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

The specific surface area (SSA) of snow is defined as the area

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

In the context of this workshop, the present study focuses on the measurement
of density and SSA derived from 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

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.

Snow sampling, preparation and scanning were conducted at SLF, Davos, Switzerland, during the Snow Grain Size Workshop in March 2014.

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.

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

The greyscale images were obtained by a commercial microcomputer tomograph
(Scanco Medical

Grayscale image (400

As shown in Fig.

Figure

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 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.

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

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

Different intensity models based on the greyscale distribution.

After the denoising step, a global threshold

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 (

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

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

The energy function

Proximity functions to air (

The spatial regularisation term

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

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

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

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.

An oblate spheroid (or ellipsoid of revolution) with symmetry axis along

Surface area of an oblate spheroid, obtained by different
calculation methods. The spheroid has a horizontal (

The different surface area computation methods were then evaluated on the
entire set of snow images segmented with the energy-based method
(

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 (

The binary image resulting from the sequential filtering approach depends on
(1) the standard deviation

The threshold

As shown in Fig.

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

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

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

Sequential filtering segmentation: relative variation of density

The sensitivity of density and surface area to the standard deviation

Depending on the sample, density varies in the range [

Sequential filtering segmentation: relative variations of density

The computed surface area significantly decreases when

Sequential filtering segmentation: relative variations of density

Energy-based segmentation: relative variations of density

Figure

The binary image resulting from the energy-based approach depends on
the parameter

As shown in Fig.

As shown in Fig.

Relative importance of the artefacts due to noise (

The sensitivity of surface area to the parameters

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

Figure

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

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

Lastly, systematically larger density and SSA values are found on the images
scanned with a 10

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

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

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

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.

A few recommendations to derive density and SSA from microtomographic data
are summarised below.

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