Image processing of X-ray-computed polychromatic cone-beam micro-tomography
(

Advances in the technological (image resolution) and computational (image
size) aspects of X-ray computed micro-tomography (

The research aim of image classification optimization is to obtain representations of structures that can be automatically used for categorization of samples into a finite set of labels (i.e. phases in geological materials). As part of the recent development of computer performance and advanced automated computer algorithms, the classical machine learning technique provides a methodology for (non-)linear function estimation and classification problems (Vapnik, 1995). In general, the supervised machine learning approach involves the construction of a convincing model of the distribution of class labels in terms of predictor features for the whole image on the basis of a reduced example (i.e. training) data set (Alpaydin, 2004; Kotsiantis, 2007). The resulting classifier is then used to optimize a performance criterion and assign class labels to the testing data. Representative generalization is an important property of a classifier or classification algorithm, because it offers information about as yet unknown data. The support vector machine (SVM) algorithm as one capable example of such classifiers was first developed by Vapnik (1995) as an extension of the generalized portrait algorithm. The SVM algorithm is firmly grounded in the framework of statistical learning theory, which improves the generalization ability of learning machines for unknown data.

The success with any classification technique depends on the quality of the

A variety of both hard- and software measures have been developed to
eliminate BH artefacts. These include physical pre-filtering to pre-harden
the X-ray photon spectrum, the dual-energy approach, and a variety of
computational pre- and post-processing image corrections (Schlüter et
al., 2014). A major prerequisite for success with the latter software
approaches is that the correction method should not rely on any prior
knowledge of the material properties; i.e. it should not depend on known
attenuation coefficients. Commonly, there is no prior quantitative
information available on the number and distribution of phases present in a
geological sample. The most common technique for BH correction in
pre-processing is linearization, but this is preferable for monophasic
material cases. Although the most commonly used algorithm for the
reconstruction of

In principle, the idea of a surface-fitting approach in

The custom-built

The reconstruction of the 3-D data set was performed on the fly by the
Feldkamp filtered back-projection algorithm (Feldkamp et al., 1984). This
classical 3-D cone-beam reconstruction algorithm follows three main steps:
(i) pre-weighting the projection rays according to their position within the
beam cone; (ii) filtering the projections along horizontal detector lines
using a discrete filtering kernel; and (iii) performing a weighted
back projection of the filtered projections along the cone with a weighting
factor. Raw projections were corrected for dark current and flat-field
variations, followed by non-local mean filtering, ring removal, and filtered
back-projection reconstruction using the imaging software package Octopus
(

The

Our post-reconstruction method corrects the BH artefact by fitting a 2-D
polynomial, i.e. a quadratic surface to the reconstructed

Once BH correction of a

The idea of the maximum-margin hyperplane is obtained from statistical
learning theory and provides a probabilistic test error bound that is
minimized when the margin is maximized (see graphical representation of
NL-SVM, Fig. 1). The parameters of the maximum-margin hyperplane are derived
by solving a quadratic programming (QP) optimization problem. Suykens and
Vandewalle (1999) proposed the idea of modifying Vapnik's SVM formulation by
adding a least-squares term to the cost function, which transformed the
problem from solving a QP problem to the practically more convenient solving
of a set of linear equations. This modification significantly reduces the
effort in complexity and thus the computational cost, which may otherwise
become excessive. Consider the feature (kernel) function

Graphical presentation of the support vector machine classifier
with a non-linear kernel,

These can be written as a linear system:

Hence,

In our model approach, only the Gaussian radial basis function (RBF) kernel
is implemented in the LS-SVM classifier due to its high accuracy in function
estimation and data set classification (Van Gestel et al., 2002; Selvaraj et
al., 2007; Caicedo and Van Huffel, 2010; Ghorbanzade and Fatemi, 2012):

Workflow chart of our proposed

In the presence of a BH artefact, the reconstructed grey-scale values vary
across the rock core from higher values at the periphery to lower values in
the central region for the same mineral phase. Visual inspection of the
images A, B, and C of our three samples showed that the grey-scale values of
the minerals in the central region may overlap with the grey-scale values of
other minerals at the periphery (Fig. 3), which would significantly hamper
the correct differentiation between both minerals. In order to adjust
unequivocally a unique grey-scale level for each mineral phase, we applied
the quadratic 2-D polynomial function (Eq. 1) to our images. This polynomial
approximation constructs the surface that best fits the cloud of data points
subject to the coefficients determined by Eq. (3). The residual data values
were extracted as the difference between the values of the original data and
those of the fitted surface (Fig. 4c). Due to the BH effect, the attenuation
cross-section function across a sample is a parabolic curve rather than a
linear line (Fig. 5, light-grey colour lines). For a better visualization of
the fitted surface to the frequency (distribution) of grey-scale values
(cloud data) of each phase in an image, we illustrated here grey-scale
values row-wise at the centre of each image in

Reconstruction of a

BH correction after noise filtering, where

Upon successful removal of the BH artefacts, the LS-SVM algorithm was tested
for the multi-classification task. The performance of the LS-SVM algorithm
was ultimately evaluated also in the same image but with uncorrected BH
artefacts. The two

Plot of grey values of BH correction by our proposed
method. The curves in plot

Due to the nature of the BH artefact present in

Pixel-based image classification using LS-SVM, where

ROC curve analysis of LS-SVM classifier performance on the basis
of training set.

From the performance classification plot in Fig. 7a and b, the calculated parameters of AUC and accuracy were 0.989 and 99.41 % for the BH-corrected image (Fig. 6b), but as low as 0.901 and 81.10 % in the presence of a BH artefact (Fig. 6c). Similarly, the parameters of AUC and accuracy were calculated from the performance classification plot shown in Fig. 7c and d. The AUC and accuracy were 0.999 and 99.82 % for the BH-corrected image (Fig. 6e) and 0.963 and 88.71% in the presence of a BH artefact, respectively (Fig. 6f). Therefore, the performance measure results based on the pixel-based grey value training data set demonstrate that the probabilistic bias rate was higher in the BH-affected images, and this consequently caused misclassification of the test data. This finding provides evidence that BH correction is an important prerequisite in obtaining a good classifier performance, e.g. by using the LS-SVM approach. For an optimal classification result, it is always desirable to include the full grey-scale range (“pixel value”) of each individual phase to be trained in order to avoid misclassification, i.e. an undecided data classification as undesired output.

In this study, polychromatic cone-beam X-ray-source-generated

The advanced least-squares support vector machine (kernel-based learning)
method is proposed as an efficient routine to segment the

function [M_corr Surfacefit]

% Quadratic surface equation of second-order polynomial

%

% To find coefficients “

% M_corr

% Surfacefit

% First convert

% Image input parameters

nX

nY

limitval

zshift

% Main function

[r,c,v]

M

M(:,1)

M(:,2)

M(:,3)

M(:,4)

M(:,5)

M(:,6)

cyl

R

[m,n,f]

a

p

corr

S

M_corr

p1

Surfacefit

M_corr

end

The concepts and information presented in this paper are based on research and are not commercially available. This work was supported partially by BMBF grant 02C15262 and by the German National Science Foundation under the priority research program DFG SPP 1315. Edited by: H. Steeb