Download Quantitative evaluation of light microscopes based on image

Bioimaging 6 (1998) 138–149. Printed in the UK
PII: S0966-9051(98)95425-9
TECHNICAL NOTE
Quantitative evaluation of light
microscopes based on image
processing techniques
L R van den Doel†, A D Klein, S L Ellenberger, H Netten,
F R Boddeke, L J van Vliet and I T Young
Pattern Recognition Group, Faculty of Applied Sciences, Lorentzweg 1,
Delft University of Technology, NL-2628 CJ Delft, The Netherlands
Submitted 5 June 1998, accepted 25 June 1998
Abstract. In this note we will present methods based on image processing techniques
to evaluate the performance of light microscopes. These procedures are applied to three
different ‘high-end’ light microscopes. Tests are carried out to measure the
homogeneity of the illumination system. From these tests it follows that Köhler
illuminated images can have an exceedingly high amount of shading. Another result
found from the illumination calibration is that traditional lamp housings are not
designed to make fine-tuning easy. Next, the (automated) stage is considered. Several
tests are performed to measure the stage motion in the xy-plane and in the axial
direction to address accuracy, precision, and hysteresis effects. Finally, the entire
electro-optical system is characterized by measuring the optical transfer function (OTF)
at wavelengths 400 nm, 500 nm, 600 nm, and 700 nm. The results of these
experiments show that there is a consistent deviation from the theoretical OTF at
wavelengths around 400 nm. The final conclusion is that modern light microscopes
perform better than their five-to-ten-year-old predecessors.
Keywords: quantitative microscopy, calibration methods, digital image processing
1. Introduction
When using a (light) microscope as a quantitative
instrument, careful calibration and testing are essential
to achieve proper results. A complete calibration of an
advanced microscope system would include, besides the
calibration of the microscope system itself as presented in
this note, a calibration of the charge coupled device (CCD)
camera mounted on the microscope as well [1]. This study
concentrates on the illumination system, the stage, and the
electro-optical system of a microscope. For fluorescence
microscopy applications it is necessary that as much light
as possible is passed through the optical system and that
the field-of-view is evenly illuminated. We have developed
image processing tools to calibrate the illumination system
in terms of the signal-to-noise ratio, shading, and the
position of maximum intensity. These measurements use a
quadratic intensity profile as an approximation to the light
† E-mail address: [email protected]
c 1998 IOP Publishing Ltd
0966-9051/98/030138+12$19.50 intensity distribution in the field of view. A series of
experiments have been performed to measure the motion
characteristics of the (automated) stage. The calibration of
the stage is performed in terms of hysteresis (in both planar
and axial directions), the focus position as a function of
the stage position, and stability. The last two tests exploit
focus functions [2]. A complete calibration procedure of the
automated z-axis, including axial resolution, can be found
in [3]. Finally, the electro-optical system is characterized
for various objectives and for four test wavelengths in terms
of the optical transfer function (OTF).
All experiments described in this note are performed
on three different ‘high-end’ microscope systems. All three
systems will be described in section 2.1. In order to perform
all the experiments two different microscope test slides
were needed. These will be described in section 2.2. All
test procedures, including the results and the conclusions,
will be described in section 3.
Quantitative evaluation of light microscopes
2. Materials
2.1. Microscope systems
All calibration and test procedures are applied to three
different light microscopes: a Zeiss Axioskop (in our
laboratory since 1993), a Zeiss Axioplan (1997), and a
Leitz Aristoplan (in our laboratory since 1991). The Zeiss
Axioskop has a fully automated xyz-stage (Ludl Electronics
Products Ltd, Hawthorne, NY, USA). The Zeiss Axioplan
can be regarded as a completely digital microscope:
objective revolver, reflector turret, and condenser aperture,
for instance, can be adjusted by computer. The Zeiss
Axioplan used in this study, however, has only an
automated z-axis. This implies that a number of procedures
could neither be implemented nor automated for this
microscope. The Leitz Aristoplan has a fully automated
xyz-stage (Mac4000, Märzhäuzer, Wetzlar, Germany). A
KAF Photometrics Series 200 CCD camera can be mounted
on both Zeiss microscopes. The array of the CCD
element contains 1317 × 1035 pixels with a pixel size of
6.8 × 6.8 µm2 . This camera is Peltier-cooled to −42 ◦ C.
Due to this cooling and a slow readout rate (500 kHz),
this camera is photon limited. The characteristics of
this camera in terms of signal-to-noise ratio are excellent
[1]. This camera is connected via a NuBus interface
to a computer. A Sony XC-77RRCE CCD camera is
mounted on the Leitz Aristoplan. The CCD array contains
756 × 581 pixels with a pixel size of 11.0 × 11.0 µm2 , and
this camera is not cooled. The Sony camera is connected
to a computer via a Data Translation frame grabber. An
Apple Macintosh Quadra 840AV computer takes care of
the microscope control and the image acquisition. Both
the cameras and the microscopes can be controlled from
within the environment of an image processing package;
this package is SCIL Image (TNO Institute of Applied
Physics (TPD), Delft, The Netherlands, e-mail address:
[email protected]).
2.2. Test slides
Two different test slides were used in this study. For
the calibration of the epi-illumination an ALTUGLAS
127.34000 Fluo green (Atochem Departement ALTUGLAS,
4 cours Michelet - La Defense 10 Cedex 42-92091 Paris,
France) was used. This slide is 75 × 25 mm2 and about
3 mm thick and is a homogeneous distribution of fluorescent molecules embedded in a plastic matrix. As a result, a
uniform distribution of excitation light produces a uniform
distribution of fluorescent, emission light. The second test
slide, which will be referred to as the HCM slide (Opto-Line
Corporation, Andover, MA, USA) is used for various tests.
This slide has a number of features.
• A grid of fiducial marks. The HCM slide contains
four rows labeled from A to D, each row containing a series
of 12 fiducial marks labeled from A to L. The distance
AE
AF
AG
AH
AI
AJ
BE
BF
BG
BH
BI
BJ
CE
CF
CG
CH
CI
CJ
DE
DF
DG
DH
DI
DJ
Figure 1. Part of the HCM slide. The top row, A, contains
transparent density filters with squares and circles, the middle
row, B, contains opaque density filters with squares and circles,
and the bottom row, C, contains empty density filters.
Furthermore, the fiducial marks are shown labeled AE to DJ. At
CF and BG the fiducial marks are replaced by stage micrometers.
between two successive fiducial marks on a row is 4.5 mm.
The distance between two successive fiducial marks on
a column is 6.0 mm. The fiducial marks consist of two
perpendicular lines intersecting each other in their middle.
The lines have a length of 0.2 mm and are 1 µm thick. The
horizontal lines of the fiducial marks along a row lie on the
same line. The same holds for the vertical lines. Two stage
micrometers have replaced the fiducial marks at ‘coordinate
positions’ BG and CF. The fiducial marks are used to check
the alignment of the camera, and the slide with respect to
the stage. Furthermore, this grid of fiducial marks is used
to measure if the xy-plane of the stage is perpendicular to
the optical axis, and to test the motion characteristics of the
stepper motors.
• A row of optical density filters without pattern. These
filters are of size 3 × 3 mm2 , have transmission values in
the range 5%, 10%, . . . , 90%, and 100%, and are used for
the calibration of the trans-illumination.
• Two rows of density filters with patterns. These
filters are of size 3 × 3 mm2 , their upper half containing
squares with sizes 10 × 10 µm2 with an inter-square
distance of 5 µm and their lower half containing circles
with diameter 10 µm with an inter-circle distance of
5 µm. In row A the background pattern is transparent
and the squares and circles have transmission values in
the range 5%, 10%, . . . , 90%, and 100%. In row B the
background pattern is opaque and the squares and circles
have transmission values in the range 5%, 10%, . . . , 90%,
and 100%. The two-dimensional patterns are used to
measure the sampling density in pixels µm−1 .
A part of the HCM slide is shown in figure 1.
139
L R van den Doel et al
-400
-200
0
400
200
400
300
3400
200
200
3200
0
3000
-200
2800
100
100
2600
-400
-400
-200
0
(a)
200
400
-400
-200
0
200
-200
400
0
200
(c)
(b)
Figure 2. (a) The image of an empty field showing the shading, (b) the quadratic surface fit through the image, (c) the statistical
distribution of the error, which is the distribution of the difference between the empty image and the quadratic surface. (The histogram
is based on a reduced image (100 × 100 pixels).)
2.3. Transmission filters
The characterization of the electro-optical system by means
of measuring the OTF (λ) requires near-monochromatic
light. For this purpose four narrow-bandpass interference
filters are used, each with a bandwidth of 10 nm. The center
wavelengths of these filters are 400 nm, 500 nm, 600 nm,
and 700 nm (Andover Corporation, Salem, NH, USA).
3. Procedures
In this section the variety of test procedures that use
digital image processing techniques will be described and
the results of these procedures applied to the different
microscope systems will be shown.
3.1. Illumination calibration
For fluorescence (epi-illumination) as well as for bright
field (trans-illumination) microscopy it is essential that
the field of view is evenly illuminated. Furthermore, for
fluorescence microscopy it is necessary that as much light as
possible is passed through the optical system. The second
demand can be measured by computing the signal-to-noise
ratio (SNR) of the total electro-optical system. The SNR is
defined as [1]:
µ
SNR = 20 log
dB
(1)
σ
in which µ is the average brightness in the digital image
and σ is the standard deviation of the image brightnesses
defined by
σ2 =
140
N
M X
1 1 X
(I1 [m, n] − I2 [m, n])2 .
2 MN m=1 n=1
(2)
In formula (2) it is assumed that both images consist of
M × N pixels. Since these images are not yet corrected for
possible non-uniform shading effects, two digital images,
I1 [m, n] and I2 [m, n], are recorded and the pixel-to-pixel
difference is used to estimate the noise variance in the
images. Doing this, systematic spatial variations are
‘subtracted out’. The average value can be estimated either
by taking the average over the M × N pixels in the image
or by taking the maximum brightness in the image. The
former approach will be affected by the possible shading
and will, in general, lead to an underestimate of the average.
The latter approach will be more reliable, but occasionally
subject to outliers or defective camera pixels.
In practice, it is difficult to achieve uniform illumination
as is illustrated in figure 2(a). The distribution of the light
intensity, however, can be assumed to have a quadratic
form given by
Iqs (x, y) = β1 + β2 x + β3 y + β4 xy + β5 x 2 + β6 y 2 . (3)
The approach in the illumination calibration is to fit
this light distribution through an empty image or a socalled white image as shown in figure 2(a). This fit
is based on least-squares estimators. Given an image
I [m, n] it is possible to derive an analytical expression for
the parameters β1 , . . . , β6 . In order to find least-squares
estimators for the parameters β1 , . . . , β6 the following
expression must be minimized:
N
M X
X
(I [m, n] − Iqs [m, n])2 .
(4)
m=1 n=1
The parameters then follow by differentiating this
expression with respect to the parameters β1 , . . . , β6 and
setting the result equal to zero. This results in a set of six
equations, which look like (formula (4) differentiated with
Quantitative evaluation of light microscopes
Table 1. Results of the intensity check for microscopes and
modalities.
respect to β4 )
β1 S(m, n) + β2 S(m2 , n) + β3 S(m, n2 ) + β4 S(m2 , n2 )
N
M X
X
+β5 S(m3 , n) + β6 S(m, n3 ) =
mnI [m, n] (5)
Absorption
m=1 n=1
where
S(mp , nq ) =
M
X
N
X
mp
m=1
nq
(6)
n=1
is the product of two independent summations, for which
expressions can be found in terms of M and N . This
set of equations can be written in matrix notation β =
A(M, N )I 0 , with β = (β1 , . . . , β6 )T , A(M, N) is a 6 × 6
matrix containing elements of the form of (6) and is just a
function of the image size (M ×N pixels), and I 0 is a vector
of the form of the right-hand side of (5). The description
until now has been the standard procedure for quadratic
regression [4]. Before inverting the matrix A(M, N ),
however, it is illustrative to have a closer look at its
elements. Instead of summing over an asymmetric interval
from 1 to 2K +1 (2K +1 elements), one can also sum over a
symmetric interval from −K to K (2K + 1 elements). This
corresponds to a translation of the origin in the image from
the upper left corner to the center of the image. Notice the
following:
K
X
k s = (−K)s + (−K + 1)s + · · ·
k=−K
· · · + (K − 1)s + K s = 0
K
X
s = 1, 3
(7)
k s = (−K)s + (−K + 1)s + · · ·
k=−K
· · · + (K − 1)s + K s = 2
K
X
ks
s = 2, 4.
(8)
k=0
In (7) the terms disappear pairwise, whereas in (8) the terms
appear exactly twice. The consequence of summing over a
symmetric interval is that a number of elements of A(M, N )
become zero: each element with p or q odd. For p or q
even, expressions can be found [5]. The inverse matrix
A−1 (M, N ) now contains only ten non-zero elements. It is
now possible to compute the parameters β1 , . . . , β6 .
The result of such a fit is shown in figure 2(b).
Also shown in this figure is the histogram of the error
between the camera image and the fitted quadratic intensity
distribution.
Having fitted a quadratic intensity distribution, the
highest intensity (Imax ) can be found by solving for the
extremum of formula (3). The lowest intensity (Imin )
can be computed along the border of the fitted intensity
distribution. Furthermore, the amount of shading in the
image and the root-mean-square (RMS) residual error can
be calculated. The shading gives information about the
homogeneity of the illumination in the image and the RMS
is a measure for the quality of the fit. The amount of
Fluorescence
Microscope
CV
(%)
SNR
(dB)
Leitz Aristoplan
Zeiss Axioskop
Zeiss Axioplan
Leitz Aristoplan
Zeiss Axioskop
Zeiss Axioplan
0.76
0.64
0.73
1.80
1.14
0.74
42.0
43.9
42.7
34.9
38.8
42.7
shading in the recorded field of view can be calculated
from the highest and lowest intensities as follows:
shading =
Imax − Imin
× 100%.
Imax
(9)
In the ideal situation Imax is positioned in the center of the
image at coordinates (x = 0, y = 0). The position of Imax
can be computed by manipulating (3). The goal of uniform
illumination is to have as little shading as possible, i.e. Imin
is as close as possible to Imax .
The RMS is given as a percentage of the maximum
intensity in the fit and is defined as
PN
P
2 1/2
((1/MN ) M
m=1
n=1 (I [m, n] − Iqs [m, n]) )
RMS =
Imax
×100.
(10)
Note that noise (e.g. photon noise and/or electronic noise)
also has its effect on the RMS value. Using the positioning
controls on the lamp housing it should be possible to center
the maximum intensity and to improve the homogeneity
of the illumination by reducing the shading as defined in
formula (9).
The calibration of the illumination starts with an
intensity check for which the results are shown in table 1.
The microscope lamps are centered and focused by using
the Köhler procedures as described in the respective
manuals of the microscopes. In this study 40× objectives
are used. The standard deviations, σ , as defined in
formula (2) are used for the calculation of the coefficient of
variation (CV = σ/µ×100%) and are given as a percentage
of the respective maximum intensities, µ, in each image.
After the intensity check a uniformity check is
performed and some typical results are given in table 2.
Note that the shading percentages, as they are shown
here, should not be directly compared as they do not
represent the same-size field of view. Two choices are
available: one option is to standardize the size of the
field of view to compare the different microscopes. The
second option is to describe each microscope in terms
of its specific shading. In table 2 the latter option is
shown. In figure 3 an illustrative comparison is shown.
The intensity distributions (given as a percentage of the
maximum intensity) have about the same amount of shading
in the recorded images (1000 × 1000 pixels) as defined
141
L R van den Doel et al
Table 2. Results of the uniformity check.
Absorption
Fluorescence
Microscope
Image size
(µm2 )
Max. position
Shading
(%)
RMS
(%)
Leitz Aristoplan
Zeiss Axioskop
Zeiss Axioplan
Leitz Aristoplan
Zeiss Axioskop
Zeiss Axioplan
122 × 122
172 × 172
68 × 68
122 × 122
172 × 172
68 × 68
(−1.20; 0.03)
(−0.45; 4.66)
(−0.43; −0.30)
(5.11; −1.20)
(2.37; −3.03)
(3.49; −0.63)
4.52
17.48
16.31
6.85
27.70
16.67
1.96
1.64
1.71
2.05
1.53
1.69
-500
-500
0
0
500
100
100
500
100
90
90
90
500
m pos. (pixels)
m pos.
)
els
0
0
80 -500
500
(pix
)
els
(pix
-500
(pixels)
os.
os.
np
np
m pos.
-500
-50
0
100
50
100
75
90
75
0
50
x pos. (µm)
m)
x pos. (µ
)
(µm
)
(µm
50 -50
50
os.
os.
m)
0
50
100
yp
yp
x pos. (µ
0
80
500
0
-50
-50
(pixels)
-50
0
50
50
Figure 3. The top-left intensity distribution in an image with 1000 × 1000 pixels has about the same amount of shading as the top-right
intensity distribution. The left image corresponds to a field of view of 68 × 68 µm2 , whereas the right image corresponds to a field of
view of 172 × 172 µm2 . When the image size is converted to a standard field of view of 100 × 100 µm2 as shown in the bottom-left
and bottom-right distributions, it is clear that the left distribution shows much more shading than the right distribution. The center
images (top and bottom) show the value of the fitted surface along the line Iqs [m, n = 0].
in formula (9), but in a standardized field of view of
100 × 100 µm2 the left distribution has much more shading
than the right distribution.
The RMS values in table 2 show that the quadratic fit
to the shading is quite good. The value that is calculated
according to formula (10) can be regarded as the standard
deviation of the histogram shown in figure 2(c). The
standard deviation in this histogram, which is composed
of the error in the quadratic model plus the inherent noise
142
in the image, can be almost entirely attributed to the image
noise as it is comparable to the noise value found in the
SNR calculation.
As suggested above, measurements were also performed to follow the position of maximum intensity as the
lamp was moved. This was achieved by turning the lamp
positioning controls on the lamp housing and following the
position of Imax . A characteristic graph of these measurements is shown in figure 4.
60
y-position
y-position
Quantitative evaluation of light microscopes
50
max. shading:
40
60
50
40
min. shading:
-60
-50
-40
-30
-20
30
30
20
20
10
10
-10
(a)
10
20
x-position
-30
-20
-10
10
(b)
20
30
x-position
Figure 4. Effects of changing the lamp adjustment controls. The graphs show the movement of the position of the maximum intensity
in the xy-plane when either the lateral (a) or the vertical (b) position is manually changed. The gray values of the arrows indicate the
amount of shading.
3.2. Sampling density and component alignment
When measuring the sampling density in the x and y
directions it is important that the coordinate system of
the camera and the pattern on the slide are aligned.
Furthermore, measuring the stage characteristics requires
that the coordinate system of the camera and the coordinate
system of the stage are aligned. Three rotation angles can
be defined: first, the angle of the slide with respect to the
camera, denoted by αcam-sl , second, the rotation angle of
the stage with respect to the camera, αcam-st ; and finally,
the rotation angle of the slide with respect to the stage,
αst-sl . By definition,
αcam-st + αst-sl = αcam-sl .
(11)
Two rotation angles can be measured with respect to the
frame of the camera: αcam-sl and αcam-st . The first angle
can be measured in a single image of one of the fiducial
marks of the HCM slide as shown in figure 5. First, the
fiducial mark is skeletonized into a one-pixel thick object
with one branch point. This branch point and its eightconnected neighbors are removed, resulting in two pairs
of lines. Two straight lines are then fitted to the two
remaining pairs of lines of the skeleton using least-squares
estimators. Doing this the fiducial is completely defined
by the subpixel intersection point of the two lines and both
slopes. The slopes of the lines define the rotation angle of
the slide with respect to the camera. In order to measure
the rotation angle of the stage relative to the camera, the
movements of the stage need to be related to the movements
of the intersection point of the fiducial mark in a series of
images. The remaining rotation angle follows then from
formula (11). When the camera and the slide are aligned,
the sampling densities in the x and y directions can be
measured. For this purpose an area with circles or squares
of the HCM slide is used.
Figure 5. A 150 × 150 image of part of a fiducial mark. The
white lines are the result of the skeleton operation. The
computed intersection point is (75.14, 85.74), and the slopes of
the two lines are 0.000 34 (θ = 0.02◦ ) for the horizontal line and
−495.3 (θ = −89.88◦ ) for the vertical line. (The origin is in the
top left corner.)
First, after acquisition of an image with either squares
or circles, the objects connected to the border of the
image are removed. The coordinates of the centers of
gravity are computed for the remaining objects. From
the description of the HCM slide the distance between
two objects is known to be 15 µm in both the x and
y directions. From the list of coordinates the sampling
143
L R van den Doel et al
Table 3. Results of the sampling density measurements.
Microscope
Objective
Magnification
relay lens
Sampling density
(pixels µm−1 )
Nyquist frequency
(λ = 400 nm)
(pixels µm−1 )
Leitz Aristoplan
Zeiss Axioskop
Zeiss Axioskop
Zeiss Axioplan
40×/1.30/oil
40×/1.30/oil
40×/0.75/—
40×/0.75/—
1×
1×
2.5×
2.5×
4.09(±0.1%)
5.57(±0.02%)
14.11(±0.004%)
14.58(±0.007%)
13.0
13.0
7.5
7.5
density in both the x and y directions in pixels µm−1
can be computed multiple times in a single image. This
sequence is repeated a (chosen) number of times and each
time the stage is shifted a small random distance either
automatically or manually. The results are then averaged.
This method is fast, because it uses a very simple algorithm,
complete, because it measures the sampling density in both
directions per image, and accurate, because the sampling
density can be computed multiple times per image. Some
of the results of this measurement procedure are given
in table 3. The measurements were performed with two
different relay lenses and two different 40× objectives.
The sampling density values in both directions were
equal indicating ‘square pixels’. For a diffraction-limited
objective the maximum spatial frequency can be determined
and the minimum sampling density (frequency) according
to the Nyquist sampling theorem [6–8] is given by
fs > fNyquist =
4NA
.
λ
(12)
As can be seen in table 3, the combination oilimmersion objective, the 1× relay lens, and the camera
do not lead to a sufficient sampling density. Both of
the sampling frequencies are less than the critical Nyquist
frequency of 13.0 pixels µm−1 .
3.3. Stage characterization
3.3.1. Planar stage motion. One of the procedures for
the characterization of the stage is the measurement of the
planar xy motion of the stage. The basic procedure as
well as the results are described here. The measurements
were performed using a 40× objective. The goal of this
experiment is to measure the hysteresis of the stepper
motors. This is done by moving the stage back and forth
over an increasing distance 1x and measuring the error
between the defined distance and the measured distance.
The experiment starts with one of the fiducial marks at
the left-hand side of an image. It is known that the final
movement of the stage has been in the positive x direction.
In that image the coordinates of the intersection point of
the fiducial mark are determined. Then the stage is moved
over a distance 1x in the positive x direction such that
the fiducial mark remains in the camera field of view. The
144
Table 4. Planar hysteresis for the Leitz Aristoplan and the Zeiss
Axioskop.
Microscope
Hysteresis (blue)
(µm)
Hysteresis (red)
(µm)
Leitz Aristoplan
Zeiss Axioskop
1.13
0.39
1.27
0.35
subpixel crosspoint is again determined. Note that this is a
hysteresis-free movement in the positive direction. Now the
stage is moved in the negative x direction and once again
the crosspoint in the image is determined. A certain amount
of hysteresis (in the negative x direction) is expected
in this movement. The algorithm is also performed in
the negative x direction to determine the hysteresis in
the positive direction. The results of these experiments
are shown in figure 6. Since the Ludl stepper motors
(and the Märzhäuser stepper motors) are not perfectly
identical, these experiments should also be performed in
the y direction. The Zeiss Axioplan microscope is only
equipped with a motor in the axial direction, therefore
no hysteresis measurements in the planar directions could
be performed. In figure 6 the hysteresis can be seen as
the average distance between the blue and the red curve.
The trend indicated by the dashed lines is the result of a
positioning error. When the stage moves over a defined
distance of 1 µm, it actually moves a little less. This error
is cumulative and the slope is 0.5 µm in 60 µm or 0.8% in
the top graph of figure 6. Another feature that can be seen
in figure 6 is the periodic behavior of the error. The period
of 20 µm equals the length of one full motor revolution
of the stepper motor. Similar graphs were obtained for
the Leitz Aristoplan. The measured planar hysteresis for
both directions for the two microscope systems is given in
table 4. The hysteresis in the first column corresponds to
the blue curve in the top graph of figure 6, whereas the
hysteresis in the second column of table 4 corresponds to
the red curve in the bottom graph of figure 6.
3.3.2. Axial position. By moving the stage along every
fiducial mark on the HCM slide and measuring the in-focus
position of every mark, a z(x, y) surface can be found [3].
Figure 7 shows the results for the three microscope systems.
measured error in µm.
Quantitative evaluation of light microscopes
0.75
0.5
0.25
∆x
20
-0.25
60 ∆x in µm.
40
−∆x
-0.5
measured error in µm.
-0.75
0.75
0.5
0.25
∆x
20
-0.25
60 ∆x in µm.
40
−∆x
-0.5
-0.75
Figure 6. For the Zeiss Axioskop with the Ludl motorized stage, the two graphs show errors with and without hysteresis. In the top
graph the initial position is always approached in the positive x direction. In the bottom graph the initial position is approached in the
negative x direction. The red curve is the error made in the positive direction (bottom graph, with hysteresis) and the blue graph the
error made in the negative direction (top graph, with hysteresis). The dashed lines show the trends.
18 mm
CD
AB
EF
G
60
z
y
x
C
B
A
a) Leitz Aristoplan
CD
AB
EF
G
60
CD
EF
G
60
40
40
40
20
20
20
0
D
C
B
A
b) Zeiss Axioskop
0
D
C
60 µm
AB
27 mm
0
D
B
A
c) Zeiss Axioplan
Figure 7. Stage tilt (z) as a function of lateral position. z(x, y) surfaces found for the three microscope systems. For all three graphs
the z-range is 60 µm.
The best results were found for the Zeiss Axioskop, but, as
can be seen in figure 7(c), the z(x, y) surface of the Zeiss
Axioplan is almost perpendicular to the optical axis of the
system.
3.3.3. Axial hysteresis. To assess the hysteresis in the
z direction we measured a focus function [2] twice, once
moving the stage in the positive z direction and once in
the negative z direction. The results of this experiment
are shown in figure 8. Each graph shows the two focus
functions, F (z) against the z position, slightly shifted.
This shift is caused by the hysteresis in the motor in
the axial direction. For two of the microscopes the shift
can be clearly seen; the shift is negligible for the Zeiss
Axioplan. This stage apparently incorporates a mechanism
that corrects for axial hysteresis.
3.3.4. Stability of the z-axis. If the focus function
has been measured as a function of position, v = F (z),
then the axial position relative to the in-focus position
(z = 0) can be deduced from the z position as z =
F −1 (v). By acquiring images over a period of time without
intentionally moving the stage, the stability of the z-axis
can be determined in the presence of mechanical vibrations,
disturbances to the motor electronics, temperature changes,
and gravity.
145
L R van den Doel et al
1.6 µm
2.4 µm
1
1
0.5
0.5
0.5
F(z)
1
-5
-2.5
2.5
a) Leitz Aristoplan
5 -5
-2.5
2.5
5 -5
b) Zeiss Axioskop
-2.5
2.5
5
z-position
c) Zeiss Axioplan
F(z)
Figure 8. Two focus functions, one moving upwards and one moving downwards, were measured for each microscope system. The
distance between the in-focus positions is the axial hysteresis. The two focus functions almost completely overlap in (c).
0.5
0.4
0.3
0.2
0.1
2.5
5
7.5 10
z in µm
30
60
90
120
t in min.
Figure 9. Results for a stability measurement on the Zeiss Axioskop. The left-hand side shows the focus function F (z). The right-hand
side shows the focus value as a function of time. These focus values can be related to the distance from the in-focus position.
The Zeiss Axioskop gave reproducible results for this
experiment. Typical results are shown in figure 9. The
Leitz Aristoplan and the Zeiss Axioplan did not. Over
a period of 60 min the stage of the Zeiss Axioskop
dropped about 5.7 µm or 96 nm min−1 . To understand the
implications of this consider a microscope objective with a
depth of focus of 1 µm [9]. If a series of images is to be
acquired for dynamical studies, then the images will remain
in focus for up to 10 min.
3.4. Optical transfer function
The quality of images acquired through a properly adjusted
microscope is inevitably limited by the lenses. The entire
electro-optical system of a microscope can be characterized
by means of its OTF. For an ideal circularly-symmetric,
diffraction-limited objective the OTF is given by [10]

p
2

(2/π
)
arccos(f/f
)
−
(f/f
)
1
−
(f/f
)

c
c
c


OTF(f ) =
|f | ≤ fc




0
|f | > fc
(13)
146
where f is the spatial frequency expressed in cycles per
unit length, and fc is the cut-off frequency, representing
the maximum spatial frequency that can be passed through
the lens [7, 11, 12]. The cut-off frequency is given by
fc =
2N A
.
λ
(14)
There are a variety of techniques for measuring the OTF.
Some involve the use of ‘impulse-like’ objects, for example
microspheres, while others use bar patterns to determine a
contrast modulation transfer function (CMTF), which can
then be transformed into the OTF [13–15]. Our procedure
for measuring the OTF requires an image of a ‘knife-edge’
acquired with a known objective at a specific wavelength.
A ‘knife-edge’ is simply a dark–white transition as shown
in figure 10(a). It is essential that the image is sampled
at a sampling frequency that meets the Nyquist criterion
from formula (12). The first step in the procedure is to
correct for shading in the image. This can be done either
by the procedures described in section 3.1 or by a flatfield correction procedure. The latter procedure uses a
dark image Idark [m, n], which represents the dark current.
Furthermore, it uses a white image Iwhite [m, n], which is an
Quantitative evaluation of light microscopes
empty image with only the illumination. This white image
includes the dark image as well. The flat-field correction is
then based on the following equation:
If lat-f ield [m, n] = A
a)
b)
c)
Figure 10. (a) The ‘knife-edge’ image. The red full curve is the
intensity in grey values along a single line in this image. The
region between the red dashed lines contains the dark–white
transition. (b) A cubic spline interpolation algorithm with eight
interpolation points is applied to the dark–white transition in the
top image. The red curve shows the intensity along a row in the
interpolated image. (c) The derivative of the mid image is
computed and the resulting set of PSFs is aligned to produce the
line response.
I [m, n] − Idark [m, n]
Iwhite [m, n] − Idark [m, n]
(15)
where A is a constant to scale the image If lat-f ield . After
the flat-field correction a 32-pixel-wide region around the
edge is interpolated by a factor of eight using a cubic spline
interpolation algorithm [4] resulting in a 256-pixel-wide
image. The interpolated image is shown in figure 10(b).
The result is convolved with a 7-pixel-wide 1-D derivativeof-Gaussian kernel with σ = 1.5 to obtain a smoothed
estimate of the knife edge. This result can be regarded
as the line response of the electro-optical system. To
obtain an improved estimate of the PSF the maxima are
arranged line by line in the center column of the image,
as shown in figure 10(c). These lines are then averaged
columnwise to provide the 1-D PSF. The averaging over
N
√ lines improves the signal-to-noise ratio by a factor of
N . The Fourier transform of the PSF is the OTF. We have
measured the OTF for a collection of objectives for all three
microscope systems and for four different test wavelengths.
In figure 11 the OTFs are shown for three 40× objectives:
one with N A = 0.75, and two with N A = 1.30. The
sampling densities of the objectives with N A = 1.30, as
shown in table 3, are below the Nyquist frequency from
formula (12). However, the performance of these two
particular objectives is so bad that the undersampling of
the knife edge had no effect on the measurement of the
OTF.
Several aspects of these curves are striking. First,
the measured OTF cannot exceed the theoretical OTF, but
several OTFs appear to do so. This is an artifact of the
scaling of the curves. Each curve is scaled so that its
maximum value is 1 and thus some OTFs are increased
in amplitude. In an ideal microscope lens system there
is no energy loss (photons) between input and output.
In mathematical terms this translates to the condition
OTF(f = 0) = 1.
Second, the 40×/0.75 objective achieves almost ideal
performance at the test wavelengths λ = 500 nm,
600 nm, and 700 nm. At λ = 400 nm, however,
no objectives perform near the diffraction-limited OTF:
apparently, objectives are not corrected for wavelengths in
the deep blue (400 nm) part of the spectrum.
Finally, all of the OTFs have been plotted along the
horizontal axis in normalized units of N A. This permits
easy comparison of each OTF per wavelength against the
ideal behavior, described by equation (13).
We suspect that the bad performance of the 40×/1.30
objective is caused by prolonged use of this objective
for multi-wavelength studies [16, 17]. This has led to a
deterioration in the OTF performance of the lens.
147
1
OTF(f)
OTF(f)
L R van den Doel et al
Leitz Aristoplan
40x/1.30
0.5
1
Zeiss Axioskop
40x/1.30
λ=400nm
λ=500nm
λ=600nm
λ=700nm
0.5
theoretical OTF
1
2NA/λ
NA/λ
OTF(f)
OTF(f)
NA/λ
Zeiss Axioplan
40x/0.75
1
2NA/λ
Zeiss Axioskop
40x/0.75
0.5
0.5
NA/λ
NA/λ
2NA/λ
2NA/λ
Figure 11. OTFs for wavelengths λ = 400 nm, 500 nm, 600 nm, and 700 nm for three microscopes and three objectives.
4. Conclusions
The purpose of this study was on the one hand to provide
a systematic approach, a methodology, for the evaluation
of microscopes as quantitative imaging devices, and on the
other hand to compare three specific microscope systems.
The digital image processing techniques presented in this
note are straightforward to implement and can be applied
to other scanning devices such as flatbed scanners. With
respect to the comparison of microscopes, it is important
to understand that the measurements presented here are for
one microscope of each type, one lens of each specific type,
and one specific set of experiments. The Leitz Aristoplan
microscope has been in constant use in our laboratory since
1991, and the Zeiss Axioskop since 1993. These factors
should be taken into account when weighing issues such
as illumination, planar and axial stage motion, and spatial
resolution in terms of the OTF.
Acknowledgments
We would like to express our thanks to Dr Heinz Gundlach
of Carl-Zeiss in Jena, Germany for making the Zeiss
Axioplan microscope available to us and to Dr Françoise
Giroud of the Université Joseph Fourier, Grenoble for
providing us with the HCM test slide.
This work
was partially supported by the Netherlands Organization
for Scientific Research (NWO) grant 900-538-040, the
Foundation for Technical Sciences (STW) project 2987, the
Concerted Action for Automated Molecular Cytogenetics
148
Analysis (CA-AMCA), the Human Capital and Mobility
Project FISH, the Rolling Grants program of the Foundation
for Fundamental Research in Matter (FOM) and the
Delft Inter-Faculty Research Center Intelligent Molecular
Diagnostic Systems (DIOC-IMDS).
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