High-resolution, large field-of-view label-free imaging via
aberration-corrected, closed-form complex field reconstruction:
Supplementary Notes
Ruizhi Cao
1,*,
†
, Cheng Shen
1,
†
, and Changhuei Yang
1
1
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA,
USA
*
rcao@caltech.edu
†
These authors contribute equally to this work
Contents
1 System calibration
2
2 Result of FPM and APIC using reduced dataset
3
3 Resolution quantification
5
4 Reconstruction of a hematoxylin and eosin stained sample
6
5 Result of FPM and APIC when imaging a phase target
7
6 Reconstruction time
8
7 Aberration correction
9
8 Comparison under different signal-to-noise ratios
11
9 Number of NA-matching measurements required in APIC
14
10 Inaccurate illumination angle estimates
15
11 Result of spatial-domain Kramers-Kronig method and APIC
16
12 Derivation of APIC
17
12.1 The forward model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
12.2 Reconstruction under the NA-matching angle illumination . . . . . . . . . . . . . . . . . . . .
18
12.3 Aberration extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
12.4 Reconstruction using darkfield measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1
1 System calibration
To do reconstruction using the Angular Ptychograhic Imaging with Closed-form method (APIC), we need
to know the transverse illumination vector
k
i
for each measurement, as it tells us which area of the whole
sample’s spectrum is measured (Eq. 6). This indicates that we need to determine the angle of each tilted
illumination. To do that, we used the previously developed circle-finding algorithm to find the exact illu-
mination angle for the NA-matching measurements [1]. The brightfield measurements whose illumination
angles were below the acceptance angle of our imaging system were collected as well. These brightfield mea-
surements were used for geometrically calibrating the angles associated with our darkfield measurements.
We note that these brightfield measurements are only for calibration purpose and not for reconstruction in
APIC.
The illumination unit consisted of a LED ring and a LED array. The LED ring was attached on top of
the LED array and was used for the NA-matching measurement. This unit was mounted on a motorized
transnational stage for height adjustment. We adjusted its tilt and height to exactly match the illumination
angle of the ring LED and the acceptance angle of our imaging system. To find the exact height, we first
moved the LED unit close to our sample such that the ring LED produced the darkfield measurement. Then,
we gradually increased the separation between the LED and the sample until we saw the image under the ring
LED illumination transited from darkfield to brightfield. The transition point is our desired height. Once
the height and tilt of the system were fixed, we acquired all calibration data and calibrated the illumination
angles for all LEDs. We also used a high NA objective to measure the relative intensity of each LED with
a blank slide. The high NA objective was selected such that the incident light from any LED can directly
enter the system. It needs to be emphasized that the intensity calibration is done only once with a high NA,
small field-of-view objective, we acquired all our actual experiment data with a low NA, large field-of-view
objective. In our experiment, we normalized the measurements using the measured relative intensities and
then conducted the reconstruction in APIC.
F
F
-1
APIC reconstructed spectrum
Brightfield measurement
Fourier transform
Circle-finding based calibration
Estimated by circle-finding algorithm
Geometrically calculated
Tuned with grid search
Initial optimized angles
Fine tuned using grid search
Cropped spectrum
Image given by the model
Measurement
Calculate intensity
in real space
Correlation
Maximize
correlation
Reconstruction
Crop
Effective
CTF (tuned)
Figure S1: Calibrating the illumination k-vector. By locating the center of the circle in the Fourier transform
of the measurement, we extract the corresponding
k
i
in the spatial frequency domain. Using the separation of
the LEDs on the array and the estimated brightfield LED illumination angles, we can calculate the darkfield
LED illumination angles using such geometry. With those, APIC could reconstruct sample’s complex field,
which can be used to further optimize the illumination angles by maximizing the correlation between the
real measurement and the image obtained with the forward model. CTF: coherent transfer function.
We note that we can reconstruct the calibration data with the geometrically calculated darkfield LED
illumination angle and then use the reconstructed complex field to further optimize the illumination angle
by searching over a pre-defined finer grid. Once this is done, we fix the calibrated angles and use them for
all other measurements. The entire process is illustrated in Fig. S1.
2
2 Result of FPM and APIC using reduced dataset
In our main manuscript, we acquired 316 images for one sample. As we explained in our main manuscript,
this large redundancy was chosen to show the best performance of Fourier Ptychographic Microscopy (FPM).
Here, we reduced the dataset so that there are 9 bright field measurements, 8 NA-matching measurements
and 28 darkfield measurements in this reduced dataset. In our reconstruction, FPM used all these 45
images while APIC used 36 images (8 NA-matching measurements and 28 darkfield measurements). The
arrangement of the LEDs is shown in Fig. S2.
a
b
full set
reduced set
LED unit
Illumination vector
k
x
k
y
+
Figure S2: Arrangement of the LEDs in APIC.
a
, Image of the ring LED on top of a LED array. A black
tape is covered on the ring LED to prevent stary light goes into the system due to the scattering of its
white shell. The wires for the ring LED was glued in between the LED pairs on the LED array so that they
do not block the LEDs.
b
, Illumination k vector for the full and reduced dataset. The ring LEDs sit in
between the two black circles in the figure. We note the reduced dataset is a subset of the full set. The dots
in orange shows the illuminations covered in the reduced dataset while the orange and blue together show
the illuminations in the full dataset. To form the reduced set, the LEDs are chosen so that they are more
uniformly distributed in k-space.
k
x
and
k
y
denote the spatial frequency coordinates.
To prevent the stray light entering the system, we covered the ring LED with a black tape. We note that
there are some LEDs on the LED array being blocked by the ring attached onto it. These LEDs were not
used in our experiment and thus we see a gap in the spatial frequency domain in Fig. S2b. We additionally
note that such blocking has no obvious impact on the final reconstruction as the smallest overlap is still over
70% when these LED are dropped.
To construct the reduced dataset, we first uniformly sampled the illumination angle in the continuous
spatial frequency space for the region corresponding to the brightfield, NA-matching, and darkfield measure-
ments. For each of the sampled illumination angle (the ideal uniformly distributed illumination angle), we
selected the LED with the smallest angle difference with respect to the desired one. By doing that, we made
the LEDs distribute as uniformly as possible for our reduced dataset. The selected LEDs for the reduced
dataset are shown in orange in Fig. S2b.
The reconstruction results using this reduced dataset are shown in Fig. S3. For comparison, we also
included the reconstruction results when feeding in the entire dataset.
When FPM is not given the privilege of having a highly redundant dataset, its reconstruction result can
be severely disturbed by the aberration of an imaging system whose phase variation exceeds
2
π
5
(NA of the
objective: 0.25). We see that although FPM partially reconstructed the high spatial frequency information of
the Siemens star target using the full dataset, it failed to maintain even the low spatial frequency information
when a dataset with approximately 7 times fewer measurements was provided. In contrast, APIC, retrieved
both the high and low spatial frequency information in either case. As we can see from Fig. S3, APIC
3