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Understanding Magnifi cation

magnifi cation from a stereo microscope for an image of the sample projected onto the camera sensor is:

max magnifi cation onto sensor = 2× (objective) · 16× (zoom) · 1× (C-mount) = 32×.

T e pixel ratio value corresponding to a total magnifi cation of 30,000:1 with the above magnifi cation of 32× onto the sensor is:

T e pixel size of the camera’s 5 MP sensor is 2.35 µm. Using the pixel ratio value of 938:1 and a 1-to-1 camera-to-monitor pixel correspondence, the monitor pixel size must be:

monitor pixel size = 938 · 0.00235 mm = 2.2 mm

T erefore, to achieve a total magnifi cation of 30,000:1 with a 5 MP image sensor containing 2,592 horizontal pixels, the monitor horizontal size (assuming 1,920 pixels (HD) of 2.2 mm) would be 4.2 meters (13.8 feet)! T is width is more than 2.2 times larger than that of the largest monitor (1.882 m, UHD/4k) in Table 2 .

But that is not the whole story. T e distance the viewer is from a large monitor is also important for assessing what the viewer will see. If the viewer is too close, it is likely that the large pixels will be annoyingly visible. On the other hand, if the viewer is too far away, the fi ner details may be too small to see.

Range of Useful Magnifi cation for Digital Microscopy

Now one must ask the question if this level of magnifi - cation, 30,000:1, is simply beyond the useful range, meaning it is empty magnifi cation. How do we determine a range of useful magnifi cation for digital microscopy when an image is observed on a monitor? First, it is important to understand the resolution or resolving power of the microscope system and the viewing distance from the monitor. Microscope system resolution . T e system resolution for a digital microscope (or stereo microscope with a digital camera) is infl uenced by three main factors: (1) Diff raction-limited light microscope resolution, using Rayleigh’s criterion and the numerical aperture [ 9 ]:

(6)

where M DIS is the total lateral magnifi cation (Equation 4) and the monitor pixel size is in mm. T e basis for the camera sensor and display monitor resolution limit is the Nyquist sampling theorem for digital signal processing (see Figure 2b ) [ 14 , 15 ]. T is theorem assumes that at least two pixels are needed to resolve one line pair. One can simply imagine a microscope image of a sample showing line pairs projected onto the camera sensor and then displayed on the monitor. If a single line in the image corresponds to a single line of pixels, as shown in Figure 2b , say for the monitor, then to determine the minimum line pair spacing on the sample resolvable by the monitor, one can just divide the size of 2 pixels by the total lateral display magnifi - cation ( M DIS ). If the same exercise is done now for the camera sensor, that is, a single line in the image corresponds to a single line of pixels, then dividing by the total projection magnifi cation from sample to sensor ( M TOT PROJ ) determines the minimum line pair spacing resolvable by the camera. To write these calculations out for more clarity:

where M TOT PROJ is the magnifi cation from the sample to the sensor (Equation 3), the “sensor bin mode” refers to the binning mode which is 1 for full frame (1 × 1) and 2 for 2 × 2 pixel binning, (see Figure 2a ), and “pixel size” refers to the sensor pixel size in μ m. (3) Display (monitor) resolution [ 9 ]:

where NA is the numerical aperture and λ is the wavelength of light in nm.

(2) Image sensor (camera sensor) resolution [ 9 ]:

Table 3 : Pixel size ratios (Equation 5) for HD monitors ( Table 2 ) and image sensors used in digital microscopes and cameras supplied by Leica Microsystems ( Table 1 ).

Image sensor type

85” 79” 75”

Monitor Size (inch) 65”

48” 32” Pixel Ratio 27”

24” 21.5”

2.5 MP 147:1 135:1 258:1 222:1 165:1 108:1 93:1 81:1 75:1 5.04 MP 209:1 192:1 366:1 315:1 234:1 153:1 132:1 115:1 106:1 9.98 MP 293:1 270:1 515:1 443:1 329:1 216:1 186:1 162:1 150:1 19.96 MP 204:1 188:1 358:1 308:1 229:1 150:1 129:1 113:1 104:1

24

T e reciprocal of the minimum resolvable line pair spacing gives the resolution limits shown above in Equations 7 and 8. Simply add a conversion (µm to mm) and pixel binning factor for the camera sensor to arrive at the fi nal form of Equation 7. For this article, as stated above, the scenario of a 1-to-1 correspondence is assumed between the pixels of the sensor and monitor. For this specifi c case, using Equation 4b and converting the monitor pixel size units from mm to μ m, it becomes clear that the resolution limit of the sensor and monitor are identical. Example calculations demonstrating this will be given in a later section of this article.

www.microscopy-today.com • 2018 July

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