TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-1039-2016Sensitivity of snow density and specific surface area measured by microtomography to different image processing algorithmsHagenmullerPascalpascal.hagenmuller@meteo.frhttps://orcid.org/0000-0002-6581-2048MatzlMargretChambonGuillaumehttps://orcid.org/0000-0002-9812-9683SchneebeliMartinhttps://orcid.org/0000-0003-2872-4409Météo-France – CNRS, CNRM-GAME, UMR3589, CEN, 1441 rue de la piscine, 38400 Saint Martin d'Hères, FranceUniversité Grenoble Alpes, Irstea, UR ETGR, 2 rue de la Papeterie – BP 76, 38402 Saint Martin d'Hères, FranceWSL Institute for Snow and Avalanche Research SLF, Fluelastrasse 11, 7260 Davos Dorf, SwitzerlandPascal Hagenmuller (pascal.hagenmuller@meteo.fr)19May20161031039105426November201525January201611April20162May2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/1039/2016/tc-10-1039-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/1039/2016/tc-10-1039-2016.pdf
Microtomography can measure the X-ray attenuation coefficient in a 3-D volume
of snow with a spatial resolution of a few microns. In order to extract
quantitative characteristics of the microstructure, such as the specific
surface area (SSA), from these data, the greyscale image first needs to be
segmented into a binary image of ice and air. Different numerical algorithms
can then be used to compute the surface area of the binary image. In this
paper, we report on the effect of commonly used segmentation and surface area
computation techniques on the evaluation of density and specific surface
area. The evaluation is based on a set of 38 X-ray tomographies of different
snow samples without impregnation, scanned with an effective voxel size of
10 and 18 µm. We found that different surface area computation
methods can induce relative variations up to 5 % in the density and SSA
values. Regarding segmentation, similar results were obtained by sequential
and energy-based approaches, provided the associated parameters were correctly chosen.
The voxel size also appears to affect the values of density and SSA,
but because images with the higher resolution also show the higher noise
level, it was not possible to draw a definitive conclusion on this effect of
resolution.
Introduction
The specific surface area (SSA) of snow is defined as the area S of the
ice–air interface per unit mass M, i.e. SSA =S/M expressed in
m2 kg-1. We use this definition of SSA which can be directly
related to surface area per ice volume (SSAv) via
SSAv=ρice SSA with ρice
the density of ice. This quantity is essential for the modelling of the
physical and chemical properties of snow because it is an indicator of
potential exchanges with the surrounding environment. For instance, SSA can
be used to predict snow electromagnetic characteristics such as light
scattering and absorption (albedo in the near infrared)
e.g. or microwave radiation
e.g., and snow metamorphism
e.g.. Precise knowledge of this quantity is
required in numerous applications such as cold regions hydrology, predicting
the role of snow in the regional/global climate system, optical and microwave
remote sensing, snow chemistry, etc.
In the last decade, numerous field and laboratory instruments were developed
by different research groups to measure snow grain size. One possible
definition of this grain size is the equivalent spherical radius req,
computed from the SSA as req=3/(ρice× SSA) with ρice
the density of ice. This definition is actually equivalent to the optical
radius, i.e. the radius of a collection of spheres with the same infrared
albedo as that of the snow microstructure . In contrast,
the definition of grain size used in traditional snow classification as the
mean of the longest extension of disaggregated particles is
correlated to SSA only for a few snow classes. The crystal size as
stereologically measured by is another potential
definition for grain size, a priori, independent of the other definitions
mentioned above. Because of the co-existence of these inconsistent grain
size definitions and of different associated measurement methods (optical,
gas adsorption, tomography, stereology), an intercomparison of different
grain size measurement methods was organised by the International Association
of Cryospheric Sciences (IACS) working group, “From quantitative stratigraphy
to microstructure-based modelling of snow”. One of the main objectives of
this intercomparison was to determine the accuracy, comparability and quality
of existing measurement methods.
In the context of this workshop, the present study focuses on the measurement
of density and SSA derived from microtomographic data .
Microtomography measures the X-ray attenuation coefficient in a
three-dimensional (3-D) volume with high spatial resolutions of a few to a
few tens of microns. To extract the microstructure from the reconstructed 3-D
image, this image has to be filtered to reduce noise and subsequently
analysed to identify the phase relevant for the investigation, in our case,
ice. This step, called binary segmentation, affects the subsequent
microstructure characterisation, especially when the image resolution is
close to the typical size of microstructural details. Different algorithms
exist to calculate density and SSA from these images. Here we
investigate the effects of binary segmentation and surface area calculation
methods on density and SSA estimates, in order to provide guidelines for the
use of snow sample microtomographic data.
Comparative studies of processing techniques for images obtained via X-ray
microtomography have already listed the performance of several segmentation
methods with respect to different quality indicators
e.g.. These studies
emphasised the importance of using local image information such as spatial
correlation to perform suitable segmentations and highlighted the superior
performance of Bayesian Markov random field segmentation ,
which consists in finding the segmentation with minimum boundary surface and
which at the same time respects the grey value data in the best possible way.
However, none of the studies mentioned were interested in snow. This material
exhibits specific features such as a natural tendency, induced by
metamorphism, to minimise its surface energy
e.g.. Moreover, these former studies tended to
focus on properties linked to volumetric material contents, while less
attention was paid to the surface area of the segmented object.
applied an energy-based segmentation method on images
of impregnated snow samples, which is a three-phase material (impregnation
product, ice and residual air bubbles). This method is based on the same
principles as the Bayesian Markov random field segmentation but the
optimisation process is performed differently. It explicitly takes advantage
of the knowledge that the local surface energy of snow tends to be
limited due to inherent snow metamorphism that occurred before sampling, and
was shown to be accurate in comparison to a segmentation method based on
global thresholding. However, the set of snow microtomographic images used by
was limited to impregnated samples and to a few
different snow types. Moreover, no independent SSA measurements were
available to provide a reference or at least a comparison. Here, the flexible
energy-based segmentation method was adapted to two-phase images (air–ice)
and applied to 38 images on which SSA measurements were conducted with
independent instruments. Note that comparisons with these independent SSA
measurements are beyond the scope of this paper and will be reported in a
synthesis paper of the working group to be published in the present special
issue of The Cryosphere.
First, the sampling and X-ray measurement procedures to obtain greyscale
images are described. Attention is paid to the fact that the parameters used
for binary segmentation also depend on the scanned sample and not only on the
X-ray source set-up. Second, two different approaches of binary segmentation
are presented. The first one, commonly used in the snow science community,
consists of a sequence of filters: Gaussian smoothing, global thresholding
and morphological filtering. The second one is based on the minimisation of a
segmentation energy. Third, different methods to compute surface area from
binary images are presented. Finally, the different methods of binary
segmentation and area computation are applied to the microtomographic images
and the results are compared to provide an estimation of the scatter in
density and SSA measurements due to numerical processing of the greyscale
image and area calculation.
Material and methodsData set
Snow sampling, preparation and scanning were conducted at SLF, Davos,
Switzerland, during the Snow Grain Size Workshop in March 2014.
Sampling
Thirteen snow blocks of apparently homogeneous snow were collected in the
field or prepared in a cold laboratory. These blocks span different snow
types (decomposing and fragmented snow, rounded grains, faceted crystals and
depth hoar, Fig. ). Smaller specimens were taken out of these
blocks to conduct grain size measurements with different instruments. Two
snow cylinders of radius 35 mm and height of 60 mm and one snow cylinder of
radius 20 and 60 mm height were extruded from each block to perform
microtomographic measurements.
The different snow types and microstructural patterns used in this
study. The 3-D images shown have a side length of 3 mm and correspond to a
subset of the images analysed in the present study. The grain shape is also
indicated in brackets below the images, according to the international snow
classification .
X-ray scanning
The greyscale images were obtained by a commercial microcomputer tomograph
(Scanco Medical μCT40) operating in a cold room at -15 ∘C. The
X-ray source was set to an energy of 55 keV. The two samples with a radius of
37 mm were scanned with a nominal resolution of 18 µm, and the smaller
sample with a radius of 20 mm was scanned with a nominal resolution of
10 µm. To avoid edge effects, a sub-image of about
10003 voxels in size was extracted from each image, which corresponds to a volume of 103 mm3
for the 10 µm resolution and 183 mm3 for the 18 µm resolution.
These volumes are larger than the previously established representative
elementary volumes on the order of 2.53 mm3 for SSA
and density . In the following, the
images corresponding to a resolution 18 µm are identified by the suffixes
“s1” and “s2”, and the 10 µm by “10 µ”. The output of the
tomograph is a 3-D greyscale image with values encoded as unsigned short
integers (16 bits) (Fig. ). The greyscale value or
intensity (I) quantifies the X-ray attenuation coefficient.
Grayscale image (4002 pixels) representing the X-ray attenuation
coefficient and its corresponding greyscale histogram. The 2-D slice is
extracted from image G2-s1. The image exhibits two materials: air (dark grey)
and ice (light grey). The contour of ice resulting from binary segmentation
is plotted in red. The zoom panel (top left) was enlarged eight times to
emphasise the fuzzy transition between air and ice.
Images artefacts
As shown in Fig. , air and ice can be distinguished by their
respective attenuation coefficient, i.e. by their greyscale value or
intensity. However, the greyscale distributions in ice and air are not
completely disjoint, as there are always pixels (voxels in 3-D) that consist
of both materials. In addition to this fuzzy transition between air and ice,
the image is also noisy, which does not make the binary segmentation straightforward.
Figure shows the greyscale distributions obtained on all
scanned images. The exact position of the attenuation peaks and the scatter
around the peaks depend both on resolution and snow sample. Slight
differences are also observed between the greyscale distribution of the two
images coming from the same snow block and scanned with the same resolution.
This may be due to slight variations in the temperature of the X-ray
source during successive scans. Hence, it is doubtful whether binary
segmentation parameters “optimised” for one image can be used to segment
other images even of the same sample and with the same resolution. It appears
necessary to determine the segmentation parameters on each image independently.
Segmentation methods
In this section, two binary segmentation methods are presented: (1) the
common method based on global thresholding combined with denoising and
morphological filtering, hereafter referred to as sequential filtering, and (2) a method based on the minimisation of a segmentation
energy, referred to as energy-based segmentation.
Sequential filtering
Sequential filtering is commonly used to segment greyscale microtomographic
images of snow because it is simple, fast and is implemented in packages of
several different programming languages. It consists of a sequence of
denoising, global thresholding and post-processing, the input of each step
coming from the output of the previous step.
Denoising with a Gaussian filter
Numerous filters exist to remove noise from images, the most common being the Gaussian filter, the median
filter, the anisotropic diffusion filter and the total variation filter
. The objective of denoising is to smooth intensity
variations in homogeneous zones (characterised by low-intensity gradients)
while preserving sharp variations of intensity in the transition between
materials (characterised by high-intensity gradient). In snow science, the
most popular denoising filter is the Gaussian filter
e.g., which
consists in convoluting the intensity field I (i.e. the greyscale
value) with a Gaussian kernel of zero mean N(0, σ) defined as
N(0,σ)(I)=1σ2πexp-I22σ2,
with σ the (positive) standard deviation. The support of the Gaussian
kernel can be truncated (here to ⌊4σ⌋) to speed up the
calculations. This filter is very efficient in smoothing homogeneous zones.
However, it fails to preserve sharp features in the image by indifferently
smoothing low-intensity and high-intensity gradient zones, and therefore
reduces the effective resolution of the image.
Grayscale distributions for all images. The solid lines and dashed
lines respectively represent the distributions for the images scanned with
the resolution 10 and 18 µm. The greyscale distribution is
computed on 1000 bins of homogeneous size in the intensity range
[30 000, 50 000] (arbitrary units).
Different intensity models based on the greyscale distribution.
(a) Mixture model composed of two Gaussian distributions to reproduce the
greyscale distribution on the low-intensity gradient zones. L1 error
between the masked and reconstructed histograms is 0.006. (b) Mixture
model composed of two Gaussian distributions to reproduce the whole greyscale
distribution. L1 error between the initial and reconstructed
histograms is 0.14. (c) Mixture model composed of three Gaussian
distributions to reproduce the whole greyscale distribution. L1 error
between the initial and reconstructed histograms is 0.03. For all
figures, the L1 error is the integral of the absolute difference between
the measured and the modelled greyscale distribution. Note that the area
under the greyscale distribution of the entire image is 1.
Global thresholding
After the denoising step, a global threshold T is determined for the entire
image in order to classify voxels as air or ice, depending on whether their
greyscale value is smaller or greater, respectively, than the threshold. The
choice of this threshold is generally based on the greyscale histogram
without considering the spatial distribution of greyscale values. Different
methods exist, an exhaustive review of which can be found in
. Here the focus is limited to methods commonly used for
snow, namely (1) local minimum, (2) Otsu's method and (3) mixture modelling.
Local minimum: a simple way to determine the threshold is to define it
as the local minimum in the valley between the attenuation peaks of ice and
air (Figs. and )
e.g.. However, the histogram may
be noisy, resulting in several local maxima and minima, which makes the
method inapplicable. In some cases, the attenuation peaks of ice and air can
also be too close, which results in a unimodal histogram without any valley
e.g.. Moreover, the position of the local minimum is
generally affected by the height of the attenuation peaks in the histogram:
the less ice in the image, the closer the local minimum is to the ice
attenuation peak. The threshold obtained by this method is denoted
Tvalley in the following.
Otsu's method: another popular method, first introduced by
, is to find the threshold that minimises the intra-class
variance σw defined as
σw2=nairσair2+niceσice2, with nair and
nice the numbers of voxels classified as air and ice respectively, and
σair and σice the standard deviations of the greyscale
value in each segmented class. This method is generic and does not require
any assumption of the greyscale distribution. However, this also represents a
drawback of the method, in that knowledge of the origin of the image
artefacts can help to find the optimal threshold. The threshold obtained by
this method is denoted Totsu in the following. This method is less used
in the snow community but is widely used for other porous materials
e.g..
Mixture modelling: the classification error induced by the
thresholding can be also minimised by assuming that each class is
Gaussian-distributed. From there, different methods can be considered to
decompose the greyscale histogram in a sum of Gaussian distributions.
The greyscale histogram computed on the image masked on high-intensity
gradients can usually be perfectly decomposed into two Gaussian distributions
centred on the attenuation peaks of air (μair) and ice (μice),
respectively, and with identical standard deviations σ
(Fig. a). Masking the high-intensity gradients enables
suppression of the fuzzy transition zones between ice and air.
Therefore, on the corresponding histogram, the scatter around the attenuation
peaks can be attributed to instrument noise only, and appears to be Gaussian-distributed. The optimal threshold value derived from this method is
Tmask= (μice+μair)/2. Note that the ratio
Qnoise=σ/(μice-μair) provides a quantitative estimate of
the quality of the greyscale images with regard to noise artefacts. In
practice, however, masking the greyscale image on high-intensity gradient
zones is time-consuming and not straightforward with existing segmentation
software. Moreover, in the case of very thin ice structures, homogeneous ice
zones are almost non-existent.
directly fitted the sum of two Gaussian distributions to
the complete greyscale histogram (Fig. b). Note that the partial
volume effect at the transition between materials changes the position of the
attenuation peaks and the agreement between the fit and the histogram remains
partial in comparison to the fit on the histogram of the masked image. The
threshold, defined as the mean of the centre of the two Gaussian
distributions obtained by this fit, is denoted Tkerbrat.
fitted the greyscale histogram with the sum of
three Gaussian distributions to take into account the fuzzy transition between
materials. The fitting function isF̃=λair⋅Nμair,σ+λice⋅Nμice,σ+1-λair-λice⋅N(μ‾,μ̃),where λair, λice, μair, μice and σ
are five adjustable parameters (representing the proportions of
non-fuzzy air and ice in the image, the attenuation peaks of air and ice, and
noise, respectively), μ‾= (μair+μice)/2, and
μ̃= (μice-μair)/4. The two first terms of the sum model
the greyscale distribution in low-intensity gradient zones, while the last
term models the greyscale distribution in high-intensity gradient zones. The
choice of μ̃ value is arbitrary but was found to provide a good
fit to the transition zone. The agreement of this model with the greyscale
histogram is generally very good, although no additional free parameter is
added in comparison to the two-Gaussian distributions model (Fig. c).
The optimal threshold value derived from this method is
Thagen= (μice+μair)/2. Note that the value
Qblur= 1 -λair-λice provides a quantitative estimate of
the quality of the greyscale image with regards to the fuzzy transition artefact.
Post-processing
In general, the binary segmented image needs to be further corrected to
remove remaining artefacts. This can be done manually for each 2-D section,
but it is extremely time-consuming . The continuity of the
ice matrix can also be used to correct the binary image by deleting ice zones
not connected to the main structure or to the edges of the image
. Among generic and
automatic post-processing methods, the morphological operators erosion and
dilation are the most popular. The combination of these operators enables
deletion of small holes in the ice matrix (closing: erosion then
dilation) or small protuberances on the ice surface (opening: dilation then
erosion). In the following, the support size of these morphological filters
is denoted d.
Energy-based segmentation
Energy-based segmentation methods consist in finding the optimal segmentation
by minimising a prescribed energy function. These methods are robust and
flexible since the best segmentation is automatically found by the
optimisation process, and the energy function can incorporate various
segmentation criteria. In general, the optimisation of functions composed of
billions of variables can be complex and time-consuming. However, provided
that the variables are binary and some additional restrictions on the form of
the energy function, efficient global optimisation methods exist. In
particular, functions that involve only pair interactions can be globally
optimised in a very efficient way with the graph cut method
. Using this method, the typical computing time of the
energy-based segmentation of a 10003 voxel image is 5 h on a desktop
computer with a single processor (2.7 GHz).
The energy function E used in the present work is composed of two
components: a data fidelity term Ev and a spatial regularisation term Es.
The definition of E is similar to that proposed by
for the binary segmentation of impregnated snow
samples (air/ice/impregnation product), except that the data fidelity term
is, here, adapted to the processing of air/ice images. This term assigns
penalties for classifying a voxel into ice or air, according to its local
greyscale value. Qualitatively, assigning to air a voxel with a greyscale
value close to the attenuation peak of ice “costs more” than assigning it
to ice. Quantitatively, we define Ev as follows:
Ev(L)=v⋅∑i1-Li⋅P1Ii+Li⋅P0Ii,
where Li is the segmentation label (0 for air, 1 for ice) for voxel i,
Ii is its greyscale value, P0 is the proximity function to air and
P1 is the proximity function to ice. This energy is scaled by the volume v
of one voxel. The proximity functions quantify how close a greyscale
value is to the corresponding material. They are defined from the
three Gaussians fit (Eq. ) adjusted on the greyscale histogram as follows:
P0(I)=1ifI<μairmin1,eNμair,σ(I)elsewhereP1(I)=1ifI>μicemin1,eNμice,σ(I)elsewhere,
with e=exp(1) (Fig. ).
Proximity functions to air (P0) and to ice (P1) computed from
Eqs. () and (), and the three
Gaussians histogram fit obtained on image G2-s1.
The spatial regularisation term Es(L) is defined as r⋅S(L), with
S(L) the surface area of the segmented object L and r (r≥ 0) a
tunable parameter with the dimension of a length. Accounting for this
regularisation term in the energy leads to penalising large interface areas:
a voxel with an intermediate grey value is segmented so that the interface
air/ice area is minimised. The parameter r assigns a relative weight to the
surface area term in the total energy function E, and can be interpreted as
the minimum radius of protuberances preserved on the segmented object
. This regularisation term minimising the ice/air
interface is of particular interest for materials such as snow where
metamorphism naturally tends to reduce the surface and grain boundary energy.
Such processes are known to be particularly effective on snow types resulting
from isothermal metamorphism. For other snow types, such as precipitation
particles, faceted crystals or depth hoar, the surface regularisation term is
expected to perform well in recovering the facet shapes, but may induce some
rounding at facet edges.
Surface area computation
evaluated three different approaches to compute the area of
the ice–pore interface from 3-D binary images: the stereological approach
e.g., the marching cubes approach
e.g. and the voxel projection approach
. These authors showed that the three approaches provide
globally similar results, but each possesses its own inherent drawbacks: the
stereological approach does not handle anisotropic structures properly, the
marching cubes tends to overestimate the surface and the voxel projection
method is highly sensitive to image resolution. In the present work, in order
to estimate whether variations of SSA due to different surface area
computation approaches are significant compared to the effect of binary
segmentation, we tested three different methods to quantify the surface area:
the stereological approach, the marching cubes approach and the Crofton
approach. We did not evaluate the voxel projection method
because its implementation is sophisticated and the required computation of
high-quality normal vectors is excessively time-consuming if used only for
surface area computation.
Model-based stereological approach
Stereological methods derive higher dimensional geometrical properties, as
density or SSA, from lower dimensional data. The key idea is to count the
intersections of the reference material with points or lines. Prior to the
development of X-ray tomography, so-called model-based methods were used.
These models assume certain geometric properties of the object being studied,
such as the isotropy of the material . They are now replaced
by design-based methods that do not require any prior information on the
studied object but require denser sampling of the object .
Here, we used two variants of the stereological method by measuring the
intersection of lines in a 3-D volume. The first method consisted in counting the
number of interface points on linear paths aligned with the three orthogonal
directions. The surface area is then twice the number of intersections
multiplied by the area of a voxel face. A surface area value is obtained for each
direction. With the microtomographic data presented in this paper, the 2-D
sections are virtual and do not correspond to physical surface sections of
the sample. This corresponds to a model-based stereological method since
isotropy of the sample is assumed; we call it “stereological” in the following.
In addition, we used the mathematical formalism provided by the
Cauchy-Crofton formula that explicitly relates the area of a surface to the
number of intersections with any straight lines . Instead
of using only three orthogonal directions of the straight lines, we used
13 and 49 directions, and the improved approximations based on Voronoi diagrams
proposed by . This method comes close to a design-based
stereological method, as the volume (and direction) is almost exhaustively
sampled. We refer to this area computation method as the Crofton approach.
Marching cubes approach
The marching cubes approach consists in extracting a polygonal mesh of an
isosurface from a 3-D scalar field. Summing the area
contributions of all polygons constituting the mesh provides the surface area
of the whole image. We used a homemade version of the algorithm developed by
. It computes the area of the 0.5 isosurface of the
binary image without any further processing of the image. Our version of the
algorithm is adapted to compute only the surface area without saving all the
mesh elements that are required for 3-D visualisation.
Results
In this section, the methods to compute the area of the ice–air interface are
evaluated first, since this evaluation can be performed on reference objects
whose area is theoretically known, without accounting for the interplay with
the binary segmentation method. The Crofton approach, which is shown to
perform best, is selected for the rest of the study. The sensitivity of
density and SSA to the parameters of the sequential filtering and
energy-based segmentations on the entire set of snow images is then
investigated. Finally the variability of SSA due to numerical processing is
compared to the variability of SSA due to snow spatial heterogeneity and
scanning resolution.
Surface area estimation
An oblate spheroid (or ellipsoid of revolution) with symmetry axis along z
was chosen as a reference object to compare the different surface area
computation methods. An anisotropy of 0.6 was considered (ratio between the
dimensions of z and (x, y) semi-axes), and spheroids of different sizes
were used to evaluate the impact of the discretization on the surface area
computation. Figure shows that the surface area calculated with
the Crofton approach is in excellent agreement with the theoretical area: for
sufficiently large spheroids, i.e. a surface area larger than 200 voxels2,
the relative error is less than 1 % for the Crofton approach with
49 different directions and 2 % for the Crofton approach with 13 different
directions. Adding more directions does not significantly improve the
accuracy of the Crofton approach while it increases the computation time. The
marching cubes approach systematically overestimates the surface area by
about 5 % due to the presence of artificial stair steps in the triangulation
of the isosurface. As expected, the stereological method shows scatter in the
results obtained between the z and the (x, y) components. The mean value
of the three components provides a fair estimation of the surface area, with
a relative error of about 2 % compared to the theoretical value. These
observations on the stereological and marching cubes approaches corroborate
previous results obtained by on snow images.
Surface area of an oblate spheroid, obtained by different
calculation methods. The spheroid has a horizontal (x, y) semi-axis a
varying in [3, 19] voxel and a vertical (z) semi-axis c= 0.6 ⋅a. The
reference surface area of the spheroid is computed analytically. The Crofton
approach is computed with 13 or 49 different directions. As in Figs. ,
, and , the relative
error is calculated as the computed value minus the reference value, divided
by the reference value. Note that due to symmetry of the oblate
spheroid, the values of stereological y are superimposed with the values of
stereological x.
The different surface area computation methods were then evaluated on the
entire set of snow images segmented with the energy-based method
(r= 1 voxel). According to the results obtained on the spheroid, the
Crofton approach with 13 directions was chosen as a reference. As shown in
Fig. , the SSA obtained by the direction-averaged stereological
method is in excellent agreement with the value provided by the Crofton
method. The results of the marching cubes method are in fair agreement but
show a systematic over-estimation of the SSA (+6 % average relative deviation).
In summary, all presented area computation methods showed consistent
results. The Crofton approach showed the best accuracy on an artificial
anisotropic structure whose surface area is theoretically known. The
stereological approach is negatively affected by strong anisotropy of the
imaged structure. However, on the tested snow images, the structural
anisotropy is low and this method is in excellent agreement with the Crofton
approach. The simple marching cubes approach presented here (without
additional filtering or pre-smoothing of the binary image) overestimates the
specific surface on the order of 5 %. For the following analysis of the
sensitivity to binary segmentation, the SSA is computed via the Crofton
approach with 13 directions.
Comparison of the SSA obtained by different surface area
calculation methods on all images binarised with the energy-based
segmentation (r= 1 voxel). The root mean square difference between the SSA
computed as the direction average of the stereological method and the SSA
computed with the Crofton approach is 0.008 m2 kg-1. This difference
is 1.13 m2 kg-1 for the marching cubes approach. The black line
represents the 1 : 1 line.
Sequential filtering
The binary image resulting from the sequential filtering approach depends on
(1) the standard deviation σ of the Gaussian filter, (2) the threshold
value T and (3) the size d of the post-processing morphological filters
(opening/closing). As shown in Fig. , both SSA and density
are sensitive to these segmentation parameters. The relation between SSA and
density, on the one hand, and σ and d, on the other hand, depends
significantly on the chosen threshold. Thus, in the following, we first
investigate the dependence of SSA and density on the threshold, and then
analyse the effects of σ and d with a threshold obtained by the
mixture model of .
Choice of threshold
The threshold Tmask obtained by the two-Gaussian fit of the greyscale
histogram computed on the low-intensity gradient zones is chosen as a
reference, since this value is not affected by the fuzzy transition artefact.
This reference threshold ranges between 38 800 and 39 500 for the different
scanned images (Fig. ). The mean values of the attenuation
peaks of air and ice are μ‾air= 35 800 and
μ‾ice= 42 600, respectively. Hence, the variations of the
reference threshold value remain small compared to the contrast between the
two attenuation peaks μ‾ice-μ‾air= 6800.
However, these variations clearly indicate, once again, that a unique
threshold value cannot be used for all images. These variations could be
explained by slight variations in the X-ray source energy level due to slight
temperature changes, or to deviations from the Beer–Lambert attenuation law,
depending on the total ice content of the sample.
As shown in Fig. , the computed threshold depends
significantly on the determination method. These variations in turn affect
the density extracted from the binary image (Fig. a). Note
that the scatter on density due to the choice of the threshold remains the
same even if a Gaussian filter is applied on the greyscale image before
thresholding (Fig. a). The SSA values are also affected by
the threshold determination method, but to a smaller extent since the
threshold value tends to affect density and total surface area in the same
proportion (Fig. b). The variation of SSA due to smoothing
is much more important than those due to the choice of the threshold (Fig. b).
SSA and density of image G2-s1 obtained by sequential filtering
for different segmentation parameters.
In detail, the valley method systematically overestimates the threshold
value, leading to a systematic underestimation of the snow density by about
10 kg m-3 on average. Otsu's method tends to underestimate the threshold
value, leading to an overestimation of the snow density by about
6 kg m-3 on average. Kerbrat's method tends to underestimate the
threshold value, leading to an overestimation of the snow density by about
4 kg m-3 on average. Note that the density overestimation with Kerbrat's
method is more pronounced on low-density snow samples scanned with a
18 µm resolution. Lastly, the method introduced by
slightly underestimates the threshold value, and
therefore overestimates the snow density with a mean absolute difference of
about 2 kg m-3 compared to the reference.
In summary, the threshold value obtained by the valley method, a method
widely used in the snow community, clearly leads to an underestimation of
snow density. The mixture models of or
, which assume that noise is Gaussian-distributed,
provide a threshold value in good agreement with the reference method. The
model of , which explicitly accounts for the fuzzy
transition between materials, yields the threshold which is the closest to
the reference value obtained on the masked image.
Sequential filtering segmentation: threshold values obtained on the
entire set of images by the different intensity models. The black line
represents the 1 : 1 line. The obtained thresholds T are also expressed as a
function of the mean threshold T‾ and the mean contrast
μ‾ice–μ‾air between air and ice, obtained by
the reference method.
Sequential filtering segmentation: relative variation of density (a)
and SSA (b) computed with different threshold determination methods, with
respect to the reference values computed with Tmask. Void (solid) markers correspond to values obtained without any smoothing
(with a Gaussian filter of standard deviation σ= 1 voxel).
The legends apply to both subplots.
Gaussian filtering
The sensitivity of density and surface area to the standard deviation σ
of the Gaussian smoothing kernel is shown on Fig. . The
segmentation was performed with the threshold Thagen derived using the
method of .
Depending on the sample, density varies in the range [-8, +2] % (compared to
the value obtained without smoothing) when σ is increased from 0 to
20 µm (Fig. a). Density appears to be insensitive to σ
when σ is much lower than the voxel size. For larger values of σ,
an average decrease of density with σ is observed due to the
fact that snow structure is generally convex and smoothing tends to erode
convex zones. Slight increase of density with σ is observed for
σ> 5 µm for samples M1-1, M1-3 and M2-2. These samples are the most
faceted snow samples exhibiting a large proportion of flat surfaces
(Fig. ), which explains the different variation of density
with σ. Systematic differences can also be noted between the images with a
resolution of 10 and 18 µm. At a resolution of 10 µm, a fast
decrease of density is observed for σ in the range [3, 6] µm. This
regime is absent at a resolution of 18 µm. For larger values of σ,
the evolution of density is then similar for the two resolutions, and depends
on the snow type. This difference is attributable to a stronger noise in the
10 μm images, which results in local greyscale variations that are
generally smoothed out when σ> 6 µm.
Sequential filtering segmentation: relative variations of density (a)
and surface area (b) with the size σ of the Gaussian filter. The
variations are calculated with respect to density and surface area obtained
without a smoothing filter (σ= 0).
The computed surface area significantly decreases when σ increases
(Fig. b). Relative variations up to 50 % are observed. On the
10 µm images and with σ in the range [3, 6] µm, the surface
area decreases rapidly when σ increases. These variations probably
correspond to the progressive smoothing of noise-induced fluctuations on the
interface. For larger values of σ, the surface area decreases much
more slowly, which corresponds to the progressive smoothing of real
microstructural details. On the 18 µm images, these two regimes cannot be
distinguished because the overall surface area is less affected by noise
artefacts and only the smoothing of real structural details is observed. The
same variations with σ can be observed on SSA since the variations of
density with σ are small compared to the variations of the surface
area. Note that the absolute values of density and SSA for the different
scanned snow images are indicated on Fig. .
Sequential filtering segmentation: relative variations of density (a)
and surface area (b) as a function of the morphological filter size d
for different values of the Gaussian filter size σ. The legends apply
to both subplots.
Energy-based segmentation: relative variations of density (a) and
surface area (b) as a function of parameter r.
Morphological opening/closing
Figure shows the relative variation of density and surface area
obtained by morphological filters of different sizes d. Note that the
values of d are constrained by the voxel grid and are thus discrete (1,
2, 3, 2 voxel, etc.). The opening and closing filters
delete holes in the ice matrix or ice elements in the air, of a typical size
d. Therefore, the surface area decreases when d increases. Density, on
the contrary, is not very sensitive to d. When no Gaussian filter is
applied to the greyscale image, thresholding yields a lot of small details in
the binary image, which enhances the effect of morphological filters
(Fig. b). When the image is already smoothed by Gaussian filtering, the
morphological filters have less effect on the overall density and specific
surface area. However, note that using a Gaussian filter of standard
deviation σ does not guarantee the complete absence of details
“smaller” than σ. Certain algorithms based on the binary images,
such as grain segmentation e.g., are
highly sensitive to the presence of residual artefacts in the ice matrix and
require the use of these additional morphological filters.
Energy-based approach
The binary image resulting from the energy-based approach depends on
the parameter r which controls the smoothness of the segmented object. The
other parameters involved in the volumetric term Ev of the segmentation
energy are directly derived from the three Gaussians mixture model (see Sect. ).
As shown in Fig. a, the density of the segmented object slightly
varies with r. On the 10 µm images, the evolution of density with
r is not monotonic but relative variations remain limited in the range [-4,
+1] %. On the 18 µm images, density clearly decreases when
σ increases. This higher sensitivity of density to r on the 18 µm images
can be explained by the fact that the fuzzy transition appears to be larger
than on the 10 µm images, which leads to a higher indetermination of the
exact position of the interface between ice and air in this moderate
intensity gradient zone (Fig. ).
As shown in Fig. b, surface area is more sensitive to r than
density, and decreases significantly when r increases. Two regimes can be
distinguished. For low values of r in the range [0, 10] µm, the surface
area decreases rapidly when r increases. For larger values of r in the
range [10, 20] µm, the decrease of the surface area with r is much
slower, and displays an almost constant slope. As discussed by
, this second regime is due to real details of the
snow structure being progressively smoothed out, and is indicative of a
continuum of sizes in structural details of snow microstructure. The
distinction between the two regimes is more pronounced on the 10 µm
images which are more affected by noise (Fig. ).
Relative importance of the artefacts due to noise (Qnoise) and
to the fuzzy transition (Qblur). Qnoise corresponds to the ratio
between the standard deviation of the Gaussian distributions fitted on the
attenuation peaks and the difference between the peak attenuation intensity
of ice and air (see Sect. ). Qblur corresponds to
the area of the Gaussian representing the fuzzy transition in Hagenmuller's
mixture model (see Sect. ).
Comparison between images and methods
The sensitivity of surface area to the parameters σ and r is
similar, but the energy-based and sequential filtering approaches are
conceptually different (Figs. and ). The Gaussian
filter smoothes high-frequency intensity variations with a small amplitude,
independently of the subsequent binary thresholding. Small details due to
noise artefacts remaining in the binary image are then deleted independently
of the initial greyscale value by applying morphological filters. The
energy-based approach smoothes the segmented object so that the ice–air
interface area is minimised while respecting at best the greyscale intensity
model. Hence, the greyscale smoothing and morphological filtering are somehow
done simultaneously with thresholding in the energy-based approach. In
addition, the Gaussian filter is “grid-limited”: as shown in Fig. b,
this filter does not affect the segmented object if σ is
to small compared to the voxel size. In contrast, in the energy-based
approach, smoothing of the ice–air interface occurs even for very low values
of r (Fig. b) because voxels with a greyscale value close to
the threshold between ice and air can be segmented as air or ice without much
change in the data fidelity term Ev but with a clear change in the surface
term Es. The parameter r defines the largest equivalent spherical radius
of details preserved in the segmented image, whereas σ does not
directly correspond to the size of the smallest detail.
Specific surface area as a function of density for the entire set of
images and the two different binary segmentation methods. The sequential
filtering (“Sequ.” in the legend) was applied with σ= 1.0 voxel,
T=Thagen and d= 1.0 voxel. The energy-based approach (“Ener.” in the
legend) was applied with r= 1.0 voxel. The surface area was computed with
the Crofton approach.
Figure shows density and specific surface area computed on
the entire set of snow images segmented with the sequential filtering
approach (σ= 1.0 voxel, T=Thagen, d= 1.0 voxel) and the
energy-based approach (r= 1.0 voxel). The “smoothing” parameters (σ
and r) were chosen equal to 1.0 voxel since this value roughly corresponds
to the transition beyond which the computed surface area starts to vary
slowly with σ and r (Figs. b and b), and
therefore provides the segmentations that best preserve the smallest snow
details while deleting most of noise-induced protuberances. As already
pointed out, this transition is clear on the 10 µm images, but is less
evident on the 18 µm images. To be consistent, however, and to ensure
that all noise artefacts are smoothed out, values of r, σ= 1 voxel
were used in all cases.
It is observed that the two approaches generally produce similar results in
terms of density (root mean square deviation between the two segmentation
methods is 6 kg m-3) and specific surface area (root mean square
deviation of 0.7 m2 kg-1). The largest differences are observed for
the snow types presenting the highest SSA. In general, the density provided
by the sequential filtering is slightly larger than that computed with the
energy-based approach. The opposite difference is observed for SSA.
Scatter can be observed even between the density and SSA derived from images
coming from the same snow block, probably due to the existence of spatial
heterogeneities with the blocks and the difference of image quality. The
averages of standard deviations calculated for each snow block are
10.7 kg m-3 and 1.1 m2 kg-1 for density and SSA, respectively
(calculated with the energy-based approach). This intra-block variability
nevertheless appears to be limited compared to the inter-block variability
(46.5 kg m-3 for density and 4.7 m2 kg-1; see Fig. ),
and is on the same order as the variability due to the image processing technique (see above).
Lastly, systematically larger density and SSA values are found on the images
scanned with a 10 µm resolution, compared to the images with a 18 µm
resolution. It could be argued that this difference is due to a better
imaging of small details with a lower voxel size. However, as already
noticed, the 10 µm images also present stronger noise artefacts
(Fig. ), and it is difficult to assess whether the effective
resolution of these images is, in practice, finer than the one of the
18 µm images. Note that the root mean square difference between the
density (respectively SSA) computed on the images s1 and s2 is
3.8 kg m-3 (respectively 0.15 m2 kg-1), which is much lower
than the intra-block variability (including the 10 µm images). This
observation indicates that the hardware set-up of the tomograph and the
subsequent image quality or resolution can significantly affect the measured
density and SSA.
Conclusions and discussions
We investigated the effect of numerical processing of microtomographic images
on density and specific surface area derived from these data. To this end, a
set of 38 X-ray attenuation images of non-impregnated snow were analysed with
different numerical methods to segment the greyscale images and to compute
the surface area on the resulting binary images.
The segmentation step is not straightforward because the greyscale images
present noise and blur. It is shown that noise artefacts can significantly
affect the computed SSA, and that the fuzzy transition between ice and air
can have a strong impact on the computed density.
The sequential filtering approach critically depends on the threshold used to
separate ice and air. The greyscale histogram on low-intensity gradient zones
presents two disjoint attenuation peaks, whose characteristics are not
affected by blur. The threshold derived from this method was used as a
reference to evaluate other methods based on the analysis of the greyscale
histogram of the entire image. The mixture models which consist in
decomposing the histogram into a sum of Gaussian distributions are shown to
be accurate. On the contrary, the local minimum method is shown to be
unsuitable in general.
Smoothing induced by the Gaussian and morphological filters in the sequential
approach, or by accounting for the surface area term in the energy-based
method, efficiently remove noise artefacts from the segmented binary image.
Morphological filters applied on the binary image in the sequential approach
miss the initial grey value information. However, it seems that their effect
is negligible if the applied Gaussian filter is strong enough. The smoothing
can also induce the disappearance of real structural details, contributing to
the overall SSA. The transition between smoothing of noise and smoothing of
real details can be well estimated on the curve, showing the evolution of SSA
as a function of σ or r. However, due to the influence of noise, it
remains difficult to assess the potential contribution to the SSA of
structural details of size smaller than the voxel size. It has previously
been shown that the SSA measured with the gas adsorption technique, which has a
molecular resolution, is in good agreement with the SSA measured with
microtomography for aged natural snow . This observation
corroborates the idea that the surface of aged snow is smooth up to a scale
of about tens of microns, and that if smaller structures are present, they do
not contribute significantly to the overall SSA . To
further investigate this issue on recent snow and to disregard any additional
influence of the measuring technique except the resolution, the use of new
tomographic systems with very high resolutions of about 1 µm would be necessary.
The formalism of the energy-based segmentation could enable more advanced
criteria in the segmentation process, such as the maximisation of the
greyscale gradient at the segmented interface , the
minimisation of the curvature of the segmented object
or the spatial continuity in time series of 3-D images . In
this study, only criteria on the local greyscale value and on the surface
area of the segmented object were used. The advantage of this method is that
the parameter r formally defines an effective resolution of the segmented
image. In contrast, the standard deviation σ of the Gaussian smoothing
kernel in the sequential approach does not explicitly define the smallest
structural detail in the segmented image. In practice, however, both methods
provide very similar results on the tested images in terms of density and
SSA, provided appropriate parameters are chosen.
Comparison between the presented area computation methods showed similar
results when applied to a synthetic image or to the set of snow images. On
the synthetic image (oblate spheroid), the Crofton approach computes the
surface area with highest accuracy (less than 2 % for sufficiently large
spheroids), whereas the stereological approach is negatively affected by
strong anisotropy of the imaged structure and the unfiltered marching cubes
approach overestimates the specific surface on the order of 5 %.
Stereological methods using more complex test lines, such as cycloids, can
compensate for the effect of anisotropy if the snow sample exhibits isotropy
in a certain plane, which is often the case for the stratified snowpack
. However, on the tested snow images, the surface anisotropy
is low and the stereological method is in excellent agreement with the
Crofton approach. The unfiltered marching cubes approach still overestimates
the specific surface on the order of 5 %. Note that methods have been
developed to overcome this overestimation problem of the marching cubes
approach, such as the use of grey levels or smoothing .
However, these methods may create other artefacts, depending on the image
considered, such as systematic underestimation of the surface
, and were not evaluated here.
The comparison of the sequential filtering and energy-based methods shows
that density and SSA can be estimated from X-ray tomography images with a
“numerical” variability of the same order as the variability due to spatial
heterogeneities within one snow layer and to different hardware set-ups.
Recommendations
A few recommendations to derive density and SSA from microtomographic data
are summarised below.
Surface area computation: the unfiltered marching cubes approach
systematically overestimates the surface area and should thus be avoided.
Counting intersections with test lines of different orientations (at least in
the three axes x, y and z) provides an efficient way to compute the
surface area and properly accounts for structural anisotropy.
Threshold determination: the value of the threshold depends on the
tomograph configuration but also potentially on the scanned sample. A constant
value for a time series does not necessarily prevent density deviations due to
beam hardening. Visual inspection of the histogram or the valley method do
not always provide consistent threshold values. The fit of Gaussian distributions
on the histogram provides an automatic and satisfactory method to determine an
appropriate threshold value. However, all methods need visual inspection and
comparison with the greyscale image.
Smoothing of greyscale image: smoothing of the greyscale image,
such as the convolution with a Gaussian kernel, is required to reduce noise
artefacts but also reduces the effective resolution of the image by deleting
structural details that could contribute to the overall SSA. The filter, and
in particular the standard deviation σ of the applied Gaussian kernel,
expressed in µm, should be systematically mentioned if SSA values derived
from tomographic data are presented. Indeed, SSA is a decreasing function of
the effective resolution even if the resolution is larger than the nominal
voxel size. This function is expected to become constant only for
sufficiently small resolutions, depending on the snow type.
Acknowledgements
CNRM-GAME/CEN and Irstea are part of Labex OSUG@2020 (Investissements
d'Avenir, grant agreement ANR-10-LABX-0056). Irstea is member of Labex TEC21
(Investissements d'Avenir, grant agreement ANR-11-LABX-0030). We thank two
anonymous reviewers for their positive feedback.
Edited by: P. Marsh
References
Baddeley, A. and Vedel Jensen, E. B.: Stereology for statisticians, Chapman & Hall/CRC,
Boca Raton, 395 pp., 2005.Berthod, M., Kato, Z., Yu, S., and Zerubia, J.: Bayesian image classification
using Markov random field, Image Vis. Comput., 14, 285–295, 10.1016/0262-8856(95)01072-6, 1996.
Boykov, Y. and Jolly, M.-P.: Interactive graph cuts for optimal boundary & region
segmentation of objects in ND images, Int. Conf. Comput. Vis., 1, 105–112, 2001.Boykov, Y. and Kolmogorov, V.: Computing geodesics and minimal surfaces via
graph cuts, in: Proceedings Ninth IEEE Int. Conf. Comput. Vision 2003,
vol. 1, IEEE, Nice, France, 26–33, 10.1109/ICCV.2003.1238310, 2003.Brucker, L., Picard, G., Arnaud, L., Barnola, J.-M., Schneebeli, M., Brunjail,
H., Lefebvre, E., and Fily, M.: Modeling time series of microwave brightness
temperature at Dome C, Antarctica, using vertically resolved snow temperature
and microstructure measurements, J. Glaciol., 57, 171–182, 10.3189/002214311795306736, 2011.Calonne, N., Flin, F., Geindreau, C., Lesaffre, B., and Rolland du Roscoat, S.:
Study of a temperature gradient metamorphism of snow from 3-D images: time
evolution of microstructures, physical properties and their associated anisotropy,
The Cryosphere, 8, 2255–2274, 10.5194/tc-8-2255-2014, 2014.Cnudde, V. and Boone, M. N.: High-resolution X-ray computed tomography in
geosciences: A review of the current technology and applications, Earth-Sci.
Rev., 123, 1–17, 10.1016/j.earscirev.2013.04.003, 2013.Coléou, C., Lesaffre, B., Brzoska, J.-B., Lüdwig, W., and Boller,
E.: Three-dimensional snow images by X-ray microtomography, Ann. Glaciol.,
32, 75–81, 10.3189/172756401781819418, 2001.Danek, O. and Matula, P.: On Euclidean Metric Approximation via Graph Cuts,
in: Comput. Vision, Imaging Comput. Graph. Theory Appl., vol. 229 of Communications
in Computer and Information Science, edited by: Richard, P. and Braz, J., Springer,
Berlin, Heidelberg, Angers, France, 125–134, 10.1007/978-3-642-25382-9_9, 2011.Domine, F., Taillandier, A.-S., and Simpson, W. R.: A parameterization of the
specific surface area of seasonal snow for field use and for models of
snowpack evolution, J. Geophys. Res., 112, 1–13, 10.1029/2006JF000512, 2007.Ebner, P. P., Schneebeli, M., and Steinfeld, A.: Tomography-based monitoring
of isothermal snow metamorphism under advective conditions, The Cryosphere,
9, 1363–1371, 10.5194/tc-9-1363-2015, 2015.
Edens, M. and Brown, R. L.: Measurement of microstructure of snow from surface
sections, Def. Sci. J., 45, 107–116, 1995.El-Zehiry, N. and Grady, L.: Fast global optimization of curvature, IEEE
Conf. Comput. Vis. Pattern Recognit., San Francisco, 3257–3264, 10.1109/CVPR.2010.5540057, 2010.
Fierz, C., Durand, R., Etchevers, Y., Greene, P., McClung, D. M., Nishimura,
K., Satyawali, P. K., and Sokratov, S. A.: The international classification
for seasonal snow on the ground, Tech. rep., IHP-VII Technical Documents in
Hydrology N83, IACS Contribution N1, UNESCO-IHP, Paris, 81 pp., 2009.Flanner, M. G. and Zender, C. S.: Linking snowpack microphysics and albedo
evolution, J. Geophys. Res., 111, D12208, 10.1029/2005JD006834, 2006.
Flin, F.: Snow metamorphism description from 3D images obtained by X-ray
microtomography, PhD thesis, Université de Grenoble 1, Grenoble, 188 pp., 2004.Flin, F., Brzoska, J.-B., Lesaffre, B., Coléou, C., and Pieritz, R. A.:
Full three-dimensional modelling of curvature-dependent snow metamorphism:
first results and comparison with experimental tomographic data, J. Phys. D.
Appl. Phys., 36, A49–A54, 10.1088/0022-3727/36/10A/310, 2003.Flin, F., Brzoska, J.-B., Lesaffre, B., Coléou, C., and Pieritz, R. A.:
Three-dimensional geometric measurements of snow microstructural evolution
under isothermal conditions, Ann. Glaciol., 38, 39–44, 10.3189/172756404781814942, 2004.Flin, F., Budd, W. F., Coeurjolly, D., Pieritz, R. A., Lesaffre, B.,
Coléou, C., Lamboley, P., Teytaud, F., Vignoles, G. L., and Delesse,
J.-F.: Adaptive estimation of normals and surface area for discrete 3-D
objects: application to snow binary data from X-ray tomography, IEEE Trans. Image
Process., 14, 585–596, 10.1109/TIP.2005.846021, 2005.
Flin, F., Lesaffre, B., Dufour, A., Gillibert, L., Hasan, A., Roscoat, S.
R. D., Cabanes, S., and Pugliese, P.: On the Computations of Specific
Surface Area and Specific Grain Contact Area from Snow 3D Images, in: Phys.
Chem. Ice, edited by: Furukawa, Y., Hokkaido University Press, Sapporo, Japan, 321–328, 2011.Hagenmuller, P., Chambon, G., Lesaffre, B., Flin, F., and Naaim, M.:
Energy-based binary segmentation of snow microtomographic images, J.
Glaciol., 59, 859–873, 10.3189/2013JoG13J035, 2013.Hagenmuller, P., Chambon, G., Flin, F., Morin, S., and Naaim, M.: Snow as a
granular material: assessment of a new grain segmentation algorithm, Granul.
Matter, 16, 421–432, 10.1007/s10035-014-0503-7, 2014.Haussener, S.: Tomography-based determination of effective heat and mass
transport properties of complex multi-phase media, PhD thesis, ETH
Zürich, http://e-collection.library.ethz.ch/view/eth:2424 (last access: 1 December 2015), 2010.Heggli, M., Frei, E., and Schneebeli, M.: Snow replica method for
three-dimensional X-ray microtomographic imaging, J. Glaciol., 55, 631–639,
10.3189/002214309789470932, 2009.Hildebrand, T., Laib, A., Müller, R., Dequeker, J., and Rüegsegger,
P.: Direct three-dimensional morphometric analysis of human cancellous bone:
microstructural data from spine, femur, iliac crest, and calcaneus, J. Bone
Miner. Res., 14, 1167–1174, 10.1359/jbmr.1999.14.7.1167, 1999.Iassonov, P., Gebrenegus, T., and Tuller, M.: Segmentation of X-ray computed
tomography images of porous materials: A crucial step for characterization
and quantitative analysis of pore structures, Water Resour. Res., 45, W09415,
10.1029/2009WR008087, 2009.Kaestner, A., Lehmann, E., and Stampanoni, M.: Imaging and image processing in
porous media research, Adv. Water Resour., 31, 1174–1187, 10.1016/j.advwatres.2008.01.022, 2008.Kerbrat, M., Pinzer, B. R., Huthwelker, T., Gäggeler, H. W., Ammann, M.,
and Schneebeli, M.: Measuring the specific surface area of snow with X-ray
tomography and gas adsorption: comparison and implications for surface
smoothness, Atmos. Chem. Phys., 8, 1261–1275, 10.5194/acp-8-1261-2008, 2008.Kolmogorov, V. and Zabih, R.: What energy functions can be minimized via graph
cuts?, IEEE Trans. Pattern Anal. Mach. Intell., 26, 147–159, 10.1109/TPAMI.2004.1262177, 2004.Lomonaco, R., Albert, M., and Baker, I.: Microstructural evolution of
fine-grained layers through the firn column at Summit, Greenland, J. Glaciol.,
57, 755–762, 10.3189/002214311797409730, 2011.Lorensen, W. E. and Cline, H. E.: Marching Cubes: A High Resolution 3D Surface
Construction Algorithm, SIGGRAPH Comput. Graph., 21, 163–169, 10.1145/37402.37422, 1987.Matzl, M. and Schneebeli, M.: Stereological measurement of the specific
surface area of seasonal snow types: Comparison to other methods, and
implications for mm-scale vertical profiling, Cold Reg. Sci. Technol., 64,
1–8, 10.1016/j.coldregions.2010.06.006, 2010.Otsu, N.: A threshold selection method from gray-level histograms, Automatica,
20, 62–66, 10.1109/TSMC.1979.4310076, 1975.Pinzer, B. R., Schneebeli, M., and Kaempfer, T. U.: Vapor flux and
recrystallization during dry snow metamorphism under a steady temperature
gradient as observed by time-lapse micro-tomography, The Cryosphere, 6,
1141–1155, 10.5194/tc-6-1141-2012, 2012.Riche, F., Schreiber, S., and Tschanz, S.: Design-based stereology to quantify
structural properties of artificial and natural snow using thin sections,
Cold Reg. Sci. Technol., 79–80, 67–74, 10.1016/j.coldregions.2012.03.008, 2012.Schleef, S. and Löwe, H.: X-ray microtomography analysis of isothermal
densification of new snow under external mechanical stress, J. Glaciol., 59,
233–243, 10.3189/2013JoG12J076, 2013.Schleef, S., Löwe, H., and Schneebeli, M.: Hot-pressure sintering of
low-density snow analyzed by X-ray microtomography and in situ microcompression,
Acta Mater., 71, 185–194, 10.1016/j.actamat.2014.03.004, 2014.
Schlüter, S. and Sheppard, A.: Image processing of multiphase images
obtained via X-ray microtomography: A review Steffen, Water Resour. Res.,
50, 3615–3639, 10.1002/2014WR015256, 2014.Sezgin, M. and Sankur, B.: Survey over image thresholding techniques and
quantitative performance evaluation, J. Electron. Imaging, 13, 146–165, 10.1117/1.1631315, 2004.Theile, T. and Schneebeli, M.: Algorithm to decompose three-dimensional
complex structures at the necks: tested on snow structures, Image Process.
IET, 5, 132–140, 10.1049/iet-ipr.2009.0410, 2011.Theile, T., Szabo, D., Luthi, A., Rhyner, H., and Schneebeli, M.: Mechanics of
the Ski-Snow Contact, Tribol. Lett., 36, 223–231, 10.1007/s11249-009-9476-9, 2009.Torquato, S.: Random heterogeneous materials: microstructure and macroscopic
properties, vol. 16, Springer, New York, 703 pp., 10.1007/978-1-4757-6355-3, 2002.Vetter, R., Sigg, S., Singer, H. M., Kadau, D., Herrmann, H. J., and
Schneebeli, M.: Simulating isothermal aging of snow, Eur. Lett., 89, 26001,
10.1209/0295-5075/89/26001, 2010.Warren, S.: Optical properties of snow, Rev. Geophys., 20, 67–89, 10.1029/RG020i001p00067, 1982.Wolz, R., Heckemann, R. A., Aljabar, P., Hajnal, J. V., Hammers, A.,
Lötjönen, J., and Rueckert, D.: Measurement of hippocampal atrophy using
4D graph-cut segmentation: application to ADNI, Neuroimage, 52, 109–118,
10.1016/j.neuroimage.2010.04.006, 2010.