Band Limit and Appropriate
Sampling in Microscopy
I. INTRODUCTION
A. What Is "Sampling?"
In microscope images obtained with the help of
electronic devices (e.g., CCD camera) or in a scanning
microscope system, data are usually obtained
(sampled) at equidistant coordinates in object space.
The distance between these measurement positions is
usually denoted as the "sampling distance." Current
technology samples on orthogonal coordinates (rectilinear
sampling) with appropriate sampling distances
along the inplane coordinates
X,
Y and the axial coordinate
Z (for threedimensional imaging).
For a deeper understanding of how to choose these
distances correctly, it is useful to assume the intensity
distribution near the detector (translated back to
sample coordinates) to be a continuous function
(Fig. 1). The equidistant sampling coordinates, at
which the values of the image function are recorded,
are depicted as dashed lines in Fig. 1. The fact that
pixels have a discrete physical size is not considered
here to keep the discussion simple. This pixelsize
effect is described by a convolution and can be considered
as a multiplicative modification of the optical
transfer function (Sheppard
et al., 1995). In the considerations
given in this article, noise resulting from
photons and/or the detector system is neglected.

FIGURE 1 Diagram of a microscope image as a continuous function
over space (here indicated as the X coordinate). Dashed lines
indicate the points at which the function is sampled (its value is
recorded). 
B. Spatial Frequencies
Any physical function (e.g., as shown in Fig. 1) can
be decomposed into a sum of sine waves, each with a
different frequency, amplitude, and zero position (socalled
phase). To exactly represent a function, this sum
generally contains an infinite amount of components.
Such decomposition is useful especially in optics, as
images of (linear) optical systems have the interesting
property of only transmitting sine waves of the sample
up to a fixed limiting frequency. A periodic fluorescence
distribution with a count of maxima per µm
above this limit frequency will appear as a uniform
constant in the image. Such systems are often called
"bandlimited" systems, as only a compact, limited
band of frequencies (from zero to the maximal transmittable
frequency) is transferred. How well a defined
frequency is transferred is depicted in the "optical
transfer function (OTF)" as outlined in Fig. 2 for inplane
imaging in a fluorescence widefield microscope.
The "spatial frequency" given on the X axis of this
graph denotes the number of maxima per meter for
each sine wave. The sum of all sine waves yields the
perfect image. The transfer strength of the OTF (Y axis)
denotes how well a sample consisting of only one
spatial frequency (one such sine wave) would be transferred
to the detector.

FIGURE 2 The approximate shape of the inplane optical transfer
function of a fluorescence widefield microscope. For spatial
frequencies above the limiting frequency (k_{xy,max}), no information
can be transferred from the object to the microscope image. 
C. Band Limit of Optical Systems
1. Fluorescence Microscopy
When considering the sampling properties of
microscopic imaging, a physical model (Young, 1985)
of the microscope system has to be constructed. Due
to the laws of physics, this OTF is zero beyond a welldefined
limiting frequency (see Sheppard
et al., 1994). In the case of widefield fluorescence
microscopy, the limiting frequency is as follows:
k_{xy,max} = 
2NA 
, 
λ_{em} 
with
k_{xy,max} denoting the maximal inplane spatial frequency, λ
_{em} the shortest detected vacuum wavelength of the fluorescence emission light, and
NA =
nsin(α)
the numerical aperture defined by the refractive index
n of the medium immersing the sample and the objective
detecting light at an aperture halfangle of α.
Similarly, a limiting maximal transmittable axial
spatial frequency
k_{z,max} can be given for sine waves
with
Z components as:
k_{z,max} = 
NA 

[1  cos(α)] 
. 
λ_{em} 

sin(α) 
In confocal fluorescence microscopy, the illumination
is confined to a diffractionlimited spot. At very
small pinhole size, detection is also diffraction limited.
A fluorophore needs to be excited and detected by the
system, leading to a multiplicative process of excitation
and detection. As a result, the aforementioned frequency limits have to be extended for the case of
confocal fluorescence microscopy.
It should be noted that for larger pinhole sizes (>0.3
Airy discs, definition see Section III,B), the inplane
transfer function reaches values very close to zero
above the spatial frequency stated by the widefield
limit. This explains why the inplane resolution of
confocal images achievable in practice is rarely substantially
superior to widefield images.
2. Aliasing
Because the images can be decomposed into the sine
waves, it is useful to consider how an individual sine
wave is sampled by the equidistant measurements.
Figure 3 shows such a sine wave (solid black line)
along with its values marked at equidistant sampling
positions. Does such a measurement determine the
frequency of the measured sine wave uniquely? As
evident from Fig. 3 (dotted line), this is not the case, as
multiple sine waves (these are called aliases) could
possibly give the same measured values. However, if
it is known that a maximal transmittable frequency
exists, frequencies of sine waves in the image beyond
that limit can be excluded as possible sources for this
measurement. To guarantee a unique interpretation of
a given measurement, all that needs to be done is to
ensure all aliasing frequencies lie beyond the a priori
known transmittable frequency limit.

FIGURE 3 Aliasing. Two sine waves with different spatial frequencies
yield identical sampled values. A distinction based on the
measured values is only possible when the dashed alternative wave
can be excluded, as its frequency lies above the transfer limit. This
gives rise to the Nyquist theorem stating that the maximal possible
frequency in the image has to be sampled with more than two positions
per wavelength. 
In an acquired image, multiple frequencies are
measured simultaneously, and the aforementioned should be true for all spatial frequencies possibly
present in the image. The highest frequencies in the
image (just below the frequency limit, e.g.,
k_{img} <
k_{max} =
5µm
^{1}) turn out to be most critical, as their lowest frequency
alias (
k_{alias} =
k_{s} 
k_{img}, e.g., at
k_{s} = 11µm
^{1},
k_{alias} is
6µm
^{1}) appears at lower frequencies than aliases of
other lower frequency image contributions. If the
image is sampled at least at twice its highest contained
spatial frequency (
k_{s} > 2
k_{img}), aliasing can be avoided,
as interpretation of the sampled values is unique
because kalias is above the limiting frequency kma x. In
other words, the sampling frequency (
k_{s} =
1/D), as
defined by the sampling distance
D, has to exceed the
maximal transmittable frequency by a factor of two
(
k_{s} >
k_{Nq} = 2
k_{max}). This limiting sampling frequency
kNq is called the Nyquist frequency.
This means that all values of the continuous bandlimited
function in between the sampled values can
theoretically be predicted solely based on the knowledge
of the band limit of the system and the sampled
values (assuming an infinite amount of them and not
considering noise). Sampling denser than this requirement
stated earlier is termed oversampling; sampling
sparser than the limit is termed undersampling.
II. INSTRUMENTATION
Selection of the appropriate sampling distances
is crucial for almost all optical systems with computerized
image acquisition. In the field of optical
microscopy, one has to discriminate among
fluorescence,
transmission, and reflectionbased systems.
III. PROCEDURES
In the procedures suggested here, equations are
stated for different microscopy arrangements.
A. CCDBased Imaging Systems (in Plane)
Many imaging systems do not allow for any direct
control over the size of the detector bins. CCD camerabased
systems and digitizing micrographs on a
scanner fall in this category. To select the appropriate
sampling distance:
 Obtain the pixel pitch (P) of your CCD camera
from the manufacturer/supplier. The pixel pitch is the
distance in X and Y directions between successive pixels. Usually this is in the range of 5 to 20 µm. When
binning is used during image acquisition, the pixel
pitch has to be multiplied with the binning factor (e.g.,
with 2 × 2 binning: multiply the pixel pitch by 2). When
digitizing micrographs, a corresponding value is given
by either the pixeltopixel distance after scanning or
the resolution of the micrograph, whichever value is
bigger. When a micrograph is scanned, care has to
be taken to account for all magnification factors to
finally obtain a pixel size corresponding to the image
plane.
 Make a list of available objectives. Note magnification
(e.g., 40 or 63×) and numerical aperture (e.g., 0.9
or 1.3) for each of them. These values are usually
engraved on the side of the objectives. If any postmagnification
system (e.g., Zeiss Optovar) is available
on your microscope, multiply the objective magnification
with the appropriate factor. The obtained total
magnification is called M.
 Calculate the inplane sampling distance (D_{xy}) as
dictated by the detector pixel pitch in the object plane
for each objective, dividing the pixel pitch by the
magnification:
 Calculate the maximum sampling distance (dmax)
in the object plane from the numerical aperture of the
objective and the vacuum wavelength of light used for
imaging:
Wide  field fluorescence : 
d_{xy,max }= 
λ_{em} 
4NA_{obj} 
Transmission or phase microscopy: 
d_{xy,max }= 
λ 
2(NA_{obj }+ NA_{cond}) 
In widefield illumination microscopy systems, λ_{em} should be the shortest detected emission wavelength
(e.g., FITC λ_{em}= 500nm); in transmission microscopy,
the shortest transmitted wavelength should be used
(e.g., blue at λ = 400nm). Note that the numerical
aperture of the condensor (NA_{cond}) contributes equally
as the objective numerical aperture to the final resolution,
although the contrast may suffer in transmission
microscopy with a high condensor numerical aperture.
Also note that the value NA_{cond} is taken for a fully open
condensor aperture; a closed condensor aperture
enhances the contrast but reduces NA_{cond} and thus the
resolution.
 Ensure that D_{xy} < d_{xy,max} by selecting the appropriate
objective and/or postmagnification optics from
the list.
CCD Example
Let's say the pixel pitch of your camera is 7 µm and
you are using no binning. The question is whether
using a 100×, 1.3 NA oil (
n = 1.516) immersion objective
with the standard microscope tube lens (i.e.,
M = 100),
the sampling distance in the focal plane corresponding to
satisfies the Nyquist limit. The
maximum sampling distance for widefield fluorescence
microscopy (e.g., detecting FITC at 500nm shortest detection wavelength) is
d_{xy,max} = 
500 nm 
≈ 96.15 nm > 70 nm, 
4 · 1.3 
thus your system respects the
Nyquist limit. A 2 × 2 binning on the CCD, however,
would undersample the image.
B. Confocal Systems (in Plane)
Confocal microscopy usually allows for free control
over the sampling distance in the object plane by
selecting an appropriate magnification ("zoom") and
image size in pixels. A notable exception to this is a
Nipkovtype discscanning system employing a CCD
camera. For such systems, the maximum sampling distances
(
d_{xy,max} and
d_{z,max}) corresponding to a confocal
system have to be selected, although the protocol for
the CCD camera should be followed.
 Calculate the maximum sampling distance
(Sheppard, 1989; Wilson, 1990; Sheppard et al., 1994)
from the parameters of the objective (see CCD procedure
for definitions):
Confocal fluorescence (small pinhole): 
d_{xy,max} = 
λ_{eff} 
, λ_{eff} = 
1 
4NA_{obj} 
1 
+ 
1 


λ_{ex} 
λ_{em} 
Confocal fluorescence (large pinhole): 
d_{xy,max} = 
λ_{eff} 
4NA_{obj} 
Confocal twophoton fluorescence (no pinhole): 
d_{xy,max} = 
λ_{ex} 
, 
8NA_{obj} 
with λ_{ex} being the irradiating wavelength (usually in
the infrared).
Confocal reflection: 
d_{xy,max} = 
λ 
4NA_{obj} 
The strict theoretical limit even for large pinholes is
the value given for small pinhole size. However, because the lateral highfrequency content for larger
pinholes is negligible and usually lies well below the
noise level, the "widefield fluorescence" inplane
equation (as given earlier) can be applied, replacing
the emission wavelength with the excitation wavelength.
As a rule of thumb, consider pinhole sizes
below 0.5 Airy discs in the pinhole plane as being small
and sizes above 1.5 Airy discs as being large. In object
space coordinates one Airy disc diameter (the first
dark ring of a diffraction limited spot, assuming low
NA) is
To compare with actual pinhole sizes, the pinhole
has to be translated to the object space coordinates by
the appropriate demagnification factor, if the pinhole
size is not stated in Airy disc units in the microscope
operating software.
 For beamscanning or objectscanning confocal
systems, the inplane sampling distance (D_{xy}) is
usually stated somewhere on the screen. It can also
be calculated from the image size in the object plane
(S_{img}) and the number of pixels (N_{pix}) along X or Y:
D_{xy} = 
S_{img} 
. 
N_{pix} 1 
This distance should be selected to be below the dxy, max
calculated earlier.
Confocal Example
With the microscope parameters as stated in the
widefield example in Section III,A, confocal microscope
illuminating at 488 nm
with a small detection pinhole setting, would allow a
maximum inplane sampling distance of
d_{xy,max }≈ 
247 nm 
= 47.5 nm 
4·1.3 
and the pixeltopixel spacing
D_{xy} should be adjusted
to a smaller value. For large pinholes, the following
sampling distance should be acceptable:
d_{xy,max }= 
488 nm 
≈ 93.8 nm. 
4·1.3 
C. Focus Series
For the acquisition of focus series (
Z stacks), an
additional sampling limit along the axial direction also
needs to be obeyed by choosing an appropriate distance
between neighboring image planes. To calculate
the corresponding maximum planetoplane distance
(
d_{z,max}), the aperture halfangle of the objective (α
_{obj}) has
to be known. Because this value is often not stated, it
has to be calculated from the numerical aperture and
the refractive index (
n) of the immersion medium:
 Calculate the aperture halfangle of the objective as
Approximate values for the refractive index are
stated in Table I.
 Once the aperture halfangle is known, the
maximum planetoplane distance is calculated as
Widefield fluorescence: 
d_{z,max} = 
λ_{em} 

sin(α_{obj}) 
2NA_{obj} 
(1  cos(α_{obj})) 
Transmission or phase microscopy: 
d_{z,max} = 
λ 

sin(α_{max}) 
2NA_{max} 
(1  cos(α_{max})) 
Confocal fluorescence: 
d_{z,max} = 
λ_{eff} 

sin(α_{obj}) 
, λ_{eff} = 
1 
2NA_{obj} 
(1  cos(α_{obj})) 
1 
+ 
1 


λ_{ex} 
λ_{em} 
Confocal twophoton fluorescence: 
d_{z,max} = 
λ_{ex} 

sin(α_{obj}) 
4NA_{obj} 
(1  cos(α_{obj}))' 
with λ_{ex} being the irradiating wavelength (usually
in the infrared).
Reflection confocal: 
d_{z,max} = 
λ 
4n 
For transmission or phase microscopy, NA_{max} and
α_{max} denote the corresponding values of the greater
of the condensor and objective numerical aperture.
 Ensure that the distance between successive image
planes (Dz) in object coordinates is belowd_{z,max}.
Focus Series Examples
For the parameters of the widefield fluorescence
example just given, the appropriate spacing between
successive image planes should be chosen to be below
d_{z,max }≈ 
500 nm 

sin(59°) 
≈ 339.6 nm 
2·1.3 
(1 cos(59°)) 
with the aperture half angle being estimated from the
NA (1.3) and the refraction index of oil (
n  1.516) :
Accordingly, the maximum spacing for confocal fluorescence
microscopy (parameters as given in Section
III,B) is calculated as
d_{z,max }≈ 
247nm 

sin(59°) 
≈ 167.8 nm 
2·1.3 
(1 cos(59°)) 
IV. PITFALLS: CONTRAST AND
SAMPLING
Suppose the object and its image consist of a sine
wave with a single spatial frequency (e.g., 200nm
distance between two maxima). The sampling limit
requires sampling at more than the double frequency
(i.e.,
D_{xy} < 100nm). If this image is sampled too close
to the limiting frequency, there may be a problem.
Depending on the exact position of the object, one may
be fortunate enough to sample a maximum, then a
minimum, a maximum again, and so on or, if unlucky,
always sample half the maximum (see example in Fig.
4). In the latter case, the result would be indistinguishable
from an object with zero frequency. Even when
sampling at distances slightly below the required
minimum distance, a very low contrast can result for
images of small size. Therefore, one should oversample,
such that at least one full period of the resulting envelope
amplitude modulation is captured. In other words,
with
M pixels along a spatial direction, the stated
maximum sampling distance should be lowered by
multiplication with the factor
M/(
M + 1). For common
image sizes (e.g., 512 x 512), this factor is negligible, but
it can be substantial when acquiring only few Z sections
(in which case M is the number of sections).

FIGURE 4 For images containing only a few measured values,
the measured contrast can be dependent on the exact phase of the
sine wave to perform measurements even when the Nyquist limit
frequency is obeyed (compare the contrast of closed and open
circles). To ensure that at least one full variation of contrast is
recorded, oversampling is recommended, as it leads to a required
sampling frequency enhanced by a factor of (M + 1)/M with M measured sampled points. 
A. When to Use Undersampling
For some applications the sampling limit does not
need to be obeyed. If, for example, the task is to count
homogeneously filled cells with a fluorescent dye
using a 20x, 0.9 NA objective, it makes a lot of sense to
seriously undersample the data. For a cell to be identified
it might suffice to detect two adjacent bright
pixels. Thus the required sampling can, in some cases,
depend on the size of the object structure to image.
Undersampling (e.g., by binning) can reduce the
amount of acquired data (important in screening
applications), reduce the readout noise, dark current,
and sometimes enlarge the field of view. In the
mentioned application the introduced aliasing effects
should be tolerable.
In other applications (e.g., neuronal imaging of
dendrites) it may be very tempting to undersample
the data. However, thin structures (such as dendrites)
may occasionally be lost in some pixels because they
happen to lie between two sample points. Such
missing gaps can then render a computerbased analysis
difficult, as the structures appear ruptured and will
also cause serious problems when successive computerized
deconvolution is applied to data. In these cases
it can even be better to reduce the numerical aperture
of the objective than to seriously undersample at
high
NA.
B. When to Use Oversampling
For some gray valuebased image processing tasks,
oversampling is recommendable (Young, 1996, 1988; Verbeek, 1985; Verbeek and vanVliet, 1993); e.g., if the
aim is the precise determination of object positions,
one should oversample data by a factor of at least 1.5
(sample at a 1.5 times smaller pixel pitch as compared
to the limits given earlier). Simulations revealed
(Heintzmann, 1999) that determination of the center of
mass can result in a significant systematical error even
when sampled according to the stated sampling limits.
Furthermore, a smaller sampling distance determines
more precisely where each photon hits the detector,
thus leading to a slightly more precise estimate of the
particle position (Heintzmann, 1999). Oversampling
can also be useful when successive deconvolution of
data is planned (see below). Note that with oversampling,
the photon dose delivered to the sample can still
be kept constant. Acquired images may look inferior
at a first glance, but they still contain all the necessary
information. Such data always allow for successive
binning, resampling, and/or smoothing to enhance the
visual appearance.
C. Deconvolution: OutofBand
Reconstruction Possibilities
When the computerized deconvolution is applied to
data, it is often useful to oversample data during data
acquisition. The reason is that constrained deconvolution
is capable of "guessing" highfrequency components
in the object structure that have not been
acquired. This is enabled by the use of prior knowledge
about the object, e.g., its positivity or smoothness
(Sementilli
et al., 1993). In principle, acquired data can
be resampled, but this always involves loss of information
about the photon statistics, or even interpolation
errors can result. Although software might be able
to reconstruct on a denser grid than raw data, there
may be (depending on the algorithm) some resampling
involved, which in turn can skew the statistics of
the deconvolution procedure, thus leading to inferior
results. As a rule of thumb, a twofold oversampling
(two times smaller sampling distances as compared to
the given limits) should suffice even for advanced
deconvolution software.
Appendix: Derivation of CutOff Frequencies
To obtain the equations given in the text, an expression
for the cutoff spatial frequency was derived
and the Nyquist theorem applied, calculating the
maximum sampling distance as half the distance corresponding
to this cutoff spatial frequency. Note that
this derivation is valid for high NA vector theory.
Electric field components of a plane wave can be
described by a single point in Fourier space. Its distance to the origin is proportional to the inverse wavelength.
A lens forms its image by the constructive interference
of converging parallel beams. It can thus be
described by a "cap" (Gustafsson
et al. 1995) in Fourier
space (Fig. 5a).

FIGURE 5 Derivation of the region of support of the fluorescence
widefield optical transfer function (b) from an autocorrelation of
interfering plane waves (a), all depicted in Fourier space. 
For incoherent fluorescence widefield imaging, the
intensity in focus describing the point spread function
(PSF) is obtained as the square of the absolute magnitude
of the electric field distribution. Because the
optical transfer function (OTF) is the Fourier transformation
of the PSF, it can be obtained as an autocorrelation
of Fig. 5a, which is identical to the convolution
with itself mirrored at the origin in Fourier space. The
resulting region of support is depicted in Fig. 5b. The
corresponding distances are the reciprocal values,
which have to be halved, yielding the respective
maximum sampling distance in
XY and
Z directions.

FIGURE 6 Support of the optical transfer function
for a confocal
fluorescence microscope
obtained by autocorrelation of Fig. 5b. 
In the case of confocal microscopy, photons have to
be excited and detected. This leads to a multiplication
of the probabilities of excitation PSF (corresponding to
the OTF in Fig. 5b, but for λ
_{ex}) and the emission PSF
(corresponding to the OTF in Fig. 5b). In Fourier space,
this translates to a convolution of Fig. 5b as depicted in
Fig. 6, yielding the appropriate equations for the confocal
closed pinhole case. The size and shape of the
pinhole is described by a multiplicative modification
of the detection OTF with the Fourier transformed
pinhole. The twophoton (no detector pinhole) derivation
follows in a similar fashion, as its PSF is the square
of the excitation PSF, which can be described by autocorrelation
of Fig. 5b for λ
_{ex}.
When dealing with widefield transmission, the
situation is different. The image can no longer strictly
be described as a convolution of the object with a point
spread function. However, for incoherent imaging in
transmission, such approximation still holds. Nevertheless
it is preferable to think of this as a scattering
problem. Neglecting multiple scattering, one can look
for the scattering object vectors in Fourier space, which should be imaged by the system. Scattering theory
states that the incoming wave vector plus the object
vector yields the outgoing wave vector. Incoming and
outgoing vectors are restricted by the Ewald sphere
and the numerical aperture of the condensor and
objective, respectively. As depicted in Fig. 7a the range
of possible scattering vectors depends on both, the
incoming light as defined by the numerical aperture of
the condensor and the outgoing light as restricted by
the objective.
The equation for confocal reflection microscopy is
obtained by considering the possible scattering vectors
when illuminating and detecting through the same
objective (Fig. 7b).

FIGURE 1 Shape of object frequencies (scattering vectors) that
can possibly be imaged with an incoherent
widefield transmission
microscope (a) and a confocal reflection microscope (b). 
Acknowledgment
B. Rieger, S. H6ppner, E. Lemke, I. T. Young, and
T. M. Jovin are thanked for their help in revising this
manuscript and C. J. R. Sheppard for fruitful discussions
on sampling.
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