Optimization of image quality and acquisition time for lab-based X-ray microtomography using an iterative reconstruction algorithm

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Contents lists available at ScienceDirect

Advances

in

Water

Resources

journal homepage: www.elsevier.com/locate/advwatres

Optimization

of

image

quality

and

acquisition

time

for

lab-based

X-ray

microtomography

using

an

iterative

reconstruction

algorithm

Qingyang

Lin

a , ∗

_,

_Matthew

_Andrew

b

_,

_William

_Thompson

b

_,

_{Martin J.}

_Blunt

a

_,

_Branko

_Bijeljic

a

a Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom b Carl Zeiss X-ray Microscopy Ltd., Pleasanton, CA 94588, United States

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 11 September 2017 Revised 23 February 2018 Accepted 3 March 2018 Available online 8 March 2018 Keywords: X-ray mictomography Iterative reconstruction In situ imaging Image quality

a

b

s

t

r

a

c

t

Non-invasivelaboratory-basedX-raymicrotomographyhasbeenwidelyappliedinmanyindustrialand research disciplines.However, the main barrier tothe use oflaboratory systems comparedtoa syn-chrotronbeamlineis itsmuchlongerimageacquisition time(hours perscan comparedto secondsto minutesatasynchrotron),whichresultsinlimitedapplicationfordynamicinsituprocesses.Therefore, the majority ofexistinglaboratory X-raymicrotomography islimited tostaticimaging; relativelyfast imaging (tensofminutesperscan) can onlybeachievedby sacriﬁcingimaging quality, e.g.reducing exposuretimeornumberofprojections.Toalleviatethisbarrier,weintroduceanoptimized implemen-tation ofawell-knowniterativereconstructionalgorithmthatallowsuserstoreconstructtomographic imageswithreasonableimagequality,butrequireslowerX-raysignalcountsandfewerprojectionsthan conventionalmethods.Quantitativeanalysisandcomparisonbetweentheiterativeandtheconventional ﬁlteredback-projectionreconstructionalgorithmwasperformedusingasandstonerocksamplewithand withoutliquidphasesintheporespace.Overall,byimplementingtheiterativereconstructionalgorithm, therequiredimageacquisitiontimeforsamplessuchasthis,withsparseobjectstructure,canbereduced byafactorofupto4withoutmeasurablelossofsharpnessorsignaltonoiseratio.

1. Introduction

Non-invasive high resolution X-ray microtomography has found a wide range of applications in materials science and engineering ( Bertei et al., 2016; Eastwood et al., 2014; Kinney et al., 1994; Lan- dis and Keane, 2010; Maire and Withers, 2014; Puncreobutr et al., 2012; Stock, 2008 ), including geological-related studies ( Gouze and Luquot, 2011; Ketcham and Carlson, 2001; Lin et al., 2016b; Saif et al., 2017 ). For example, the oil and gas industry has seen a trans- formation of the application of pore scale imaging and modelling from a primarily research focussed pursuit concerned with the in- vestigation of fundamental ﬂow and transport properties in porous media to an increasingly crucial industrial tool for the characterization of a range of geological and petrophysical properties, including porosity, pore connectivity, permeability, mineralogy, ge- omechanical response, geological facies type and diagenetic history ( Andrä et al., 2013; Blunt et al., 2013; Cnudde and Boone, 2013; Lin et al., 2016a; Lindquist et al., 20 0 0 ). Recent years have seen the rapid technological development of the ability to not only

∗ _{Corresponding author.}

E-mail address: [email protected] (Q. Lin).

image rock structures, but also in situ fluid flow and heteroge- neous reaction through these pore structures at conditions of temperature and pressure representative of subsurface processes, applied to measure fluid saturations, saturation distributions, capillary trapping, localized capillary pressure measurement, and even spatially resolved contact angle ( Aghaei and Piri, 2015; AlRatrout et al., 2017; Andrew et al., 2014a, 2014b; Armstrong et al., 2014; Arns et al., 2005; Berg et al., 2013; Blunt, 2017; Lin et al., 2017; Norouzi Apourvari and Arns, 2016; Schlüter et al., 2014 ).

When using X-ray microtomographic techniques, two crucial parameters are image quality and acquisition time. Frequently, however, these two parameters are juxtaposed; better image quality can be achieved by increasing image acquisition time (prin- cipally by increasing the number of projections, or by increasing individual projection exposure time). Absolute image acquisition speed is also of critical importance when performing in situ experimentation, as more rapid image acquisition enables the imaging of dynamic processes. There are an increasing number of studies and applications requiring fast and dynamic imaging in different disciplines ( Andrew et al., 2015; Berg et al., 2013; Gao et al., 2017; Garcea et al., 2017; Reynolds et al., 2017; Robinson et al., 2014; Saif et al., 2016; Singh et al., 2017 ), which can be achieved by us- https://doi.org/10.1016/j.advwatres.2018.03.007

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Q. Lin et al. / Advances in Water Resources 115 (2018) 112–124 113 ing a synchrotron-based X-ray tomography with shorter total to-

mography times (approximately 1–30 s in modern beamline light sources).

The main barrier for laboratory-based high resolution dynamic X-ray microtomography is its much longer acquisition time (which can be on the order of several hours) when compared to a synchrotron beamline. To significantly reduce the acquisition time when using a laboratory instrument most studies to date have sacrificed image quality; for example reducing exposure time, reducing the number of projections or using a coarser voxel size ( Al-Khulaifi et al., 2018, 2017; Bultreys et al., 2016; Menke et al., 2015 ). Significant increases in throughput (decrease in acquisition time) can be achieved without sacrificing these if we consider each step in the tomographic workflow, typically sample mount- ing, scan setup, image acquisition, 3D computational reconstruction, image processing and segmentation, and quantitative analysis. To optimize the entire imaging workflow, each step in the process must be examined. The simplest way to increase image throughput while maintaining image quality is to increase the power of the X-ray source. This allows for the same number of X-rays to be acquired in a shorter period of time, allowing for exposure times to be decreased and, so, acquisition is sped up. This is the fundamental reason why much of the work on dynamic tomography has focussed on extremely powerful synchrotron light sources. The availability of beamtime at synchrotrons, however, is very limited and expensive. In a laboratory setting, source power is fundamen- tally limited by spot size – for a given (spot size limited) system architecture a smaller spot will give an improved resolution. An alternative approach is to improve the reconstruction algorithm such that it requires less information (i.e. a lower signal to noise image at the detector, achieved using a shorter exposure time, or a smaller number of acquired projections) when compared with conventional reconstruction algorithms (i.e. Feldkamp, Davis and Kress (FDK) Filtered Back Projection) ( Feldkamp et al., 1984 ).

Iterative reconstruction has the potential of greatly increasing reconstructed image quality for rapid acquisitions at low projection numbers or short exposure times. Iterative techniques have been investigated extensively for reducing radiation dose in medical CT applications ( De Man and Fessler, 2010; Fessler, 20 0 0; Pan et al., 2009; Sidky and Pan, 2008 ). More recently, there has been much active research into iterative algorithms for microtomography, with a particular focus on techniques for time-resolved dynamic imaging ( Kazantsev et al., 2016, 2015 ; Myers et al., 2015, 2011; Van Eyn- dhoven et al., 2015 ). A signiﬁcant reduction of the number of projections has been demonstrated through techniques such as discrete tomography ( Zhuge et al., 2016 ) for samples consisting of a ﬁnite number of materials, and neural networks ( Pelt and Baten- burg, 2013 ). The development of freely available open-source software packages such as ASTRA ( van Aarle et al., 2016 ) and TomoPy ( Gürsoy et al., 2014 ) has speeded the development and application of these techniques.

The main focus of our work is to develop an implementation of iterative reconstruction to enable it to be used routinely and re- liably as part of a scientific or industrial workflow. Several chal- lenges must be overcome before this can happen. The first of these is that iterative reconstruction is much more computationally demanding than traditional analytical reconstruction techniques; an iterative algorithm is typically at least an order of magnitude slower than an analytical algorithm such as FDK. This means that while the acquisition stage of the workflow may be sped up dramatically, this comes at the cost of significantly increased reconstruction times. Secondly, iterative reconstruction requires the optimization of several tuneable parameters for effective reconstruction, which can be challenging for medium or low skill operators.

In this paper, we will demonstrate recent advances in the application of iterative reconstruction techniques, applied on the

Table 1

The signal counts and number of projections studied. Signal counts

Low ∼20 0 0 (exposure time 0.25 s)

Medium ∼50 0 0 (exposure time 0.65 s)

High ∼10,0 0 0 (exposure time 1.30 s)

Number of projections

Few 401

Many 1601

laboratory-based Versa 520 X-ray microscope (Carl Zeiss X-ray Mi- croscopy). To solve the challenge of the computational cost of iterative reconstruction we implemented both algorithmic and computational optimizations. We use a statistical iterative algorithm including ordered subsets (OS) and Nesterov’s momentum technique, similar to that described by Kim et al. (2015) . The combination of the quadratic convergence rate provided by the momentum term, and the additional linear convergence rate increase provided by OS, reduces the number of iterations required for acceptable results from many hundreds to approximately 10–20. On the computational side, we use a highly parallelized unmatched forward and back projection code implemented on nVidia GPUs using CUDA.

We compare this implementation of iterative reconstruction to traditional analytical reconstruction for a homogenous reservoir sandstone, both in a “dry” state (with its pore space filled with only air) and in a “wet” state (with its pore space partially saturated with oil and water). The comparison is mainly made of quantitative metrics of image quality, including signal to noise ratio and edge sharpness. We show that, for a defined processing and segmentation workflow, equivalent or better results can be obtained using as few as one quarter the projections (and so one quarter the scan time) increasing throughput by up to a factor of 4. Al- though the case studies demonstrated in this work are in geological samples with multiphase fluid in the pore spaces, the iterative reconstruction algorithm presented here can be applied to other tomographic applications with a similarly sparse image structure. It should be noted that this includes many common applications in materials science and non-destructive testing. The exact amount of throughput increase that is possible in any given application de- pends on the object’s structure, and also on what information is sought from the reconstructed volume. Quantifying this is still a largely open problem, which has been studied in the framework of compressed sensing ( Jorgensen et al., 2013; Jørgensen et al., 2017; Jorgensen and Sidky, 2015 )

2. Materials and methods 2.1. Materials

To demonstrate the comparison between conventional FDK ﬁl- tered back-projection and iterative reconstruction algorithms we selected a Bentheimer sandstone sample, which contains 95% quartz, 4% feldspar and approximately 1% ﬁne clay ( Andrew et al., 2014c ). A cylindrical sample was prepared having 4.95 mm in di- ameter 10 mm in length. The total porosity measured by Helium porosimetry is around 20% ( Andrew et al., 2014c ). Bentheimer sandstone was selected due to its low mineralogy complexity and high level of porosity resolvable by micro-CT – this is ideal to compare and assess different reconstruction algorithms.

2.2.Imageacquisition

In this study, two cases were designed to compare the different reconstruction algorithms: dry scans (pore space ﬁlled with air), and multiphase scans (both brine doped with 3.5 wt% Potassium

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Fig. 1. Standard Euclidian 2-norm of the difference ( EM diff) between successive iterations in the reconstructed volume for the few projection, high count datasets, normalized

to the value at the ﬁrst iteration. Table 2

Scanning time with different combinations of signal counts and number of projections.

Stage stops at each projection

Number of projections

Few Many

Signal counts Low ∼11 min ∼41 min

Medium ∼13 min ∼52 min

High ∼17 min ∼69 min

Theoretical acquisition time without projection to projection time overheads Number of projections

Few Many

Signal counts Low ∼2 min ∼7 min

Medium ∼4 min ∼17 min

High ∼9 min ∼35 min

Iodide and decane exist within the pore space). Both dry scans and multiphase scans for the sample were performed by placing the sample in a carbon fibre core holder with a confining pressure of 20 MPa to avoid fluid bypass when injecting brine and decane. The same region of the sample was scanned at an energy of 80 keV, power of 7 W, and a voxel size of 6.0 μm. The dimension for each reconstructed image is 1024 _{× 1024}_{× 1024} voxels. To assess the two reconstruction algorithms, the exposure time for each scan was controlled to quantify the impact of the average signal counts and the number of projections on image quality. The counts and number of projections for each scan are listed in Table 1 , and the scanning time for each combination is shown in Table 2 . The scans with high counts and many projections that were reconstructed using the FDK algorithm, were used as reference scans against which other scans were compared.

Two sets of scanning times are reported in Table 2 . First, the acquisition time for a scan where the sample is stopped during projection acquisition, then repositioned between projections. This is

optimized to minimize image motion and ring artefacts. Second, a theoretical acquisition time is reported, corresponding to the case where the sample is rotated during each projection.

2.3. Descriptionoftheﬁlteredback-projectionanditerative reconstructionalgorithms

The conventional technique for reconstruction of X-ray microtomographic datasets is the algorithm of Feldkamp, Davies and Kress (FDK) ( Feldkamp et al., 1984 ). This is based on a contin- uous mathematical model of the X-ray projection process, which is then solved analytically using transform methods. This analytical solution is then discretized to yield a filtered back projection algorithm. This is a “one-shot” process that can be implemented highly efficiently. The FDK algorithm is known to perform well at low cone angles, and with low-noise datasets consisting of a suf- ficiently large number of projections ( Brokish and Bresler, 2006 ). However, the performance of the algorithm with noisy and/or under-sampled data may reduce dramatically, with reconstructed images suffering from increased noise, reduced resolution and po- tentially severe artefacts ( Boas and Fleischmann, 2012; Joseph and Schulz, 1980 ).

Statistical iterative reconstruction (SIR) algorithms ( Fessler, 20 0 0 ) offer an alternative to analytical algorithms such as FDK, and are known to offer increased tolerance to noisy or under-sampled data ( De Man and Fessler, 2010 ). Beginning with a discrete representation of the projection process, this is used to formulate a cost function, which is then minimized using an iterative algorithm. Our implementation uses a cost function of the form:

(

x

)

=1

2

Ax − b

2

W+

α

R

(

x

)

(1)

where the vector x represents the reconstructed volume, vector b represents the post-log corrected (absorption) projection data, and matrix A represents the projection process. The diagonal matrix W provides a statistical weighting of the projection data, to more ac-

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Q. Lin et al. / Advances in Water Resources 115 (2018) 112–124 115

Fig. 2. An example slice from images with the lowest image quality (low counts and few projections) and the highest image quality (high counts and many projections) using both the FDK and the iterative reconstruction algorithms. The images after applying non-local means edge preserving filters are also displayed for comparison. The region used for the line profile and for the quantification of phase boundary sharpness (A to B) is also highlighted.

curately reﬂect noise statistics. Using a Gaussian noise model for the absorption data, the weighting factor for the ith ray is given by w_i₌exp( _{− b}_i). The effect of this is to more heavily weight rays with higher signal-to-noise ratio during the reconstruction, thus reducing the impact of the noise.

The ﬁrst term in the cost function measures the data ﬁt, while the function R( x) acts as a regularization penalty, rewarding smoothness on the reconstructed volume. The parameter

α

controls the balance between data ﬁt and regularization. The regularization penalty is deﬁned as:

R

(

x

)

= Nr r=1 1 d r

φ

(

[Cx ]r

)

, where[Cx ]r= Nv j=1 c r jx j, (2)

and the matrix C is a ﬁnite difference operator, encapsulating differences between each voxel and its 26 ﬁrst-order neighbours, in

such a way as each difference is counted only once. The matrix C is of size Nr× Nv, where Nvis the total number of voxels, and Nr is the total number of first-order finite difference pairs over the entire reconstructed volume. dris the Euclidean distance between each finite-difference pair; this provides the correct spatial weighting. For the penalty function

φ

, we use the edge-preserving Huber function ( Huber, 1964 ), deﬁned as:

φ

(

t

)

=

₁ 2t 2,

|

t

|

<

δ

|

t

|

−δ2

, otherwise (3)

The Huber parameter

δ

controls the trade-off between edge- preservation and noise reduction; for suﬃciently small

δ

, the Huber penalty approximates the well-known total-variation (TV) penalty. For differences between voxels of magnitude greater than

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Fig. 3. (a) Example slice for the scan with many projections and high counts. The image was reconstructed by the FDK algorithm. (b) The oil, brine and grain phases were segmented into green (oil), blue (brine) and red (grain). The boundary of the oil phase is shown in black. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

tively less heavily than the quadratic portion applied to differences smaller than

δ

. The assumption is that small differences are due to noise, while larger differences are most likely edges (and so implicitly that the edges are well resolved in the reconstructed image). Therefore, in our implementation, the value of

δ

is set based on the noise level in a test FDK reconstruction of the data.

For minimization of the cost function, we use the separable quadratic surrogates (SQS) algorithm, combined with ordered subsets (OS) and Nesterov’s momentum ( Kim et al., 2015 ). Direct minimization of the cost function of Eq. (1) is not possible, so the SQS algorithm instead minimizes a sequence of much simpler quadratic surrogate functions, each of which is solved using a single Newton iteration, yielding a gradient descent-type algorithm. This has an update step:

x (n+1)₌_x(n)₊_D−1

_AT_W

_A_x(n)_{− b}

₊

_α∇

_R

_x(n)

₍₄₎ where x(n) _represents_{the reconstructed}_volume_{at the}_n_{th iter-}

ation. The matrix D is known as a diagonal majorizing matrix; its inverse can be pre-calculated and stored prior to the iteration sequence. Convergence of SQS is mathematically proven to be monotonic, but convergence is slow; the convergence rate is 1/ n, where n is the number of iterations ( Fessler, 20 0 0 ). Therefore, it may take many hundreds of iterations to achieve satisfactory results. To accelerate this, SQS is combined with Nesterov’s momentum ( Nesterov, 1983 ), which increases the convergence rate to O(1/ n2_{), giving satisfactory results in approximately 100 iterations.}

Additionally, we apply the ordered subsets technique ( Hudson and Larkin, 1994 ), which sub-divides each iteration into approximately equally-sized sets of projection angles, and increases convergence rate of the early iterations to O(1/( Mn) 2_{), where}_M_{is the number}

of subsets.

Using ordered subsets no longer guarantees monotonic convergence of the algorithm, but in practical use cases, it is not necessary to obtain a solution to a high degree of accuracy. In most cases, using 8 subsets, satisfactory results are achieved in approximately 10–20 iterations, though the exact number required de- pends on the individual dataset. To demonstrate this, Fig. 1 shows the standard Euclidian 2-norm of the difference ( EM_diff) between successive iterations in the reconstructed volume for the few projection, high count datasets for both the wet and dry cases. At the nth iteration, EM_diff is given by:

E M diff

(

n

)

=

Nv j=1

x (_jn)− x(n−1) j

2 , (5)

where x(_jn) and x(_jn−1) are the reconstructed grey-scale values for the current and previous iterations respectively, and Nv is the total number of voxels. EMdiff is a measure of how much the recon-

structed image changes at each iteration, and therefore how close it is to convergence. Note that differences after around iteration 20 are of the order 10 − 2compared to the differences after initial iterations, and contribute no signiﬁcant visible change to the image. One of the reasons iterative algorithms have not yet found widespread use is their computational expense. The main computationally intensive steps are the matrix-vector products of the form Ax and AT_b_{, which represent respectively forward and back} projection. Each iteration requires one of each operation; in contrast, analytical algorithms such as FDK typically require only a single back projection.

In practice, the projection matrix A is not calculated explic- itly; instead, its coefficients are implicitly calculated on the fly as needed during forward and back projection. For forward projection, we use a ray-driven approach, which is effectively a 3D ex- tension of Joseph’s algorithm ( Joseph, 1982 ). Although this algorithm is known to cause a systematic variation in the sampling rate along the rays, this has not been observed to cause any problems ( Turbell, 2001 ). Our back projection uses a voxel-driven approach, with coefficients calculated using bilinear interpolation on the detector plane.

Using different algorithms for forward and back projection results in a so-called unmatched implementation ( Guedouar and Zarrad, 2010; Zeng and Gullberg, 20 0 0 ), where the matrix implicitly used in back projection is not the exact transpose of that used in forward projection. Although this is theoretically not compli- ant with the derivation of the algorithm, in practice it allows for a much more computationally eﬃcient implementation. We have also found that it improves stability and signiﬁcantly reduces alias- ing artefacts in the reconstructed volume.

Our iterative reconstruction software is an optimized, heteroge- neous solution, capable of running across multiple CPU threads and multiple nVidia GPUs. The computationally expensive operations of forward and back projection, and calculation of the regularization term are performed on the GPUs, while less demanding operations such as updates of the reconstructed volume run in parallel in CPU threads. The GPU code is written in CUDA, making full use of asyn- chronous operations to hide memory transfer latency behind com- putation. Our code makes use of built-in hardware operations for bilinear interpolation in the GPU texture pipeline.

The reconstruction time for a typical dataset, consisting of 400 projections of size 1024 × 1024 pixels, and a reconstructed volume of size 1024 _{× 1024}_{× 1024} voxels, is approximately 6 seconds per iteration, plus a 25 s setup time. The test hardware consisted of 2 Intel Xeon E5-2650 v3 CPUs clocked at 2.3 GHz, 192 GB RAM and 2 nVidia P60 0 0 GPUs, running Windows 7 Professional. The timing scales approximately linearly with the number of projections.

2.4. Workﬂowforselectionofoptimaliterativereconstruction parameters

To obtain optimal reconstructed image quality using the pro- posed iterative algorithm, it is necessary to tune two parameters; the total regularization amount

α

, and the Huber parameter

δ

. The optimal parameter choice is dependent on the individual dataset, and is highly variable. To enable parameter tuning by inexperi- enced operators, we have defined a parameter optimization workflow implemented in the GUI of our iterative reconstruction software. Our workflow begins by setting the value of

δ

based on the noise level in a test FDK reconstruction of the data. Once this is done, a series of images are reconstructed for a range of

α

values and presented to the user. The choice of ﬁnal

α

value is left to user

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Fig. 4. Histograms of the grey-scale levels for the oil, brine and grain phases for each scan with different numbers of projections, signal counts, and different reconstruction algorithms.

preference and therefore somewhat subjective. To save computational cost, the

α

scan series is performed on a reduced size version of the dataset, with projections truncated to the central 128 slices in the vertical direction. For further speedup, it is possible to compute the scan series for the different

α

values in parallel. It should be noted that for applications where similar samples of the same class are repetitively imaged, once the optimal parameters have been determined for a single sample, they can be reused for other samples of the same class.

3. Results and discussion

We demonstrate the superior performance of the iterative reconstruction algorithm by comparing it to the conventional FDK filtered back-projection algorithm. FDK reconstructions were performed with the vendor’s software, using the conventional Ram- Lak filter, with a Gaussian pre-filter applied to the projection data with a standard deviation of 0.5. Iterative reconstructions used 8 subsets and 15 iterations. The images analysed by both methods

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Table 3

Signal to noise ratio for multiphase scans.

Signal counts Number of Projections Signal to noise ratio (SNR)

Without NLM ﬁlter With NLM ﬁlter Reconstruction methods FDK Iterative FDK Iterative Low 401 2.42 9.58 9.50 10.51 1601 4.60 12.50 13.51 13.93 Medium 401 3.71 11.94 13.04 13.18 1601 6.71 13.05 13.68 14.83 High 401 4.97 12.58 13.62 14.09 1601 8.42 14.19 14.65 15.89 Table 4

The k values computed for all multiphase scans.

Signal counts Number of projections kmean value for phase boundary sharpness

Without NLM ﬁlter With NLM ﬁlter Reconstruction methods FDK Iterative FDK Iterative Low 401 N/A a _1.66 _1.13 _1.36 1601 N/A 2.63 1.17 2.00 Medium 401 N/A 2.52 1.20 2.02 1601 N/A 2.77 1.21 2.10 High 401 N/A 2.84 1.23 2.17 1601 N/A 3.66 1.24 2.68

a N/A: the image quality is not suﬃcient to compute a k

mean value (e.g. Fig. 5 a).

were registered and resampled to have the same orientation and a sub-volume of a ﬁeld of view (500 3_{cube voxels) from the origi-}

nal 1024 3 _{cube voxels was used for comparison and assessment. In}

addition, we compare the reconstructed images from both methods with and without using non-local means edge preserving ﬁl- ter, typical of traditional image processing and analysis workﬂows ( Schlüter et al., 2014 ). We perform two case studies: multiphase scans in Section 3.1 , and dry scans, which show similar results, can be found in the supporting information. Finally, in Section 3.2 , we propose optimization of fast scanning with the iterative algorithm.

3.1.Casestudy1:multiphasescans

In this section, scans (500 3 _{cube voxels sub-volume) with two}

phases in the pore space are examined. The scans with theoretically the lowest quality (low counts and few projections) and highest image quality (high counts and many projections) are highlighted in Fig. 2 . The images showing larger ﬁeld of views are shown in Figs. S1 and S2 in the Supporting information. Once again this corresponds to the best and worst case, rather than the optimized case, which is examined in Section 3.2 . It can be seen, when multiple ﬂuid phases exist in the pore space, the image quality is much worse compared to the corresponding dry scans with the same number of projections and counts. From a visual assessment, generally the images reconstructed by the iterative algorithm have lower noise levels. The difference in image quality between the two reconstruction algorithms is much more obvious when the image quality is low.

3.1.1. Quantitativeanalysis:signaltonoiseratio

Signal to noise ratio within an image is a key quantitative performance metric when determining image quality and the ease and accuracy for image segmentation. In Fig. 3 the oil, brine and grain phases were segmented into green (oil), blue (brine) and red (grain); the boundary of the oil phase is shown in black. The segmented image is generated by applying a histogram based thresh-

olding for each phase, followed by applying an erosion ﬁlter. The phase boundary sharpness is quantiﬁed in Section 3.1.2 .

In this study, we deﬁne the phase contrast as the phase signal level for the oil phase, brine phase and the grain phase, which are determined by the histogram plot of the grey-scale values. Fig. 4 shows the histogram for oil, brine and grain grey-scale values for each scan with different numbers of projections, signal counts, and the two reconstruction algorithms.

The image quality (reflected by width of the histogram) of the image reconstructed by the iterative algorithm is generally much better than for the one that uses the FDK algorithm when the counts and number of projections are the same. Although applying a non-local means filter can significantly improve the image quality for the images with the FDK algorithm, generally the image quality is still not as good as when using the iterative algorithm. Moreover, the non-local means filter does not have a strong impact when using the iterative reconstruction algorithm. This indicates that this portion of the distribution in X-ray grey-scale values is not attributable to random noise, but rather to point-to-point grey- scale variation, either due to mineralogical variation in the sample, or other artefacts. This image quality phenomenon with different scanning conditions and different reconstruction algorithms is generally applicable to almost all scans. For the low quality images, e.g. images with few projections and low counts, after applying the non-local means filter, the shape of the histogram plot for the brine phase becomes skewed. This is mainly due to the high noise level and poor phase contrast.

In the multiphase scans, the signal to noise ratio, SNR, is de- ﬁned as follows: SNR = ₁

μ

oil−

μ

grain

2

σ

oil+

σ

grain

(6)

where

μ

oil is the mean grey-scale value for the oil phase,

μ

grain is the mean grey-scale value of the grain phase,

σ

oil and

σ

grain are the standard deviation values for the oil phase and the grain phase. The computed signal to noise ratio values for all the multiphase scans are shown in Table 3 . The SNR value from iterative al-

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Fig. 5. The grey-scale values along a line (A to B in Fig. 2 ) which passes through the oil, brine and grain phases. The values of k mean were found using the image after

applying non-local means ﬁlter in (f).

gorithms without applying filters are much higher than those from FDK values, and are comparable to those from FDK values with filters. This indicates that the phase noise (which also reflects the general image quality) from the iterative algorithm is much lower than the noise from the FDK algorithm, and is at similar level compared to the noise from the FDK algorithm with image filters.

3.1.2. Quantitativeanalysis:phaseboundarysharpness

Fig. 2 shows slices from the highest (high counts and many projections) and the lowest (low counts and few projections) quality images using both FDK and iterative reconstruction algorithms. The images after applying non-local means edge preserving ﬁlters are also displayed for comparison. To assess the phase boundary

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Fig. 6. The line plot between points A and B as indicated in Fig. 2 for an image with few projections and low counts, reconstructed by the FDK algorithm followed by applying a non-local means ﬁlter. The average phase boundary sharpness which is deﬁned by the average k value is 1.24.

sharpness, the grey-scale values along a line (selected between points A and B in Fig. 2 ) which passes the oil, brine, and the grain were plotted, as shown in Fig. 5 .

From Fig. 5 , it can be observed that when using the FDK algorithm, if the image quality is not suﬃcient, e.g. Fig. 5 a, the bound-

ary between two phases is not clear. Generally, applying filters can reduce the image noise and the phase boundary becomes visible. However, applying a filter can also decrease the boundary sharpness, e.g. Fig. 5 e and f. To quantify the sharpness of the phase boundary, which is indicated by the steepness of the slope, a logistic type of distribution can be used. This is a novel application of this function, allowing for a quantitative estimation of the length scale over which a phase transition within an image takes place. A general expression of a logistic distribution can be defined as:

f

(

x

)

=C + L

1+e −k(x−x0) (7)

where L corresponds to the difference between two phases, C is the grey-scale value of the least attenuating phase, x0 is the value

of the sigmoid’s midpoint, and k is a parameter determining the steepness of the slope. The sharpness of the phase boundary increases when the value of k increases, and allows for the spatial length scale for an interface transition to be quantitatively and ob- jectively measured. The graph shown in Fig. 5 f is used to demonstrate how to compute k values for a multiphase scan by fitting portions of the plot to Eq. (6) . The graph was divided into three sections, including three sub-profiles passing from the grain phase to the oil phase ( k1), the oil phase to the brine phase ( k2), and the brine phase to the grain phase ( k3). Each profile was fitted by a logistic distribution, Eq. (6) , and the mean value of k1, k2,

and k3 was used to quantify the phase average boundary sharp-

ness ( kmean), shown in Fig. 6 .

Fig. 7. Example slices from images with theoretically the lowest quality (low counts and few projections) and the highest quality (high counts and many projections) using both the FDK and the iterative reconstruction algorithms. The images after applying non-local means edge preserving ﬁlters are also displayed for comparison. The region used for line proﬁle and phase boundary sharpness (A to B) is also highlighted. More detailed analysis for the dry scans can be found in the supporting information.

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Fig. 8. Histograms for each phase for the images reconstructed by the iterative reconstruction algorithm with few projections but high counts, and many projections but low counts are compared with a reference scan, reconstructed by the FDK algorithm with high counts and many projections, for both dry and multiphase scans. More detailed analysis for the dry scans can be found in the supporting information.

The k values with different scanning and reconstruction conditions are shown in Table 4 . Generally, the images reconstructed by the iterative algorithm are characterized by a sharper phase boundary compared with the images reconstructed by the FDK algorithm. Although applying theoretically edge preserving ﬁlters can signiﬁcantly reduce the image noise, this generally has a neg- ative effect on the sharpness of the phase boundary.

3.2. Optimizationoffastscanningwiththeiterativealgorithm The key parameters affecting the scanning time are the number of projections and the exposure time. As source brightness for a given system is usually ﬁxed, faster imaging can be achieved by sacriﬁcing one or both of these parameters. This also has the effect of reducing image quality. To obtain the same or better quality reconstructed images from the same input projection data, a different reconstruction algorithm can be used.

The assessment for the multiphase ( Section 3.1 ) images and dry scans (see Supporting information) with different reconstruction algorithms shows that by using an iterative reconstruction algorithm image quality can be significantly improved, as compared to the image with the same scanning settings which was reconstructed by the conventional FDK filtered back-projection algorithm. Therefore, iterative reconstruction has the potential to significantly reduce scanning time while maintaining image quality. In this section, the image reconstructed by the iterative algorithm with either few projections but high counts, or many projections but low counts, are compared with a reference scan reconstructed by the FDK algorithm with many projections and high counts. Both images reconstructed by the iterative algorithm were about 4 times faster to acquire than the reference scan.

Fig. 7 shows a comparison of the scans (both dry and partially saturated scans), reconstructed using the iterative technique. The images showing a larger ﬁeld of view can be found in Fig. S7 in

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Fig. 9. The number of counts between point A and B for the images in Fig. 7 . More detailed analysis for the dry scans can be found in the supporting information.

the Supporting information. The reference scan (many projections, high counts, reconstructed using the FDK algorithm) is also shown. It can be observed that by using the iterative reconstruction algorithm, the image is similar to the reference scan after either sig- niﬁcantly reducing the number of projections (using 401 projections, rather than 1601), or lowering the X-ray counts (using 20 0 0 counts, rather than 10,0 0 0).

The grey-scale histograms for both dry and multiphase scans show that the noise levels (indicated by the SNR values) from the images using the iterative reconstruction algorithm are much lower than the reference scan reconstructed by the FDK algorithm with many projections and high counts ( Fig. 8 ). The noise levels of the images using the iterative algorithms reconstructed using either few projections but high counts, and many projections but low counts, are similar.

The detailed images showing the phase boundary are displayed in Fig. 7 and the line proﬁle analysis with the corresponding k values for phase sharpness is shown in Fig. 9 . It can be seen that the iterative reconstruction algorithm can signiﬁcantly reduce the scanning time and the image quality can be maintained or even improved compared to a standard high-quality scan.

The two scans with an intermediate quality of input data (many projections, low counts, or few projections, high counts) represent somewhat of a “sweet spot” for image acquisition. Signiﬁ- cant acquisition time decreases can be seen, while still maintaining both subjective and objective image quality. The comparison be-

tween these two cases (where image quality is approximately the same when reconstructed using the iterative algorithm) is interest- ing as the effective throughput of each technique is not the same ( Table 2 ). Overall scan time is not simply proportional to exposure time; each projection requires the sample to be rotated, translated and for data to be read off the X-ray detector. All this creates a projection to projection overhead, which is still present, even at a short exposure time. Scan time is, however, simply related to the number of projections. If acquisition speed is the priority, reducing the number of projections is the best strategy, as this will have the biggest impact on resulting acquisition throughput.

4. Conclusions

The imaging of dynamic processes using laboratory-based X-ray microtomography is one of the primary emerging areas of ongo- ing scientific interest in different disciplines, for example, material science and engineering, and single/multiphase flow and transport in porous media. However, its application is limited by the time required to acquire images. In this paper, we have shown how iterative reconstruction techniques can be used to achieve the same benchmark image quality (as measured by quantitative image performance metrics of signal to noise and edge sharpness) in one quarter of the time taken compared to traditional reconstruction methods. This was the case both for scanning of just the rock structure and the pore space containing multiple fluid phases. It should be noted that both of these cases have a sparse im-

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Q. Lin et al. / Advances in Water Resources 115 (2018) 112–124 123 age structure, with the reconstructed volume consisting mainly of

piecewise-constant regions. We would expect to be able to achieve comparable reduction of acquisition time in other use cases with similarly sparse image structure, but the exact amount of achievable reduction is highly application-dependent. In particular, in use cases where preservation of ﬁne detail is important, the amount of achievable throughput increase may be less.

This quadrupling of acquisition throughput has significant im- plications for the processes accessible to laboratory examination. In the academic sphere, research in different areas such as processes reaction, and fluid dynamics are available with a coupled temporal and spatial resolution never before accessible. In the industrial field (particularly that associated with digital rock analysis for the oil and gas industry), these developments have the potential of greatly reducing the acquisition time per sample, and so associated cost for the analysis. In a price constrained environment, this could be critical for the future adoption of this technology.

There are still many opportunities to improve iterative reconstruction, both in terms of reconstruction quality, with new regularization techniques giving the same quality image for noisier input data, ease of use, completely automating the reconstruction process, and computational eﬃciency.

Acknowledgements

We acknowledge the Engineering and Physical Science Research Council for ﬁnancial support through grantEP/L012227/1 . The images acquired in this study can be downloaded from https://doi. org/10.6084/m9.ﬁgshare.5962834 .

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.advwatres.2018.03. 007 .

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