximum useful magnification that you can get. Pushing the magnification beyond this point will only increase blur size (comparable to pixilation). The maximum useful magnification is computed by dividing the size of the camera sensor pixel size by the resolved dimension of the lens.
The resolved dimension of the lens can be computed from the diffraction limited formula:
r = 1.22ƛ(f/no.) = 0.611ƛ/(N.A.) [Numerical Aperture, N.A., is 1/2(f no.)]
where r, the resolved dimension, and ƛ, the wavelength, are in microns.
EXAMPLE: For a 35 mm APS-C 12 megapixel sensor the pixel size is 5.65 microns in my Nikon D300. With a f/2.8 lens (N.A. = 0.18) and ƛ = 0.5 micron (green) the resolved dimension is 1.71 microns. This gives a maximum usable magnification of 3.32x. Larger magnifications than this will simply enlarge the blur circle and fuzz up the image. NOTE: Higher megapixel sensors reduce the useful magnification further, e.g., a 25 megapixel APS-C sensor will have a maximum useful magnification of about 2.2x. A F format sensor of the same number of megapixels will increase the useful magnification by a factor of 1.5, e.g., the 35mm SLR, 12 megapixel F format sensor has a usable magnification of 4.8x.
To achieve larger useful magnification, a second lens is required to increase the apparent size of the camera pixel (one of the reasons that a microscope has an eyepiece). One way to do this is to use a wide-angle lens on the camera along with the microscope objective (an infinity focus objective makes it easier to couple the microscope objective to the camera). The short focal length (wide-angle) lens will essentially magnify the pixel size in the camera to increase the maximum usable magnification. This will be especially useful for the high megapixel cameras, i.e., > 12 megapixels, where the camera pixel size is smaller.
An example of this problem is shown here:
where I used a Nikon D300 (12 megapixel APS-C sensor) with 7x objective on a Mikon trinocular microscope and the image appears slightly fuzzy due to pushing the magnification above the maximum 3.32x. Compare this image with this one:
where I used the same setup as above but with the 4x objective. Both images have the same pixel density for comparison purposes.
Visual Magnification
The maximum usable magnification, referred to above, is with reference to the camera sensor plane. For viewing purposes on some sort of display, the total usable magnification occurs when;
1) A pixel in the image being viewed contains information from one pixel in the camera sensor.
2) The person viewing the image is at an optimum distance from the image, i.e., the person viewing the image can just barely resolve a single pixel.
EXAMPLE: If I view an image in Photoshop at 100% on my HD monitor that is 20” wide, a single pixel is 257 microns in size and the actual magnification is 45.5x relative to the camera. If the camera was setup to provide an image at the maximum usable magnification as described above, the total magnification from the object to the monitor is 151x. If the viewer is further away than the optimal distance for viewing (violating the second criteria above), then the magnification can be increased, either optically or electronically, e.g., in Photoshop setting the image size greater then 100%. NOTE: increasing the magnification electronically beyond 100% can cause pixilation if the software does not use interpolation. With interpolation the image just gets fuzzier.
As the camera pixel size changes, e.g., switching from a 12 megapixel camera to a 25 megapixel camera with the same sensor format, e.g., APS-C,, choosing and objective magnification to maintain maximum usable magnification does not change the visual magnification! The maximum usable visual magnification will always be the viewed image pixel size divided by the objective lens resolved dimension, e.g., a 257 micron image pixel divided by 1.7 micron resolved dimension from the previous example gives a total magnification of 151x. In other words, all intervening factors like camera sensor size, secondary lenses, electronic magnification, etc. cannot improve the visual usable magnification. They can only make things worse not better.
Using Magnification Below Maximum
When the camera setup is such that a single camera pixel size does is greater than the lens resolved dimension, the image will be subsampled, i.e., there is information incident on the pixel that is averaged over the sensor pixel. This subsampling can cause aliasing error and/or moiré. Most digital SLR cameras having less than 36 megapixels use an antialiasing optical filter to alleviate this problem. When the camera pixel is equivalent to the resolved dimension of the lens, the need for an antialiasing filter goes away. This condition satisfies the Nyquist or Shannon condition from information theory and should be considered the optimal camera setup as this condition retains sharpness and limits aliasing errors.
Microscope Objective vs. Reversing Lens
The discussion and calculations used above assume that the lens being used to image the object (objective lens) is diffraction limited and that higher magnifications are being used than 1:1. At magnifications close to 1:1, the diffraction limit formula becomes;
r = 1.22ƛLo/D
where Lo is the distance from the center of the lens to the object being imaged. As the magnification becomes higher Lo approaches the focal length of the lens thereby approaching the diffraction limited formula. In general this object distance is greater then the lens focal length and the resolved dimension increases slightly, e.g., a maximum of a factor of 2 at a magnification 1:1. At less than 1:1 magnification the Lo term becomes Li, the image distance.
Another factor is the lens aberrations that can be greater than the diffraction limit at small f/no. The f/no that is optimal is where the aberration blur equals the diffraction blur and usually referred to as the “sweet spot”. This sweet spot is usually around 2x the minimum f/no of the lens, e.g., a f/2.8 lens will usually have its sweet spot at f/5.6 to f/8. Some lenses are notorious for the excellance of their resolving power even at maximum aperature, e.g., the older Micro-Nikkor 55mm macro lens. The lenses are designed to correct the aberrations at the focal plane where the image size is smaller than the object size. In the case of macrophotography, i.e., magnification greater than 1:1, it makes sense to reverse the lens and place the object in the near field.
Microscope objective lenses are specially designed for minimizing the resolution dimension at full aperature. Since the aperture is not adjustable, f/no does not have the same usefulness as for camera lenses and they are defined by their numerical aperture (defined above with respect to f/no). The higher the N. A. the better resolving power and much effort is put into achieving the maximum resolving power thereby increasing their price considerably. Adding more correction for color dispersion in the lens materials can raise the price even more, e.g., apochromats cost more than achromats.
Finite focus objective lenses are usually designated by a magnification power to facilitate the calculation of overall magnification with an eyepiece. This magnification is the ration between the image distance and the object distance relative to that lens. An eyepiece magnifies the real image to be viewed. However, a camera sensor can be placed in the image plane directly which can limit the usable magnification. Another lens can be used to magnify the camera sensor pixels for use with higher magnification objective lenses.
Infinite focus objective lenses are designed specifically for use with another lens to focus the image. The second lens can be an eyepiece or a camera lens.
Conclusion
The micro/macro setup of lens and viewing image should be such that a pixel in the viewed image corresponded to the resolved dimension in the objective lens.
MUMs (Maximum Usable Magnification) the word!
Ron,
When you say, "you only need to consider the one lens focused on the specimen," do you mean the point at which it in focus? I use the 105 focused on it's closest setting. I then fine-tune focusing with Adorama Focusing Rail moving the assembly closer or further to achieve my starting point for the stack.
What does, "with specimen at the focal plane position." mean? Does it mean the point at which it is in focus?
The pixel dimensions of the sensor on the Rebel XSI is 4272 by 2848. I am trying to process
the final image to those dimensions. I have started using PS CC2017 and don't have it under control
as yet. It changes things and doesn't tell me. I have gone back and checked my procedure for the
image I shared. I found one thing it pulled on me. I am processing it again. This time the number
of pixels on the sensor are equal to the number in the image.
What if I were to set my 7D to capture at those dimensions?
Why is a bellows so much better than extension tubes? I have two sets each equaling 68mm. I can't
imagine a bellows, at least those I have seen, extending the lens further than that. Is it because
it is infinitely variable between the extremes. I don't see bellows for sale listing the amount
of extension available. I seem to understand that your suggestion is for me to simplify my setup.
I have made images using only extension tubes and the 105mm. I have tried using two sets of
extension tubes (136mm). I don't think I was recording the FOV and X factor on the sensor back than. The images attached was created using a sigma 105mm mounted on 136mm of extension, (no filter and no 1.4X tele converter). The series contains 45 captures and was shot on an 18mp 7D.
I have looked into microscope objectives. In fact I own a microscope from which I could steal
objectives. I think that I would need objectives with an aperture adjustment built in unless
I mounted it on my 105mm.
My sensor measures 23,500 microns on the long side. I don't know how to relate this to the wave
length. Don't all colors have different wave lengths? Do you average the wave lengths of the
colors RGB?
FYI, the other attached images is supposed to be the new and improved version. Can you see a difference? I think I can.
Regards,
Larry Eicher
When you say, "you only need to consider the one lens focused on the specimen," do you mean the point at which it in focus? I use the 105 focused on it's closest setting. I then fine-tune focusing with Adorama Focusing Rail moving the assembly closer or further to achieve my starting point for the stack.
What does, "with specimen at the focal plane position." mean? Does it mean the point at which it is in focus?
The pixel dimensions of the sensor on the Rebel XSI is 4272 by 2848. I am trying to process
the final image to those dimensions. I have started using PS CC2017 and don't have it under control
as yet. It changes things and doesn't tell me. I have gone back and checked my procedure for the
image I shared. I found one thing it pulled on me. I am processing it again. This time the number
of pixels on the sensor are equal to the number in the image.
What if I were to set my 7D to capture at those dimensions?
Why is a bellows so much better than extension tubes? I have two sets each equaling 68mm. I can't
imagine a bellows, at least those I have seen, extending the lens further than that. Is it because
it is infinitely variable between the extremes. I don't see bellows for sale listing the amount
of extension available. I seem to understand that your suggestion is for me to simplify my setup.
I have made images using only extension tubes and the 105mm. I have tried using two sets of
extension tubes (136mm). I don't think I was recording the FOV and X factor on the sensor back than. The images attached was created using a sigma 105mm mounted on 136mm of extension, (no filter and no 1.4X tele converter). The series contains 45 captures and was shot on an 18mp 7D.
I have looked into microscope objectives. In fact I own a microscope from which I could steal
objectives. I think that I would need objectives with an aperture adjustment built in unless
I mounted it on my 105mm.
My sensor measures 23,500 microns on the long side. I don't know how to relate this to the wave
length. Don't all colors have different wave lengths? Do you average the wave lengths of the
colors RGB?
FYI, the other attached images is supposed to be the new and improved version. Can you see a difference? I think I can.
Regards,
Larry Eicher
