Published in Soil Sci. Soc. Am. J. 68:1508-1514 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—SOIL PHYSICS
Mapping Material Distribution in a Heterogeneous Sand Tank by Image Analysis
Thomas Gimmia,* and
Nadia Ursinob
a Rock-Water Interaction Group, Institute of Geological Sciences, Univ. of Bern, Baltzerstrasse 1-3, CH-3012 Bern, Switzerland, and Paul Scherrer Institut, CH-5232 Villigen, Switzerland
b Univ. of Padova, Dep. IMAGE, Padova, Italy
* Corresponding author (gimmi{at}geo.unibe.ch)
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ABSTRACT
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Water flow and solute transport through soils are strongly influenced by the spatial arrangement of soil materials with different hydraulic and chemical properties. Knowing the specific or statistical arrangement of these materials is considered as a key toward improved predictions of solute transport. Our aim was to obtain two-dimensional material maps from photographs of exposed profiles. We developed a segmentation and classification procedure and applied it to the images of a very heterogeneous sand tank, which was used for a series of flow and transport experiments. The segmentation was based on thresholds of soil color, estimated from local median gray values, and of soil texture, estimated from local coefficients of variation of gray values. Important steps were the correction of inhomogeneous illumination and reflection, and the incorporation of prior knowledge in filters used to extract the image features and to smooth the results morphologically. We could check and confirm the success of our mapping by comparing the estimated with the designed sand distribution in the tank. The resulting material map was used later as input to model flow and transport through the sand tank. Similar segmentation procedures may be applied to any high-density raster data, including photographs or spectral scans of field profiles.
Abbreviations: BRDF, bidirectional reflection distribution function • CV, coefficient of variation
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INTRODUCTION
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WATER FLOW AND TRANSPORT OF SOLUTES through soils or other porous media are largely influenced by the spatial heterogeneity of the medium, as has been shown by numerous experimental and numerical investigations in recent years (e.g., Wildenschild et al., 1994; Hammel et al., 1999; Vervoort et al., 1999; Walter et al., 2000; Kasteel et al., 2000). Unraveling the relations between the structural1 and chemical properties of a soil and its flow and transport behavior is still one of the main research areas of soil physics. It seems to be commonly believed that flow and transport could be simulated successfully, if only the heterogeneity of the soil was known in sufficient detail. However, detailed mapping of the heterogeneity of the subsurface is a major problem.
Our goal was to develop an image analysis procedure to obtain two-dimensional material maps at high resolution from photographs of exposed profiles. Specific or generic material maps obtained in this way will allow flow and transport simulations, provided the required material properties are known (Kasteel et al., 2000). Detailed material maps are also needed, if the distribution of a dye shall be quantified by image analysis using material specific calibration functions (Vanderborght et al., 2002), or if local material properties shall be upscaled to regions of increasing size.
The materials defining the heterogeneous structure of a soil may differ with respect to various properties like bulk density, porosity, water content, aggregate or grain size, mineralogy, organic-C content, or wettability. Various methods exist for determining some of these properties in a nondestructive way. Three dimensional density patterns may be obtained from X-ray absorption tomography, but an exact quantification of densities is difficult (e.g., because of beam hardening). Neutron scattering is well suited to probe water contents, but requires rather sophisticated equipment. In contrast, measurement of broadband or spectral light reflection is relatively easy, and allows good spatial resolution. Mineralogy or organic-C content possibly linked to wettability, as well as surface roughness and aggregate or grain sizes influence the reflection properties. The reflection of light or other radiation has long been used—at much larger scales—in remote sensing to investigate various features of the surface of the earth. Methods based on scattering and reflectance of light are related to near-surface properties and usually cannot resolve the three-dimensional distribution of the properties of interest. Nevertheless, their sensitivity to specific attributes makes them a very valuable tool in finding precious statistical information about the arrangement, density, or anisotropy of the envisaged properties.
In this article, we describe a technique to determine the quasi two-dimensional material distribution of a laboratory sand tank based on light reflection. The sand tank was previously used for a series of transport experiments, where the spreading of a dye tracer at different average water saturations was studied (Ursino et al., 2001a, 2001b). After some definitions of technical terms and a short overview on image segmentation procedures, we describe the sand tank as well as our segmentation approach in detail. The results of the classification are presented and summarized in the last two sections. The material map was later used as input for a flow and transport model. The results of these simulations are presented in Ursino and Gimmi (2004).
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THEORY
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Definitions
We are faced with the problem of segmentation and classification of a noisy, nonuniformly illuminated image. Image segmentation is the process of dividing an image into the smallest number of mutually exclusive subsets that are homogeneous with respect to certain features, that is, whose elements share the same characteristics (e.g., Pal and Pal, 1993). Features may be gray values, color, (image) texture, position, or other properties. Classification is the process of labeling the subsets, that is, of assigning each subset to one class out of a set of classes.
Image segmentation is one of the oldest problems in digital image analysis, but still a difficult task. Lau and Levine (2002) state it concisely: "The seemingly straightforward and effortless human task of segmenting objects from their background is extremely difficult to simulate in the computer." No universal algorithms exist that work equally well for all conceivable problems. For best performance, all of the proposed techniques must usually be adapted in a more or less heuristic way to a specific problem.
The many different segmentation approaches can be classified in various ways—in fact, the classification of segmentation approaches is itself a problem as difficult as the segmentation and classification of an image. Segmentation techniques are based on, among other things, histogram thresholding, clustering, edge detection, region growing, or watershed algorithms, or rest upon statistical models describing the local neighborhood of pixels. They may rely on crisp or fuzzy decisions, and possibly make use of neural networks. Overviews of different segmentation techniques can be found, for instance, in Pal and Pal (1993), Atkinson and Lewis (2000), Cheng et al. (2001), and Egmont-Petersen et al. (2002).
Segmentation Based on Local Information
The simplest thresholding techniques ignore the contextual (spatial) information, that is, the information provided by the neighboring pixels. They operate in the feature (spectral) space on a pixel-by-pixel basis and use, for instance, in a univariate case only the histogram of local gray values. The problem is then reduced to finding the valleys between different peaks in the histogram, where each peak corresponds to a class. Especially for multivariate cases, estimating the number and statistical properties of the classes is a nontrivial task, which is often approached with methods of cluster analysis. Segmentation ignoring the contextual information works well for objects with distinct colors or gray values, as long as the images are not too noisy.
Segmentation Based on Neighborhood Information
The segmentation becomes much more robust, if contextual information is taken into account as well. Local textural features can be extracted in various ways. One way is to estimate statistical properties like spectral moments (e.g., mean, variance, skewness) within a filter window (neighborhood) and use them directly as additional image features. This is computationally relatively simple; we therefore used this approach in this study. Alternatively, the pixel values in a given filter window can be analyzed with a model that defines the spatial dependence of neighboring pixels (e.g., Dubes and Jain, 1989; Atkinson and Lewis, 2000). Typically, the task then consists of extracting model parameters, which are used as additional image features in the segmentation process.
Pre- and Postprocessing
Image segmentation generally consists of several steps, namely: preprocessing (filtering, noise reduction, possibly feature extraction), coarse and refined segmentation, and postprocessing. The success of the overall segmentation depends to a large extent on the success of the preprocessing steps like noise reduction. To preserve edges and borders during the filtering is essential for the segmentation, but it naturally conflicts with the aim of noise reduction or smoothing. Median filters, morphological filters, or filters based on diffusion are especially suited for this difficult task.
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MATERIALS AND METHODS
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Sand Tank
The laboratory sand tank was constructed to run dye tracer experiments at different water flow rates (Ursino et al., 2001a). It had a height of 40 cm, a length of 75 cm, and a width of 5 cm. The polyethylene frame was covered on both sides by a glass plate 12 mm thick. Three different sands were used to fill the tank, namely a very fine sand (grain size 0.08–0.2 mm), a fine sand (grain size 0.1–0.5 mm), and a coarse sand (grain size 0.3–0.9 mm). A total of 1050 elements, each 5 mm high, 50 mm long, and 50 mm wide, were placed at an angle of 45° into the tank, leading to a quasi two-dimensional heterogeneous structure. The filling was accomplished in a cold room, where wet cubes of 50 by 50 by 50 mm3, each composed of 10 randomly chosen layers and stabilized by freezing the pore water, were placed at random into the frame. The gaps between frame and layers were filled with a mixture of the three sands. A sprinkling device and a filter plate at the bottom allowed to adjust water saturations and perform flow and transport experiments, which are described elsewhere (Ursino et al., 2001a, 2001b).
Black-and-white pictures (16-bit gray scale) of the saturated sand tank were taken with a CCD camera. The tank was illuminated from the side with a Xenon lamp (angle of about 45°). This produced approximately oval distributions of light intensities centered in the middle of the tank. The recorded pixel values were corrected for an offset due to so-called dark noise of the camera by subtracting the average value obtained with closed shutter. Two pictures, which varied slightly in resolution and illumination conditions, were analyzed. We will focus mainly on the first, which we call main image. The second one served as a control image to check the reliability of the proposed procedure. In the main image, the tank area corresponded to roughly about 680000 pixels with a size of 0.67 by 0.67 mm2 each, whereas in the control image it corresponded to about 620000 pixels with a size of 0.70 by 0.70 mm2 each. We analyzed the images with codes written in IDL (Research Systems, Inc., Boulder, CO, available at www.rsinc.com/idl/; the specific program codes used can be obtained from the authors upon request), a software for data analysis and visualization.
General Segmentation Approach
Our sands had different particle sizes and, due to different surface roughness and thus different shadowing, slightly different average colors. Thus, a segmentation based on thresholds for two image features, namely an averaged color or gray value and a texture measure, seemed to be most appropriate. The thresholds were chosen based on calibration areas within the sand tank. Soil color was estimated from the median of gray values within a filter window. As texture feature, we used the coefficient of variation (CV) of gray values within the same filter window,
 | [1] |
where 
is the standard deviation and µ is the mean gray value within the window.
The segmentation included the following steps, which will be described in detail below:
- Correction for spatial inhomogeneity of illumination and reflection.
- Feature extraction from image with adaptive, layer shaped filter.
- Calibration, leading to mean feature values for the three sands and approximate thresholds.
- Preliminary classification of pixels according to chosen thresholds, where possible.
- Morphological filtering (closing) of binary map of each sand with rhomboid structure (Fig. 1c).
- Elimination of overlapping classifications.
- Repeating Steps E and F.
- Classification of yet unclassified pixels according to nearest mean calibration values.
- Repeating Steps E, F, and G.
- Repeating Step H.
- Smoothing of image with distribution of all three sands with a layer shaped median filter (Fig. 1b).
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Fig. 1. Shapes of filters (filter windows) used in the classification procedure: (a) Window for the adaptive, layer shaped filter to smooth the gray values and estimate local coefficients of variation in Step B; (b) window for the adaptive, layer shaped filter to smooth the classified sand pattern in Step K; (c) structuring element centered at x for the closing operations in Step E.
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Steps A to C can be considered as preprocessing steps, whereas Steps E, F, and K are postprocessing steps. Instead of using the true distance in H, a distance weighted by the standard deviations could also be used, if the latter differ largely for the different materials.
Correction for Spatial Inhomogeneity of Illumination and Reflection
The inhomogeneity of illumination and reflection was corrected in two steps. First, the value of each pixel of the tank image was divided by median filtered (window of 5 by 5 pixels) and normalized local intensities of an image of a gray cardboard covering the tank, taken at identical light settings. This so-called flat fielding is usually the only correction procedure. It removed probably most of the illumination inhomogeneity, but clearly not all of the reflection inhomogeneity. The latter arises also because the reflected intensities of a rough surface generally depend on angle of incidence of light and angle of reflection, that is, angle of observation, as given by a bidirectional reflection distribution function (BRDF) (e.g., Despan et al., 1999). The BRDF depends, among other things, on the surface roughness. Consequently, the gray cardboard has a different BRDF than the sand surface, and flat fielding with the cardboard cannot flatten the image perfectly. We removed the inhomogeneity of reflection in a second step by a bootstrap procedure, where the flat-fielded reflection image of the tank itself was used to estimate a second flat field. The image was reduced (with interpolation) to 1/8 of the original size, smoothed (mean) 100 times with a window of 3 by 3 pixels, and enlarged back to the original size. The flat-fielded image was then corrected by the resulting normalized values, which were interpreted as relative reflection intensities of the sand tank. Such a bootstrap procedure requires that no large (tank) scale trends of soil color are present. In our case, the structure elements were small compared with the total image and the smoothing size, such that the color differences averaged out. The corrections for intensity and reflection inhomogeneities were essential for classification of the sands with identical criteria across the whole tank area, as can be seen from Fig. 2, which shows the values of calibration samples for different degrees of preprocessing.
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Fig. 2. Features (coefficient of variation, median of gray values) of calibration samples for various degrees of preprocessing. For (a) to (c) the features were extracted from a five by five square filter window, for (d) from the layer shaped window shown in Fig. 1a. (a) Uncorrected image; (b) image corrected for inhomogeneous illumination by flat fielding with a gray cardboard; (c) image (b) corrected also for inhomogeneous reflection by bootstrap method; (d) same as image (c), but the features were evaluated for the layer shaped filter window of Fig. 1a. The large symbols indicate means and the horizontal and vertical bars standard deviations; the contour lines encompass about 75, 50, and 25% of the data of each sand. The standard deviations and contour lines are drawn in white for coarse and in black for the other sands.
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Feature Extraction and Smoothing with Adaptive, Layer Shaped Filter
To extract the color features some averaging must be done. We tested median filters, wavelet filters, and fast Fourier transform (FFT) filters to reduce small-scale (within the same sand) variability of gray values while keeping the delineation between different sands as sharp as possible. For our purposes an adaptive median filter with an approximately rectangular filter window (23 pixels, about 3 times 8, see Fig. 1a) oriented in the direction of the sand layers turned out to be suited best. Choosing a filter window similar in shape to the structure elements to be detected is a way to include prior knowledge in the filtering process. Adaptive means that the filter was only applied when the CV within the filter window was below a certain threshold. In this way, filtering could be tuned such that it was only active within relatively homogeneous regions (e.g., within a layer of one sand), and the smearing of boundaries between different sands or the smearing of artifacts could be minimized. The color feature was extracted with this filter with a relatively large threshold (CV smaller than 0.6) to exclude just smearing of clear artifacts. The texture feature was obtained as CV of gray values within the identical layer shaped window. For comparative purposes, a square window (5 by 5 pixels) was also used. For postprocessing (Step K), the same adaptive filter with a slightly larger filter window (29 pixels, about 3 times 10, see Fig. 1b) was applied to the preliminarily classified image. Again, a relatively large threshold (CV < 0.6) was used.
Calibration
For each sand, 25 small, evenly distributed calibration areas were chosen manually within the image of the tank. The image features (gray levels and CV) were determined for the calibration areas and plotted against each other (Fig. 2). This defined the two-dimensional feature space allowing a discrimination of the three sands. As a starting point for the classification procedure, nonoverlapping (main image) or partly overlapping thresholds (control image) that approximately separated the three sands in this parameter space were chosen visually, and the mean values for each sand were determined.
Figure 2 shows the local pixel features of the calibration samples for different degrees of corrections and noise reduction. It demonstrates the importance of the preprocessing Steps A and, to a lesser degree, B in enhancing the separation of the different materials. Even after all corrections and using the layer shaped filter, the pixel features of the different sands occupied partly overlapping regions, which indicates that postprocessing will also be important.
Morphological Filtering
Morphological filters (e.g., Serra, 1982; Vogel, 1997; Horgan, 1998) are powerful in removing noise or artifacts, especially if one has some prior knowledge about the morphology of the true structures. Basic morphological operators are erosion and dilation: A structuring element (a predetermined geometrical structure like, for instance, a rhombus of five pixels) is either morphologically subtracted from (erosion) or added to (dilation) an object in an image. Formally, erosion and dilation are defined for the set X of pixels of a binary image in the following way:
 | [2] |
 | [3] |
where Bx is the structuring element centered at x that is used for filtering, 
means subset, and 
intersection. In words, X eroded by B is the set of all points x such that Bx is fully included in X, and X dilated by B is the set of all points x such that Bx hits X (i.e., Bx and X have a nonempty intersection). The choice of the structuring element is important and usually requires some prior knowledge. From the basic operators, the following morphological filters can be defined:
 | [4] |
 | [5] |
Thus, a morphological opening is an erosion followed by a dilation with the same structuring element, and a closing is a dilation followed by an erosion. The effect of a closing is to fill up small holes and to join objects, whereas an opening removes peaks and small objects, and has a tendency to split objects. Other filters may be created by combining opening and closing operations, like X 
B = 1/2(X · B + X 
B), where X B means X filtered by B.
We applied closing operations with the rhomboid structuring element shown in Fig. 1c to binary maps of each sand obtained in Step D. Since the closing operations were performed individually on maps of each sand, overlaps could occur that had to be eliminated subsequently. This was also the reason why a second closing operation (G) with the same structuring element was still beneficial.
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RESULTS AND DISCUSSION
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Figure 2 shows the effect of different degrees of preprocessing on the local features of the calibration samples. Clearly (Fig. 2b), the flat fielding with the gray cardboard did not produce acceptable results; only after the additional correction (Fig. 2c)—accounting mostly for inhomogeneous reflection properties of the porous medium in response to a BRDF—the three sands separated more or less well in the feature space. For Fig. 2a, b, and c, the features were extracted from a square five by five window, whereas for Fig. 2d the adaptive filter with the layer shaped window (Fig. 1a) was used.
In Step D, about 89% (main image) or 78% (control image) of all pixels could be classified directly. For the control image, about 11% were in regions of overlapping thresholds. The relatively high percentages of directly classified pixels resulted largely from the preprocessing, that is, the elimination of illumination and reflection inhomogeneities, as is evident from Fig. 2. The speckled material map after Step D, however, was not yet satisfying. Larger contiguous areas of the same sand were obtained with the closing operation E. After Steps H and I, the percentage of classified pixels was about 98% (main image) or 97% (control image). The remaining 2 or 3% of all pixels were then assigned to one of the three sands again according to the distance-to-mean classification (J). The filtering with the layer shaped filter (K) was finally successful in linking previously isolated segments.
The final map estimated from the main image is shown in Fig. 3, together with the uncorrected reflection image of the tank and the designed filling pattern. Generally, the visually checked correspondence is rather good, although the classified image is much less regular than the designed one and obviously some small-scale structures disappeared during the classification procedure. The discrepancies originate partly from erroneous classification, but to a large degree also from differences between realized and designed filling pattern, as can be seen by comparing the picture of the tank in Fig. 3a with the designed distribution in Fig. 3c. Erroneous classification occurred at locations with artifacts like air bubbles between glass and sand, as well as in regions with very thin layers.
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Fig. 3. (a) Uncorrected reflection image of the sand tank; (b) estimated material map; (c) designed filling pattern. Coarse sand is indicated as dark gray, fine sand as white, and very fine sand as medium gray. Areas not classified in (b), or with no sand or a mixture of all sands in (c), are black. The white frames show the inner sections that were analyzed for averages. Note the slight horizontal stretching and vertical offset of the realized as compared to the designed pattern.
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A quantitative comparison between the designed sand distribution and the map obtained by image analysis can be made based on overall averages and horizontal and vertical average distributions of the different sand fractions. The estimated fractions of very fine, fine, and coarse sand for the analyzed sections of 0.32, 0.33, and 0.35 (main image) or 0.36, 0.31, 0.32 (control image) correspond well to the designed ones (0.32, 0.34, 0.34). The agreement is somewhat worse in the control image, where the fine and very fine sands were more difficult to distinguish. The differences of the calibration values between main and control image demonstrate the importance of specific calibration for each image, even for very similar illumination conditions. A good correspondence between designed and estimated values was also obtained for the horizontal and vertical average distributions of the sands, as shown in Fig. 4 and 5 for the main image. The correlation coefficients for the horizontal average distributions are 0.85 (coarse), 0.74 (fine), and 0.82 (very fine), for the vertical average distributions 0.89 (coarse), 0.89 (fine), and 0.76 (very fine). Note that for comparison the horizontal axis of the designed sand distribution was stretched with a factor of 1.02, since the realized sand pattern was obviously somewhat wider than intended. From the control image, very similar horizontal and vertical averages were estimated.
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Fig. 4. Comparison of horizontal average profiles of estimated and designed sand fractions for (a) coarse, (b) fine, and (c) very fine sand.
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Fig. 5. Comparison of vertical average profiles of estimated and designed sand fractions for (a) coarse, (b) fine, and (c) very fine sand.
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CONCLUSIONS
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We have presented an image analysis procedure to map the material distribution in a very heterogeneous sand tank based on image color and texture, which were related here to soil color and texture. The procedure was very successful, as is evident from comparison of the estimated map and the designed material distribution, and from the good correspondence between estimated and designed horizontal, vertical, and two-dimensional average fractions of the three sands. The following points, mostly related to preprocessing, turned out to be crucial for the success:- Correction for inhomogeneous reflection intensity of a porous medium. It was done by a bootstrap method using a strongly smoothed version of the image. That works well as long as the structures to be detected are small compared with the total image size and with the smoothing window.
- Use of an adaptive filter, which filters only if the local CV is below a given threshold, and which has a filter window similar in shape to the relevant structures. In this way, prior knowledge was used as additional information during the filtering process.
- Use of a morphological closing filter with a rhomboid structuring element to fill up mostly interior holes within one segment.
- Specific calibration for image under investigation, even though objects and illumination conditions were identical or similar.
A material map as obtained here can serve many purposes. It can be used to simulate flow and transport, allowing a detailed description of interface effects and local dispersion that lead, at a larger scale, to mixing and dilution (Ursino and Gimmi, 2004). It offers also the possibility to characterize upscaled parameters to be used in a coarse grid analysis. It enables generally to apply material specific calibration functions for any quantity obtained by image analysis. This significantly reduces the estimation error of, for instance, local dye concentrations during transport experiments (Vanderborght et al., 2002).
The proposed technique can be applied to a wide range of two-dimensional raster data. Which type of heterogeneity is mapped in a particular case depends on the scale and resolution of the image. The technique may even be extended to images of field profiles, where, depending on the resolution, the local CV (the textural feature of the image) is not linked to soil texture, but to aggregate sizes, mottling, or any other visual aspect that leads to local variability of pixel values. To avoid corrections for inhomogeneous illumination and reflection in a field case, a diffuse light source should preferably be used, or the bootstrap correction should be performed individually for horizons differing in color. Increasing the feature space by using three color channels or other spectral bands offers additional possibilities to map physical and chemical heterogeneity of a soil at various scales.
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NOTES
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1 The term ‘structural properties’ is used in a general sense for properties that are linked to the spatial arrangement of different soil materials. Thus, it is defined at a larger scale than the term ‘soil structure’, which denotes the arrangement of primary particles into secondary units. 
Received for publication July 10, 2003.
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ABSTRACT
INTRODUCTION
