Image Anal Stereol 2018;37:191-204 doi:10.5566/ias.1869
Original Research Paper
VOLUME ESTIMATION FROM SINGLE IMAGES: AN APPLICATION
TO PANCREATIC ISLETS
JIŘÍ DVOŘÁKB,1,2, JAN ŠVIHLÍK2,3,4, JAN KYBIC2, BARBORA RADOCHOVÁ5, JIŘÍ
JANÁČEK5, JAROMÍR KUKAL6, JIŘÍ BOROVEC2 AND DAVID HABART7
1Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles
University in Prague, Sokolovská 83, Prague, Czech Republic; 2Biomedical Imaging Algorithms (BIA) group,
Center for Machine Perception, Faculty of Electrical Engineering, Czech Technical University in Prague,
Technická 2, Prague, Czech Republic; 3Department of Computing and Control Engineering, Faculty of
Chemical Engineering, University of Chemistry and Technology, Technická 5, Prague, Czech Republic;
4Multimedia Technology Group, Department of Radioelectronics, Faculty of Electrical Engineering, Czech
Technical University in Prague, Technická 2, Prague, Czech Republic; 5Department of Biomathematics,
Institute of Physiology of the Czech Academy of Sciences, Vı́deňská 1083, Prague, Czech Republic;
6Department of Software Engineering, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Trojanova 13, Prague, Czech Republic; 7Institute for Clinical and Experimental Medicine,
Vı́deňská 1958/9, Prague, Czech Republic
e-mail: dvorak@karlin.mff.cuni.cz, jan.svihlik@fel.cvut.cz, kybic@fel.cvut.cz, barbora.radochova@fgu.cas.cz,
jiri.janacek@fgu.cas.cz, jaromir.kukal@fjfi.cvut.cz, jiri.borovec@fel.cvut.cz, david.habart@ikem.cz
(Received December 12, 2017; revised June 1, 2018; accepted June 27, 2018)
ABSTRACT
The present paper deals with the problem of volume estimation of individual objects from a single 2D view.
Our main application is volume estimation of pancreatic (Langerhans) islets and the single 2D view constraint
comes from the time and equipment limitations of the standard clinical procedure.
Two main approaches are followed in this paper. First, two regression-based methods are proposed, using
a set of simple shape descriptors of the segmented image of the islet. Second, two example-based methods are
proposed, based on a database of islets with known volume. For training and evaluation, islet volumes were
determined by OPT microscopy and a stereological volume estimation using the so-called Fakir probes.
The performance of the single image volume estimation methods is studied on a set of 99 islets from human
donors. Further experiments were also performed on a stone dataset and on synthetic 3D shapes, generated
using a flexible stochastic particle model. The proposed methods are fast and the experimental results show
that in most situations the proposed methods perform significantly better than the methods currently used in
clinical practice, which are based on simple spherical or ellipsoidal models.
Keywords: Fakir probes, pancreatic islets, single image, volume estimation, 2D projection.
INTRODUCTION
Pancreatic islets can be isolated from a surgically
removed pancreas, to be used to treat diabetes by
either allogeneic or autologous islet transplantation,
respectively (Shapiro et al., 2000; Suszynski et al.,
2014), which is a less invasive treatment than a full
pancreas transplantation. Clinical outcome of such
treatment is known to depend on several variables,
including the total islet volume in the graft relative
to the body weight of the recipient and the islet size
distribution (Lehmann et al., 2007; Suszynski et al.,
2014). It is therefore desirable to characterize these
properties before the transplantation, ideally using an
automatic method (Buchwald et al., 2016; Habart et
al., 2016).
The standard approach is to take a few samples
from the graft, stain them for insulin producing cells,
acquire 2D images using a standard microscope,
identify the islets, and estimate their volumes
assuming a spherical or ellipsoidal shape. The
accuracy is not very high, as the actual islet shape
is more variable. While it is not possible to acquire
3D images of the islets during the surgery because of
logistical, financial, and time constraints, it is possible
to acquire such 3D images for a limited number of
islets for the purpose of method development. We
applied optical projection tomography (Alanentalo et
al., 2007) to individual islets and used the acquired
images to obtain ground truth islet volumes.
From the stereology point of view, the task is
to estimate the volume of an object from a known
population given its one 2D image. This is a very
challenging and ill-defined task, and only a few
methods were proposed in the literature (see below).
In particular, we assume that binary segmentations
191
DVOŘÁK J ET AL: Islet volumes from single images
of the individual islets are available and that these
2D segmentations correspond to thresholded weighted
projections of the 3D objects (see below for details).
In this study we use manual segmentations for the
estimation and compare the results with those based
on automatic segmentations of the islet images (see
below).
There are several non-standard aspects of this
problem in comparison to standard stereology
tasks (Baddeley, Jensen, 2005; Howard, Reed, 2005).
First, the population of objects is very heterogeneous,
we cannot assume they are multiple observations
of essentially the same object. Second, the islet
orientation is not isotropically random but some
orientations are preferred due to islets floating in
the liquid medium or lying on the glass bottom
(depending on the relative density). Consequently,
some parts of the islets may never be observed. Third,
we observe neither a parallel projection, nor a cross-
section but a convolution with the microscope point
spread function, which is something between the two.
It resembles a parallel projection but with increasing
blur and decreasing influence of points away from
the focal plane. Moreover, as the islets are relatively
thick and opaque, the interior of the objects cannot
be observed. We therefore assume the objects to be
homogeneous and only use information from the shape
of the 2D projection boundary.
The classical Ricordi method (Ricordi et al., 1990)
used in clinical practice quantizes manually estimated
islet diameters into bins. A volume corresponding to
each bin is calculated assuming a spherical shape.
More advanced approaches fit an ellipse to the 2D
shape and assume the islet to be a prolate ellipsoid
with the longest axis parallel to the observation
plane (Girman et al., 2003). However, the real shapes
of the pancreatic islets are not always well represented
by these simple models.
A regression approach similar to ours but with
different descriptors was used to estimate limestone
particle masses in the construction industry (Banta
et al., 2003). Chaarani et al. (2013) report
a correlation between volume and area measurements
for brain ventricles but do not attempt a regression.
Multicellular cancer spheroid volumes are estimated
using a method similar to the classical vertical
planar rotator method (Piccinini et al., 2016). Other
applications of single image volume estimation
include food volume estimation, see, e.g., Xu et
al. (2013) or Zhang et al. (2011), using 3D stereo
reconstruction or virtual reality model fitting. Stereo
reconstruction cannot be used for our application
because only one image is available, while islet shapes
are not sufficiently distinguished and too variable for
the model fitting.
MATERIAL AND METHODS
PANCREATIC ISLET DATASET
A sample of human pancreatic islets was obtained
from 8 different donors. The total number of islets in
this study was 99. The islets were chosen to cover
wide range of sizes and shapes and do not constitute
a random sample from an islet population.
The islets were isolated using collagenase based
method (Berkova et al., 2005) and immediately placed
in a standard cell culture. Fresh individual islets
(up to 4 hours after the isolation) were briefly
stained with Dithizone (which stains the insuline
producing beta cells red) and gently washed in
Hank’s buffered solution (HBSS) supplemented with
0.2% albumin. Washed islets were imaged using
inverted microscope CXK41 (Olympus), 4× or 10×
magnification objective, and a 3MP CMOS camera
(Infinity1 or Bresser), in order to obtain a single 2D
image (see Fig. 1).
Immediately after the microscopy image
acquisition, each individual islet was gently placed
into a 2% agarose with low gelling temperature
(A9414, Sigma) at 37◦C and allowed to solidify at
room temperature. The agarose block was mounted
on the sample holder of a custom made OPT
(optical projection tomography) scanner (Politecnico
di Milano) equipped with an EM CCD camera
(Andor) and PlanApo Infinity-Corrected objective
(10x, Edmund Optics). The scanner is designed to
rotate a specimen around the vertical axis and acquire
a set of images during one turn. For every islet,
400 image projections were obtained using bright
field illumination with a rotation step of 0.9◦ and
a resolution of 1004×1002 pixels.
3D volumes were created by a tomographic
reconstruction using a free software package
NRecon (Skyscan), implementing a convolution-
backprojection reconstruction algorithm (Feldkamp
et al., 1984). Islet volume was estimated from the 3D
reconstruction to serve as ground truth as described in
section “Volume estimation from 3D volumes”.
STONE IMAGE DATASET
We obtained colour images of 100 small stones
(one image per stone) lying in a natural position
on a flat white surface. The images were taken by
a digital camera from a fixed position above the stones.
192
Image Anal Stereol 2018;37:191-204
Fig. 1. Examples of the original dark field microscopy images of dithizone-stained individual islets (left), their
manual segmentation (middle) and their automatic segmentation (right). The white horizontal bar has length
100 µm.
193
DVOŘÁK J ET AL: Islet volumes from single images
Fig. 2. Examples of stone images (top) and the corresponding segmentations (bottom).
Examples of such image are given in Fig. 2. The true
volume of each stone was determined by placing the
stone into a graduated cylinder and measuring the
volume of water necessary to fill the cylinder to a given
level. The volume of a stone used as a ground truth was
the mean of three measurements.
SIMULATED DATASETS
We generated islet-like artificial particles from
a flexible stochastic particle model inspired by (Ziegel
et al., 2015). The model was based on ellipsoidal
particles being inflated by a Lévy random field on the
unit sphere. We considered star-shaped particles, i.e.,
we assumed that all the points on the boundary of the
particle can be seen directly from the origin O of the
coordinate system. Hence, the boundary of the particle
K was fully determined by the so-called radial function
RK(u),u ∈ S2, measuring the distance from O to the
boundary of K in the direction u. Here S2 denotes the
unit sphere in R3.
The model for the radial function of the random
particle considered in (Ziegel et al., 2015) is given by
RK(u) = RK0(u) · ε(u),u ∈ S2, where RK0 is the radial
function of a fixed star-shaped particle K0 and ε > 0
is an isotropic Lévy random field on S2, constructed
using a Gamma Lévy basis with parameter θ > 0 and
the von Mises-Fisher kernel with parameter κ > 0
(Ziegel et al., 2015). In our study, we used a slightly
modified version of this model, namely RK(u) =
RK0(u) · (1+ ε(u)),u ∈ S2, where K0 was an ellipsoid
(with semiaxes lengths equal to 1,1,and a) and ε was a
Lévy random field as above. In this model it holds that
K0 ⊂ K which seems more appropriate for modeling
of pancreatic islets. Note that by fixing two of the
three semiaxes lengths, we refrained from evaluating
the effect of the absolute size of the particles, making
it easier to study the effect of the shape.
In cooperation with two medical experts, we
identified four combinations of the model parameters
(a,κ,θ) for which the shapes of the simulated
particles resemble the shapes of pancreatic islets
encountered in practice. The four combinations of
parameter values correspond to four sub-populations
of real pancreatic islets, described as “small islets”
with (a1,κ1,θ1) = (6/5,5/6,5/6), “medium-sized
islets” with (a2,κ2,θ2) = (6/5,2,2), “large and flat
islets” with (a3,κ3,θ3) = (7/10,5/2,5) and “large and
elongated islets” with (a4,κ4,θ4) = (18/10,5/4,2).
We disregarded other types of real islets which are
difficult to describe or simulate, e.g., “large and
irregular islets”. For each of the four combinations
of parameter values we generated 1000 simulated
particles, recording their true volumes.
In order to assess the performance of the volume
estimation methods also outside the context of smooth
particles resembling the shape of pancreatic islets
we simulated also 1000 rough particles with sharp
points and edges. The particles were generated by
forming a convex hull of a random number of points
(the number being uniformly distributed between
20 and 50) uniformly distributed in a cube, hence
obtaining random polyhedra. They somewhat resemble
the limestone particles studied in Banta et al. (2003).
In order to mimic the 2D imaging of real objects
(islets, stones), we determined for each simulated
particle its orientation in the three-dimensional space
with the lowest possible potential energy with respect
to the horizontal plane. With this orientation we
194
Image Anal Stereol 2018;37:191-204
(a) (b) (c) (d) (e)
Fig. 3. Examples of projections of the simulated particles. Column (a) “small islets” population, (b) “medium-
sized islets” population, (c) “large and flat islets” population, (d) “large and elongated islets” population and
(e) “limestone particles” population. Note that the relative sizes of the first four populations of the simulated
particles do not reflect the relative sizes of the corresponding populations of human islets. Hence the particles in
different columns above look similar in size.
recorded a binary image of the projection of the
particle to the horizontal plane. Examples of such
images for the four populations of smooth particles and
one population of rough particles are given in Fig. 3.
VOLUME ESTIMATION FROM 3D
VOLUMES
To provide a ground truth estimate of the
pancreatic islet volumes, we used the Fakir method
based on a randomized virtual spatial grid of lines
(Kubı́nová, Janáček, 2001) to find islet volumes from
their 3D OPT reconstructions. Islet volumes estimated
by this method will be denoted VF .
The Fakir method is based on the fact that the mean
length of the grid lines inside the objects multiplied by
grid length density is equal to the volume of the object.
The mean is assessed with respect to the random
position of the grid. The method is available as an
interactive program.
We also tried to estimate the volume directly
from the segmented 3D reconstruction using level set
segmentation as implemented in ImageJ (Yoo, 2004).
However, the islets were very large and dense and
not enough light was transmitted through, violating
the reconstruction assumptions and leading to many
reconstruction artifacts, making the segmentation
very challenging. Alternatively, the volume can be
estimated directly from the projections (Švihlı́k et al.,
2017). This method performed better but was again not
sufficiently reliable to serve as the ground truth for our
purposes.
AUTOMATIC VOLUME ESTIMATION
FROM 2D IMAGES
All subsequently described methods assume that
binary masks of individual objects have been extracted
from the images. The segmentation of stone images
was performed by thresholding the gray-scale image
while the binary masks are available directly for the
simulated particles.
Pancreatic islets can be segmented automatically
(Habart et al., 2016; Švihlı́k et al., 2016). In this
work we used expert manual segmentations and an
automatic supervised segmentation method (Borovec
et al., 2017) based on superpixels, color features,
random forest classifier, and GraphCut regularization.
This enabled evaluation of the effect of segmentation
errors. The automatic method was trained on
expert manual segmentations using a cross-validation
procedure (described in the Results section). Note that
because the reference OPT method cannot distinguish
between islets and exocrine tissue (red and yellow in
195
DVOŘÁK J ET AL: Islet volumes from single images
Fig.1), both were considered as foreground for the
purpose of this work.
For easier reference, Table 1 summarizes the
names of the volume estimation methods considered in
this paper together with their respective abbreviations.
Table 1. Names and abbreviations of volume
estimation methods. VF is used here as the ground
truth. VS,VE were proposed earlier for pancreatic
islets. VP,VB were introduced in other contexts. The
remaining methods are proposed here.
Method Abbreviation
Fakir method VF
Spherical model VS
Ellipsoidal model VE
Piccinini method VP
Banta method VB
Regression method for logarithm of volume VLS
Regression method for volume VNLS
Database method with scaling VD1
Database method without scaling VD2
Spherical model
The most simple approach to volume estimation
from a single image is based on the assumption of
a spherical shape of the islets (Niclauss et al., 2008).
This is a slight and natural modification of the original
Ricordi approach (Ricordi et al., 1990). First, the area
A of the segmented object is automatically determined
and the diameter d of a circle with the same area
is calculated as d =
√
4A/π . Then, the volume of
the islet is estimated as the volume of a sphere with
diameter d. Islet volume estimated by this method will
be denoted VS:
VS =
π
6
d3 =
4
3
√
π
A3/2. (1)
Ellipsoidal model
This approach generalizes the shape model to
a prolate ellipsoid and assumes that its maximal
projection is observed. Given the segmented object we
first calculate the major and minor axes lengths a and
b, a < b, of an equivalent fitted ellipse – an ellipse
with the same moments up to the second order which
is then uniformly scaled to have the same area as the
segmented object (as implemented in ImageJ, see also
Cramér (1946, p. 283)).
The volume of the islet is then estimated as
the volume of a prolate ellipsoid with axes lengths
(a,a,b), see Girman et al. (2003). Islet volume
estimated by this method will be denoted VE :
VE =
π
6
a2b. (2)
Regression methods
After fitting the ellipse to the 2D shape as
described previously and obtaining axes lengths a and
b, a < b, the 3D volume is sought in the form
V = γaαbβ . (3)
This formula generalizes the spherical and ellipsoidal
models described above (thanks to the relation A =
πab/4) and can describe also other situations, such
as ellipsoids with the longest axis perpendicular to
the imaging plane. We shall describe two different
methods of determining the parameters α,β ,γ (other
approaches were tested but did not bring any further
improvement).
Given a training dataset of M objects described by
3D volumes Vi and axis lengths ai, bi, the simplest
way to estimate the parameters α , β , γ is to take the
logarithm of (3) and minimize the sum of squares:
JLS =
M
∑
i=1
(
logγ +α logai +β logbi− logVi
)2
. (4)
This is a standard linear least squares problem in
variables α , β and logγ .
While the previous method is simple and fast, the
criterion JLS being minimized does not necessarily
correspond to any practically relevant quantity. As an
alternative, we have minimized the sum of squares of
the relative volume estimation error:
JR =
N
∑
i=1
(
Vi− γaαi b
β
i
Vi
)2
. (5)
This is a non-linear least squares problem which needs
to be solved iteratively, for example by the Levenberg-
Marquardt method.
Derivatives can be calculated analytically but even
the finite difference approximation seems to work
well. In practice the optimization procedure converges
quickly and reliably and sometimes brings a small
accuracy improvement over the linear regression (4).
Object volumes calculated by plugging in the
estimated values of α,β ,γ minimizing (4) and (5)
into (3) will be denoted VLS and VNLS, respectively.
196
Image Anal Stereol 2018;37:191-204
Database method
In the database method, the estimate of the
unknown volume of a new object is calculated from
the known volume of the most similar object in the
database, with a correction based on the object area
in the 2D image. The database contains not only the
original images but also their reflections along the
horizontal axis, vertical axis, and both axes.
Let the binary database images be denoted fi,
i = 1, . . . ,M, with known volumes Vi. To estimate
a volume of an object from a new image f, we find
a database image fi with the lowest dissimilarity,
corresponding to the relative number of pixelwise
differences between the two binary images, i =
argmin j=1,...,M
∥∥f− f j∥∥/‖f‖, where ‖f‖ corresponds to
the area of the object in image f.
The volume of the new islet is then estimated as
VD =Vi
(
‖f‖
‖fi‖
)3/2
. (6)
The scaling factor in (6) relates the volume to
cross-section area. The exponent 3/2 is exact for the
spherical case, see (1). It corresponds to the situation
when the (unobserved) height of the object grows at the
same speed as the other two (observed) dimensions.
We consider two versions of the database method.
In the first version, both the database and the input
images are normalized by rotation and scaling, so
that the major axis of the best-fitting ellipse is
aligned horizontally and the area of the segmented
object is constant. The scaling factor in (6) needs to
be calculated from the images before scaling. The
estimates obtained by this method will be denoted VD1.
In the second version, the images are only rotated but
not scaled. The estimates obtained by this method will
be denoted VD2.
For illustration of the database method VD1 see the
top part of Fig. 4: (a) segmented image of the islet
whose volume is to be estimated, (b) the same, rotated
and scaled to the reference position, (c) segmented
image of the islet identified to be the most similar, (d)
the same, rotated and scaled to the reference position.
Bottom part of Fig. 4 shows an illustration of VD2
where (e-h) corresponds to (a-d) above with the only
difference that scaling is not used.
Other reference methods
For the sake of comparison we also consider
two volume estimation methods based on a single
2D image that were proposed in other contexts and
were not used so far for pancreatic islets. The first
method (Piccinini et al., 2016) is used in the context
of cancer spheroid volume estimation and it is closely
related to the classical vertical planar rotator (Jensen,
Gundersen, 1993). Details of the method can be found
in the original paper (Piccinini et al., 2016). We used
the publicly available source code for the Piccinini
method and the estimates obtained in this way will
be denoted VP. Second, we consider the method of
Banta et al. (2003). It is a regression-based method
with descriptors combining the axes lengths of the
best-fitting ellipse and the mean and variance of the
distances from the centroid to regularly sampled points
on the particle boundary. Details of the method can be
found in the original paper (Banta et al., 2003). The
estimates obtained by this method will be denoted VB.
RESULTS
Both the regression and database methods need
a training dataset of 2D images of objects with
known 3D volumes. To evaluate the performance of
the estimation methods, we applied a cross-validation
scheme, repeatedly dividing the full dataset into
a training set and a testing set. For the human islets
dataset (99 islets, Fig. 1) we used a 9-fold cross-
validation scheme. For the stones dataset (100 stones,
Fig. 2) and the simulated particles (1000 particles in
each population, Fig. 3) we applied a 10-fold cross-
validation scheme.
We compared the performance of different volume
estimation methods based on the relative bias (RB)
and the mean relative squared error (MRSE) defined
as follows:
RB(VM) =
1
N
N
∑
i1
VM(i)−VF(i)
VF(i)
, (7)
MRSE(VM) =
√√√√ 1
N
N
∑
i1
(
VM(i)−VF(i)
VF(i)
)2
, (8)
where VM is the volume estimation method in question
(i.e., VS,VE , . . .), VF(i) is the reference volume of
the i-th object and N is the number of objects in
the dataset. Note the square root in the definition
of MRSE; in this way the values are given in the
same units as the measurements and the values of RB.
Additional robust error measures were calculated but
are not reported here since they did not provide any
new insight.
We also compared the performance of the
volume estimation methods using formal hypotheses
testing comparing pairs of different methods. When
197
DVOŘÁK J ET AL: Islet volumes from single images
original normalized
(a) (b)
(c) (d)
original normalized
(e) (f)
(g) (h)
Fig. 4. Top part (a-d): illustration of the database method VD1. Bottom part (e-h): illustration of the database
method VD2. For details see the main text.
198
Image Anal Stereol 2018;37:191-204
comparing, e.g., VLS against VS we performed the sign
test on the relative error differences
Z(i) =
|VLS(i)−VF(i)|
VF(i)
− |VS(i)−VF(i)|
VF(i)
,
for i = 1, . . .N. (9)
The reason for choosing the sign test over, say,
Wilcoxon test was that the individual values of Z(i)
are independent but we do not consider them to be
identically distributed – the performance of the volume
estimation methods may vary across different types of
objects (size, shape). Also, the human islets in this
study did not constitute a random sample but were
chosen in a somewhat systematic way, see section
“Pancreatic islet dataset”.
The assumptions of classical methods such as
ANOVA were not fulfilled either. We used the Holm
correction to adapt the set of the resulting p-values
for multiple testing (Holm, 1979). For each dataset
we performed two groups of tests. The first group
consisted of 16 one-sided sign tests, with the Holm
correction, of the methods proposed in this paper
(VLS,VNLS,VD1,VD2) against the reference methods
(VS,VE ,VP,VB). One-sided tests were chosen because
they enable us, when rejecting the null hypothesis, to
establish superiority of our proposed method over a
reference method. The second group consisted of 6
two-sided sign tests, with the Holm correction, of all
pairs of the methods proposed in this paper. Two-sided
tests were chosen because we see no apriori reason
why any of the methods should outperform the others.
All tests were performed at a 0.05 significance level.
EXPERIMENTS ON THE PANCREATIC
ISLET DATASET
The relative classification error of the automatic
segmentation method was 1.5% with respect to all
pixels, which corresponds to 5% with respect to
the number of object (islet) pixels, using the cross-
validation scheme described above.
For the model (3), the regression methods provided
the following parameters, estimated from the whole
set of 99 islets: α = 1.645,β = 1.097,γ = 1.778 for
the least-squares criterion (4) (the VLS method) and
α = 1.667,β = 1.076,γ = 1.722 for the relative error
criterion (5) (the VNLS method), respectively. It is
instructive to see that these values are actually quite
far from the classical ellipsoidal model (2) which
corresponds to α = 2, β = 1, γ ≈ 0.523. This means
that an ellipsoid is a poor approximation of the islet
shape.
Tables 2 (manual segmentation) and 3
(automatic segmentation) summarize the numerical
characteristics of the different estimators, determined
by cross-validation where appropriate. Most of the
estimations method perform better on the manual
segmentations in terms of MRSE. A notable exception
is the Banta method (VB).
For both manual and automatic segmentation, it
is clear that the spherical model (VS) is too simple
to provide useful estimates of islet volumes – it had
a large positive bias (approx. 30%) and, as a result,
large MRSE. Compared to the spherical model, the
ellipsoidal model (VE) had a smaller bias (14–15%)
but the variability was still rather high. The Piccinini
method (VP) performed similarly to VE while the Banta
method (VB) performed slightly better than VE .
The regression methods (VLS, VNLS) were
essentially unbiased and their relative error was
comparable. Both regression methods outperformed
the spherical and ellipsoidal models and the Piccinini
and Banta methods in terms of both RB and MRSE.
The method with the smallest value of MRSE was
VNLS.
The database methods did not perform as well
as the regression methods. They were slightly better
in comparison with the ellipsoidal model in terms of
MRSE and they had a smaller bias (4–6%). They were
slightly outperformed by the Banta method.
Fig. 5 shows the estimates for individual islets
plotted against their ground truth volume VF for the
proposed methods VLS,VNLS,VD1,VD2 and the reference
methods VS,VE . This provides graphical comparison of
the bias and variability of the estimates. To provide
even clearer perspective we report here the number
of islets (out of 99) for which it holds that |VM(i)−
VF(i)|/VF(i) ≤ 0.15, i.e., for which the achieved
precision is 15% or better. Here VM(i) is the volume
estimate for the i-th islet obtained by a particular
method and VF(i) is its ground truth volume. The
numbers corresponding to the respective methods are
19 (VS), 46 (VE), 67 (VLS), 71 (VNLS), 58 (VD1) and 51
(VD2).
For the manual segmentation, in the first group
of sign tests all methods proposed here were found
to perform significantly better than VS; VLS and
VNLS were found to perform significantly better than
VE . In the second group of sign tests a significant
difference was found between the pairs (VLS,VD2) and
(VNLS,VD2), the former performing better. No other
differences between two methods were significant. The
observations made in this paragraph for the manual
segmentation apply also to the case of automatic
segmentation with a single exception: in the first
199
DVOŘÁK J ET AL: Islet volumes from single images
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
S
 [n
L]
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
E
 [n
L]
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
LS
 [n
L]
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
N
LS
 [n
L]
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
D
1 
[n
L]
0 20 40 60 80 100
0
20
60
10
0
VF [nL]
V
D
2 
[n
L]
Fig. 5. Volume estimates for individual islets plotted against their ground truth volume VF for the proposed
methods VLS,VNLS,VD1,VD2 and the reference methods VS,VE . The dashed line represents the identity mapping
y = x.
group of sign tests VNLS was not found to perform
significantly better than VE .
For the two database methods (with manual
segmentation) we also investigated the approach of
choosing the exponent in (6) based on minimizing the
MRSE on training data instead of fixing it at 1.5. For
VD1 the optimal scaling factor was 1.43, resulting in
MRSE = 0.188 instead of 0.206 for 1.5. For VD2 the
optimal scaling factor was 1.25, resulting in MRSE =
0.217 instead of 0.218 for 1.5. However, the limited
size of our dataset does not allow us to generalize these
observations and we therefore kept the exponent at 1.5.
We also investigated the possibility to modify
the database method (with manual segmentation) by
using K > 1 nearest neighbors instead of just one
and obtaining the volume estimate as a weighted
average. The best results were obtained for K = 10 and
exponentially decreasing weights. For the VD1 method
we obtained MRSE = 0.192 instead of 0.206. For the
VD2 method we obtained MRSE = 0.183 instead of
0.218. We remark that even with this modification the
two database methods did not perform as well as the
regression methods in terms of both MRSE and RB.
EXPERIMENTS ON THE STONE
DATASET
Table 4 summarizes the numerical characteristics
of the different estimators considered here. While
VS and VE exhibited large positive bias and large
variability, the regression methods were virtually
200
Image Anal Stereol 2018;37:191-204
Table 2. Comparison of the volume estimation methods for the islet dataset (manual segmentation).
VS VE VLS VNLS VD1 VD2 VP VB
RB 0.304 0.140 0.012 -0.017 0.052 0.055 0.114 0.030
MRSE 0.388 0.242 0.155 0.151 0.206 0.218 0.214 0.180
Table 3. Comparison of the volume estimation methods for the islet dataset (automatic segmentation).
VS VE VLS VNLS VD1 VD2 VP VB
RB 0.313 0.152 0.012 -0.019 0.061 0.036 0.117 0.029
MRSE 0.392 0.243 0.159 0.154 0.222 0.228 0.225 0.173
Table 4. Comparison of the volume estimation methods for the stones dataset.
VS VE VLS VNLS VD1 VD2 VP VB
RB 0.858 0.630 0.0194 -0.035 0.028 0.074 0.634 0.051
MRSE 0.935 0.730 0.199 0.190 0.380 0.348 0.734 0.221
Table 5. Comparison of the volume estimation methods for the simulated datasets. Population 1 = small islets; 2
= medium-size islets; 3 = large and flat islets; 4 = large and elongated islets; 5 = rough (limestone) particles.
Population 1 VS VE VLS VNLS VD1 VD2 VP VB
RB 0.162 0.049 0.000 -0.001 0.001 0.002 0.047 0.001
MRSE 0.165 0.057 0.023 0.023 0.032 0.032 0.055 0.024
Population 2 VS VE VLS VNLS VD1 VD2 VP VB
RB 0.365 0.161 0.002 -0.004 0.008 0.009 0.157 0.005
MRSE 0.380 0.186 0.067 0.067 0.091 0.092 0.182 0.071
Population 3 VS VE VLS VNLS VD1 VD2 VP VB
RB 0.798 0.605 0.003 -0.006 0.013 0.014 0.605 0.007
MRSE 0.813 0.622 0.078 0.077 0.113 0.114 0.622 0.085
Population 4 VS VE VLS VNLS VD1 VD2 VP VB
RB 0.669 0.072 0.001 -0.001 0.002 0.003 0.070 0.001
MRSE 0.677 0.085 0.036 0.036 0.046 0.049 0.083 0.077
Population 5 VS VE VLS VNLS VD1 VD2 VP VB
RB 0.355 0.257 0.020 -0.027 0.043 0.043 0.258 0.027
MRSE 0.428 0.335 0.214 0.164 0.227 0.217 0.335 0.176
201
DVOŘÁK J ET AL: Islet volumes from single images
unbiased and showed rather small MRSE. The
database methods again exhibited small positive bias
and values of MRSE about twice as high as the
regression methods but smaller than VS and VE . The
Piccinini method VP performed similarly to VE ; the
Banta method VB performed slightly worse than the
proposed regression methods.
In the first group of tests all methods proposed here
were found to perform significantly better than three
of the reference methods (VS,VE ,VP). No significant
differences were found between the proposed methods
and the VB method. In the second group of tests no
significant difference was found between any pair of
methods proposed in this paper.
EXPERIMENTS ON SIMULATED
DATASETS
Table 5 summarizes the numerical performance
characteristics of the different estimators on the five
populations in the simulated datasets.
We first discuss the populations 1 to 4 (smooth,
resembling the shapes of real islets). In general, the
spherical and ellipsoidal models again exhibited large
positive bias and large variability, higher when the
population deviated more from the shape assumptions
of the model (e.g., in population 3 with large and
flat islets the ellipsoidal model overestimated the
volume systematically because the unobserved height
of the particles was smaller than the observed minor
semiaxis length). On the other hand, population 4
with large and elongated particles fitted the assumption
of the ellipsoidal model rather well and resulted in
only a small positive bias (approx. 7%) and a rather
low variability. Again, the VP method performed
very similarly to the VE method in all situations.
Another general observation is that the regression
methods, the database methods and the VB method
were virtually unbiased and that the two proposed
regression methods outperformed the two database
methods and the VB method in terms of MRSE. The
VB method outperformed the database methods in
terms of MRSE except for the population 4. The
differences in numerical characteristics of VLS and
VNLS were negligible for all the four populations. The
same applies to VD1 and VD2. It is interesting to note
that the most challenging population for the estimation
methods was population 3 with flat particles.
In the first group of sign tests all methods proposed
here were found to perform significantly better than
three of the reference methods (VS,VE ,VP) for all
populations of smooth simulated particles. The VLS
method was found to perform significantly better than
VB in all populations. The VNLS method was found to
perform significantly better than VB in populations 3
and 4. The database methods VD1,VD2 were found to
perform significantly better than VB for population 4.
The second group of sign tests identified no
significant difference between VD1 and VD2 (all
populations) and between VLS and VNLS (populations
1,3 and 4). In all other cases the difference between
the methods was significant. However, bear in mind
that the sample size was 10 times larger than for the
islets and stones datasets. In cases where the difference
was found to be significant, the regression methods
always outperformed the database methods and VLS
outperformed VNLS in terms of the number of times the
Z(i)’s were positive/negative, see (9).
Concerning the rough particles (population 5), we
observed again that the spherical and ellipsoidal model
(methods VS,VE) did not capture well the shape of
the particles, resulting in large positive bias and large
variability of the estimates. The same holds true for
the VP method. The database methods VD1,VD2 were
again slightly outperformed by the regression methods
VLS,VNLS. It is interesting to note that the VB method
performed very well in this case. This is not surprising
as it was developed for volume estimation of rough
limestone particles. Nevertheless, the VB method was
outperformed by the VNLS method proposed here.
In the first group of sign tests, for the rough
particles, all methods proposed here were found
to perform significantly better than three of the
reference methods (VS,VE ,VP) and they did not
perform significantly better than VB – they in fact
performed worse. In the second group of sign tests
VNLS performed significantly better than VLS,VD1,VD2
and VD2 performed significantly better than VD1. No
other significant differences were found.
DISCUSSION
In this study we proposed four methods for volume
estimation of individual objects from their single 2D
images. All proposed methods (VLS,VNLS,VD1,VD2)
outperformed the currently used method for islet
volume estimation (VE) in terms of relative bias and
mean relative squared error (MRSE) on all three
datasets. On the pancreatic islets dataset, as well
as on the stones dataset and the simulated particles
dataset, we demonstrated a significant performance
improvement over VE for the two proposed regression
methods methods (VLS,VNLS) by means of sign tests
with a correction for multiple comparisons. The
database methods (VD1,VD2) performed significantly
better than VE in the stones and simulated particles
datasets, but not on the pancreatic islets.
202
Image Anal Stereol 2018;37:191-204
From the regression methods we suggest using
VNLS as it explains the relevant quantity (volume)
directly, as opposed to VLS which explains the
logarithm of the volume instead. The experiments
on the stone dataset and simulated particles indicate
that the VNLS method is able to capture enough
information about the population in question to
provide reliable estimates. However, if speed and
simplicity is preferred, then VLS can also be used with
a negligible loss of accuracy. The database methods
were outperformed by the regression methods in all
situations. We assume this might be because of the
high heterogeneity of the objects and the relatively
small database size.
ACKNOWLEDGEMENTS
This work has been supported by the grant
17-15361S of the Czech Science Foundation,
by the project LM2015062 of the Ministry of
Education, Youth and Sports, by the projects
CZ.02.1.01/0.0/0.0/16 013/0001775 (funded by
OP RDE) and CZ.02.1.01/0.0/0.0/16 019/0000765
(Research Center for Informatics).
REFERENCES
Alanentalo T, Asayesh A, Morrison H, Lorén CE,
Holmberg D, Sharpe J, Ahlgren U (2007).
Tomographic molecular imaging and 3D
quantification within adult mouse organs. Nat
Methods 4:31-3.
Baddeley A, Jensen EBV (2005). Stereology for
statisticians. Boca Raton: Chapman & Hall/CRC.
Banta L, Cheng K, Zaniewski J (2003). Estimation of
limestone particle mass from 2D images. Powder
Technol 132:184–9.
Berková Z, Křı́ž J, Girman P, Zacharovová K, Koblas
T, Dovolilová E, Saudek F (2005). Vitality of
pancreatic islets labeled for magnetic resonance
imaging with iron particles. Transplant Proc
37:3496–8.
Borovec J, Švihlı́k J, Kybic J, Habart D (2017).
Supervised and unsupervised segmentation using
superpixels, model estimation, and graph cut. J
Electron Imaging 26:061610.
Buchwald P, Bernal A, Echeverri F, Tamayo-Garcia
A, Linetsky E, Ricordi C (2016). Fully automated
islet cell counter (ICC) for the assessment of islet
mass, purity, and size distribution by digital image
analysis. Cell Transplant 25:1747-61.
Chaarani B, Capel C, Zmudka J, Daouk J, Fichten A,
Gondry-Jouet C, Bouzerar R, Balédent O (2013).
Estimation of the lateral ventricles volumes from
a 2D image and its relationship with cerebrospinal
fluid flow. BioMed Res Int 2013:215989.
Cramér H (1946). Mathematical methods of statistics.
Princeton: Princeton University Press.
Feldkamp LA, Davis LC, Kress JW (1984). Practical
cone-beam algorithm. J Opt Soc Am A 1:612–9.
Girman P, Křı́ž J, Friedmanský J, Saudek F (2003).
Digital imaging as a possible approach in
evaluation of islet yield. Cell Transplant 12:129-
33.
Habart D, Švihlı́k J, Schier J, Cahová M, Girman
P, Zacharovová K, Berková Z, Křı́ž J, Fabryová
E, Kosinová L, Papáčková Z, Kybic J, Saudek F
(2016). Automated analysis of microscopic images
of isolated pancreatic islets. Cell Transplant
25:2145–56.
Holm S (1979). A simple sequentially rejective
multiple test procedure. Scand J Stat 6:65-70.
Howard CV, Reed MG (2005). Unbiased stereology,
2nd Ed. Liverpool: QTP Publications.
Jensen EBV, Gundersen HJG (1993). The rotator. J
Microsc 170:35–44.
Kubı́nová L, Janáček J (2001). Confocal microscopy
and stereology: Estimating volume, number,
surface area and length by virtual test probes
applied to three-dimensional images. Microsc Res
Tech 53:425-35.
Lehmann R, Zuellig RA, Kugelmeier P, Baenninger
PB, Moritz W, Perren A, Clavien PA, Weber M,
Spinas GA (2007). Superiority of small islets in
human islet transplantation. Diabetes 56:594-603.
Niclauss N, Sgroi A, Morel P, Baertschiger R, Armanet
M, Wojtusciszyn A, Parnaud G, Muller Y, Berney
T, Bosco D (2008). Computer-assisted digital
image analysis to quantify the mass and purity
of isolated human islets before transplantation.
Transplantation 86:1603–9.
Piccinini F, Tesei A, Bevilacqua A (2016). Single-
image based methods used for non-invasive
volume estimation of cancer spheroids: a
practical assessing approach based on entry-
level equipment. Comput Methods Programs
Biomed 135:51-60.
Ricordi C, Gray DW, Hering BJ, Kaufman DB,
Warnock GL, Kneteman NM, Lake SP, London
NJ, Socci C, Alejandro R et al (1990). Islet
isolation assessment in man and large animals.
Acta Diabetol Lat 27:185-95.
Shapiro AM, Lakey JR, Ryan EA, Korbutt GS, Toth
E, Warnock GL, Kneteman NM, Rajotte RV
(2000). Islet transplantation in seven patients with
203
DVOŘÁK J ET AL: Islet volumes from single images
type 1 diabetes mellitus using a glucocorticoid-
free immunosuppressive regimen. N Engl J Med
343:230-8.
Suszynski TM, Wilhelm JJ, Radosevich DM,
Balamurugan AN, Sutherland DE, Beilman GJ,
Dunn TB, Chinnakotla S, Pruett TL, Vickers SM,
Hering BJ, Papas KK, Bellin MD (2014). Islet size
index as a predictor of outcomes in clinical islet
autotransplantation. Transplantation 97:1286-91.
Švihlı́k J, Kybic J, Habart D (2016). Automated
separation of merged Langerhans islets. In:
Medical Imaging 2016: Image Processing. Proc
SPIE 9784:978438.
Švihlı́k J, Kybic J, Habart D, Hlushak H, Dvořák
J, Radochová B (2017). Langerhans islet volume
estimation from 3D optical projection tomography.
In: Chen CS, Lu J, Ma KK (eds). Computer Vision
– ACCV 2016. Lect Not Comput Sci 10117:583–
94.
Xu C, He Y, Parra A, Delp E, Khanna N, Boushey
C (2013). Image-based food volume estimation.
In: Proc ACM 5th Int Works Multimed Cooking
Eating Activ (CEA ’13) 75–80.
Yoo TS (2004). Insight into images: Principles and
practice for segmentation, registration, and image
analysis. New York: CRC Press.
Zhang Z, Yang Y, Yue Y (2011). Food volume
estimation from a single image using virtual
reality technology. In: Proc 2011 IEEE 37th Ann
Northeast Bioeng Conf (2 p.).
Ziegel JF, Nyengaard JR, Jensen EBV (2015).
Estimating particle shape and orientation using
volume tensors. Scand J Stat 42:813–31.
204