# Designing and simulating realistic spatial frequency domain imaging systems
using open-source 3D rendering software

Jane Crowley  1,* and George S.D. Gordon  1  1Optics & Photonics Group,
Department of Electrical and Electronic Engineering, University of Nottingham,
Nottingham, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Spatial frequency domain imaging (SFDI) is a low-cost imaging technique that
can deliver real-time maps of absorption and reduced scattering coefficients,
offering improved contrast over conventional reflectance methods for imaging
important tissue structures such as tumours. However, there are a wide range
of imaging geometries that practical SFDI systems must cope with including
imaging flat samples _ex vivo_ , imaging inside tubular lumen _in vivo_ such
as in an endoscopy, and measuring tumours or polyps of varying shapes, sizes
and optical properties. There is a need for a design and simulation tool to
accelerate design and fabrication of new SFDI systems, and to validate
performance under a wide range of realistic imaging scenarios. We present such
a system implemented using open-source 3D design and ray-tracing software
_Blender_ that is capable of simulating turbid media with realistic optical
properties (mimicking healthy and cancerous tissue), a wide variety of shapes
and size, and in both planar and tubular imaging geometries. Because Blender
is a full end-to-end rendering package, effects such as varying lighting,
refractive index changes, non-normal incidence, specular reflections and
shadows are naturally accounted for, enabling realistic simulation of new
designs. We first demonstrate quantitative agreement between Monte-Carlo
simulated scattering and absorption coefficients and those measured from our
Blender system. Next, we show the ability of the system to simulate
absorption, scattering and shape for flat samples with small simulated tumours
and show that the improved contrast associated with SFDI is reproduced.
Finally, to demonstrate the versatility of the system as a design tool we show
that it can be used to generate a custom look-up-table for mapping from
modulation amplitude values to absorption and scattering values in a tubular
geometry, simulating a lumen. As a demonstrative example we show that
longitudinal sectioning of the tube, with separate look-up tables for each
section, significantly improves accuracy of SFDI, representing an important
design insight for future systems. We therefore anticipate our simulation
system will significantly aid in the design and development of novel SFDI
systems, especially as such systems are miniaturised for deployment in
endoscopic and laparoscopic systems.

††journal: boe††articletype: Research Article

##  1 Introduction

Optical properties, specifically scattering and absorption, and shape are
important potential indicators of cancer within the gastrointestinal (GI)
tract[1, 2]. Conventional white light endoscopes and capsule endoscopes are
the standard method of imaging the GI tract but provide limited information
about tissue properties that are hallmarks of a range of potential tumours[3],
leading to low five-year survival rates of oesphageal cancer (15%[4]) and
colon cancer (63%[5]). SFDI is a well-established, low-cost imaging
technique[6], with applications for imaging blood oxygenation[7], burn
depth[8], dental caries[9], bowel ischaemia[10], and indicators of cancer[11].
A range of commercial[12] and research[13, 14, 15] SFDI systems are now
available. However, these existing systems are almost exclusively designed for
_planar_ imaging geometries, where the sample is flat and the camera and
projector are located above it at near-normal incidence. However, many
important clinical applications exhibit _non-planar_ geometries: for example
imaging inside tubular lumen such as the GI tract, blood vessels, biliary
system (Fig 1). SFDI imaging _in vivo_ in such organs is challenging due to
miniaturisation needs, and because the surfaces are cylindrical, creating non-
planar illumination conditions and sample geometries. This means that
illumination and imaging may no longer be normal (or nearly normal) to the
surface being imaged so different scattering behaviour will be observed [16],
and specular reflections will be altered.

Figure 1: Future SFDI systems, especially those for _in vivo_ clinical use,
may require significantly different geometries from conventional SFDI: a)
conventional ‘planar’ SFDI imaging geometry with projector at a small angle to
flat sample, with real-world application of measuring diabetic foot shown in
inset [17], b) SFDI operating in a tubular (lumen) geometry, that may be
required for use in future endoscope systems where projection is no longer
approximately flat, with example usage for imaging polyps in the colon shown
inset [18], c) screenshot of our _Blender_ SFDI model applied to a planar
geometry, with reconstructed scattering properties of tumour like sample shown
inset, d) a screenshot of our _Blender_ model applied to a non-planar tubular
geometry, with reconstructed scattering properties shown inset.

To aid in the design of novel SFDI systems under these constraints, we have
created an SFDI design and simulation tool in the open source $3$D modelling
software _Blender_ (v 2.93) using the built-in ray-tracing engine Cycles (Fig
1). Cycles is a physically based path tracer, in which rays of light are
traced from each camera pixel into the scene and either absorb into the world
background, reflect, refract or reach their maximum bounce limit, typically
$1024$ bounces. To increase accuracy, Cycles produces randomised light rays
and averages the results from a singular pixel over time, analogous to a Monte
Carlo simulation[19]. Cycles simulates volume scattering inside objects using
a Henyey-Greenstein Phase function, which is commonly also used in Monte-Carlo
simulations of tissue [20, 21]. _Blender_ has previously been used for three-
dimensional shape measurement of additive manufacturing parts with complex
geometries[22], for the development anatomically accurate meshes to use in
Monte Carlo light simulations[23], and for the generation of SFDI image data
sets to train neural networks [24, 25]. By using Blender for both geometry
specification (i.e. design) and simulation (via ray-tracing with Cycles), we
are able to simulate realistic optical properties and geometries while
naturally accounting for realistic features of SFDI systems such as stray
light, specular reflections and shadows.

Conventional imaging in the spatial frequency domain is demonstrated in Fig 2
and consists of projecting a known set of structured illumination patterns
onto a sample at a small angle ($\lesssim 6^{\circ}$) to the normal to
minimise specular reflections recorded by the camera [26]. The structured
illumination set typically consists of 2D sinusoids at 3 equispaced phase
offsets but recent work has shown the successful use of randomised speckle
patterns as an alternative illumination scheme [27]. The amplitude of the
projection pattern is modified by the absorbing and scattering nature of the
sample. A camera placed orthogonal to the sample captures images which are
processed to determine the modulation amplitude of the reflected illumination
pattern as a function of spatial frequency, via the equations[28]:

$M_{AC}(x_{i},f_{x})=\frac{\sqrt{2}}{3}\sqrt{(I_{1}(x_{i})-I_{2}(x_{i}))^{2}+(I_{2}(x_{i})-I_{3}(x_{i}))^{2}+(I_{3}(x_{i})-I_{1}(x_{i}))^{2}}$
(1)
$M_{DC}(x_{i})=\frac{1}{3}\left(I_{1}(x_{i})+I_{2}(x_{i})+I_{3}(x_{i})\right)$
(2)

where $I(x_{i})$ is intensity of each pixel (position $x_{i}$) value in the
captured images for sinusoidally modulated illuminations with a spatial phase
shift of $0^{\circ}$ ($I_{1}$), $120^{\circ}$ ($I_{2}$) and $240^{\circ}$
($I_{3}$). These results can also be obtained from a single image, termed
_single snapshot of optical properties (SSOP)_ , by the use of Fourier-domain
filtering [29] or convolutional neural networks [30] to separate the AC and DC
components. In either approach, the AC and DC modulation amplitudes need to be
calibrated with the modulation transfer function (MTF) of the system in order
to produce _diffuse reflectance_ values that represent the next intermediate
step toward obtaining scattering and absorption.

Figure 2: Conventional SFDI process: A sample is illuminated with a sinusoidal
pattern then an image is captured. This is then demodulated and compared with
a reference material of known optical properties. The optical properties of
the material of interest are then estimated using a look-up table. Finally,
maps of absorption and reduced scattering coefficient are produced.

Conventional calibration for the MTF is achieved by imaging a reference
material of known optical properties and computing diffuse reflectance values
for these properties using a light propagation model: Monte Carlo simulation
or the Diffusion Approximation[28, 31]. The difference between the computed
and measured diffuse reflectance is used to infer the MTF, which can then be
applied to obtain diffuse reflectance values from the modulation amplitudes of
the sample of interest. Finally, the absorption, $\mu_{a}$, and reduced
scattering, $\mu_{s}\prime$, coefficients are then determined via a look-up
table (LUT) generated from the chosen light-propagation model. An alternative
method of generating this LUT is to measure a large data set of materials of
known optical property values and interpolate between values to create an LUT
unique to the imaging system in use. This has been achieved previously by
calculating the reflectance and modulation of each material by comparison to a
reflectance standard [32].

Another important property that SFDI imaging can extract is height
information, i.e. shape in flat geometries. This can be done via fringe
projection profilometry[33, 34]. A structured illumination pattern is
projected onto a sample of interest and a camera captures how the pattern is
distorted by the presence of the sample to determine sample shape [11]. This
information can be useful in clinical settings for quantifying the size and
shape of polyps, which is linked to their pathology [2]. Previous studies have
shown that structured illumination significantly reduces error in shape
measurement compared to visual assessment, and is comparable to measurement
with biopsy forceps and ruled snare [35]. Combining shape information with
optical property information would give valuable diagnostic information to the
clinician to determine optimal treatment for patients.

Developing an SFDI system to determine optical properties and shape in
clinical environments, both _ex vivo_ and _in vivo_ , has many challenges
associated with it, such as determination of optimum illumination source
placement and determination of optimum illumination patterns. Here, we present
a design and simulation system using free, open-source 3D modelling and
rendering package _Blender_ , that can implement SFDI for simulation of
absorption, scattering and shape. We first show how to use _Blender_ to model
a customisable scattering and absorbing material with in-built _Blender_
material nodes. We then show how to construct a virtual characterisation
system for the absorption density, $A_{\rho}$, and scattering density,
$S_{\rho}$, of this material using two approaches: a double integrating sphere
(DIS)[36] and an SFDI system. For both approaches, we validate the accuracy of
retrieved optical properties and show how this can improved by generating an
empirically derived LUT from the DIS in-situ data. Next, we present two
illustrative example use cases for our system. First, we show that the
simulated SFDI system enables reconstruction of scattering, absorption and
shape of planar geometry samples mimicking cancerous and pre-cancerous
conditions such as squamous cell carcinoma and Barrett’s Oesophagus
respectively. Second, we demonstrate, for the first time, a novel illumination
scheme tailored for non-planar, tubular geometries (such as inside a lumen)
where the spatial frequency is constant throughout the length of the tube such
that the optical properties can be accurately obtained. To improve accuracy,
we longitudinally section the tube and create separate look-up tables for each
section, a straight-forward task in our system. We show that this customised
illumination can detect changes in absorption and scattering properties within
a tube of biologically relevant material, providing a potential design for
future SFDI systems.

##  2 Methods

###  2.1 Material simulation

The use of 3D modelling or CAD software to simulate conventional optical
imaging with a light source, 3D objects and a camera is well-established. The
key challenges for designing an SFDI simulation system in such software is
ensuring accurate, calibrated simulation of optical scattering and absorption,
and designing appropriately structured illumination patterns.

To achieve the most physically realistic ray-traced renderings in _Blender_ ,
some optimisation of the render settings is required. Within the ray-tracing
engine Cycles, which physically traces rays of light, the maximum number of
bounces a light ray can travel before the simulation terminates can be set. We
set this value to $1024$, the highest allowed. The number of samples to render
per pixel in the image was set to $1000$. Clamping of direct and indirect
light, which limits the maximum intensity a pixel can have, was disabled by
setting both to $0$. Colour management, which is typically used to make
visually appealing images but introduces unwanted artefacts such as gamma
correction, was disabled by setting the display device to _‘None’_. View
transform was set to _‘Standard’_ to ensure no extra conversions were applied
to the resulting images. The sequencer, which sets the the colour space, was
set to _‘Raw’_ to avoid unwanted colour balancing or further gamma correction.
For all images rendered, the camera exposure is adjusted to avoid saturation
while maximising power of detected signal, but the images must then have their
intensities corrected by following the equation:

$I_{output}(x,y)=I_{render}(x,y)\times 2^{t_{exposure}}$ (3)

where $I_{output}$ is the exposure-corrected intensity we required,
$I_{render}$ is the raw value obtained following the render, and
$t_{exposure}$ is the exposure setting.

Previous work has used a weighted mixture between transparent, sub-surface
scattering and absorbing materials to create a composite material with the
desired optical properties [24]. Though this approach works in many realistic
operating regimes, it is limited because the sub-surface approximation applies
only at surfaces and not in the entire material volume. Here, we therefore
model the material more accurately using a volume shader, instead of surface
shader, exploiting Blender’s built-in volume absorption and volume scattering
shaders. The absorption and scattering was varied by changing the density
parameters of the nodes, $A_{\rho}$ and $S_{\rho}$ respectively. The
anisotropy, _g_ , in the volume scatter node was set to $0.8$, in line with
the anisotropy values measured of the tissue within the GI junction[1] and
with the value of _g_ set in the Virtual Photonics Monte Carlo simulation
software[37], which was used to generate a LUT. When capturing image data, we
extract the red channel of the RGB colour images. However, because Blender
supports tri-colour operation it can also provide physically realistic
scattering at green and blue wavelengths if desired.

In order to use a LUT generated from a Monte-Carlo simulation or the Diffusion
Approximation, the semi-infinite thickness requirement must be met [38]. To
set an appropriate thickness for the material to meet this property, a red
sphere was placed behind the material and the parameters were varied until the
sphere was not visible, i.e. the material was not transparent for a fixed
thickness. For a material of $2$ m thickness, this red sphere was not seen for
$A_{\rho}\geq 2$ when $S_{\rho}=0$, and for $S_{\rho}\geq 3$ when
$A_{\rho}=0$. These are therefore the limitations of the material in our
system when used for SFDI. However, it is noted that this limitation could be
circumvented by using an empirically derived LUT that is calibrated to a
particular physical thickness.

Our aim is to create a simulation of an SFDI system with biologically relevant
samples, and so we have identified two disease states relevant for detection
of cancer in the upper GI tract that have distinctive scattering and
absorption properties: squamous cell carcinoma (SCC) and Barrett’s Oesophagus
(BO). SCC is formed of neoplastic cells invading the submucosal layer of
tissue[39]. For simplicity, we model SCC with a spheroid but the flexibility
of Blender allows for generation of arbitrary shapes. We modelled tumour
spheroids using sphere meshes scaled to be $40$ mm diameter. We note that the
_‘scale’_ parameter of the object in Blender should be reset when the desired
size is reached to ensure proper behaviour with regard to scattering length
scales along different dimensions.

BO can be a pre-cursor to oesophageal adenocarcinoma, and occurs when the
epithelium of the oesophagus begins transforming into a structure mimicking
the the lining of the stomach[40]. To examine contrast, two materials were
placed adjacent to one another: one with the optical properties of healthy
oesophageal tissue and the other with the optical properties of BO. To
simulate realistic gastrointestinal imaging, we consider two imaging
geometries. The first simulates an ‘up-close’ view of a tumour on the wall of
a large lumen and can be approximated by a flat geometry. However, to identify
such structures during a typical endoscopy or to examine such structures in a
smaller lumen, it is also necessary to consider a tubular geometry with a wide
field-of-view. We therefore also consider the scenario of an SFDI system
pointing down a tube, shown in Figure 1b.

###  2.2 Calibration of material optical properties

For SFDI measurements, a reference material of known optical properties is
required to correctly calibrate the system response (as discussed in Sect 1).
This requires determining the relationship between the material parameters in
_Blender_ and the recovered absorption and reduced scattering coefficients.
This can be done directly with an SFDI system through a ‘trial and error’
approach – the material parameters are adjusted until plots of diffuse
reflectance vs. spatial frequency agree with theoretically derived curves
[24]. However, this approach is imprecise and laborious. We therefore
developed a more accurate approach which involves simulating a double
integrating sphere (DIS) system in _Blender_ [36]. This system consists of two
hollow spheres, termed the ‘reflectance’ sphere and ‘transmittance’ sphere,
each with $100$ mm diameter and $10$ mm wall thickness. The material of these
spheres is set to be highly reflective using the diffuse bi-directional
scattering distribution shader with $0$ roughness and colour of pure white.
The reflectance sphere has an entry port and an exit port, with the sample
located at the exit port, which are rectangular in shape with a $10$ mm side
length. The transmittance sphere has only an entry port, where the sample is
located, of the same shape and size as the reflectance sphere exit port. The
sample placed at the exit (sample) port of the reflectance sphere and the
entry (sample) port of the transmittance sphere has a thickness of $1$ mm. The
material of the sample is that of the material described in Sect. 2.1.

The input light source is a spot light of power $5$ W, with a beam radius of
$0.5$ mm and a spot size of $6^{\circ}$. The light is placed at the entry port
of the reflectance sphere. Cameras were placed at the base of each of the
spheres to act as detectors, with all pixels summed together (i.e. integrated
over the detector area) to give a power value. For our initial tests, only the
red channel is considered. A baffle is placed between the sample ports and the
cameras to block specularly reflected light from the sample entering the
camera detector. To perform normalisation, it is necessary to collect
transmission and reflectance measures with a reflectance standard sample, no
sample and with the light beam blocked. The reflectance standard sample is
simulated using the diffuse bi-directional scattering distribution function
(BSDF) shader in _Blender_ with a purely white colour and roughness set to
$0$. It is assumed that the normalised reflectance of this material is $0.9$.
For each captured image, the camera exposure was varied until the average
intensity was approximately in the middle of the 0-255 range (i.e. 8-bit
colour). This exposure was noted and corrected for using Eqn 3.

To determine the reduced scattering and absorption coefficients, a series of
images is taken in the reflectance sphere and the transmittance sphere, and
the normalised reflectance and transmittance are calculated respectively for
varying sample material properties using the equations:

$M_{R}=r_{std}\frac{R_{2}(r_{s}^{direct},r_{s},t_{s}^{direct},t_{s})-R_{2}(0,0,0,0)}{R_{2}(r_{std},r_{std},0,0)-R_{2}(0,0,0,0)}$
(4)
$M_{T}=\frac{T_{2}(r_{s}^{direct},r_{s},t_{s}^{direct},t_{s})-T_{2}(0,0,0,0)}{T_{2}(0,0,1,1)-T_{2}(0,0,0,0)}$
(5)

where $r_{std}$ is the normalised reflectance of the reflectance standard,
$R_{2}(r_{s}^{direct},r_{s},t_{s}^{direct},t_{s})$ and
$T_{2}(r_{s}^{direct},r_{s},t_{s}^{direct},t_{s})$ are reflectance and
transmittance measurements respectively when the sample material is in place,
$R_{2}(r_{std},r_{std},0,0)$ is a reflectance measurement when the standard
reflectance sample previously described is in place of the material,
$R_{2}(0,0,0,0)$ is a reflectance measurement when there is no sample present
and the transmittance sphere is removed, $T_{2}(0,0,1,1)$ is a transmittance
measurement when light passes straight through the reflectance sphere when no
sample is present into the transmittance sphere and $T_{2}(0,0,0,0)$ is a
transmittance measurement when the incident beam is blocked and there is no
sample in the port. These normalised values were then input into an inverse
adding doubling (IAD) algorithm to determine the optical properties[41].

We used nine data points in the ranges $A_{\rho}:1-100$ and
$S_{\rho}:5000-20000$. These values were chosen as when input into the IAD
algorithm, they gave optical properties within our range of interest. We
captured images in our SFDI set-up for these same material parameters, with a
camera placed $0.5$ m above the sample of interest and a $5$ W spot light
source, acting as the projector, placed at a $4^{\circ}$ offset to the camera
to reduce any specular reflections. The camera and projector were placed at
the same height from the sample, at $0.035$ m apart. The optical properties in
the up-close flat geometry were calculated using two different LUTs: a Monte
Carlo generated LUT and an empirically-derived LUT.

####  2.2.1 Monte Carlo LUT

The Monte Carlo (MC) LUT was generated using Virtual Photonics Monte Carlo
simulation software [37], with absorption coefficients ranging from $0-0.5$
mm-1, and reduced scattering coefficients ranging from $0-5$ mm-1. The optical
properties of the nine material values were calculated using a reference
material of $A_{\rho}=1$ and $S_{\rho}=20000$ with the corresponding reference
optical properties determined from the IAD algorithm for this specific
material.

####  2.2.2 Empirically-derived LUT

The empirically-derived LUT is able to correct for discrepancies between the
SFDI and IAD measurements, which arise from the way anisotropy is implemented
when mixing the absorption and scattering materials, leading to slight
deviations in effective anisotropy constant ($g$). For our first simulation of
an ‘up-close’ view of the wall of a large lumen, approximated as a flat
geometry, we generated a modulation vs reflectance LUT as described by
_Erickson et. al._ [32]. We started with the same nine data points as before,
captured the modulation and reflectance of these densities, and then did a
first linear interpolation between these data points to increase the LUT from
9 data points to 100 $\times$ 100 data points, improving granularity of final
optical properties. A second interpolation step, this time using bicubic
interpolation, was then carried out to determine the optical properties of a
sample of interest. We also performed an inverse calculation, such that if one
has desired optical properties of interest, they may determine what $A_{\rho}$
and $S_{\rho}$ are required to produce these optical properties. For our
second simulated situation of a wide field-of-view tubular geometry, we
generated a LUT of AC modulation amplitude (normalised to reference) vs DC
modulation amplitude (normalised to reference). This type of LUT that uses
normalisation was chosen because in the tubular geometry there is a difference
in intensity from the distal to proximal end of the tube, and therefore large
intensity variation within the material. Dividing by the reference material
with similar variation in intensity reduces this intensity variation in the
resultant image and makes for a more accurate optical property calculation.

###  2.3 Robust shape determination

In addition to measuring optical properties, we would also like to exploit the
fringe-projection approach to reconstruct 3D shape via fringe profilometry. To
do this, we will consider a fringe projection pattern of the form:

$\psi(x,y)=\sin(\omega y+\phi)$ (6)

where $\omega=2\pi f$ is the angular frequency of the projected pattern with
spatial frequency $f$. The sinusoidal pattern must be rotated $90^{\circ}$
from the optical property measurements (which are of the form
$\psi(x,y)=\sin(\omega x+\phi)$) such that the fringes show maximum
sensitivity to surface variations [42]. This is because a change in vertical
height now corresponds to a displacement in one axis from the centre of the
projector, which in turn results in a phase shift of the sinusoid. This is a
consequence of the small angle of the projector relative to the camera. If
this displacement were instead along the fringes there would be no shift in
phase observed.

Typically, a single fringe image may be used to reconstruct height but, as
with SSOP, this may require spatial filtering and hence incur a reduction in
resolution. While single-shot methods may offer a speed advantage, their noise
performance is typically worse. We therefore use a generalised approach for
using $N$ phase-shifted images to reconstruct height maps [43]. If the
geometry of the system is precisely known, this phase can then be converted to
height for each pixel in the image via the equation[33]:

$h(x,y)=\frac{l_{0}\Delta\phi(x,y)}{\Delta\phi(x,y)-2\pi f_{0}d}$ (7)

where $l_{0}$ is the distance from the projector to the reference material,
$\Delta\phi$ is the phase difference between the actual phase (calculated) and
the phase of the background reference plane, $f_{0}$ is the spatial frequency
of the projected pattern and $d$ is the separation distance of the projector
and camera. Because of the geometrical assumptions made in mapping phase to
height, this approach cannot be straightforwardly applied to non-planar
geometries for shape reconstruction. In non-planar geometries, reconstruction
of exact physical height could instead be approximately deduced by comparison
with a reference phantom, e.g. perfectly straight tube for a lumen geometry,
or by applying advanced techniques such as deep-learning [44].

###  2.4 Development of projection pattern for tubular geometry

Conventional SFDI systems with planar sinusoidal projections are suitable for
planar samples, such as _ex vivo_ resected tissue specimens. However for _in
vivo_ use, an SFDI system would typically need to be operated inside a tubular
lumen, e.g. the gastrointestinal tract if in an endoscope. Using our _Blender_
simulation it is very simple to explore such a situation. We began by
simulating a tube of length $250$ mm with an outer diameter of $80$ mm and an
inner diameter of $20$ mm. A $120$ mW spot light source was placed at a
distance of $100$ mm from the top of the tube and projected a flat sinusoidal
pattern down the tube. This naive approach creates a non-uniform spatial
frequency pattern throughout the length of the tube which makes reconstructing
accurate optical properties challenging (see Fig 3a). Therefore, we developed
a process to create a more suitable illumination pattern for other imaging
geometries and demonstrated for the test case of a tube.

Figure 3: Comparing tubes with (a) planar sinusoidal pattern and (b) our novel
illumination pattern. Top left shows image being projected from projector
(highlighted yellow in complementary images). Bottom left shows view of shown
projection pattern down tube with red dashed line indicating where tube is cut
to view cross section on right panel

First, the material of the tube was set to be highly reflective using a pre-
existing material node of diffuse BSDF with a roughness of $0$ and a shade of
pure white. Next, the surface of the tube was ‘unwrapped’ within _Blender_
using the UV mapping tool, resulting in a flattened map of the inside of the
tube. A sinusoidal pattern of the desired phase and spatial frequency was then
applied to this flat surface. Once applied, the material is then wrapped, such
that the inside of the tube now has a uniform spatial frequency throughout its
length. $1$ W light sources were placed equally throughout the tube such that
the illumination intensity is uniform looking down the tube at the top. Here,
we evenly distributed 40 point sources down the $250$ mm tube. A camera placed
$110$ mm above the top of the tube then captured an image of the concentric
circle illumination pattern. This image was then exported to _Python_ where a
normalisation was applied to ensure that the sinusoid pixel values vary across
the maximum range for projection ($0-255$). This process was carried out for
sinusoidal patterns of a fixed spatial frequency at 3 different phase shifts.

Then, the material of the tube can be reverted to the original
scattering/absorbing material described in Sect 2.1. Finally, the normalised
images of the patterned tube are then used as the new projection patterns,
which are projected onto the tube with a $5$ W light source. The new
projection pattern and resultant tube cross section is shown in Fig 3b. This
process can be considered a ‘pre-distortion’ of the projected pattern to
produce more uniform spatial frequencies and could alternatively be computed
using analytically-derived formulae, or by direct inverse computation using a
ray-tracing engine. However, our approach here is very simple to implement and
is highly effective. These modified projection patterns can then be used for
SFDI imaging as there is a now a uniform spatial frequency pattern within the
geometry length.

However, the tubular geometry inherently allows less light to reach the distal
end of the tube and also less light to be reflected back as only a small range
of angles can escape the tube via the opening. The projector placement, at a
large angle to the normal of the tube surface, also creates different
incidence angles along the length of the tube. The empirically-derived LUT
approach still therefore produced substantially different optical properties
along the length of the tube. As the LUT is constructed by taking the mean of
pixels on the tube wall, this is an inevitable result using a single LUT for
the whole tube. To improve these results, we developed a sectioning approach.
Instead of using a single LUT as before, we sectioned the tube (length wise)
into five different longitudinal subsections and used a different LUT for each
section. The five sections were selected as regions that showed a mean
intensity difference $>10$ relative to other sections.

##  3 Results and Discussion

###  3.1 Material simulation

In the DIS simulation, $A_{\rho}$ and $S_{\rho}$ were varied, the reflectance
and transmission values obtained, and input into the IAD algorithm which
computes the optical properties of the material. This was repeated until
optical properties within our range of interest were obtained. Following this
process, it was found that realistic ranges for these parameters were $1\leq
A_{\rho}\leq 100$ and $5000\leq S_{\rho}\leq 20000$. We selected material of
$A_{\rho}=1$ and $S_{\rho}=20000$ with optical properties
$\mu_{a}=0.026\textrm{ mm}^{-1}$ and $\mu_{s}\prime=4.75\textrm{ mm}^{-1}$ to
be the reference material for the SFDI measurements using a Monte Carlo
generated LUT as described in Sect 2.2.1.

Figure 4: (a) Absorption and (b) reduced scattering coefficient vs scattering
density, $S_{\rho}$, calculated for varying absorption densities, $A_{\rho}$,
via IAD algorithm (solid line), SFDI Monte Carlo LUT (dashed line) and SFDI
empirically derived LUT (dotted line). The error bars represent the standard
deviation across the calculated $500\times 500$ pixel optical property map.

The results from the SFDI measurements are shown in Fig 4. We note from Fig
4(a) that there is a discrepancy between the results from the IAD and the SFDI
Monte Carlo LUT calculations, particularly at low $S_{\rho}$ values. We also
note a small cross-coupling between absorption and scattering: increased
scattering density reduces absorption coefficient. The reason for the first of
these two irregularities can be seen by examining the implementation of the
material in Blender. The scattering and absorbing materials are combined using
an ‘add’ shader which equally weights both materials and adds them together.
Because the absorbing component of the material does not display anisotropy,
unlike the scattering component, the combination of the two components
slightly modifies the ‘effective’ anisotropy of the whole material to a value
different from the nominal $g$ of the scattering component. This is the cause
of the small discrepancy between the two methods of measuring scattering. The
cross-coupling between scattering and absorption may arise in part because
increased absorption reduces the accuracy of scattering measurements as there
will be fewer ‘scattering’ events simulated for each ray before it is
absorbed. The effect observed here is comparatively small and so for the
purposes of designing SFDI systems may be neglected. However, we note that
such cross-coupling is observed, often more strongly, when imaging
experimental phantoms.

To account for these discrepancies, we introduce the empirically-derived LUT
described in Sect 2.2.2 with resultant calculated optical properties displayed
in Fig 4. Using this we are able to select optical properties we wish to
simulate and input the associated $A_{\rho}$ and $S_{\rho}$ values. We can
then simulate materials of various shape and give them specific optical
properties, which is ultimately the most relevant feature required for
designing and testing new SFDI systems.

###  3.2 Simulation of typical gastrointestinal conditions in up-close planar
geometry

Fig 5 shows the optical property and height maps generated for a $40$ mm
diameter simulated polyp, with an absorption coefficient higher than that of
surrounding healthy tissue and a reduced scattering coefficient lower than
that of surrounding healthy tissue, simulating squamous cell carcinoma. At
$635$ nm, the absorption coefficient of squamous cell carcinoma is $0.12$
mm-1, which is much greater than that of healthy oesophageal tissue, $0.058$
mm-1, and the reduced scattering coefficient of $0.64$ mm-1, is less than that
of healthy oesophageal tissue, which is typically $0.75$ mm-1[39]. To recreate
this behaviour we therefore simulated healthy background tissue with
$A_{\rho}=25$ and $S_{\rho}=4236$, and we simulated a polyp with $A_{\rho}=96$
and $S_{\rho}=3721$. The simulated polyp has a height (from the base material)
of $40$ mm. Fig 5e shows a successful height map generation from fringe
profilometry measurements. Fig 5c,d,g,h demonstrate successful recovery of
optical properties, demonstrating the success of this method to simulate and
image objects. We note that the empirically derived LUT produces results
closer to the expected values, which is because it accounts for some of the
discrepancies in our tissue simulation as described earlier. However,the Monte
Carlo LUT still provides high contrast between the squamous cell carcinoma and
background, which is arguably more important for wide-field diagnostic
applications. We note that because the surface profile information is
available, the optical property accuracy may be improved by the addition of
surface profile correction for optical property determination[42].

Figure 5: Simulated squamous cell carcinoma as a spheroid on a background of
healthy oesophageal tissue showing (a) white light image and (e) reconstructed
height map with (b) expected absorption coefficient, $\mu_{a}$, (c) $\mu_{a}$
recovered with MC LUT (d) $\mu_{a}$ recovered with empirically derived LUT (f)
expected reduced scattering coefficient, $\mu_{s}\prime$, (g) $\mu_{s}\prime$
recovered with MC LUT and (h) $\mu_{s}\prime$ recovered with empirically
derived LUT. Scale bar = 20mm. Figure 6: Simulated Barrett’s Oesophagus with
mild chronic inflammation (left half of sample) adjacent to healthy
oesophageal tissue (right half of sample) showing (a) white light image (b)
expected absorption coefficient, $\mu_{a}$, (c) $\mu_{a}$ recovered with MC
LUT and (d) $\mu_{a}$ recovered with empirically derivied LUT (e) expected
reduced scattering coefficient, $\mu_{s}\prime$, (f) $\mu_{s}\prime$ recovered
with MC LUT and (g) $\mu_{s}\prime$ recovered with empirically derived LUT.
Scale bar = 20mm.

Fig 6 shows the optical property maps generated for a segment of Barrett’s
oesophagus next to a segment of healthy oesophageal tissue. The tissue
properties are designed to exhibit similar absorption coefficients, while the
reduced scattering coefficient of the simulated BO is less than that of the
adjacent healthy oesophageal tissue. At $635$ nm, the absorption coefficient
of Barrett’s oesophagus with mild chronic inflammation, $0.057$ mm-1 is
similar to that of healthy oesophageal tissue, while the reduced scattering
coefficient of Barrett’s oesophagus with mild chronic inflammation, $0.51$
mm-1 is much less than that of healthy oesophageal tissue[39]. We simulated
healthy oesophageal tissue as before with $A_{\rho}=25$ and $S_{\rho}=4236$,
and Barrett’s oesophagus with $A_{\rho}=22$ and $S_{\rho}=3215$. Fig 6c,d,f
and g show these optical properties are recovered as expected, demonstrating
the capability of the simulation system to differentiate between tissue types.
We observe a close match between the Monte Carlo and empirically derived LUT
results, which may arise due to the ideal geometric condition in which samples
are totally flat, thereby reducing artefacts. We note that at the intersection
region of the two simulated tissue types, there is a spike in both the optical
properties, which results from effects at the interface and a small air gap
that is present.

###  3.3 Simulation of optical property variation in tubular geometry

The results presented here were taken using our modified projection pattern
such that there was a uniform spatial frequency at the material surface
enabling accurate optical property determination. This novel approach allows
for the simulation of a geometry of any size to be generated and the desired
illumination patterns of interest obtained for demodulation at the material
surface. We initially recovered optical properties of the tube material via
nearest neighbour interpolation using our Monte Carlo generated LUT described
in Sect 2.2. We use nearest-neighbour interpolation in this situation for
robustness to points outside the convex hull of the LUT. This gives the
results of Figure 7 and 8 a quantized appearance. However, because it is
relatively easy to generate more sample images using _Blender_ , the sample
set could be expanded such that cubic interpolation could be reliably used
with sufficient computing simulation time available. For the measurements
here, the measured AC modulation amplitude was higher than expected, resulting
in overestimation of scattering coefficients for low $S_{\rho}$ value
materials. This difference arises because of the different geometry in the
tubular case compared to the planar case. Here, the camera and projector are
placed directly at the proximal end of the tube at no offset angle to one
another, meaning the camera may be more prone to detecting specular
reflections. This results in a higher AC modulation amplitude and reduced
scattering coefficient offset.

Figure 7: Comparison of sectioned an un-sectioned empirically derived LUT for
tube wall material of $\mu_{a}=0.107$ mm-1 and $\mu_{s}\prime=4.64$ mm-1 (a)
white light image of tube (b) expected absorption coefficient, $\mu_{a}$, (c)
measured $\mu_{a}$ using un-sectioned LUT (d) measured $\mu_{a}$ using
_sectioned_ LUT (e) expected reduced scattering coefficient, $\mu_{s}\prime$,
(f) $\mu_{s}\prime$ measured using un-sectioned LUT, (g) $\mu_{s}\prime$
measured from _sectioned_ LUT. Tube inner diameter = 20mm.

To overcome this issue, we introduced the empirically derived LUT described
for tubular geometry in Sect 2.2. Nearest neighbour interpolation was again
used in this case. This was somewhat successful however it was not applicable
along the length of the tube. Significant improvement was achieved when
sectioning the LUT as described in Sect 2.4, shown in Fig 7. We calculated,
over six varying material values, that the sectioned LUT method reduced the
calculated absorption coefficient relative error by $50\%$ and reduced the
calculated reduced scattering coefficient relative error from $8\%$ to $3\%$.

Figure 8: Imaging different material types within complex geometry, analogous
to a lumen showing (a) white light image of the tube (b) expected absorption
coefficient of tube material (c) measured absorption coefficient (d) expected
reduced scattering coefficient of tube material and (e) measured reduced
scattering coefficient. Tube inner diameter = 20mm.

Differentiating between optical properties inside lumen can be useful for
detecting different diseases. To show the capability of the method to
differentiate between optical properties inside a lumen, we simulated a tube
with two different materials; three quarters had material properties
$A_{\rho}=100$ and $S_{\rho}=5000$ for expected optical properties of
$\mu_{a}=0.121$ mm-1 and $\mu_{s}\prime=0.93$ mm-1, and the top right quarter
had material properties $A_{\rho}=50$ and $S_{\rho}=20000$ for expected
optical properties $\mu_{a}=0.066$ mm-1 and $\mu_{s}\prime=4.69$ mm-1. The
results are shown in Fig 8. We note that the there is a distinct difference in
material properties from the top right quarter and the rest of the tube.
However, there are artefacts present in the main part of the tube that do not
match the optical properties. Due to the geometry of the system, it is likely
that light is reflecting off multiple surfaces before reaching the camera
which may contribute to these artefacts.

##  4 Discussion

These results demonstrate the capability of the _Blender_ system to simulate
varying tissue types of various shapes, in different imaging geometries, and
the capability of the SFDI imaging system to successfully determine their
optical properties. This can be useful for many applications. Software such as
_OptogenSIM_[45], _FullMonte_[46] and _ValoMC_[47] model Monte Carlo
simulations in biologic al relevant samples, however they suffer from a
variety of limitations; such as incapability to generate realistic, complex
sample geometries within the software and lack of full consideration for
lighting conditions or camera positions. The presented SFDI simulation model
can overcome many of the limitations of existing software by enabling custom
configuration of illumination source and camera position and orientation,
spatial frequency, and illumination pattern. This allows the introduction of
real-world artefacts that help to test the limitations of a new system design.

SFDI can have various sources of errors[48], such as errors arising from
assumptions made with selected light propagation model, differences in optical
properties dependent on depth, divergence of the projection beam and how the
spatial frequency may change with distance from projector to sample. These
sources of error can be simulated in _Blender_ and their optimum solutions
determined in a cost-effective, timely manner. Once solutions are found they
can be applied to current systems to improve system accuracy.

We envisage the use of this system to simulate novel systems, as well as
current systems, as a means to optimise an SFDI system to speed up the
progression of system development and test the possibilities and limitations
of the technique in different situations.

Another potential application of this system could be to generate large data
sets using SFDI, which may then be used in lieu of experimental data,
eliminating the time and money necessary to go into data collection. These
could be used to improve optical property uncertainty measurements by making
large look up tables for specific system setups [49], to correct the optical
property profile measurements using deep learning[50], and has previously
shown promise to train neural networks for SFDI demodulation[25].

##  5 Conclusion

We have further developed an SFDI and fringe profilometry imaging system in
the open-source graphics software _Blender_ which enables the simulation of
typical gastrointestinal conditions with specific absorption and reduced
scattering coefficients in tubular imaging geometries relevant to that of the
gastrointestinal tract. We have shown simulation of objects of specific shape,
size and optical properties and successful imaging of these objects to recover
maps of height, absorption and scattering. We anticipate our results will aid
in the design of a future SFDI systems, e.g. miniaturised systems, by enabling
the testing of different illumination geometries and patterns.

Funding The authors acknowledge support from a UKRI Future Leaders Fellowship
(MR/T041951/1) and an ESPRC Ph.D. Studentship (2268555).

Acknowledgments We acknowledge open-source software resources offered by the
Virtual Photonics Technology Initiative (https://virtualphotonics.org), at the
Beckman Laser Institute, University of California, Irvine.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement The data presented in this study are available
from the the following source: [DOI to be inserted later].

## References

  * [1] C. Holmer, K. S. Lehmann, J. Wanken, C. Reissfelder, A. Roggan, G. Mueller, H. J. Buhr, and J.-P. Ritz, “Optical properties of adenocarcinoma and squamous cell carcinoma of the gastroesophageal junction,” Journal of Biomedical Optics 12, 014025 (2007).
  * [2] D. K. Rex, A. Shaukat, and M. B. Wallace, “Optimal Management of Malignant Polyps, From Endoscopic Assessment and Resection to Decisions About Surgery,” Clinical Gastroenterology and Hepatology 17, 1428–1437 (2019).
  * [3] G. Cummins, B. F. Cox, G. Ciuti, T. Anbarasan, M. P. Desmulliez, S. Cochran, R. Steele, J. N. Plevris, and A. Koulaouzidis, “Gastrointestinal diagnosis using non-white light imaging capsule endoscopy,” (2019).
  * [4] CRUK, “Oesophageal Cancer Statistics,” (2020).
  * [5] Cancer.net, “Colon Cancer Statistics,” (2020).
  * [6] N. Dögnitz and G. Wagnières, “Determination of Tissue Optical Properties by Steady-State Spatial Frequency-Domain Reflectometry,” Tech. rep., Swiss Federal Institute of Technology (1998).
  * [7] S. Gioux, A. Mazhar, B. T. Lee, S. J. Lin, A. M. Tobias, D. J. Cuccia, A. Stockdale, R. Oketokoun, Y. Ashitate, E. Kelly, M. Weinmann, N. J. Durr, L. A. Moffitt, A. J. Durkin, B. J. Tromberg, and J. V. Frangioni, “First-in-human pilot study of a spatial frequency domain oxygenation imaging system,” Journal of Biomedical Optics 16, 086015 (2011).
  * [8] A. Mazhar, S. Saggese, A. C. Pollins, N. L. Cardwell, L. Nanney, and D. J. Cuccia, “Noncontact imaging of burn depth and extent in a porcine model using spatial frequency domain imaging,” Journal of Biomedical Optics (2014).
  * [9] A. D. Bounds and J. M. Girkin, “Early stage dental caries detection using near infrared spatial frequency domain imaging,” Scientific Reports 11 (2021).
  * [10] M. R. Rodríguez-Luna, N. Okamoto, L. Cinelli, L. Baratelli, S. Ségaud, A. Rodríguez-Gómez, D. S. Keller, E. Zonoobi, E. Bannone, J. Marescaux, M. Diana, and S. Gioux, “Quantification of bowel ischaemia using real-time multispectral Single Snapshot Imaging of Optical Properties (SSOP),” Surgical Endoscopy (2022).
  * [11] J. P. Angelo, M. van de Giessen, and S. Gioux, “Real-time endoscopic optical properties imaging,” Biomedical Optics Express 8, 5113 (2017).
  * [12] “Spatial frequency domain imaging,” .
  * [13] S. Gioux, A. Mazhar, and D. J. Cuccia, “Spatial frequency domain imaging in 2019: principles, applications, and perspectives,” Journal of Biomedical Optics 24, 1 (2019).
  * [14] M. Erfanzadeh, S. Nandy, P. D. Kumavor, and Q. Zhu, “Low-cost compact multispectral spatial frequency domain imaging prototype for tissue characterization,” Biomedical Optics Express 9, 5503 (2018).
  * [15] M. B. Applegate, J. P. A. Jr, S. M. Tabassum, K. Tilbury, R. B. Saager, D. Roblyer, M. B. Applegate, K. Karrobi, J. P. A. Jr, W. Austin, S. M. Tabassum, E. Aguénounon, K. Tilbury, R. B. Saager, S. Gioux, and D. Roblyer, “OpenSFDI : an open-source guide for constructing a spatial frequency domain imaging system,” Journal of Biomedical Optics 25 (2020).
  * [16] R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Tech. rep. (1999).
  * [17] S. Lee, L. Mey, A. F. Szymanska, H. S. Takhar, D. J. Cuccia, A. Mazhar, and K. Yu, “SFDI biomarkers provide a quantitative ulcer risk metric and can be used to predict diabetic foot ulcer onset,” Journal of Diabetes and its Complications 34 (2020).
  * [18] J. Ribeiro, S. Nóbrega, and A. Cunha, “Polyps Detection in Colonoscopies,” in _Procedia Computer Science,_ vol. 196 (Elsevier B.V., 2021), pp. 477–484.
  * [19] R. L. Harrison, “Introduction to Monte Carlo simulation,” in _AIP Conference Proceedings,_ vol. 1204 (2009), pp. 17–21.
  * [20] L. Wang, S. L. Jacquesa, and L. Zhengb, “CML-Monte Carlo modeling of light transport in multi-la.yered. tissues,” Tech. rep. (1995).
  * [21] “Blender source code,” (2022).
  * [22] J. Eastwood, H. Zhang, M. Isa, D. Sims-Waterhouse, R. K. Leach, and S. Piano, “Smart photogrammetry for three-dimensional shape measurement,” Proceedings of SPIE p. 11 (2020).
  * [23] Y. Zhang and Q. Fang, “BlenderPhotonics: an integrated open-source software environment for three-dimensional meshing and photon simulations in complex tissues,” Journal of Biomedical Optics 27 (2022).
  * [24] J. Crowley and G. S. D. Gordon, “Simulating medical applications of tissue optical property and shape imaging using open-source ray tracing software,” Proceedings of SPIE 1165707 (2021).
  * [25] A. Osman, J. Crowley, and G. Gordon, “Training Generative Adversarial Networks for Optical Property Mapping using Synthetic Image Data,” Biomedical Optics Express (2022).
  * [26] D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Optics Letters 30, 1354 (2005).
  * [27] M. T. Chen, M. Papadakis, and N. J. Durr, “Speckle illumination SFDI for projector-free optical property mapping,” Optics Letters 46, 673 (2021).
  * [28] D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” Journal of Biomedical Optics 14, 024012 (2009).
  * [29] J. Vervandier and S. Gioux, “Single snapshot imaging of optical properties,” Biomedical Optics Express 4, 2938 (2013).
  * [30] M. T. Chen, F. Mahmood, J. A. Sweer, and N. J. Durr, “GANPOP: Generative Adversarial Network Prediction of Optical Properties from Single Snapshot Wide-field Images,” IEEE Transactions on Medical Imaging pp. 1–10 (2019).
  * [31] S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo Modeling of Light Propagation in Highly Scattering Tissues—I: Model Predictions and Comparison with Diffusion Theory,” IEEE Transactions on Biomedical Engineering 36, 1162–1168 (1989).
  * [32] T. A. Erickson, A. Mazhar, D. Cuccia, A. J. Durkin, and J. W. Tunnell, “Lookup-table method for imaging optical properties with structured illumination beyond the diffusion theory regime,” Journal of Biomedical Optics 15, 036013 (2010).
  * [33] M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Applied Optics 22, 3977 (1983).
  * [34] X. U. Su, “Fourier transform profilometry: a review,” Optics and Lasers in Engineering 35, 263–284 (2001).
  * [35] M. Visentini-Scarzanella, H. Kawasaki, R. Furukawa, M. Bonino, S. Arolfo, G. Lo Secco, A. Arezzo, A. Menciassi, P. Dario, and G. Ciuti, “A structured light laser probe for gastrointestinal polyp size measurement: a preliminary comparative study,” Endoscopy International Open 06, E602–E609 (2018).
  * [36] J. W. Pickering, S. A. Prahl, N. van Wieringen, J. F. Beek, H. J. C. M. Sterenborg, and M. J. C. van Gemert, “Double-integrating-sphere system for measuring the optical properties of tissue,” Journal of Applied Optics pp. 399–410 (1993).
  * [37] C. K. Hayakawa, L. Malenfant, J. Ranasinghesagara, D. J. Cuccia, J. Spanier, and V. Venugopalan, “MCCL: an open-source software application for Monte Carlo simulations of radiative transport,” Journal of Biomedical Optics 27 (2022).
  * [38] D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Tech. rep., Dipartimento di Fisica dell’Universita’degli Studi di Firenze (1997).
  * [39] J. A. Sweer, M. T. Chen, K. J. Salimian, R. J. Battafarano, and N. J. Durr, “Wide-field optical property mapping and structured light imaging of the esophagus with spatial frequency domain imaging,” Journal of Biophotonics (2019).
  * [40] R. Anaparthy and P. Sharma, “Progression of Barrett oesophagus: Role of endoscopic and histological predictors,” Nature Reviews Gastroenterology and Hepatology 11, 525–534 (2014).
  * [41] S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Applied Optics 32, 559 (1993).
  * [42] S. Gioux, A. Mazhar, D. J. Cuccia, A. J. Durkin, B. J. Tromberg, and J. V. Frangioni, “Three-dimensional surface profile intensity correction for spatially modulated imaging,” Journal of Biomedical Optics 14, 034045 (2009).
  * [43] C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry : A review,” Optics and Lasers in Engineering 109, 23–59 (2018).
  * [44] S. Feng, Q. Chen, G. Gu, T. Tao, L. Zhang, Y. Hu, W. Yin, and C. Zuo, “Fringe pattern analysis using deep learning,” Advanced Photonics 1, 1 (2019).
  * [45] Y. Liu, S. L. Jacques, M. Azimipour, J. D. Rogers, R. Pashaie, and K. W. Eliceiri, “OptogenSIM: a 3D Monte Carlo simulation platform for light delivery design in optogenetics,” Biomedical Optics Express 6, 4859 (2015).
  * [46] J. Cassidy, A. Nouri, V. Betz, and L. Lilge, “High-performance, robustly verified Monte Carlo simulation with FullMonte,” Journal of Biomedical Optics 23, 1 (2018).
  * [47] A. A. Leino, A. Pulkkinen, and T. Tarvainen, “ValoMC: a Monte Carlo software and MATLAB toolbox for simulating light transport in biological tissue,” OSA Continuum 2, 957 (2019).
  * [48] N. Bodenschatz, A. Brandes, A. Liemert, and A. Kienle, “Sources of errors in spatial frequency domain imaging of scattering media,” Journal of Biomedical Optics 19, 071405 (2014).
  * [49] V. Pera, K. Karrobi, S. Tabassum, F. Teng, and D. Roblyer, “Optical property uncertainty estimates for spatial frequency domain imaging,” Biomedical Optics Express 9, 661 (2018).
  * [50] E. Aguénounon, J. T. Smith, M. Al-Taher, M. Diana, X. Intes, and S. Gioux, “Real-time, wide-field and high-quality single snapshot imaging of optical properties with profile correction using deep learning,” Biomedical Optics Express 11, 5701 (2020).