IEEE Signal Processing - March 2018 - 125

extraction of such data is a problem that involves both beam-hardening correction and segmentation.
Beam-hardening effects also can cause streaks or rings in the
acquired scan [15]. Smoothing filters are used to attenuate CT
streaks. Ringing artifacts can be removed using a transformation to polar coordinates, where these rings can be identified
and eliminated.
Depending on the quality of acquired scans, other image processing steps might be needed. Examples of these operations
include denoising, corrections for brightness nonuniformity, and
morphological operations. The goal of these operations is focused
on finding a representation that separates the different minerals
and pores in the rock sample into unique gray-level values to support the forthcoming segmentation step.

Segmentation methods
Image segmentation is not a trivial problem. In DRP, our goal is to
classify CT scan voxels into one of two or more mineral and pore
classes. In an ideal situation, histograms of CT rock scan images
are multimodal, with distinct peaks representing each mineral/
void in the scans. In practice, image histograms show more complex behavior in which the range of gray levels between histogram
peaks is continuous. Simple thresholding algorithms easily can
cause inaccurate segmentation results. The complexity of image
histograms is caused by many factors, including refraction, and
partial volume effects. Refraction is caused by changes in X-ray
trajectories as they pass through dense material [16]. Partial volume effects are caused by the limited resolution of CT scans. Voxels in interphaseal contact regions can overlap in space, so they are
assigned to unique values that represent the combination of the
overlapped phases. Denoising algorithms and histogram equalization methods can increase the contrast between the different
phases. These operations should be considered before applying image segmentation algorithms. Performance comparisons
between different segmentation algorithms used in DRP were
presented in [7] and [17]. The quality of segmentation algorithms
is assessed by comparing experimental values for physical rock
properties, like permeability and porosity, obtained using laboratory measurements with numerical simulations on the segmented
data sets.
Traditional image segmentation algorithms like Otsu's method
and the watershed algorithm are commonly used in DRP studies.
Modern segmentation techniques can avoid the limitations of traditional methods and improve segmentation results. For example,
Otsu's method does not consider the spatial distribution of intensity values in the image. Furthermore, since Otsu's method is a
global pixel-based thresholding algorithm, intensity inhomogeneities within a specific region can have a negative influence on
segmentation results. In contrast, 3-D hierarchical segmentation
algorithms (e.g., [18] and [19]) produce better results than purely
pixel-based algorithms by explicitly tracking variations in the spatial distribution of pixel intensities.
Image superresolution algorithms can also be utilized to attenuate partial volume effects. The use of smaller samples allows the
extraction of higher-resolution data. Such data can provide regulatory information that guide resolution enhancement procedures.

Computation of physical properties
Physical properties of interest, such as permeability, elastic moduli, and electrical conductivity, are computed by applying numerical
simulation techniques to segmented rock volumes. In this section,
we present various methods used in estimating fluid, mechanical,
and electrical properties using CT scans of rock samples.

Permeability
The Lattice-Boltzmann method (LBM) is the most widely used
method for computing permeability from digital rock scans. The
method is a bottom-up approach that uses a collection of fictitious
particles. The flow domain consists of regular lattice shapes, and
the particles move along the lattices to interact locally with other
particles in accordance with simple rules.
Mathematically, the goal of the LBM is to find discrete-velocity
distribution function fa (x, t) . The velocity distribution is computed for each direction a, while x and t are spatial and time parameters. Each direction, a, is associated with a weight, w a, and a
basis vector, e a. Several velocity sets are used in practice. The
most common ones in 3-D are the D3Q15, D3Q19, and D3Q27. In
these expressions, the first number refers to spatial dimensions and
the second number is the number of possible velocity directions.
Figure 6 illustrates the velocity set used in D3Q15.

Electrical conductivity
In computing the electrical conductivity of a rock sample, we
assume that rock pores are filled with a fluid with conductivity
vfluid. It is also common to assume that solid grains are insulators
with conductivity values of zero. Estimates of the sample's electrical conductivity are made by using a suitable numerical solver [5].
A potential difference is applied to the segmented 3-D volume.
Next, a finite difference or finite element method is used to solve
the following Laplacian equation
d $J = 0
J = vdU,

(8)

where J is the electrical current density, v is the local electrical conductivity, and U is the electrical potential. Total current,

12
3
6

5
8

13
4
11

Figure 6. Basis vectors used in the D3Q15 formulation of the LBM [20].

IEEE Signal Processing Magazine

March 2018

125

Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018