## Abstract

We present a spectral phase unwrapping approach for grating-based differential phase-contrast data where the unwrapped interferometer phase shift is estimated from energy discriminated measurements using maximum likelihood principles. We demonstrate the method on tomographic data sets of a test specimen taken at different x-ray energies using synchrotron radiation. The proposed unwrapping technique was demonstrated to successfully correct the data set for phase wrapping.

© 2013 Optical Society of America

## 1. Introduction

For a wide range of imaging techniques, such as magnetic resonance imaging (MRI), synthetic aperture radar (SAR) or phase-contrast imaging (PCI) the phase of a measured periodic signal is the quantity of interest. However, the intrinsic periodic nature of the phase quantity leads to signal wrapping when measuring and converting the physical quantity of interest, if the original signal exceeds 2*π*. Correcting for this effect is known as phase unwrapping, and is a very active research area in digital imaging. Depending on the situation, unwrapping can become challenging, for example, because of noise. Various unwrapping solutions have been proposed for a wide range of applications, such as region growing algorithms, statistical models and Bayesian approaches among others [1–4].

In grating-based differential phase imaging (gbDPI) phase wrapping occurs when the beam deflection, due to a strong refraction by the sample, exceeds the ratio of the analyzer grating period to its distance from the point of deflection [5, 6]. Hence, to reduce or avoid phase wrapping, either the refraction angle or the sensitivity of the interferometer needs to be reduced. But in many cases a high sensitivity of the interferometer is needed to resolve small signal differences, e.g. in biological soft-tissues, and phase wrapping occurs at higher deflecting regions of the sample. For example the presence of bone or air cavities in soft tissue leads to strong phase wrapping which strongly corrupts the 3-D tomographic reconstruction [7, 8]. Existing approaches to correct or reduce phase wrapping errors in differential phase data make use of the attenuation contrast, which is simultaneously obtained [9, 10]. Those approaches are very promising, for example in non-destructive testing, but as they rely on a good attenuation contrast, they might fail in biomedical imaging where the attenuation contrast is weak.

Here we show that, for gbDPI, the phase unwrapping problem can be solved efficiently, exploiting the energy dependency (1/*E*^{2}) of the phase shift using measurements at different energies followed by a maximum likelihood estimation [11–13]. The spectral x-ray image data can be either obtained with specially designed x-ray imaging detectors, which are currently in use or under development [14–19], or by using tunable monochromatic x-ray sources. In this study we used the latter alternative, and performed the experiments using multiple selected and highly monochromatic (Δ*E/E* ∼ 10^{−1}) x-rays from a synchrotron source.

## 2. Principles

The spatial derivative of the phase shift Φ, which an x-ray wavefront experiences when it passes through an object, can be measured using an x-ray interferometer [20–22]. In Fig. 1 the three grating Talbot-Lau interferometer is illustrated, which uses the first attenuation grating *G*_{0} with period *p*_{0} to match the spatial coherence of the low brilliance x-ray source [23]. The phase grating *G*_{1} placed at a distance *D* with period *p*_{1} produces an interference pattern with a period *p*_{2} further downstream. When a sample is introduced into the beam path, the interference pattern is shifted, due to refraction caused by the varying phase [24]. Measuring this interference pattern phase shift *φ* allows to extract the spatial derivative of the phase shift introduced by the sample (i.e.
$\frac{\partial \mathrm{\Phi}}{\partial x}$). However, the period of the interference pattern and its phase shift is typically too small to be directly spatially resolved with a common x-ray detector. For this reason the second attenuation grating *G*_{2}, which has the same period *p*_{2} as the interference pattern, is placed at a distance *l* from *G*_{1}. The interference pattern is then sampled by stepping one of the gratings (in our case *G*_{1}) laterally in *x*-direction while the intensity *I* is measured at each grating position *x _{g}*. As a result, the intensity

*I*as a function of

*x*contains the information about the interference pattern phase shift

_{g}*φ*as well as the absorption signal 〈

*I*〉 which contains the information about the absorption of the x-rays and the signal

*A*

_{1}which contains the information about the small angle scattering of the x-rays within the specimen. Assuming a sinosoidal variation of the signal, those parameters can finally be obtained from the first components

*f*

_{0}and

*f*

_{1}of the Fourier transform of

*I*[25]:

*N*is the total number of regularly spaced stepping points (i.e. total number of

*x*positions) and

_{g}*I*is the measured intensity in the relevant pixel at step

_{s}*s*. From 〈

*I*〉 and

*A*

_{1}we can obtain the conventional attenuation contrast and respectively the so called dark-field contrast which arises from the small angle scattering. The interference pattern phase shift

*φ*, relative to an undisturbed interference pattern without the object in the beam, is proportional to the spatial derivative of the object phase shift Φ introduced by the sample [27, 28]:

Here *l* denotes the distance between the analyzer grating *G*_{2} and the phase grating *G*_{1}, *λ* is the x-ray wavelength, *r _{e}* the classical electron radius and

*ρ*the electron density of the sample. In Eq. (2

*c*) we combined the energy-independent parameters into $M=\frac{l}{{p}_{2}}{r}_{e}\frac{\partial}{\partial x}\int \rho \left(x,y,z\right)\text{dz}$. In the current context, phase wrapping occurs when the interference pattern phase shift

*φ*exceeds its scope of 0 to 2

*π*leading to an observed phase shift of

In an imaging experiment, we are only interested in the unknown sample parameters which we included into the energy-independent term *M*, and more specifically the electron density of the material therein, of Eq. (2*c*). Consequently, the invariance of *M* with respect to the x-ray energy allows to obtain multiple values of *M* when *φ* is measured at different x-ray energies. Hence, if *k* different energy bins *E _{j}* with

*j*= 1,...,

*k*are used we obtain

*k*values for

*M*which we can assume to be Gaussian distributed due to the different counting statistic, which is a consequence of the x-ray source statistic. Because the measurements are statistically independent, we evaluate the maximum likelihood estimator (ML) for

*M*by minimizing the following likelihood function

*L*

*M*by minimizing the following negative log-likelihood function

*ℓ*

*M*are more probable. This assumption is enforced by the Tikhonov regularization

*βR*

^{2}with a suitable factor

*β*[29].

The evaluation of the ML estimator from measurements at different energies brings several advantages:

First, the ML estimator can still be retrieved correctly even if *φ* wraps in each individual measurement. Second, we effectively increase the signal-to-noise ratio, incorporating an explicit gbDPI noise model into the ML estimator. Further, relying on more than one measurement makes this approach robust against low counting statistics and the capability is only limited to the effective signal-to-noise ratio of all measurements. Finally, in contrast to most existing 2-D phase unwrapping techniques our method does not rely on the information of neighbouring pixels. Instead, each pixel is processed individually and errors possibly arising during the unwrapping process, even though they are quite unlikely with state of the art unwrapping algorithms, do not propagate through the image. In general, the reliability of this approach and its robustness against noise improves with the number of distinct energies, which ideally should cover a wide range of the x-ray source spectrum.

## 3. Experimental results

To test our method experimentally, we performed tomographic measurements of a plastic (*PTFE*) cube test sample at different x-ray energies at the beamline HARWI II, operated by the Helmholtz-Zentrum Geesthacht, at the synchrotron DESY in Hamburg. Because of its flat edges, a cube object is difficult to reconstruct from differential phase-contrast projections since strong phase wrapping occurs when the angle between its facets and the incoming x-rays becomes small. To measure the interference pattern phase shift we used the Talbot-Lau interferometer installed at the beamline [30]. The intergrating distances were *D* = 3.0m between the source and the phase grating and *l* = 0.32m between the phase and the analyzer grating. The periods of the gratings were 22.3*μ*m, 4.33*μ*m and 2.4*μ*m for the source, phase and analyzer grating, respectively.

The interferometer showed good performance (i.e. high visibility) at energies of 24keV, 30keV and 48keV which were therefore chosen for the measurements. For each tomography scan we took 301 projections, equally spaced over 360°, while each projection was taken with 4 phase steps and an exposure time of 2s per step. To record the images a 580*μ*m thick *CdWO*_{4} scintillator lens-coupled to a CCD camera with an effective pixel size of 10*μ*m was used.

Figure 2 (second and third row) depicts the interference pattern phase shift *φ* of the cube at an x-ray energy of 24keV for different rotation angles *α*. From a cube, only the regions indicated with white and black triangles lead to a positive and negative interference pattern phase shift respectively. In the gray region the phase shift Φ induced by the object is constant and no interference pattern phase shift occurs as it derives from Eq. (2a). Due to the increasing gradient of the facets the interference pattern phase shift *φ* increases as the cube is rotated. Phase wrapping could be observed at rotation angles smaller than *α* < 8° as shown in Fig. 2(c) and Fig. 2(d). At those angles the measured interference pattern phase shift is not anymore proportional to the actual differential phase shift of the object as expressed by Eq. (3).

The attempt of a tomographic reconstruction with filtered back projection of a data set containing such phase wraps is corrupted by strong streak artifacts. To correct for those artifacts we used all measured interference pattern phase shifts at 24keV, 30keV and 48keV to estimate the unwrapped interference pattern phase shift *Mλ*^{2} by minimizing the cost function [Eq. (5)]. In order to find the global minimum, we evaluated Eq. (5) for a large number of discrete values of *M*, with constant spacing and selected the *M* which lead to a minimal *ℓ*. The precision in the evaluation of *M* is therefore dependent on the chosen spacing, and the processing time on the total number of values of *M* which are evaluated. There is no doubt that more advanced algorithms can be used to improve accuracy and to reduce processing time. The regularization was defined as
$R=\frac{M}{{M}_{\mathit{max}}}$ where *M _{max}* is the maximum value of

*M*which we tested to find the global minimum and a Tikhonov factor

*β*= 4 was used.

The result for a single pixel is depicted in Fig. 3, where the measured interference pattern phase shift *φ* for all x-ray energies as well as the corresponding estimated values *Mλ*^{2} for a rotation of the cube from 0° to 90° is shown. The angles where phase wrapping occurs are indicated by arrows. The strength of the interference pattern phase shift *φ* depends further on the x-ray energy as given in Eq. (2). Since *φ* decreases with increasing energy phase wrapping tends to occur at smaller angles for higher energies. At an rotation angle of 1.6° the signal has wrapped at even the highest energy, but a correct estimation of *M* was still obtained.

Figure 4 shows the profile of the measured interference pattern phase shift *φ* together with the estimated interference pattern phase shift *Mλ*^{2} at 4 different rotation angles for an x-ray energy of 24keV. As can be seen, the method is able to estimate the interference pattern phase shift also at angles where the deflection angle exceeds multiple times the direct detectable value of *p*_{2}/*l*. A comparison of the reconstructed electron densities from the measured and estimated interference pattern phase shifts in Fig. 5(a) and Fig. 5(b) demonstrates the quantitative improvement. Within the region marked with the dashed square we extracted mean electron densities of *ρ _{m}* = (4.79±0.93)×10

^{29}

*m*

^{−3}and

*ρ*= (5.59±0.61)×10

_{e}^{29}

*m*

^{−3}from the measured and estimated reconstruction respectively. Compared to the value in literature

*ρ*= 6.24×10

_{lit}^{29}

*m*

^{−3}the error could be reduced by a factor of 2.23 [31]. The line plot through the uncorrected slice (continuous curve in Fig. 5(c)) shows that the square profile of the cube is strongly blurred, due to the phase wrapping artifacts. On the other hand, the line plot through the corrected slice (dashed curve in Fig. 5(c)) exhibits nearly the expected square profile of the cube. The derivation from a perfect square profile and the reason for the too low electron density are mainly caused by the remaining streak artifacts which have their origin in distinct effects. First, this unwrapping approach comes to its limits at points where the phase gradient shows singularities. In addition, total external reflection of the x-rays occurred at small glancing angles. This effect can also be seen in Fig. 4(d) where the sharp spikes (indicated by arrows) are a result of those reflections which cause streak artifacts in the 3D reconstruction.

## 4. Discussion and conclusion

In conclusion we could demonstrate a method to estimate the unwrapped interference pattern phase shift *φ* from a set of gbDPI measurements performed at various energies. For that we proposed to evaluate the maximum likelihood estimator by minimizing a suitable likelihood function and were able to show its efficiency in a proof of principle experiment. In contrast to the presented experiment, the measurement of the interference pattern phase shift at different energies can simultaneously be performed using a polychromatic x-ray source in combination with an energy-sensitive detector. This is in particular of interest for experiments where the total radiation dose applied to the specimen might be an issue. The efficiency of this method depends on the properties of the sample, such as the amount of attenuation, phase shift and scattering of the x-rays. Further, the x-ray source spectrum, the number and the range of selected energies, the changing interferometer performance at different energies, as well as the regularization will have an impact on its success. A detailed analysis of the mentioned influences and a quantitative evaluation of the final signal-to-noise ratio and thus, the relative improvement of the method with respect to existing methods will be subject of following studies. We believe that this study will be of particular interest to the biomedical imaging community, which pushes for the implementation of spectral x-ray imaging detection schemes right now, and the x-ray phase-contrast modality in the near future.

## Acknowledgments

We acknowledge financial support through the DFG Cluster of Excellence Munich-Centre for Advanced Photonics (MAP), the DFG Gottfried Wilhelm Leibniz program and the European Research Council (ERC, FP7, StG 240142). This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT).

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