research papers
Fast threedimensional phase retrieval in propagationbased Xray tomography
^{a}Commonwealth Scientific and Industrial Research Organisation, Clayton, Victoria, Australia, ^{b}University of New England, Armidale, New South Wales, Australia, ^{c}University of Canterbury, Christchurch, New Zealand, ^{d}Monash University, Clayton, Victoria, Australia, ^{e}ARC Centre of Excellence in Advanced Molecular Imaging, The University of Melbourne, Parkville, Victoria, Australia, and ^{f}The University of Sydney, New South Wales, Australia
^{*}Correspondence email: darren.thompson@csiro.au
The following article describes a method for 3D reconstruction of multimaterial objects based on propagationbased Xray phasecontrast tomography (PBCT) with phase retrieval using the homogeneous form of the transport of intensity equation (TIEHom). Unlike conventional PBCT algorithms that perform phase retrieval of individual projections, the described postreconstruction phaseretrieval method is applied in 3D to a localized region of the CTreconstructed volume. This work demonstrates, via numerical simulations, the accuracy and noise characteristics of the method under a variety of experimental conditions, comparing it with both conventional absorption tomography and 2D TIEHom phase retrieval applied to projection images. The results indicate that the 3D postreconstruction method generally achieves a modest improvement in noise suppression over existing PBCT methods. It is also shown that potentially large computational gains over projectionbased phase retrieval for multimaterial samples are possible. In particular, constraining phase retrieval to a localized 3D region of interest reduces the overall computational cost and eliminates the need for multiple CT reconstructions and global 2D phase retrieval operations for each material within the sample.
Keywords: phase contrast; computed tomography; reconstruction; Xray imaging.
1. Introduction
Xray imaging is a cornerstone of modern medical imaging, with conventional 2D radiography and 3D computed tomography (CT) being common tools both in clinical and research domains (Bushberg et al., 2012). Phasecontrast imaging (PCI) is a specialized modality where image contrast is achieved by exploiting both the refractive and absorption properties of the imaged object. This technique has been widely studied and refined for over fifty years since the pioneering work of Bonse & Hart (1965). PCI has become a powerful approach for improved imaging of softtissue samples, which often exhibit poor contrasttonoise ratios due to weak absorption contrast and constraints in conventional absorptionbased radiography (Bushberg et al., 2012). With the ongoing of synchrotron and microfocus Xray sources, PCI is being adapted for medical use (Bravin et al., 2013; Tromba et al., 2016). Several PCI methods exist utilizing different mechanisms to encode phase information into projection images. In common use are analyserbased imaging (ABI) (Goetz et al., 1979; Gureyev & Wilkins, 1997; Davis et al., 1995; Nesterets et al., 2004) and gratingbased imaging (GBI) (Cloetens et al., 1997; Momose et al., 2003; Weitkamp et al., 2008; Nesterets & Wilkins, 2008) which utilize crystals and gratings, respectively, in the experimental setup. In this work, we study the socalled propagationbased imaging (PBI), also known as the inline method (Snigirev et al., 1995; Wilkins et al., 1996). In contrast to the other PCI methods, exclusive of combined approaches (Pavlov et al., 2004, 2005; Coan et al., 2005), PBI relies on the freespace propagation between the sample and detector for phasecontrast effects to manifest themselves as detectable intensity variations. Given this, PBI is simpler from an experimental perspective than other PCI methods. However, this simplicity is offset by more stringent requirements of the spatial coherence properties of the incident Xray beam (Nugent, 2010).
PBI has been shown to produce enhanced image contrast in weakly absorbing objects such as biological samples, generally in the form of Fresnel diffraction fringes at the interfaces between different materials. However, materialspecific quantitative information cannot be gleaned from PBI intensity images directly and requires the application of phase retrieval methods prior to CT reconstruction in order to recover the complex n(r) = 1 − δ(r) + iβ(r) within the sample. Several methods of phase retrieval have been developed in PBI, with different restrictions imposed on the object and imaging system. The transport of intensity equation (TIE) based methods for phase retrieval from the work of Teague (1983) and refined by others (Cloetens et al., 1999; Bronnikov, 1999, 2002) are commonly used. To reconstruct the 3D distribution of n(r), most of these methods require multiple Xray projections (at different propagation distances and/or Xray energies) to be acquired at each angular position of the object, which can be difficult to achieve under experimental conditions with time and dose constraints.
distributionA significant breakthrough was made by Paganin et al. (2002) who developed the socalled `homogeneous' TIE phase retrieval algorithm (TIEHom) that accurately reconstructs the complex of monomorphous objects (Turner et al., 2004). The algorithm requires only a single projection for each view angle and is robust to noise. As such, it has become the de facto standard for phase retrieval in PBCT. The TIEHom algorithm makes use of a spatially uniform factor or deltatobeta ratio, γ = δ/β, which defines the relative weight of the phase shift and absorption in the material of interest and results in the simplification of the reconstruction of the complex to
In reality, most samples do not consist of a single material, so use of TIEHom phase retrieval requires a compromised choice of γ for one particular material interface. Qualitatively, this choice results in the blurring of edges at the interfaces of materials where γ is overestimated and the retention of phasecontrast fringes for the underestimated case. Quantitatively, there are corresponding errors in the reconstructed distribution of n(r). These errors have been shown to be reduced by the collection of additional projections or utilizing suitable a priori information (Gureyev et al., 2013).
The TIEHom method has also been extended to enable quantitatively accurate phase retrieval of images containing nonoverlapping projections of two materials (Gureyev et al., 2002) and subsequently of m materials (Beltran et al., 2010). In both cases, a priori information is available for values of γ_{m} for each material interface. The method proposed by Beltran et al. (2010) demonstrates that a composite 3D reconstruction of n(r) can be produced by m separate applications of TIEHom phase retrieval using a different value of γ_{m} for the projection set, each followed by CT reconstruction from which localized subvolumes are spliced into the final reconstructed volume. The present work seeks to extend this method for phase retrieval of multimaterial samples, differing from the previously described approach whereby the materialspecific TIEHom phase retrieval step is performed as a post CTreconstruction filtering operation in 3D. Importantly, this 3D version of TIEHom is applied to localized subvolumes, again using a priori values of γ_{m} in each. The use of TIEHom phase retrieval discussed thus far has been applied to plane projections. This will now be referred to as prereconstruction 2D TIEHom (PreTIEHom2D). Multimaterial phase retrieval has also been attempted by the application of a 3D correction filter operation applied to the reconstructed volume in addition to conventional 2D TIEHom (Ullherr & Zabler, 2015). Also, other nonTIE based methods of 3D phase retrieval have been developed (Vassholz et al., 2016; Ruhlandt et al., 2014; Maretzke et al., 2016). For clarity, our proposed method will be referred to as postreconstruction 3D TIEHom (PostTIEHom3D) for which the derivation is given in Section 2. A numerical simulation framework has been created to simulate, evaluate and compare these methods and is discussed in Section 3.
2. Derivation – postreconstruction 3D TIEHom phase retrieval
Consider the imaging system shown in Fig. 1. Let an object be illuminated by a monochromatic plane Xray wave with wavelength λ and intensity I_{in}, I_{in}^{1/2} exp(ikz) with k = 2π/λ. The image of the object is recorded on a positionsensitive detector located at a distance R downstream from the object. In the following we assume that the dimensions of the object are small compared with the sourcetoobject distance ρ and that ρ ≫ R. Interactions of the Xrays and object matter are described via the spatial distribution of the complex n(r) = 1 − δ(r) + iβ(r), r = (x, y, z).
If the projection approximation (Paganin, 2006) is applied to the PBCT experimental setup depicted in Fig. 1, the following equations allow for the calculation of the transmitted phase and intensity,
where P_{θ}f(x′, y) represents the projection operator defined as
Here δ_{D}(x) is the Dirac delta function. The 2D Fourier transform operator
combined with the socalled filtered back projection (FBP) operator, (Natterer, 2001),
allows the construction
This inversion equation forms the mathematical basis of CT, permitting the reconstruction of the 3D distribution f(x, y, z) from a set of measured projections, P_{θ}f(x′, y), at θ angles within the interval (0, π).
One can now utilize the FBP operator to obtain expressions to separately reconstruct the real and imaginary parts of the complex ) and (3) into equation (7), we obtain the following pair of equations for β and δ, respectively,
By substituting rearrangements of equations (2Conventional Xray radiography and CT are generally concerned with measuring the intensity distribution of transmitted radiation in the object plane and reconstructing the imaginary part of the complex β, relating the absorption characteristics of the object to the measured intensity. We now seek to utilize the TIE to infer phase information from the visible diffraction fringes created upon propagation through a given distance. The finitedifference form of the TIE is given by
where and is an inline phasecontrast image in the detector plane. This method generally requires multiple (at least two) intensity measurements (at different R) in order to solve equation (10) for the unknown phase distribution (at a given λ), which may be difficult or problematic in the context of an experimental implementation. Let us assume that the sample object is monomorphous such that a spatially independent (but energydependent) proportionality constant, γ = δ/β, holds for the complex (Paganin et al., 2002; Mayo et al., 2003). This assumption is valid, for example, for objects consisting of a single material and objects composed of light elements (z < 10) when irradiated with highenergy Xrays (60–500 keV). Utilizing this property, one can link phase and intensity by rearranging equations (2) and (3) into
applying to equation (11) to obtain
and then inserting equation (12) into the TIE [equation (10)] to arrive at
From the expression above, one notes that the intensity at z′ = R includes, in addition to the contact intensity, a phasecontrast term proportional to the 2D Laplacian of the contact intensity. A further simplification can be introduced if one considers the case of weak absorption, whereby 2kP_{θ} β(x′,y) ≪ 1 in equation (2),
Inserting this approximation into equation (13) and rearranging, gives
Moreover, TIE implies weak phase contrast (Gureyev et al., 2004), . This allows the approximation of the inline contrast function, . Using this approximation and the Fourier space Laplacian identity, , the Fourier transform of equation (15) results in the following equation,
Inserting equation (16) into the FBP operator, equation (7) provides a `singlestep' phase retrieval and CT reconstruction expression for monomorphous objects (Gureyev et al., 2006; Arhatari et al., 2007, 2012),
A similar result has been derived by Bronnikov (1999, 2002) for purephase objects with negligible absorption which can be seen to correspond to equation (17) with γ → ∞. Similarly, equation (17) reduces to conventional pure absorption CT when R = 0.
Using an explicit form of the FBP operator, equation (5) and applying , where , to both sides of equation (17) and utilizing the identity
such that with some algebraic manipulation the terms in the FBP operator cancel leading to the following expression for ,
thus concluding the derivation.
Moreover, the result in equation (18) can be rewritten in the Fourier domain,
which is a direct counterpart to equation (17) and makes use of the 3D Fourier transform operator
Equation (18) is the mathematical basis of our new approach for a 3D phase retrieval method for weakly absorbing monomorphous objects. As is seen, the phase retrieval step is decoupled from the FBP operator, implying that this step can be performed in 3D after the conventional CT reconstruction. A similar derivation for purephase objects is described by Baillie et al. (2012). One of the more interesting aspects to the form of equation (18) relates to the ability of performing phase retrieval localized to a region of interest (ROI). This property potentially gives an advantage over prereconstruction phase retrieval techniques for samples which may contain a range of different materials of interest whereby it is often difficult to optimize global parameters to achieve optimal contrast across the entire sample (Gureyev et al., 2013). We will illustrate that with localization we can apply a specific value of γ chosen to `focus' on a desired material composition inside a 3D region, thus giving a quantitatively accurate phase retrieval within that region. A spatially localized form of phase retrieval also leads to some potential computational gains over existing methods, such as the ability to divide and parallelize phase retrieval over the sample reconstruction. The use of fast discrete Fourier or GPU based filters for implementing the 3D TIEHom filter on subregions would make this phase retrieval method computationally efficient, even for large datasets.
3. Numerical experiment setup
3.1. Xray CT simulation framework
In order to evaluate the accuracy and characteristics of PostTIEHom3D, a computational simulation framework was constructed for conventional absorption CT and PBCT workflows. This framework allows for the definition of an analytical 3D model consisting of multiple simple geometric primitives in space representing a material defined by its complex ITK image processing toolkit (Ibáne~z et al., 2005) and uses elements of the Astratoolbox (van Aarle et al., 2016), XTRACT (Gureyev et al., 2011) and RTK (Rit et al., 2014) packages for CTreconstruction routines.
at a given Xray energy. With such a model, the workflow can be used to generate a volume image in addition to contact or PBI projections for a given number of rotation angles with specified resolution and photon statistics. These simulated projections can then be subsequently reconstructed, including phase retrieval as part of the processing pipeline and quantified with a range of metrics. The framework was primarily constructed using the3.2. Multimaterial numerical phantom
The numerical phantom model defined for these simulations consists of an airfilled 1024 µm × 1024 µm × 256 µm cuboid containing a central 700 µmdiameter cylinder consisting of breast tissue with the composition described by Hammerstein et al. (1979) and serving as the background material. Embedded within the central cylinder are four smaller 80 µmdiameter cylinders of different materials distributed azimuthally at a fixed distance from the origin. A schematic representation of the phantom and corresponding material properties for the objects at an Xray energy of 20 keV are shown in Fig. 2 and Table 1. Values obtained for material specific β and δ were calculated using the webbased tool (Gureyev et al., 2011). The relative deltatobeta ratio versus breast tissue, γ_{rel}, has been calculated for each material as γ_{rel} = (δ − δ_{breast})/(β − β_{breast}), where δ_{breast} and β_{breast} are the values for breast tissue.

The materials modelled within the inner cylinders include organic tumorous and adipose tissue. The calcite weddellite (CaC_{2}O_{4}·2H_{2}O, 1.94 g cm^{−3}) represents one of the primary components of type I and type II breast calcifications (Ghammraoui & Popescu, 2017). Additionally, the hydrocarbon paraffin (C_{31}H_{64}, 0.9 g cm^{−3}) was included as it can be used to enclose biopsy samples and has Xray absorption and refractive properties similar to those of the other organic materials.
3.3. Simulations
This research seeks to evaluate the numerical implementation of PostTIEHom3D derived in Section 2 and compare its performance with conventional absorption CT and the Beltran et al. (2010) PreTIEHom2D method for the simulated multimaterial phantom defined in Section 3.2.
3.3.1. Projection simulation
Distributions of transmitted intensity and phase shift from a simulated Xray source (at 20 keV) were computed for N_{p} sample rotation angles analytically from the phantom model definition in Section 3.2 using equations (2) and (3). The complex amplitude of the transmitted wave was then calculated over a 2D plane sampled at a given resolution. In the case of the simulations for this article, a relatively fine of 0.25 µm was chosen for the generation of initial projections. Due to the invariance of the phantom along the rotation y axis, we can employ the simplification only requiring the generation of a singlerow projection for the phantom at each angle step, thus resulting in 1D planeprojection images of 8192 pixels.
For PBI projections, the Fresnel propagation operator (Paganin, 2006) implemented as a 2D Fourier filter was applied to each 2D projection, with the transfer function, FG(ξ′,η) = exp[−iπλR(ξ′^{2}+η^{2})]. In the case of contact projections, the above propagation step was skipped leading to the final step of projection simulation where a discrete Gaussian smoothing filter was applied (with the variance σ^{2} = 2 µm^{2}) to simulate finite resolution of the detector. This step led to the smearing of sharp edges at the interfaces between objects and to reducing potential aliasing artefacts of subsequent phase retrieval and CT reconstruction. Finally, projections were sampled with a finite square aperture of 2 µm resulting in N_{p} 1D row intensity projection images of 1024 pixels (32bit real).
3.3.2. Dose and noise
In this work we seek to model a fixed total radiation dose per scan utilizing a specified exposure time per angle. To quantify the dose, we introduce the simulation parameter `total photons per pixel' (TPP) as the total number of incident photons per pixel per scan. In practice, we simulate noisy projections by applying a
with a known mean number of photons per individual pixel.3.3.3. Prereconstruction 2D TIEHom phase retrieval
For the application of the PreTIEHom2D method, a set of N_{p} 2D projections was constructed by vertically stacking copies of the simulated, propagated and binned 1D row projections generated as per Section 3.3.1. Poisson noise was then generated as described in the previous section, followed by 2D TIEHom phase retrieval [equation (21)] applied to each 2D projection using the corresponding phantommaterialspecific value of γ_{m},
3.3.4. CT reconstruction
To recover the imaginary component β of the complex from the thus far simulated projections, the `goldstandard' FBP CTreconstruction algorithm, equation (7), was applied. Prior to performing the actual reconstructions, standard background (flatfield) correction was performed before the −ln transform. In the case of the PreTIEHom2D multimaterial method and as discussed in Section 3.3.3, N_{p} 2D phaseretrieved projections are reconstructed into a 1024^{3} voxel volume. To construct the final spliced multimaterial volume, 2D TIEHom phase retrieval followed by FBP CTreconstruction was performed for each phantom material separately with the composite volume constructed by inserting subvolumes enclosing each cylinder. For PostTIEHom3D, N_{p} 2D simulated projections were constructed following the approach described in Section 3.3.3, without the phase retrieval step and then reconstructed with FBP producing a 1024^{3} voxel volume to which postreconstruction 3D phase retrieval was applied.
3.3.5. Postreconstruction 3D TIEHom phase retrieval
PostTIEHom3D phase retrieval is implemented as a 3D Fourier filter [see equation (19)] and applied to subvolumes corresponding to ROIs contained within the whole CT reconstructed volume. Successful application of PostTIEHom3D requires that the selected 3D ROI meets the following two criteria. Firstly, the ROI should fully contain the singlematerial object under investigation. Ideally, this region should not be `polluted' with the inclusion of other objects or reconstruction artefacts which will lead to further undesirable artefacts in the phaseretrieved subvolume. Secondly, the ROI should be chosen to consider the width of the 3D TIEHom pointspread function (PSF), P_{TIE}, the Fourier transform of which is defined as
Here, A = πγλR is a positive constant (we restrict our consideration to the case of positive γ). The corresponding realspace expression is
Here, l_{TIE} = A^{1/2}/2π and r = (x^{2} + y^{2} + z^{2})^{1/2}, with the standard deviation of this distribution equal to . To overcome situations where the criteria are unable to be met with an appropriately large ROI due to neighbourhood constraints, it is possible to artificially enlarge the ROI by zeropadding to the required dimensions. Zeropadding to a power of two is also commonly applied in implementations of fast Fourier transform (FFT) based filters for optimal performance. For the simulations performed in this article, the dimensions and locations of the cylindrical phantom objects are known explicitly, therefore a 128 µm × 128 µm × 128 µm cubic ROI was chosen for each of the four cylinders which adequately extends beyond the 80 µm cylinder diameter and easily satisfies the requirement for P_{TIE}. For example, given an Xray energy of 20 keV, adipose tissue with γ ≃ 523 (Table 1) and a propagation distance of R = 60 mm results in the calculated scale 1_{TIE} ≃ 12µm, corresponding to only several pixels at the simulated detector resolution.
3.3.6. Evaluation metrics
To quantify and compare the performance of the evaluated phase retrieval methods, two metrics have been selected, the contrasttonoise ratio (CNR) (Gureyev et al., 2014) and a universal image quality index (UIQI) introduced by Wang & Bovik (2002). CNR is defined as follows,
where V is the volume of the object ROI (in voxels), 〈β_{o}〉 and σ_{o}^{2} are the mean and variance of β within the object ROI and 〈β_{b}〉 and σ_{b}^{2} are the mean and variance for a `background' ROI devoid of any phantom objects. In these simulations, this background region corresponds to a similarly sized region as the object ROI and is located outside all object ROIs in the reconstructed volume. CNR is calculated to compare the associated statistical gain between absorption only contact CT and the two phase retrieval methods which will highlight the respective noise suppression properties. CNR gain, G_{CNR}, is calculated as the ratio of the CNR in the CT reconstructions using phaseretrieved and contact projection data, for sameobject and background ROIs,
The second metric UIQI, where
requires a reference image for comparison, which is provided by the same ROI extracted from the numerical phantom model, with 〈β_{r}〉, σ_{r}^{2} and σ_{or} representing the mean, variance and covariance between the reference and sample ROIs, respectively. The UIQI produces a single numerical index that combines three factors: loss of correlation, luminance distortion and contrast distortion which its authors suggest permit the measurement of information loss as opposed to the quantification of error with other metrics (Wang & Bovik, 2002).
3.4. Results
Simulation parameters have been selected to be consistent with an experimental PBCT imaging scenario representative of a small biological biopsy. For the simulations presented and discussed here, the following series of fixed parameters have been selected: an incident monochromatic Xray beam with E = 20 keV and pixel size h = 2 µm (in both directions). A total of 900 projections were generated, N_{p} = 900 corresponding to an angular step of 0.2°. This number conforms to the Nyquist sampling condition N_{p} > (π/2)d/h, for estimating the lowerbound for the number of required projections to avoid subsampling, with d being the reconstruction diameter (Hsieh, 2009). A propagation distance of R = 60 mm was chosen to be just below the limit for the TIEHom validity condition of λR/h^{2} = 1, such that good phase contrast should be obtained in the simulated propagated projections. A range of photon statistics were applied to the simulated projections as described in Section 3.3.4. An exponentially increasing series of values in terms of the TPP and relative noise as a percentage were used, 1 × 10^{4} (1.0%), 1 × 10^{5} (∼0.32%), 1 × 10^{6} (0.1%), 1 × 10^{7} (∼0.032%) and 1 × 10^{8} (0.01%). Notably, noisefree and contact projections were also simulated for comparative purposes.
The grid of 16 images presented in Fig. 3 comparatively illustrates PBCT and the application of phase retrieval to FBP reconstructions of the simulated phantom for the imaging scenario described above in the case of 0.1% noise. Each image displays the central 2D slice through a 128 µm^{3} cubic ROI of each material cylinder. Grid rows represent the materials: weddellite, paraffin, adipose and tumorous tissue; whilst columns display contact CT (R = 0 mm), PBCT (no phase retrieval), PreTIEHom2D and PostTIEHom3D for the propagation distance R = 60 mm. Visually inspecting the reconstructions for weddellite [Figs. 3(a)–3(d)] some object contrast is achieved in the contact reconstruction [Fig. 3(a)], albeit with a reasonably high level of noise. Phasecontrast fringes are clearly visible in the reconstruction of the propagated case without phase retrieval [Fig. 3(b)] and similar highquality results are shown for both TIEHom phase retrieval methods in Figs. 3(c) and 3(d). The results for paraffin [Figs. 3(e)–3(h)] and adipose tissue [Figs. 3(i)–3(l)] are visually very similar, in both cases no apparent object contrast is perceptible in contact reconstructions. Again, noted is the presence of phasecontrast fringes in the nonphase retrieved images in Figs. 3(f) and 3(j), although they are less defined and contain significantly more visible noise than the weddellite case. Both variants of phase retrieval produced visually similar results for the two materials and achieve relatively good contrast and reduced levels of noise such that the object is clearly distinguishable from the background. In the case of tumorous tissue [Figs. 3(m)–3(p)] the results are far less revealing. It is worth noting that, at E = 20 keV, tumorous tissue has absorption and refractive properties more similar to the background breast tissue (Table 1) than the other materials in the phantom, making the task of distinguishing the two regions after reconstruction significantly more difficult. This difficulty is clear in the nonphase retrieved images; both contact and propagated images in Figs. 3(m) and 3(n) exhibit only noise. Notably, the success in the application of phase retrieval for tumorous tissue is not convincing when viewing a single reconstructed slice as in the case in Figs. 3(o) (PreTIEHom2D) and 3(p) (PostTIEHom3D), where little apparent contrast or structure is present for one to identify the reconstructed object.
3.4.1. Reconstruction line profiles
To gain insight into the relative quantitative merits of the twophase retrieval methods, line profiles of the reconstructed central slice in each of the four materials are presented in Fig. 4. Each plot presents the line profile of β values over a central line extending 40 µm horizontally across the ROI. Separate plots for the reference computational model and reconstructed values after phase retrieval with PreTIEHom2D and PostTIEHom3D for R = 60 mm and 0.1% noise are plotted.
Inspection of Fig. 4(a) (weddellite) shows a similar reconstruction result with phase retrieval in both cases resulting in the elimination of diffraction fringes which are clearly visible in the contact reconstruction [Fig. 3(c)]. Good contrast and suppression of noise have been achieved, giving rise to values of β in agreement with the model. Notably, there is a similar degree of smoothing of the edges for both methods which may be attributed to the finite resolution of the imaging system and interpolation of projection data during CT reconstruction. The noisesuppressing properties of PostTIEHom3D compared with PreTIEHom2D are illustrated for the lessabsorbing materials paraffin [Fig. 4(b)] and adipose tissue [Fig. 4(c)]. Here, both profiles and images are similar, both materials exhibit comparable absorption and refractive properties at the simulated Xray energy as shown by the values in Table 1. Again, for these simulation parameters, PostTIEHom3D displays qualitatively improved noise reduction for these materials. For the final material, tumorous tissue [Fig. 4(d)], the profiles indicate only limited success of both PreTIEHom2D and PostTIEHom3D in improving object contrast in this case. The object is barely perceptible after phase retrieval although it may be suggested that PostTIEHom3D is marginally less noisy.
3.4.2. Evaluation with respect to noise
The evaluation of the noise suppression, contrast enhancing properties and quantitative accuracy of both methods as displayed in the previous section needs to be viewed with respect to varying levels of Poisson noise. Fig. 5 presents plots of G_{CNR} and UIQI (Section 3.3.6) for each material over a range of values of Poisson noise on a logarithmic scale. In the case of UIQI, an additional plot for `contact' reconstructions without any phase retrieval is displayed as a point of reference. The relative noise suppression characteristics between PreTIEHom2D and PostTIEHom3D are quantified by the calculated values of G_{CNR} and UIQI, respectively. Almost universally, both methods exhibit improved CNR at all noise levels for all four materials with PostTIEHom3D generally superior. However, some variation in the overall trends are observed across different materials. For example, in Figs. 5(a) (weddellite) and 5(b) (paraffin) we see that the reported values of G_{CNR} are both maximal at the highest simulated noise level, TPP = 1 × 10^{4} (1.0%), and decrease monotonically approaching 1 (no gain) at the lowest simulated noise level, TPP = 1 × 10^{8} (0.01%). For adipose tissue [Fig. 5(c)], one notes quite different behaviour with both methods displaying an initial increase in gain, peaking at TPP = 1 × 10^{5} (∼0.32%) and then decreasing, approaching 1 at 0.01% noise. In this case, it is also noted that PostTIEHom3D significantly outperforms PreTIEHom2D with respect to G_{CNR} until their values align at around TPP = 1 × 10^{7} (∼0.032%). For tumorous tissue [Fig. 6(d)], one sees some similarity with that of adipose with a general increase in G_{CNR} with a peak shifted towards the lower end of the simulated noise spectrum at TPP = 1 × 10^{7} (∼0.032%) and then falling away. Again, PostTIEHom3D exhibits a near uniform improvement in G_{CNR} over PreTIEHom2D at noise levels approaching the peak, after which both methods perform similarly, as previously noted.
Turning now to the plots for UIQI, as mentioned in the definition (Section 3.3.6), this metric attempts to quantify `image quality' from a broader perspective than the purely statistical nature of CNR. Its use of a reference image in the form of the numerically accurate model ROI implies that it may provide values more representative of the quantitative deviation of the subject image from the reference. For two of the less absorbing materials, paraffin [Fig. 5(b)] and adipose [Fig. 5(c)], one observes very similar results with a monotonically increasing value of UIQI as TPP increases, with PostTIEHom3D giving a slightly higher quality result for higher noise levels until converging with PreTIEHom2D at around 0.01% noise. For these materials, one also notes that UIQI for `contact' reconstructions remains relatively uniform until around 0.032% noise, from where the quality metric increases with a corresponding decrease in noise. With tumorous tissue [Fig. 5(d)] the UIQI for both phase retrieval methods dips below that of contact reconstructions before increasing in line with decreasing noise as with the other materials. This initial degradation in UIQI at higher noise levels for tumorous tissue can be attributed to the relatively little object information available to `retrieve' relative to noise such that phase retrieval introduces a greater deviation from the model. The more highly absorbing weddellite sample [Fig. 5(a)] illustrates different behaviour from the others in that UIQI for both phase retrieval methods remains relatively uniform and similar irrespective of the noise level. In contrast, UIQI for contact reconstructions of weddellite increases monotonically (as expected) with decreasing noise, exceeding the corresponding values for both PostTIEHom3D and PreTIEHom2D for TPP > 1 × 10^{7}.
The relative improvements in image quality due to phase retrieval in comparison with contact CT as presented quantitatively via G_{CNR} and UIQI as noise levels are varied are shown in Fig. 6. Here, in a similar form to Fig. 3, central 2D FBP reconstructed slices for the ROI containing tumorous tissue are displayed for contact and PBCT with phase retrieval for three different levels of noise (rows) corresponding to 0.1%, ∼0.03% and 0.01%. Columns display contactCT, PBCT (no phase retrieval), PreTIEHom2D and PostTIEHom3D for the propagation distance R = 60 mm. Figs. 6(a)–6(d) show the same images as Figs. 3(m)–3(p), corresponding to 0.1% noise. Reviewing the images for contactCT in the first column for each noise level [Figs. 6(a), 6(e) and 6(j)] one observes no perceivable contrast; all reconstructed slices appear to display uncorrelated noise. For the second column, PBCT without phase retrieval, one notes at ∼0.03% [Figs. 6(f)] and 0.01% noise [Fig. 6(j)] the emergence of a diffraction fringe at the boundary interface between the tumorous and breast tissue in addition to reconstruction `streak' artefacts which are most likely due to sharp edges of the neighbouring weddellite object and are similar in appearance to those for adipose tissue and paraffin at the higher noise level of 0.1% as seen earlier in Figs. 3(f) and 3(j). Turning now to the phase retrieval results in columns three and four, one sees that both PreTIEHom2D and PostTIEHom3D are able to produce results with enough contrast that the circular region of tumorous tissue is clearly distinguishable from the background breast tissue for ∼0.03% noise [Figs. 6(g)–6(h)] and slightly higher perceivable contrast at 0.01% noise [Figs. 6(k) and 6(l)]. Again, noted is the appearance of streak artefacts and a marginally visually enhanced result in the case of PostTIEHom3D than PreTIEHom2D.
Overall, the results in this section illustrate the ability of TIEHombased phase retrieval in conjunction with PBCT to reveal useable image contrast in the presence of noise where conventional contactCT fails. Moreover, this research demonstrates that the level of contrast achieved is a function of the material, noise statistics and intrinsic properties of the imaging system and is consistent with theoretical derivations for the effect of phase retrieval in PBCT (Nesterets & Gureyev, 2014).
3.4.3. Evaluation with respect to propagation distance
With some insight gained as to the characteristics of the phase retrieval schemes in the presence of varying levels of simulated noise and hence dose, the focus now becomes an evaluation of the response of varying propagation distance, which serves as the key experimental parameter in PBCT. Several studies (Nesterets & Gureyev, 2014; Kitchen et al., 2017) have demonstrated that increasing propagation distance in combination with TIEbased phase retrieval results in significantly improved signaltonoise ratio in resulting images. Such improvements permit the use of a smaller radiation dose, producing images of similar quality to those produced without PBCT methods at higher doses.
For the simulations previously described at a fixed propagation distance, R = 60 mm was used and corresponded to just below the distance for maximum phase contrast in the TIE regime defined by λR/h^{2} = 1. For the analysis in this section the noise level is fixed at 0.1%, TPP = 1 × 10^{6} and the propagation distance varied with 10 mm ≤ R ≤ 70 mm at 10 mm increments. Fig. 7 presents plots of G_{CNR} and UIQI across this range of propagation distances. With respect to G_{CNR}, one sees relatively consistent trends across the four materials with gain generally increasing as propagation distance increases with PostTIEHom3D, recording greater levels than PreTIEHom2D. The magnitude of the gain does vary with material, with weddellite [Fig. 7(a)] achieving modest levels of the order 2.3 ≤ G_{CNR} ≤ 2.5 for PreTIEHom2D and 2.4 ≤ G_{CNR} ≤ 2.8 for PostTIEHom3D. The other lessabsorbing materials exhibit greater levels of G_{CNR}, with paraffin [Fig. 7(b)] exhibiting around 10 ≤ G_{CNR} ≤ 32 (PreTIEHom2D) and 15 ≤ G_{CNR} ≤ 36 (PostTIEHom3D). Adipose tissue [Fig. 7(c)] reports slightly higher levels with 15 ≤ G_{CNR} ≤ 40 for PreTIEHom2D and 20 ≤ G_{CNR} ≤ 50 for PostTIEHom3D. In this particular case, one also notes that there is a slight reduction in the magnitude of G_{CNR} for propagation distances where R > 50 mm, which is the likely result of reconstruction artefacts beginning to dominate over the random noise. The results for tumorous tissue [Fig. 7(d)] differ again from previous cases with G_{CNR} for both phase retrieval methods showing an initial rise then dip in the vicinity of 10 mm ≤ R ≤ 40 mm followed by a sustained increase for R > 40 mm where the magnitude for PreTIEHom2D is 15 ≤ G_{CNR} ≤ 60 and 45 ≤ G_{CNR} ≤ 80 for PostTIEHom3D. The graphs of G_{CNR} over all four materials reveal that PostTIEHom3D consistently produces greater contrast over PreTIEHom2D by a relatively fixed, nonconverging magnitude as the propagation distance increases.
Examination of the graphs for UIQI shows that again both paraffin [Fig. 7(b)] and adipose tissue [Fig. 7(c)] trend similarly with increasing propagation distance with nearly identical values in the range 0.2 ≤ UIQI ≤ 0.5. As evident with G_{CNR}, PostTIEHom3D tends to outperform PreTIEHom2D by a fixed magnitude with UIQI levelling off from around R ≥ 60 mm, after which the theoretical validity conditions of TIEHom are exceeded, leading to a loss of fidelity in the phase retrieval process. Tumorous tissue [Fig. 7(d)] shows some inconsistency in the evaluated UIQI over both methods as was the case with G_{CNR}. This is likely due to the lack of available phase contrast in the projections to adequately reconstruct the object as seen visibly in the line profiles in Fig. 5, and is quantified by the relatively low values of UIQI compared with the other materials. Despite this, PostTIEHom3D produces a UIQI nearly double that of PreTIEHom2D.
The plot in Fig. 7(a) for weddellite shows distinctly different behaviour from the other materials. In this case one observes a monotonic decrease in UIQI for both methods as the propagation distance increases, opposite in behaviour to the others. Additionally, one notes that PreTIEHom2D outperforms PostTIEHom3D from around R > 20 mm where the latter decreases more rapidly as the propagation distance increases. Fig. 8 illustrates line profiles plotting PostTIEHom3D β values for weddellite in the interval 10 mm ≤ R ≤ 70 mm with TPP = 1 × 10^{6} in addition to the model and contact profiles. Evident from these plots is the everincreasing degree of smoothing produced around the edges of the object as the propagation distance increases, resulting in an overall increased deviation from the reference model profile and degrading UIQI relative to the model as shown in Fig. 7(a). Such oversmoothing can be attributed to the validity conditions of the TIE being exceeded for weddellite at increasing propagation distances.
Overall, the results show that the PBCT in conjunction with the phase retrieval methods investigated result in significant improvements to image quality over contactCT imaging. The results are consistent with those reported by Gureyev et al. (2014), illustrating improvements in contrasttonoise of several orders of magnitude with increasing propagation distances. Moreover, such improvements correspond with an even greater relative reduction in the radiation dose required to obtain the equivalent quality to conventional contact imaging.
3.5. Computational analysis
Up to this point, this article has considered the merits of the two phase retrieval methods discussed purely in terms of their imaging characteristics. This section will now examine their performance from a computational perspective, which is increasingly relevant in the context of reallife applications of PBCT. Often, a realistic experimental scenario involves the use of synchrotrons or laboratorybased microCT systems leading to large datasets requiring dedicated highperformance computing (HPC) infrastructure to process within feasible timeframes. Such conditions lead to the need to consider the computation costs of the methods employed. Computationally, the application of the multimaterial method by Beltran et al. (2010) for a sample consisting of N_{m} distinct materials over N_{p} projections requires N_{m} × N_{p} 2D phase retrieval operations in addition to N_{m} FBP CT reconstructions. The computational complexity of the FBP algorithm is well studied (Natterer, 2001) and is dominated by the backprojection step. If one assumes optimal sampling conditions, the total work for reconstructing an N_{w}^{2} × N_{h} volume is proportional to N_{h}N_{w}^{3}. TIEHom phase retrieval in 2D projection space is generally implemented as a 2D Fourier filter with total computational cost for N_{p} projections proportional to N_{p}N_{w}N_{h} ln(N_{w}N_{h}). From these two components it is clearly seen that the dominant computational element of the Beltran and coworkers multimaterial method is the N_{m} FBP CT reconstructions, thus computationally bound by the number of materials to be examined and the sample dimensions, N_{m}N_{h}N_{w}^{3}. In contrast, the PostTIEHom3D method only requires a single FBP CT reconstruction with the application of phase retrieval implemented as a set of N_{o} 3D Fourier filtering operations, where N_{o} is the number of localized object ROIs examined. The complexity of each 3D Fourier filter is proportional to N^{3}_{ROI} ln(N^{3}_{ROI}), where N^{3}_{ROI} is assumed to be a cube of uniform dimensions representing the ROI. Therefore, the PostTIEHom3D method is computationally bounded by the number of objects examined and their size, N_{o} and N_{ROI}, respectively. It is evident from these observations that PostTIEHom3D requires significantly less computation when both the number of objects and their size relative to overall sample under investigation is constrained. Conversely, the multimaterial PreTIEHom2D method becomes computationally expensive as the number of materials examined increases. In terms of memory use, PostTIEHom3D requirements correspond to that for computing the forward and inverse 3D FFTs for each ROI subvolume. If an in situ FFT algorithm is considered, then the memory required mirrors that of the memory used to store the volume itself. Moreover, the Fourier filtering operation that computes the phase retrieval can also be performed inplace, thus conserving the memory footprint of the total operation. For example, a 128^{3} pixel ROI subvolume would require a total of 128^{3} memory elements to compute the inplace 3D FFT for the phase retrieval step, corresponding to around 8 KB if using 4byte floating point data.
A common experimental scenario is the case of a singlematerial and singlephase retrieval ROI that extends over the full tomography volume. This example represents the pathological `worst case' for the PostTIEHom3D method whose computational gains are derived from the ability to perform phase retrieval on one or more subvolumes in parallel with dimensions smaller than the full tomographic volume. As such, in this case PostTIEHom3D would offer no benefit over conventional preCT phase retrieval. The method is specifically targeted at applications where the full volume may contain isolated features of different materials that need to be accurately segmented or classified as part of a CT workflow and as such require materialspecific phase retrieval.
One scenario where PostTIEHom3D offers a significant advantage over prereconstruction projectionbased phase retrieval is that where the method is applied to a small ROI and varying the γ value with the aim of obtaining a subjectively optimal value of γ for a certain imaging task (for example, for feature detectability). This calculated value can be subsequently used in a global or localized CTreconstruction workflow. Obtaining γ in this way eliminates the need for projectionwide phase retrieval followed by full volume CT reconstruction as in preCT phase retrieval, thus reducing the computational cost substantially.
Synchrotron CT for mammography is another reallife case whereby successful quantitative detection and differentiation between glandular, adipose, tumorous tissue and microcalcifications are required. In such a workflow, an initial classification step is required to define the ROIs and materialspecific gamma value for phaseretrieval; this may be a manual or automated process. In the mammography usage case described, the number of ROIs detected may be of the order of tens or hundreds for which PostTIEHom3D can be computed in parallel. In the case of HPC systems, this may be achieved in a distributed approach across many compute nodes and/or cores etc. Moreover, the PostTIEHom3D algorithm can be fully implemented on GPU architectures such that both the forward/inverse FFTs in addition to the filter can be performed as a completely GPU resident operation subject to available GPU memory to contain the ROI volume. For larger volumes, a more sophisticated distributed multiGPU approach is theoretically possible. Such a GPU implementation would achieve realtime phase retrieval.
4. Conclusions
This research has derived a variant of TIEHom phase retrieval that can be applied directly to localized 3D regions of interest, consisting of isolated singlematerial objects within a greater reconstructed volume of the distribution of the complex a priori information relating to the absorption and refractive properties of the contained material. This method allows for efficient and accurate reconstruction of multimaterial samples within the TIE regime, with several marked benefits over alternative approaches. A simple numerical framework for PBCT Xray imaging has been constructed allowing for the method to be simulated and compared with the approach developed by Beltran et al. (2010) on a synthetic phantom for a range of experimental parameters.
Each ROI requiresIt is shown numerically using the contrasttonoise and universal imagequality index metrics that this new method achieves improved noise suppression, contrast enhancement and overall quantitatively correct results compared with the contact absorptiononly CT and TIEHom phase retrieval methods applied to projections prior to CT reconstruction.
It is also shown that the proposed 3D TIEHom method offers significant computational efficiencies for localized and multimaterial samples where performing postreconstruction phase retrieval eliminates the need for multiple CT reconstructions, representing the most computationally expensive element. The postreconstruction nature of the method also offers several practical benefits for tomographic workflows. Namely, by having the phase retrieval step after reconstruction, it allows for the retention of existing absorptiontomography experimental configurations, where the insertion of extra processing steps between acquisition and reconstruction may be difficult. In contrast, 3D TIEHom phase retrieval may be performed optionally on a previously acquired and reconstructed dataset without having to repeat the full CTreconstruction workflow. Finally, a practical application of this approach has been suggested by Paganin (2015), whereby PostTIEHom3D phase retrieval is applied to a conventional CTreconstruction of phasecontrast projections, interactively adjusting the TIEHom γ parameter and observing the result in order to obtain subjective `focus' for a specific material within the ROI. It is envisaged that such a method, if implemented in a software application using modern GPU hardware, could allow realtime combined reconstruction and visualization of multimaterial objects.
Acknowledgements
The authors wish to acknowledge the University of New England for the support in conducting this research.
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