Measurement of specimen volume alteration during triaxial tests
is valuable for understanding the volumetric compression or dilation characteristics
of the soil which is very important when analysing the mechanical behaviour.
In a saturated soil, assuming water is incompressible, the total volume
change of the soil specimen is commonly assumed to be equal to the change
in volume of water leaving or entering the soil specimen and is easily
measured using a twin burette or an automatic water volume change system.
However, accurate measurement of total volume changes in an unsaturated
soil specimen is much more difficult and complicated, where not only pore-water
but also pore-air volume may vary.
Geiser et al. (2000) summarized the existing techniques for measuring
volume change in unsaturated soil specimens and classified them into three
main categories: (i) measurement of the volume of the cell fluid, (ii)
measurement of the air and water volumes separately and (iii) direct measurement
of the specimen volume.
For the first category, volume change in a soil specimen is recorded
by measuring the volume change in the confining cell fluid. Several problems
are usually encountered using this method such as expansion- contraction
of the cell wall, connecting tubes and cell fluid because of pressure
and temperature variations, creep under pressure and possible water leakage
(Ng et al., 2002). To minimize the errors caused by expansion or
contraction of cell, sorts of double wall cell setup have been proposed
by Yin (2003) and Sivakumar et al. (2006).
The second category of measuring systems, is impractical in the triaxial
testing of unsaturated soils because measuring the air volume change is
somewhat difficult as the volume of air is very sensitive to the factors
such as changes in the atmospheric pressure and ambient temperature and
undetectable air leakage through the tubes and connections and air diffusion
through the rubber membrane (Ng et al., 2002).
Measurements using digital image processing fall into the third category.
Macari et al. (1997) used video images to compute the volume change
of sand during drained conventional triaxial compression test. The experimental
results were compared to the volumetric strains obtained by analysing
the front view and side view images. Two video cameras were placed orthogonally
to each other to take front and side view images. It was mentioned that
the results were in accordance with each other unless irregular shapes
occurred. Alshibli and Al-Hamdan (2001) measured the volume change of
triaxial soil specimens by locating three cameras at equal distance to
capture the entire body of the specimen. In addition to these, more recently
Gachet et al. (2007) measured the volume change of unsaturated
Sion silt using image processing method. They applied a digital plug and
A novel approach for volume measurement in triaxial tests based on image
processing is proposed in this research. Generally a black background
is placed behind the sample which makes it easier to distinguish the sample
boundaries. The selection of a suitable method for tracing the sample
boundaries is crucial for accurate measurement of specimen volume. The
triaxial sample has two vertical edges and two horizontal edges at the
Baseline and bottom. The diameter of sample in each level is a function of
the number of pixels located between the left and right boundaries. Many
methods of edge detection have been developed yet each one has its special
advantages and drawbacks. The researches on edge detection methods are
still going on. One of the novelties of this research is employing the
wavelet transform to detect the sample boundaries. This method is more
practical and powerful compared with those applied by Macari et al.
(1997), Alshibli and Al-Hamdan (2001) and Gachet et al. (2007).
The presence of water and plexiglass chamber in the line of vision between
the camera and sample causes magnification and distortion of the image.
Furthermore, lens refraction of light may produce radial distortion. It
was found that the pattern of image distortion is complicated such that
it cannot be successfully calibrated by the conventional regression approach.
Artificial neural network has been used in this research as a successful
method to calibrate these effects.
Two DV cameras were used for viewing and capturing images from triaxial
specimens. Each camera was connected by a USB port to a computer. The
software supplied by the manufacturing company enabled the user to send
the capturing command from the computer. In this mode, images with a resolution
of 2048x1536 pixels were taken. Since the cameras had not the optical
zooming feature and in order to increase the number of pixels to the real
dimension ratio, the cameras were located at a distance of 20 cm from
the centre of sample and 90° from each other. The advantage of using
this type of camera in addition to the low price was its remote shooting
capability. A simple Visual Basic script was written to execute the picture
taking command at certain time intervals. The cameras were fastened to
metal bases fixed on triaxial cell. This allowed keeping a constant distance
between the cameras and the specimen.
||Digital camera and triaxial cell
Suitable lightening enhances the quality of images and hence the quality of analyses
is improved. Therefore, two florescent lights were employed. The positions
of these lights were determined by a trial and error process such that
the best quality images were obtained. Figure 1 shows
the triaxial and imaging system setup.
EDGE TRACING SCHEME
The simplest edge detection method consists of manual tracing of
the edges by a mouse on the computer screen. This method in addition of
being tedious could come up with large errors. Therefore, it is necessary
to use a proper systematic technique for this purpose. Classical edge
detection methods are quite sensitive to noise. In some circumstances
they may not be able to distinguish the exact position of sample boundaries.
That`s why Macari et al. (1997) used a method which was a combination
of both manual and systematic edge detection. Manual tracing was used
to trace the edge in regions of uncertainty where the program identified
a false edge. Gachet et al. (2007) applied a threshold method.
They determined a threshold value for the entire image, the pixels whose
intensity were higher than this value were turned white and the pixel
with intensities lower than threshold were turned black. Finally, a binary
image containing the specimen in white pixels on a black background is
obtained. The next stage is image cleaning. In this stage the pixels which
are incorrectly set to black or white are corrected. Thus, a manual post
processing stage was included in their process.
Classical edge detectors are modified gradient operators. Since an edge
is characterized by having a gradient of large magnitude, edge detectors
are approximations of gradient operators. This group consists of well-known
edge detectors, such as Sobel, Roberts, Prewitt and etc. Their major drawbacks
are high sensitivity to noise and disability to discriminate edges versus
Because noise influences the accuracy of the computation of gradients,
usually an edge detector is a combination of a smoothing filter and a
gradient operator. An image is first smoothed by the smoothing filter
and then its gradient is computed by the gradient operator.
Wavelet transform maps a time function into a two dimensional function
of α and τ. Parameter α scales the function by compressing
or stretching it. Parameter τ corresponds to the translation of the
wavelet function along the time axis. A remarkable property of the wavelet
transform is its ability to characterize the local regularity of functions.
It enables to focus on localized signal structures with a zooming procedure
that progressively reduces the scale parameter α. Mallat and Hwang
(1992) proved that the maxima of the wavelet transform modulus can locate
the irregular structures. Edges in images can be mathematically defined
as local singularities. For an image f(i, j), its edges correspond to
singularities of f(i, j) and therefore are related to the local maxima
of the wavelet transform modulus. Therefore, the wavelet transform is
an efficient method for edge detection. In an image, all edges are not
created equal. Some are more significant and some are blurred and insignificant.
The edges of more significance are usually more important and more likely
to be kept intact by wavelet transforms. The insignificant edges are sometimes
introduced by noise and preferably removed by wavelet transforms.
In this way, rough and fine signal structures are simultaneously analysed
at different scales. Wavelet transform is defined by:
where, N stands for the length of signal s(n) and Ψ(t) is so called
mother wavelet. Mother wavelet is a prototype wavelet from which all other
wavelets are generated. Its characteristic must depend on the properties
of signal structures to be detected in a signal (Heric and Zazula, 2007).
Here Haar wavelet, Eq. 2, was adopted because it is orthogonal, compact
and without spatial shifting in the transform space. Its main property
is the ability to present the magnitude variation between adjacent intervals
in the signal as a modulus maximum on time-scale plane (Heric and Zazula,
2007). Haar wavelet is defined as:
||An image and a line cut from it
||Variation of the discrete function f(i, j) with the
j-coordinate at i = 1000
Proposed edge detection approach: In the following sections, the
proposed approach is described in details. First the acquired RGB images
were converted into gray scale. Using an image taken at the beginning
of the test, as an example, gray levels were extracted along the line
at i =1000 shown in Fig. 2 and the variation of the
gray levels with the j-coordinate is shown in Fig. 3.
From Fig. 2 it is clear that there exist two major interface
points along the i =1000 line. One of them corresponds to the left vertical
boundary and the other to the right boundary. The gray level changes abruptly
at these two positions as shown in Fig. 3.
The problem of image edge detection was then transformed into a search
of sudden amplitude changes in 1D signals representing a row pixel intensities.
Images can be corrupted by the noise which is represented as magnitude
variation of the signal s(n). Such noise appears on the time-scale plane
as additional edges.
Haar wavelet transform of intensity level along i =
1000 line for different scale values of α = 2, 3, 4 and 5
But there are two principal differences between the real edges and noise: (a) edges` modulus maxima are larger than noise
modulus maxima if the signal-to-noise ratio (SNR) is low and (b) the influence
of noise decreases with progressing toward higher scales if noise is additive
with zero mean, because Haar wavelets perform averaging. Hence, the influence
of noise is gradually filtered out going toward higher scales and its
modulus maxima become negligibly small (Heric and Zazula, 2007).
As an example, the Haar wavelet transform of gray scale along i = 1000
line for different scale values is shown in Fig. 4.
In all scale values, there exist two major modulus maxima. The left one
corresponds to the left edge of the sample and the other to the right
edge. The ratio of edge modulus maxima to the noise modulus maxima increases
with increasing scale. The sample edges could then be easily distinguished
even at the lowest scale (α = 2). As the modulus maxima at finer
scales represent the edge positions more accurately, the position of modulus
maxima between j = 1 to j = 750 was determined as the left sample edge
and between j = 750 to j = 1500 as the right edge. In order to obtain
the entire vertical boundaries, this procedure was performed for all values
Figure 5 shows the typical example of a gray scale
image of a barrel shaped sample and its vertical boundaries obtained by
the proposed method. It is evident that the edge detection method can
efficiently detect the sample boundaries. In cases where a plastic pipe
inside the cell covers somewhere the sample boundary, as in Fig.
5, the sample boundary is not visible and consequently the edge detection
method fails at the overlap region. In such cases a straight line connecting
the sample boundary above the pipe to the sample boundary just underneath the pipe is set as the boundary.
(a) A gray scale image of a barrel shaped sample (b)
Sample boundaries determined with the proposed method
Since the length
of the overlap region is very small compared with sample height, such
approximation is reasonable.
The apparent magnification of the specimen is a result of the presence
of water and plexiglass chamber in the line of vision between the camera
and the sample. The order of magnification depends upon various indices
of refraction of light in the water and the plexiglass chamber wall. An
additional phenomenon is image distortion, which causes features in an
image to be shifted from their expected location. Distortion is a result
of the fact that imaging system is actually a mapping of a three dimensional
object inside the cylinder to a two dimensional image. Lens distance from
Plexiglass surface can also affect the amount of distortion. Another cause
of image distortion is that of radial lens distortion, which is an error
associated with lens refraction of light its effect increases with distance
from the centre of the lens. Radial lens distortion becomes an increasing
difficulty for less costly lenses and most lenses with focal lengths below
8 mm. (Smith and Smith, 2005).
While it is generally preferable to eliminate the source of system errors
rather than attempting to compensate for them, this often proves to be
hard to accomplish in practice. Specially, in our imaging system, the
magnification and distortion effects caused by refraction of light in
water and plexiglass were absolutely inevitable.
The challenge for any calibration process is to model the distortion
with the intention that its effects can be reduced or eliminated from
the image data. One way of achieving this modelling is to use analytical
methods. In general terms, analytic techniques based on explicit physical
models have a number of significant shortcomings. The fact that in practice
unknown contradictions may exist as a result of, for example, randomised
manufacturing errors. Hence, these effects cannot be solved for analytically
with high precision. The requirement that in order to obtain more accurate
results, more accurate modelling is generally necessary and this brings
more complex mathematical equations and calibration procedures, often
leading to an increased computational burden.
An example of such analytic physical models is that of Macari et al.
(1997) which is a theoretical approach using optical laws. The camera
is assumed as a pinhole which is a strong hypothesis. The schematic of
their model is shown in Fig. 6. The relationship obtained to convert radial
pixels to millimetres is not very convenient to use. Furthermore, unknown
causes of distortion and magnification, such as randomized manufacturing
errors and radial lens distortion are disregarded in this model.
Regression has been used as an approach for detecting and quantifying
patterns in vision systems. In a regression analysis of non-linear functions,
it is assumed that the form of the data is known and this knowledge is
used to inform the choice of transform used (e.g., a quadratic). Unfortunately,
however, in practice, the situation is often not this simple, as the overall
error is usually formed from a combination of components and their separation
is not always practical. As a result, the type of the error is often relatively
complex and cannot be accurately modelled in terms of simple curves. In
terms of lens distortion this additional complexity can be introduced
through effects such as manufacturing errors. Previous efforts that have
been done for this error calibration were dominantly on the basis of curve
fitting. Works of Gachet et al. (2007) and Alshibli and Al-Hamdan
(2001) are some examples of application of simple regression for calibration.
To evaluate the magnification and distortion in our imaging system, four
metallic cylinders were put in the cell. Those cylinders had diameters
of 4.46, 4.94, 5.72 and 6.26 cm. After the cell was filled with water
some pictures were taken. Vertical edges of the cylinders were determined
using the approach described in the edge detection section. At each elevation
(each row) the number of pixels located between the two edges was counted.
The number of pixels between the two edges at different elevations is
shown in Fig. 7-10.
||Schematic of two dimensional model used by Macari et
No. of pixels between two edges of cylinder with diameter
of 4.46 cm along i axis
No. of pixels between two edges of cylinder with
diameter of 4.94 cm along i axis
Although the cylinder diameter is constant at all elevations, these figures indicate
that the number of pixels increases with distance from the middle elevation.
If there had been no distortion in the imaging system the number of pixels
at different elevations would have been the same.
No. of pixels between two edges of cylinder with diameter
of 5.72 cm along i axis
No. of pixels between two edges of cylinder with diameter
of 6.26 cm along i axis
If we consider d as the real diameter of the object or in other words
the real distance between the two edges and i as its vertical position
in pixel and d´ as the measured diameter in pixels in the image,
it can be said that distortion function is as follows:
In case, the function f is known, distortion can be calibrated. To achieve
this goal, data set was deciphered such that from each image specified
elevations (i) were selected and the respective d´ and i values
were determined. For each calibration cylinder d was also a constant value.
Therefore, in above mentioned data set d had four values of 4.46, 4.94,
5.72 and 6.26 cm. It was first tried to fit the following second order
function to the calibration data set.
The coefficients of the above function (C1 to C8)
were obtained by minimizing the mean absolute error using genetic algorithm.
The function obtained could not fit the data set with reasonable accuracy.
That is, the mean absolute error of measuring the diameter was about 0.45
millimetres. This was not the order of accuracy needed to follow volume
change of specimens during the tests. Considering the complex nature of
the problem, neural network algorithm was employed as a suitable alternative.
Details are described in the subsequent sections.
Vertical calibration: Parallel horizontal lines with equal distance
of 1.70 cm were drawn on all metallic cylinders which were used for calibration.
To perform vertical calibration, number of pixels between these lines
was determined in all elevations for all cylinders. It is interesting
to note that the distance between these lines in all elevations and all
cylinders was constant and equal to 291 pixels. Therefore, the vertical
calibration is absolutely linear and each pixel represents the height
of 1.70/291 centimeter.
CALIBRATION USING ARTIFICIAL NEURAL NETWORK
As stated by Ekpar et al. (2003) the formation of a distorted
2D image from the undistorted 3D real world scene by an imaging system
is considered to be an input-output mapping between an undistorted 2D
image plane and the distorted 2D image plane. The development of neural
network technology has made it possible for a novel approach to imaging
system calibration. A neural network approach is presented herein as a
pattern recognition method to calibration. This approach does not require
a complicated mathematical model be developed nor any prior knowledge
about the setup or calibration parameters. Since this is often the case,
neural networks can perhaps be considered to be the most appropriate general
solution to the error modelling involved.
An artificial neural network is a parallel computing method that provides
a simple artificial analogy to biological nervous systems. Since the neural
network stores data as patterns in a set of processing elements by adjusting
the connection weights, it is possible to realize complex mapping through
its characteristics of distributed representations (Bagherieh et al.,
2008). The neural network can automatically find the closest match through
its content addressable property, even if the data are incomplete or vague.
Even in the case that a few processing elements malfunction or fail completely,
the network can still function through its fault-tolerance attribute.
The neural network has the ability for extracting a generalized correlation
(or regularity) from many individual examples (or experiences). The computing
abilities of neural networks have been proven in the field of geotechnical
engineering (Habibagahi and Bamdad, 2003; Adeli, 2001). The ability of
the network to associate a particular output with an input pattern is
a result of the training operations, where the weights are adjusted. This
is achieved by comparing output and target values a number of times and
altering the weights so as to minimise the error. This is recognized as
supervised learning. Once trained, a neural network can provide an ability
for usefully modelling phenomena that exhibit non-linear behaviour and
appreciable noise, as is often the case in image analysis.
Network architecture: The neural network architecture used is
a feed-forward back-propagation one. More than 50 neural network models
have been devised so far and it has been found that the backpropagation
learning algorithm based on the generalized delta rule is one of the most
efficient learning procedures for multi-layer neural networks (Bagherieh
et al., 2008). This technique generally consists of many sets of
nodes arranged in layers (e.g., input, hidden and output layers). The
output signals from one layer are transmitted to the subsequent layer
through links that amplify or attenuate or inhibit the signals using weighting
factors. Except for the nodes in the input layer, the net input to each
node is the sum of the weighted outputs of the nodes in the previous layer.
An activation function, such as the sigmoid logistic function, is used
to calculate the output of the nodes in the hidden and output layers.
In the calculation, both input and output are usually normalized to give
a value between -1 and 1 incorporating various mapping schemes. This depends
on adopted activation functions. The degree of non-linearity can be changed
easily by changing the transfer function and the number of hidden layer
The proposed BPNN had three layers: an input layer, a hidden layer with
six neurons and an output layer. The architecture of back-propagation
neural network is shown in Fig. 11. In this study,
the batch training rule was used to train the network. A set of 79 data
obtained from experiments were used for neural network modelling. A total
number of 51 sets were used for training the network, 18 sets were used
to test the network and the remaining 10 sets were used for validation.
Each epoch was defined as one cycle of presentation of all training data
sets to the network. Overtraining (over fitting) may occur if a network
is trained excessively, at which point, the network starts to learn noise
contained in the training data sets.
||Predicted vs. Measured diameters using the NN model
Predicted profile assuming d` = 1000, 1100, 1200 and
1300 pixels along i axis
Beyond this point, although the network`s training error may continue to decline, the testing error increases
rapidly. In order to guard against overtraining, the network was examined
during the training process by monitoring the performance for the testing
record. Figure 12 represents the prediction of data
using proposed BPNN versus actual data. The error rates (mean absolute
error) of the proposed network were 0.024, 0.071 and 0.041 millimetres
for training data, testing data and validation data, respectively and
R2 of proposed network was 0.999, 0.999 and 0.999 for training
data, testing data and validation data, respectively.
Parametric study: In order to examine the trained neural network,
it was assumed that the distance between two edges is constant at all
elevations and then the predicted diameters were plotted along i. Figure
13 shows the predicted diameters assuming d´ = 1000, 1100, 1200
and 1300 pixels. As expected from distortion of the image, the obtained
diameter is higher at the middle elevations. It can be concluded that
the train network has learned the distortion features.
METHOD OF CALCULATING SAMPLE VOLUME
After the vertical boundaries of the samples were determined using
the above mentioned procedure, the uppermost and lowermost limits of the
sample were determined. This could be done manually on the computer screen.
In order to make the Baseline and down boundaries of the sample more obvious,
it is possible to paste a black tape on the membrane on the bottom and
Baseline of the sample. Then the position of the tape borders could easily
be detected by the edge detection method. In each row (a constant value
of i) number of pixels located between the left and right sample boundaries
were determined (d´). The actual diameter was calculated by the
neural network simulation using the corresponding d´ and i values.
As described above, each pixel had a height of Δh = 1.70/291 cm,
therefore, each increment of sample volume is:
Summing ΔVi from the lowermost to the uppermost limits
of the sample yields the total volume of the samples:
In cases where the sample remained symmetrical throughout the test, the
image from one camera would be enough to calculate the volume but if the
diameters obtained by two cameras did not agree which each other, the
averaged diameters were used.
COMPARISON WITH CONVENTIONAL METHOD
To verify the accuracy of the proposed approach an isotropic compression
test on saturated specimens of Shiraz silty clay are presented.
Volume change results from image processing
method (isotropic compression test on Shiraz silty clay)
was 5 cm in diameter and its height was 10 cm. At the beginning of the test the sample
was saturated. The initial cell pressure was 200 kPa and the pore water pressure
was 50 kPa throughout the test. The cell pressure was increased by a stepped
loading procedure. After each increment of cell pressure, the sample was allowed
to complete its primary consolidation. The amount of water expelled from the
sample was recorded throughout the test. The sample volume change was also measured
with the proposed image processing technique. Figure 14 shows
the volume change of the sample obtained using the two procedures. The results
of both methods match with a good degree of accuracy. Therefore, the proposed
method can successfully be applied for unsaturated soils in which the amount
of water expelled from the sample is not equal to the sample volume change.
In this study, a new technique for measuring the volume change of
soil samples in a triaxial test was presented. This method is especially
useful for unsaturated soil tests in which the conventional method of
using the pore-water volume exchange cannot be used to determine the total
volume change. One of the major challenges in image processing technique
is the accurate detection of the edges of the specimen. The proposed method,
based on Haar wavelet transform, could accurately trace the sample boundaries.
In comparison with conventional edge detection methods, this method can
effectively distinguish the real edges from noise or texture. Refraction
of light in water and plexiglass causes magnification and distortion of
the image. In this paper it was found that the pattern of distortion is
completely complex such that it could not be calibrated by conventional regression models. The proposed approach using neural network
model could be employed to account for this effect with a good degree
of accuracy and hence the corresponding errors are eliminated from the
test results. The results of volume change measurements with this method
showed good agreement with the amount of water exchange in saturated samples.