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Phase contrast X-ray imaging is increasingly popular in the past decade. In order to acquire phase contrast X-ray images, different types of imaging mechanisms have been proposed. Among them, in-line phase contrast X-ray imaging shows the highest potential because of its simplicity. In the study of in-line phase contrast imaging, based on different physical assumptions, many non-iterative phase retrieval methods, such as Bronnikov method, modified Bronnikov method, phase-attenuation duality (PAD) method, single-material method, and two-material method have been proposed. The main step of the non-iterative methods is a filtering process, thus different methods involve different filter design. In this paper we showed that every filter applied in the methods listed above is indeed the minimizer of a L2-norm regularization problem. In addition, two methods were proposed to overcome the over smoothing problem owing to the nature of L2-norm regularization.
© 2015 Optical Society of America
1. Introduction
X-ray phase contrast imaging system is becoming increasingly popular nowadays whether it be in biomedical study or in material science [1, 2]. Comparing to conventional x-ray imaging, phase image computed from phase contrast x-ray imaging benefits from edge enhancement and higher differentiation for soft tissue, e.g. breast. Several mechanisms have been designed to exploit phase-contrast imaging. The five major types are: in-line phase contrast imaging [3], analyzer-based method [4, 5], crystal [6–8], grating interferometry method [9–11], and coded aperture method [12, 13]. Among these methods, in-line phase contrast imaging takes advantage of the simplest arrangement of optical components and therefore it is the most robust to misalignment. This is vital when considering to design a computed tomography machine utilizing phase contrast imaging [14]. Moreover, analyzer, interferometer, and mask are not needed in in-line phase contrast imaging system, so the flux will not be lost.
Considering an in-line x-ray imaging system, an object can be described by its refractive index distribution, n(r⃗) = 1 − δ(r⃗) + iβ(r⃗), where the imaginary part β describes absorption, the real part δ describes phase shift caused by the object, and r⃗ = (x, y, z) is the spatial coordinate. Furthermore, both the x-ray attenuation A(r⃗) and phase shift ϕ(r⃗) are the projection of β(r⃗) and δ(r⃗), i.e.
and where λ is the wavelength of x-ray and z is the propagation direction.In general, in-line phase contrast imaging requires two images of the observing object to derive an unique phase image [15], since the absorption and phase shift information are tangled in every single x-ray image. Nevertheless, based on different physical assumptions of the object and x-ray source, it is possible to derive a phase image from single image; we called such methods single-shot phase retrieval methods. There are four commonly used single-shot phase retrieval methods. They are (1) the modified Bronnikov method [16], which only considers small absorption, (2) single-material method, which is for homogeneous object [17], (3) two-material method [18], and (4) phase-attenuation duality (PAD) based method [19–21], which assumes that absorption is proportion to phase shift. Although the derivations of these methods are based on different assumptions, the computation procedures are all the same [14]:
- Pre-process the measured image
- Calculate the Fourier transform of the pre-processed result;
- Apply a designed filter in the frequency domain;
- Calculate the inverse Fourier transform of the filtered result; and
- Get the phase image based on different assumptions about the relation between attenuation and phase shift.
Generally speaking, a single-shot phase retrieval method is related to an ill-posed inverse problem. However, these problems can be regularized based on different physical assumptions so that the the unique solutions of the original problem can be derived. In this paper we showed that the four previously proposed filters are actually minimizers of L2-norm regularization problems, and thus provide another point of view for designing the phase contrast imaging system. In addition, we demonstrated that phase contrast CT reconstructed from phase images computed by L2-norm single-shot phase retrieval method might suffer from over smoothing and two methods were proposed to overcome this drawback. We measured the electron density values from the phase CT images reconstructed from PAD-based method and our methods to verify that the proposed methods are reliable. In this work, we mainly focused on the PAD-based method, because of its potential for medical application and the limitation of our devices. The corresponding L2-norm regularization problems of the other three single-shot phase retrieval methods were derived in Appendix I.
2. Denoising by regularization
An inverse problem is ill-posed if the solution is nonunique or unstable. In science and engineering areas, ill-posed problems are almost everywhere since most inverse problems are subjected to some physical constraints. Regularization has been shown a powerful technique to making solution well-posed and are now widely used in many areas [22]. In general, the solutions of an ill-posed problem can be formulated as
where T is a linear operator, y is the observation, J(f) is a function or a functional of f, and τ is a regularization parameter.Denoising by regularization is classical denoising method and it is formulated as
where ‖ · ‖ is some kind of norm operator, P is a linear operator, and the value of τ can be determined heuristically or according to the imaging conditions. To reduce the noise, smoothness is a commonly adapted condition for regularization and the the norm of the gradient of the given image is considered. For a vector field v = (v1, v2) where v1, v2 ∈ ℝN, the Lp-norm is defined as where . Total variation (TV) prior, L1-norm [23–25], and Sobolev prior, L2-norm [26,27] have been successfully applied to many image processing problem. When Sobolev prior is considered, the uniformly smoothness condition is imposed [26,27]. On the contrary to the Sobolev prior, TV prior favors piecewise smoothness so it is able to take into account step edges.For a continuous smooth function f, the Sobolev prior is defined as
and the Sobolev regularization is formulated as where the gradient vector at x is defined as . Since the solution of Eq. (7) satisfies the first order condition [28] where Δ stands for the Laplacian operator, the minimizer can be computed exactly in the frequency domain where ℱ stands for Fourier transform.3. Phase-attenuation based Wu’s method
Different single-shot phase retrieval methods based on different assumptions, such as energy range, weak absorption, the relationship between attenuation and phase shift, and homogeneity of the observing target, have been proposed [14]. In this study we took the phase-attenuation duality (PAD) based method proposed by Wu et al [19–21] for example to demonstrate the relation between single-shot phase retrieval method and the Sobolev regularization.
PAD-based method proposed by Wu et al was for high energy x-rays (60–500keV). Under this circumstance, the mathematical descriptions of phase shift, ϕ(r⃗) and attenuation A(r⃗) by the soft tissues are given by
and respectively [19, 21], where re = 2.82 × 10−15m is the classical electron radius, and σKN is the total cross section for x-ray Compton scattering with a free electron: where η = E/511keV and E is x-ray energy. Moreover, Wu et al [21] derived that the relationship between detected phase contrast x-ray image, I, and the x-ray attenuation map, A, is where R2 stands for object to detector distance, M is the geometric magnification, and u⃗ is the wave-front transverse frequency vector. Clearly, when πλR2u⃗/M ≪ 1 holds, Eq. (13) is reduced to [19, 21] Combining Eq. (10), Eq. (11) and Eq. (14), one can obtain where γ = λre/σKN. The key step of phase retrieval is computing ℱ(A). By reorganizing Eq. (14) we have where then Eq. (16) is the minimizer of the Sobolev regularization problemIn previous studies, M = 2 [29–32] and M = 2.5 [33,34] were widely adopted settings. From Eq. (17) one can find that if x-ray energy is fixed, i.e. both λ and σKN are fixed, τ reaches the maximum when M = 2. In practical medical application, x-ray energy for a specific application, such as CT or radiography, is often already chosen according to the object of interest, e.g. breast, chest, abdomen, and head. In this case, the most smooth phase image is derived when M = 2. However, the derived image may suffer from over smoothing [35, 36]. A demonstration of this phenomenon was shown in the Experiment Section.
To avoid the over smoothing problem, many methods have been proposed [24, 35, 37, 38]. Nevertheless, the L2-norm regularization property of phase retrieval method is intrinsic. It is owing to the nature of phase of the Fresenal propagator, exp(iπλzu⃗2), where z = R2/M, and the phase of the propagator, i.e. πλzu⃗2, determines how strongly the wave-front exit from the object will be diffracted in x-ray propagation [19, 21]. So, not any arbitrary method is appropriate. Thus, we slightly modified the norm employed in the filter from two to p, 1 < p < 2,
so that weight of the high frequency component would decrease slower and the over smoothing phenomena was alleviated. Another variation was deriving the image by solving the general total-variation (gTV) regularization problem, where 1 < p < 2, also. The method proposed by Wohlberg et al [39] was applied to solve the gTV regularization problem. Since this method is performed in image space, the original τ is inadequate. Considering an image whose width is W and height is H, τTV was defined in this study as: where L = ⌈log2 max{W, H}⌉, and d is the detector pixel size. Compared with the filter based method, more local information was considered in the image derived from gTV regularization.A comparison of phase CT reconstructed from phase image derived by Wu’s method and the proposed methods was demonstrated in the next section. In this study, we considered the case, p = 1.9 for both proposed methods. Time for computing one phase image using Wu’s method, the proposed filter, and gTV method is 3 seconds, 3 seconds, and 35 seconds, respectively. The computational tasks were all performed on a PC equipped with Intel i3 and the operating system is windows 7. The program was implemented by Matlab 7.
4. Experiment results
All original X-ray images were taken by an in-house X-ray imaging system. The X-ray source is Hamamatsu L8121-03. The focal spot size is 50 μm and a 2 mm thick aluminum filter was equipped. The detector is a CMOS detector (Dexela 2315); the detector pixel size is 75 μm. The width and height of the output image are 3072 and 1944, respectively. The source-to-detector distance is fixed at 1.864 m in all experiments.
In the first experiment, a phantom containing several plastic rods was employed. A comparison was made between ρe derived from the phase image computed by Wu’s method and the proposed methods, since δ is highly related to ρe, according to
where ħ is the reduced Planck constant, and c is the speed of light. In the second experiment, we demonstrated the images reconstructed from the phase image computed by Wu’s method and by the proposed methods. In the former, the over smoothing phenomena can be found around the vessel like structures and tiny spots. The improvement can also be observed from the images reconstructed from the phase images computed by the proposed methods. All CT volumes were reconstructed by the Feldkamp-Davis-Kress (FDK) backprojection algorithm [40]4.1. Quantitative measurement of ρe
In this experiment, we would like to verify the proposed methods. A contrast phantom containing three types of plastic rods were used for quantitative measurement. The cross section of the phantom is shown in Fig. 1.
In this phantom, the PMMA (poly(methylmethacrylate)), LDPE (low density polyethylene), and POM (polyoxymethylene) rods are all 6 mm in diameter. The phantom chamber has an inner diameter 26 mm and the wall thickness is 2.5 mm. The tube was operated at 100 kVp; the tube current was 500 μA; and the exposure time was 2000 ms. The geometric magnification was 2.57.
The reference ρe values were calculate by the equation , where ρ, NA, Z, and A are the mass density, Avogadro’s number, atomic number and atomic mass, respectively. The electron density values measured from the CT images as well as the corresponding reference values are listed in table 1. The reconstructed phase CT images were displayed Fig. 2.
Fig. 2 The images of phase contrast CT of the phantom were illustrated. From left to right, the images are reconstructed from the phase images computed using Wu’s method, the proposed filter (p = 1.9), and gTV (p = 1.9), respectively. The display window were all [0.11 0.45]×1023cm−3.
Table 1. Quantitative measurements of the reconstructed images of phase contrast.
The ρe values of POM and LDPE were calculated based the mean ρe value of PMMA. We found the ρe values measured from the phase CT images reconstructed from the phase images computed by the proposed filter and gTV method are consistent. Furthermore, it is observable that among the reconstructed phase CT images, the phase CT image reconstructed from the phase images computed using Wu’s method is the blurriest.
4.2. Tangerine CT reconstruction
In this experiment, we would like to demonstrate the over smoothing phenomena and compare the CT images reconstructed from the phase images derived by Wu’s method and the phase images computed using the proposed methods. A tangerine is the target in this experiment. The tube was operated at 70 kVp; the tube current was 500 μA; and the exposure time was 2000 ms. The geometric magnification was 2.1.
CT slices were shown in Fig. 3. The CT slice reconstructed from projection data computed applying Wu’s method lost more details and is more blurry compared to slices reconstructed from other two sets of computed projection data (Fig. 4). The over smoothing phenomena is more observable around the vessel like structure, fibers, and tiny spot enclosed in a uniform region.
Fig. 3 CT slices reconstructed from computed results were shown. From left to right, the CT images were reconstructed from the projection data computed using Wu’s method, the proposed filter (p = 1.9) and gTV method (p = 1.9), respectively.
Fig. 4 Details of CT images reconstructed from phase image and computed results were shown. The tomography reconstructed from original projection data was illustrated in (a). Two chosen areas, indicated by red and yellow retangles in (a) were were enlarged. Images (b)–(d) corresponds to the red rectangular area and (e)–(g) corresponds to the yellow rectangular area. From (b)–(d) and (e)–(g), the images were from the CT slice reconstructed from projection data computed using Wu’s method, the proposed filter (p=1.9), and gTV (p=1.9), respectively. Compared with (b), both (c) and (d) kept more details. In comparison with (e), edges in both (f) and (g) were sharper. Yellow arrows indicated the perceptual differences.
5. Conclusion and discussion
We demonstrated the relationship between single-shot phase retrieval methods and the Sobolev regularization problem, which provided another direction to understand phase retrieval methods, that is they vary from the imposed smoothness condition. To combat the over smoothing problem owing to the nature of L2-norm regularization, a filter and a method based on general total variation functional were proposed. We suggested using the proposed filter was a better alternative for phase retrieval than applying the gTV regularization especially when computation time issue is vital.
Although the relationship of image denoising and single-shot phase retrieval method was shown in this work, we would like to emphasize that phase retrieval is not just a denoising process nor are phase images just denoised results. All single-shot phase retrieval methods work relying on different physical assumptions. The proposed filter, Eq. (19) and the original filter, Eq. (9) are much alike; moreover, it is the solution of the equation:
where the fractional Laplacian is considered in contrast to Eq. (8) involving the Laplacian. However, whether it has any physical meaning or not is beyond the scope of this work.There are other single-shot phase retrieval methods that do not correspond to the Sobolev regularization problems, e.g. Rytov method, Born method [41], and Bronnikov method [42]. Nevertheless, the common drawback of these methods is that they are unstable. The denominator of the filter possibly goes to zero. To avoid this, Tikhonov’s regularization term was imposed [41].
Previously, Wu et al [43] proposed the relative phase-contrast factor (RPF) as a quantitative measure of coherence and visibility of phase-contrast at a given spatial frequency. They also demonstrated an experimental method to determine the RPF of an x-ray imaging system [43]. Based on the RPF, a guideline for designing x-ray phase-contrast imaging system was proposed [44]. In addition to the guideline proposed by Wu et al, one can further consider the smoothness of the derived image on the basis of the Sobolev regularization.
Recently, Langer et al [46] formulated the problem of phase retrieval from Fresnel diffraction images taken at several distances as a regularization problem. However, to the best of our knowledge, there is no direct regularization formation for single-shot phase retrieval method. How to unify regularization problem and single-shot phase retrieval is an ongoing study.
A. Appendix I
According to the review by Burvall et al [14], there are three other single-shot methods, where applied filters also correspond to the Sobolev regularization problems. We briefly introduce these methods and demonstrate the corresponding Sobolev regularization problems.
A.1. The modified Bronnikov methods
The modified Bronnikov method were proposed by Groso et al [16]. The assumptions were no absorption, μ ≈ 0, and the x-ray source was monochromatic. Filter applied in the modified Bronnikov method was
When α ≠ 0, rearranging (24) then we have and the filter is the minimizer of the problem Otherwise, we have the filter proposed by Bronnikov [42], However, this does not correspond to the Sobolev regularization problem.A.2. Single- and two-material methods
The formulae for single- and two-material methods are alike; however, the assumptions are different. For the single-material method, it assumed the object was homogeneous; moreover its μ and δ were known [17]. Similarly, for the two-material method, the object was assumed to consist of air and two other materials, and one was embedded in the other [18]. Knowledge of μ and δ of both materials and the projected thickness of the object are necessary. The filter used in the two-material method is
Clearly, the filter is the minimizer of the problem Set both μ2 and δ2 zero, then we have the filter for single material method, and it is the minimizer of the following problemReferences and links
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