1
ξ 6.8 Cutoff = λ × (f/#)
227lp/mm 12.4m feature size
Group 5, Element 3
145lp/mm
67lp/mm
Spatial Frequency in Cycles per mm
www.edmundoptics.eu/imaging 41
resource guide telecentric liquid lens/specialty objectives cameras illumination targets
fixed focal length filters/accessories microscopy /
The example on the previous page also assumes that the exact camera/
sensor has not yet been chosen, therefore making the optics the
limiting component in the imaging system. If a camera sensor had been
chosen prior to the lens, the lens would need to be able to resolve the
pixel size of the sensor in use.
Continuing from the example on the previous page, if a camera had
been chosen with the Sony IMX250 sensor with 3,45 μm pixels, using
Equation 6.6 the image-space resolution would be found as 144,9lp/mm.
Section 6.4: Sensors and Lenses
The MTF curve shows that the lens achieves > 40% contrast, which is
more than enough for most applications. However, using the same calculation
as in Equation 6.7 to scale into object space, 3,45 μm pixels only
corresponds to a 45 μm object, meaning the sensor would be the limiting
component in the system, as the lens is capable of 26 μm object-space
resolution.
All of these considerations must be made when determining the
proper lens for a given application to fi nd the optimal solution to a
machine vision problem.
Imaging at the Nyquist Frequency
It can be tempting to image at what is called the Nyquist frequency, which
is defi ned in Equation 6.5, from Section 6.3. However, imaging at the Nyquist
limit is cautioned against because doing so implies that the observed feature
falls on exactly one pixel. Were the imaging system to shift by half a
pixel, the object of interest would fall between two pixels, and would blur
out completely. For this reason, imaging at the Nyquist frequency is not
recommended. Assuming no sub-pixel interpolation is being used, imaging
at half of the Nyquist frequency is generally recommended, and allows
the feature of interest to always take up at least two pixels.
Another assumption that is often improperly made is that a lens is
not appropriate for use with a particular camera unless it has substantial
(>20%) contrast at the Nyquist frequency of the sensor. This is not
the case. As previously mentioned, imaging at the Nyquist limit is illadvised,
and can create several problems. The entire system needs to be
assessed to determine whether a lens is appropriate for a given camera
sensor or not, and this is often dependent on the application. The following
section describes what happens in an imaging system when they are
used at or near the Nyquist frequency, and the consequences on overall
system resolution.
Understanding the interplay between camera sensors and imaging
lenses is a vital part of designing and implementing a machine vision
system. The optimization of this relationship is often overlooked, and
the impact that it can have on the overall resolution of the system is
large. An improperly paired camera/lens combination could lead to
wasted money on the imaging system. Unfortunately, determining
which lens and camera to use in any application is not always an easy
task: more camera sensors (and as a direct result, more lenses) continue
to be designed and manufactured to take advantage of new manufacturing
capabilities and drive performance up. These new sensors present
a number of challenges for lenses to overcome and make the correct
camera to lens pairing less obvious.
The fi rst challenge is that pixels continue to get smaller. While
smaller pixels typically means higher system-level resolution, this is
not always the case after careful consideration of the optics used.
In a perfect world, with no diff raction or optical errors in a system,
resolution would be based simply upon the size of a pixel and the size
of the object that is being viewed (see Section 2.2: Resolution). To briefl y
summarize, as pixel size decreases, the resolution increases. This increase
occurs as smaller objects fi t onto smaller pixels and are able to
resolve the spacing between the objects, even as that spacing decreases.
This is an oversimplifi ed model of how a camera sensor detects objects,
not taking noise or other parameters into account.
Lenses also have resolution specifi cations, but the basics are not quite
as easy to understand as sensors since there is nothing quite as concrete
as a pixel. However, there are two factors that ultimately determine
the contrast reproduction (modulation transfer function, or MTF) of a
particular object feature onto a pixel when imaged through a lens: diffraction
and aberrational content. Diff raction will occur any time light
passes through an aperture, causing contrast reduction (more details in
Section 2.4: The Airy Disk and Diff raction Limit). Aberrations are errors
that occur in every imaging lens that either blur or misplace image information
depending on the type of aberration, as described in Section
3: Real World Performance on pages 20-28. With a fast lens (≤f/4), optical
aberrations are most often the cause for a system departing from “perfect
performance” as would be dictated by the diff raction limit; in most
cases, lenses simply do not function at their theoretical cutoff frequency
(ξCutoff ), as dictated by Equation 6.8.
To relate this equation back to a camera sensor, as the frequency of
pixels increases (pixel size decreases), contrast decreases - every lens
will always follow this trend. However, this does not account for the real
world hardware performance of a lens. How tightly a lens is toleranced
and manufactured will also have an impact on the aberrational content
of a lens and the real-world performance will diff er from the nominal,
as-designed performance. It can be diffi cult to approximate how a realworld
lens will perform based on nominal data, but tests in a lab can
help determine if a particular lens and camera sensor are compatible.
One way to understand how a lens will perform with a certain sensor
is to test its resolution with a USAF 1951 bar target. Bar targets
are better for determining lens/sensor compatibility than star targets,
as their features line up better with square (and rectangular) pixels.
113lp/mm, 24.8m feature size
Group 4, Element 3
Contrast 24.8%
Contrast 8.8%
33lp/mm
Contrast 55.3%
Contrast: 33.6%
72lp/mm
Contrast 48%
Contrast: 24.6%
TS 0.00mm
Diff. Limit
Sensor
a) ON Semiconductor
MT9P031
2.2μm
/ Nyquist Nyquist
Contrast
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 22.7 45.4 68.1 90.8 113.5 136.2 158.9 181.6 204.3 227
b) Sony
ICX655
3.45μm
c) ON Semiconductor
KAI-4021
7.4μm
Figure 6.8: A comparison of nominal lens performance vs. realworld
performance for a high-resolution 50 mm lens on the (a) ON
Semiconductor MT9P031 with 2,2 μm pixels, the (b) Sony IXC655
with 3,45 μm pixels, and the (c) ON Semiconductor KAI-4021 with 7.4
μm pixels. The red, purple, and dark green lines show the Nyquist limits
of the sensors, respectively. The yellow, light blue, and light green
lines show half of the Nyquist limits of the sensors, respectively.
/imaging