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SUPPORTING MATERIAL
Microscope System
A 20 mW, 488 nm laser (Model PC14132, Picarro, Inc., Sunnyvale, CA) was used to excite
fluorescent labels within a sample (see Figure S1). The excitation laser intensity was modulated
with a series of neutral density filters and an electro-optic modulator (Model 350-150, Conoptics,
Inc., Danbury, CT). Excitation laser intensities were adjusted based on the specimen and were
routinely kept between 0.1 and 10 nanowatts. The relationship between excitation intensity and
emission photon count at the detector, which depends on properties of the specimen and
efficiency of the detection system, are the same as for a conventional laser-scanning confocal
microscope. The beam was scanned with two closed-loop galvanometers (Model 2615H,
Cambridge Technology, Lexington, MA) with driver circuits modified by the manufacturer to
provide feedback position signals having 10x greater resolution than originally specified. The
scanned beam was coupled with a Nikon TE-2000 microscope equipped with a 60x/1.45 NA
PLAN-APO-TIRF oil immersion objective (Nikon, Tokyo, Japan). A photo-multiplier tube
(PMT) (Model H7422P-40MOD, Hamamatsu, Hamamatsu City, Japan), in photon counting
mode, was used to collect photons emitted from the sample at a maximum rate of approximately
3x106 photons/sec. The PMT signal was transformed to 3 ns TTL pulses by a GHz amplifier
(Model HFAM-26dB-10, Becker & Hickl, Berlin, Germany) such that each pulse represented a
single photon event. GHz frequencies are required to sample the feedback position signals with
high precision. A custom circuit board was interfaced to a personal computer via a data
acquisition card (Model PCI-6115, National Instruments, Austin, TX). The board includes
software-programmable pre-amplification to maximize dynamic range and the position of each
galvanometer for every photon event pulse was asynchronously recorded using 12-bit analog-to-
digital converters. Custom acquisition software was written in LabVIEW (National Instruments,
Austin, TX).
Imaging Specimens
Fluorescent beads of different sizes (Dragon Green, 488/520 excitation/emission, Bangs
Laboratories, Inc., Fishers, IN) were immobilized on a cover slip and imaged. They were
selected for convenience because of their high photo-stability and convention as an imaging
calibration standard. Images of filamentous actin stained with Alexa Fluor 488 labeled phalloidin
were acquired from a fixed specimen (Model F36925, Molecular Probes, Eugene, OR). Actin
specimens were used to create images with more complex topography compared to the images of
individual beads.
Signal-to-Noise
The signal-to-noise ratio (SNR) is a commonly reported quantitative value indicative of image
quality (Sheppard et al., 2006; Cheng, 2006) that measures how well structural regions of an
image can be distinguished from background noise. Several definitions for a SNR exist in the
literature, in the present study the SNR is defined as the intensity of a region with structural
content divided by the standard deviation of the background intensity in an image.
The SNR was measured within manually selected regions of interest (ROI). The size and
location of each region were kept constant for each image type and for an entire image series.
SNRs were measured in COP and PEDS images over a range of photon counts. Images with
higher photon counts were constructed by compiling multiple sequential images with lower
photon counts. The average intensity was measured in a ROI placed near the center of a 2 µm
bead to minimize changes in intensity due to bead geometry (Figure S3C). The standard
deviation of the background intensity was measured within a second ROI positioned remotely
from the bead to minimize the influence of photons arising from the bead on the measurement.
Series of images were acquired at a rate of ten frames per second such that the photon count (i.e.
number of photons within the ROI near the center of the bead) in each image was less than 1,000
photons. The size of the ROI used to measure signal was selected so that it was large enough to
accurately sample the signal yet small enough to minimize changes in intensity due to the
spherical geometry of the bead. Sequential PEFs were acquired and compiled to form cumulative
PEFs. The latter were used to produce images with photon counts ranging from one-thousand to
one-million photons. PEDS, COP, and distributed COP images were formed from the same
cumulative PEF and intensity values were normalized (Figure S3B, S3C, S3D). The SNR was
measured as the mean signal, from a ROI near the center of the bead, divided by the standard
deviation of background intensity measured in a ROI remote from the bead. Results from five
images were averaged for each photon count.
Photon count was monitored throughout photon position acquisition and no significant photobleaching was observed. Focal drift was checked post-acquisition by defocusing the image then
refocusing; if, after refocusing, the photon count was equivalent to that which occurred at the end
of acquisition it was concluded that no significant focal drift occurred during acquisition. Images
were selected randomly post-measurement and visually inspected to ensure that lateral drift did
not influence the measurement. If drift or any other inconsistencies were identified the entire
image series was discarded.
The SNR from PEDS, COP, and distributed COP images increased with photon count, as
expected (Figure S3A). In conventional pixilated images, the SNR decreases as pixel size is
decreased; however, our results show that distributing each photon event with the PEDS image
formation technique not only maintains the SNR when super-resolution photon position
measurements are used, but actually increases this parameter. Similar increases in SNR could
only be observed in COP images after low-pass filtering, which suppressed noise by smoothing
pixel transitions, but also rendered a less accurate image. In contrast, PEDS retains high
resolution photon position information. At high photon counts, the amount of background noise
was small compared to the signal. Because the term for background is in the denominator of the
SNR equation small changes in values near zero cause large fluctuations in SNR. This
phenomenon was exacerbated by the noise suppression effect of PEDS and distributed COP
images and was the cause of the fluctuations seen in the results (Figure S3A).
Consistency of Image formation
Photon noise is the uncertainty associated with measuring a quantum event and produces
inconsistencies in an image from an invariant region of a specimen. The coefficient of variation
(CV) is the mathematical reciprocal of the SNR and is a measure of the effect of photon noise on
an image. Similar to SNR, this parameter would be affected in a negative manner by decreased
pixel size in conventional image formation. To test for negative consequences of PEDS, and to
test the ability of PEDS to consistently represent invariant regions within specimens, the CV was
measured within a region of a specimen with uniform fluorescence. Images of uniform
fluorescence were acquired at a rate of one frame per second from plastic fluorescent slides
(gratis from Chroma Technology Corp, Rockingham, VT). Images were acquired such that the
photon count in each image was ~500 photons per frame and final images were compiled with
photon counts that ranged from 60 to 400k photons. Results from ten images were averaged at
each photon count. The CV was measured within a single ROI near the center of an image
(Figure S4C). Within this region the standard deviation of the intensity was normalized to the
mean intensity and reported as a function of photon count. For this measurement, the photon
count reported is the total number of photons within the ROI.
In an image, spatial photon noise appears as variations or speckle in what should appear as a
uniform field. A low CV indicates that an image is more spatially consistent. The CV from
PEDS, COP, and distributed COP images decreased with photon count as expected (Figure
S4A). Results show that similar to the advantage seen in SNR, distributing each photon event
with PEDS again improves the CV parameter over that measured in images formed with COP.
As was the case for SNR, extensive low-pass filtering was required to obtain similar increases in
image CV in COP images.
Spatial resolution
Spatial resolution is an important measure of the performance of an imaging system. The spatial
resolution of a microscope is often measured by imaging calibration standards of known sizes
and testing the ability of the microscope to accurately represent the size of the standards (Pawley,
2006; Klar et al., 2000; Inoue, 2006). A diffraction-limited microscope produces accurate images
down to the diffraction limit at which point smaller standards appear to be the size of the
diffraction-limit.
The intensity distributions used to form PEDS images may blur otherwise sharp edges and
contribute to a loss in spatial resolution. The full-width at half-maximum (FWHM) intensity was
measured for images of beads with three different diameters. Images were acquired of beads, two
with a diameter larger than, and one with a diameter smaller than, the optical system PSF; 1010
nm, 520 nm, and 190 nm, respectively. For each bead diameter, PEDS and COP images were
formed and an intensity profile line was drawn across the center of the bead (Figure S5). Widths
were compared between PEDS and COP images and with the diameter specified by the
manufacturer. The position of the line from which the profiles were constructed was kept
constant between image types and the profiles were normalized to the maximum intensity.
Because COP and PEDS images have different sized pixels, intensity profiles were aligned along
the abscissa by converting the coordinates of both profile lines to real spatial units (nm) relative
to the upper left corner of their respective frames. The FWHM of each intensity profile was
similar between COP and PEDS images. In each case the PEDS image provides a smoother and
more accurate representation of the bead geometry. The hollow spot in the image of the 1010 nm
bead is due to spherical aberration caused by a refractive index mismatch at the cover
glass/specimen interface. Spherical aberrations cause round specimen features in imaging planes
near the focal plane to appear doughnut-shaped. Note that the effect of this aberration is more
accurately represented in the PEDS image and corresponding PEDS intensity profile.
Measured widths of the 1010 nm and 520 nm beads are in good agreement with the
manufacturer’s specified bead diameter. The measured width of the smallest bead is
approximately 250 nm, which is greater than the manufacturer’s specification of 190 nm. The
PSF of the optical system used for this work is therefore on the order of 250 nm.
Modified Nearest-Neighbor Measurement
As applied to images, the conventional two-dimensional NN measurement is only effective for
binary images, that is, images where a single pixel contains no more than one photon. While
photon position maps are typically binary since the resolution of the position feedback signals
made it unlikely that more than a single photon occupied the same position domain, pixels in
COP images become non-binary at very low photon counts. To make the NN measurement
equally applicable to COP images, a modified (three-dimensional) NN measurement was
developed where each pixel intensity was treated as a surrogate distance from the image floor
(zero intensity) in the third, axial, dimension. The distance between the center of each pixel and
the center of its NN in three-dimensional space was recorded and the results from all pixels in an
image were used to create a test distribution. Pixels containing zero photons were ignored. The
relationship between intensity and distance in the third dimension was selected to be 8 nm/grayscale value so that the maximum intensity difference between two pixels corresponded to two
pixels separated laterally by the maximum distance in an image frame. This intensity scale has a
direct relationship with the sensitivity of the modified NN measurement in COP images. Other
values were tested; however, an 8 nm/gray-scale value produced acceptable results across the
photon counts tested (data not shown).
Spiral Scanning
An advantage of the PEDS approach is that identification of photon positions permits images to
be formed using non-raster scanning patterns. Spiral scanning is representative of the latter.
Image formation using non-raster scan patterns leads to several fundamental differences with,
and as a consequence, important advantages over raster scanning:
1. Non-raster scanning permits greater scan and image acquisition rates than can be achieved
with raster scanning using comparable beam-steering devices.
2. Numerical methods, not a pixel clock interval, are used to determine the sampled region and
to uniformly space sample points along the spiral pattern. This is possible because, with the
exception of the turnaround region (if one exists), the laser beam is moved along the spiral
pattern at a constant velocity (Figure S2C).
3. The ability to generate uniformly sized sample domains along the entire (non-raster) scan
permits sampling over the entire region being scanned. In contrast, during raster scanning
the laser beam is accelerating or decelerating (and therefore has a non-uniform velocity)
over significant parts of the region. Since the laser is not moving at a constant velocity,
uniform pixels cannot be generated using a constant pixel clock interval and these regions
are not used.
4.
Non-raster scanning permits continuous control of the size and location of the scanned
region within a field-of-view. This permits computer-controlled, real-time panning to
specific regions; and zooming to optimize the size of the scanned region. In contrast, scan
regions and pixel dimensions have to be reset and a scan restarted with raster scanning
systems. The on-line pan and zoom capabilities of non-raster scanning provide unique and
powerful advantages for implementing particle-tracking paradigms. For example, the
position of a non-raster scan can be rapidly panned in a field-of-view (at msec rates if
necessary) to maintain a maximum (or predetermined) intensity value at the center of the
scan. Changes in the position of a non-raster pattern in a field-of-view can then be mapped
frame-by-frame to track movement of an object. Thus, dynamic systems that are changing
in location, as well in time can be imaged in an optimal manner. In addition, continuous
zoom capabilities permit systems that change dimensionally as well as location and/or time
to be imaged optimally with enhanced temporal resolution, again under complete computer
control.
5. Non-raster scanning, with its smooth, non-abrupt transitions between lines and frames will
cause much less “wear and tear” on mechanical beam-steering devices, such as closed-loop
galvanometers, than raster scanning with its repeated start/stop direction reversals.
Derivation of PEDS Localization Equation
In this context, the statistical population is defined as all photons emanating from a point source
during a specified period of time. A statistical sample of this population consists of the photons
collected by the imaging system used to form the image of the point source. Starting with the
generic MVND equation, the PEDS localization equation is derived as follows.
f x x1, x 2,..., x n  
1

1
2
n
2  2
e

1
x T 1 x  
2
(S1)
Where the MVND is a function of n variables, x is a matrix that describes the coordinate space

of each variable, and µ is a vector that describes the mean of each variable. For localization the
mean of each variable is equal to the measured position of a photon. The covariance matrix ∑
describes the pair-wise relationships between all variables:
 11  12

 22
   21
 


 n1  n 2
  1n 
  2 n 
(S2)
  

  nn 
where σij is the covariance between variables i and j. Because the uncertainty of location of each
photon in the sample is treated as a normally distributed variable with equal variance, i.e., σ1 =
σ2 = … = σn = σ, and because each photon event is independent of all other events it follows that
the covariance matrix simplifies to
 2

0

 

 0
0

 0
(S3)
  

  2 

0

2

0
Next, by replacing the variant matrix, x, with the set of vectors, xi, where i = 1:n, and parsing out
the respective mean of each variable as a scalar, such that the variable i represents the measured
position of the ith photon (i out of N total photons), and xi represents the coordinate space of the x
dimension in a Cartesian coordinate system, that is, the independent variable, for the ith photon,
the inverse covariance matrix simplifies to division by 2. Now Eq. 3 can be rewritten as
f x x1, x 2,..., x n  
1

1
2
n
2  2

1
2
2
2
  1 x1  1  x 2   2  ... x n   n 
2
e

(S4)
Taking the determinant of the covariance matrix:

   2n (S5)

1
2
  n (S6)
Finally, by recognizing that all variables share the same coordinate space, i.e., x1 = x2 = … = xn,
because all photons occur in the same coordinate system, the equation simplifies to a onedimensional description of the net summation of all variables (Eq. 3).
Background Estimation
Typically, background noise from sources such as electronic noise from the detector and
background fluorescence are uniformly distributed across an image. Provided noise is uniform
throughout an image frame, the number of background photons within the spot ROI can be
estimated by a sample of background noise from a region remote from, but still within the same
image frame as, the spot (Figure S6A). The ability to accurately estimate the number of
background photons at the spot increases with the amount of background noise (Figure S6B). An
image with 0.010 background photons per PEDS photon position domain would be considered
very noisy. For example, the images of sub-diffraction beads acquired for this work typically
have 0.003 background photons per domain while the bead produces 1,000 spot photons, which
would result in a 0.02 nm error in localizing the bead due to background estimation (Figure
S7A).
To test localization in the presence of background noise, spot images were computer generated at
four different spot photon counts: 100, 500, 1,000, and 5,000. For each spot photon count, the
center of the spot was localized with PEDS localization in the presence of increasing background
noise. Uniformly distributed background noise was increased until the isolated ROI contained
twice as many background photons as spot photons, at which point the measurement was
terminated. At each background level the spot, with location held constant, was localized in
1,000 images, and the standard deviation of the measured locations is plotted as the localization
uncertainty (Figure S7B). In each case the measured localization uncertainty is insensitive to
increases in uniform background noise.
REFERENCES
Cheng, P.C. (2006) The Contrast Formation in Optical Microscopy. In: Handbook of Biological
Confocal Microscopy. Springer. 162-206.
Inoue, S. (2006) Foundation of Confocal Scanned Imaging in Light Microscopy. In: Handbook
of Biological Confocal Microscopy. Springer. 1-19.
Klar, T.A., Jakobs, S., Dyba, M., Egner, A. & Hell, S.W. (2000) Fluorescence microscopy with
diffraction resolution barrier broken by stimulated emission. Proceedings of the National
Academy of Sciences (USA). 97:8206-8210.
Pawley, J. B. (2006) Fundamental Limits in Confocal Microscopy. In: Handbook of Biological
Confocal Microscopy. Springer.
Sheppard, C.J.R, Gan, X., Gu, M. & Roy, M. (2006) Signal-to-Noise Ratio in Confocal
Microscopes. In: Handbook of Biological Confocal Microscopy. Springer. 442-452.
FIGURE LEGENDS
Figure S1: Optical Schematic
Diagram of the optical system used for photon position map acquisition and PEDS image
formation. The laser (L1) used for excitation was aligned with the system using two mirrors (M1,
M2) and exposure was controlled with an electronically controlled shutter (S). Laser intensity
was modulated with a series of neutral density filters (ND) and an electro-optic modulator
(MOD). Alignment into the MOD was achieved with two mirrors (M3, M4). The modulated
laser intensity was measured prior to entering the objective using a flip-mirror (M5) and power
meter (PM). The lens (L1) prior to the dichroic beam splitter (DI) expanded the beam to fill the
scan mirrors, which were driven by closed-loop galvanometers (X, Y). The scan lens (L2) was
placed at the image plane of the side port of the microscope and aligned so that the internal
optics of the microscope (M6, L3) directed the beam to fill the back-aperture of the objective
(O). Emitted light was de-scanned and filtered by the DI before another mirror (M7) directed it
through a band-pass filter (EM) specific to the excitation wavelength of the specimen. An 800
µm pinhole (P) rejected out-of-focus light prior to alignment mirrors directed photons to the
photo-multiplier tube (PMT). A lens (L4), camera (CCD), and flip-mirror (M8) were used to
check alignment of the system prior to acquisition. Most components were housed in an
enclosure (ENC) which minimized stray light from entering the PMT. An avalanche photo-diode
(APD) was aligned as an alternative photon detector and multiple laser lines were aligned for
excitation of various fluorophores (L2, L3). The custom circuit board (PCB) recorded the
position of the scan mirrors every time a photon was detected and custom acquisition software
recorded them as a PEF on the computer (PC). Dashed lines indicate the optical axis and dotted
curves indicate electronic interfaces.
Figure S2: Scanning command signals, velocities and accelerations
(A) Outward and inward spirals can be combined to generate a continuous spiral scan pattern
with no turnaround region. (B) shows the frequency- and amplitude-modulated sinusoidal
waveforms used as x and y command signals to generate this pattern. (C) shows the angular
increments between consecutive command points (Δω, upper trace) and the distances between
consecutive command points (Δd, lower trace) needed to generate the spiral pattern where Δω
was chosen to produce constant Δd. Because inner spirals have smaller dimensions than outer
spirals, greater angular increments (Δω) are needed within inner spirals (at both ends of the
trace). Using our present spiral generating algorithms, the only Δd that is not equi-distant (to
within 5 decimal places) lies between the last point of the outward spiral and the first point of the
inward spiral (indicated by the downward spike at the midpoint of the Δd trace). (D) shows the
distances traveled per command point (the velocity) separately in the x direction (Δx) and y
direction (Δy) where the overall amplitudes of Δx and Δy remain constant despite changes in
frequency. (E) illustrates further the rate of change of distance traveled per command point (the
acceleration) separately in the x direction [Δ(Δx)] and y direction [Δ(Δy)]. Angular rates increase
during movements associated with the higher frequency inner spirals.
Figure S3: Signal-to-noise Ratio
(A) The average signal-to-noise ratio plotted as a function of photon count for PEDS (B, solid
black line), COP (C, gray dotted line), and distributed COP (D, gray dashed line) images. The
bead is larger than the scanned region so the images are truncated. Image intensities were
normalized prior to measurement. The squares in the COP image are indicative of the signal
(right) and background (left) ROI selected for a typical measurement. The SNRs from images of
five beads were averaged for each data point (A). Scale bar = 500 nm.
Figure S4: Consistency of Image Formation
(A) The average coefficient of variation plotted as a function of photon count for PEDS (B, solid
black line), COP (C, gray dotted line), and distributed COP (D, gray dashed line) images. Low
CV values denote high image quality. Image intensities were normalized prior to measurement.
The square in the COP image is indicative of the ROI selected from a uniformly fluorescent
specimen for a typical measurement. The CVs from ten images of uniform fluorescence were
averaged for each data point (A). Scale bar = 400 nm.
Figure S5: Spatial Resolution
Spatial resolution measured with fluorescent beads of three different diameters: 190 nm (A), 520
nm (B), 1010 nm (C). Intensity profiles (left column) of images formed using COP (gray line,
middle column) and PEDS (black line, right column) indicate no appreciable difference in
FWHM (dashed line). The locations where the intensity profiles were measured are indicated in
each image by a line segment. Scale bar = 200 nm.
Figure S6 – Background Noise Estimation
(A) The number of photons in a region containing only background noise is used to estimate the
number of photons due to background noise that reside within a spot region. (B) In a space
containing uniformly distributed photons, the uncertainty of an estimate of the number of
photons (estimation error/background count) in one region by sampling another region decreases
as the number of photons increases.
Figure S7 – Estimation of Localization Error Due to Background Noise
(A) The amount of error introduced to PEDS localization by background noise estimation
depends on the number of spot photons; the relationships for three spot counts are shown (black
lines). In typical images acquired for this work, background noise was measured as 0.003
photons/photon position domain. (B) The measured uncertainty of PEDS localization of spots in
images with background noise. Images with 100 (‘x’), 500 (squares), 1,000 (open circles), and
5,000 (closed circles) spot photons were computer generated, and background noise was
uniformly distributed. The measurements were conducted over a range of photon counts until the
ROI contained twice as many background photons as spot photons.
Figure S8: Measured PSF and PEDS Localization in Acquired Images
(A) The mean width of a spot measured in each of 30 sequential images (open circles) of 20
different sub-diffraction (92 nm) beads. The average over the 20 beads (black dash-dotted line)
was determined to be  = 122 nm (FWHM = 287 nm), which was used as the width of the
microscope PSF for PEDS localization. One standard deviation of the mean spot width is 5.7 nm
(light dotted line), which is indicative of the consistency with which the focal plane was found
prior to image acquisition. (B) 12 sub-diffraction beads were localized in each of at least 40
frames using PEDS localization (‘X’) and the standard deviations of the results are in excellent
agreement with the predicted uncertainty (dark dashed line). Background in the images was
accounted for as described above. The predicted uncertainty for localization using an ideal PSF
is plotted for reference (light dashed line).