Quantitative X-ray radiology using the absorption and scattering information

Abstract

A quantitative radiographic method uses X-ray imaging. The method uses a ratio of the absorption signal and the (small-angle) scattering signal (or vice-versa) of the object as a signature for the materials. The ratio image (dubbed R image) is independent from the thickness of the object in a wide sense, and therefore can be used to discriminate materials in a radiographic approach. This can be applied to imaging systems, which can record these two signals from the underlying object (for instance, an X-ray grating interferometer). Possible applications could be in material science, non-destructive testing and medical imaging. Specifically, the method can be used to estimate a volumetric breast density. The use of the R image and the corresponding algorithm are also presented hereafter.

Claims

1. A method for quantitative radiology of a sample, which comprises the steps of: determining a material or material composition of the sample via a radiographic approach by using a ratio of an absorption signal and a small-angle scattering signal of the sample, the absorption signal and the small-angle scattering signal are obtained from an X-ray investigation based on a grating-based interferometer, analyzer-crystal-based imaging, coded aperture imaging or an imaging technique that is enabled to record both absorption signals and small-angle scattering signals.

2. The method according to claim 1, wherein the absorption signal T and the small-angle scattering signal S of the sample being a single material sample are expressed by: $T=-\ln \left(\frac{I}{I_0}\right)={}_l{}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}},l\right)l{}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)L$ $S={}^2={}_l{s}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}},l\right)l{s}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)L,$ where .sub.eff and s.sub.eff are an effective attenuation coefficient and effective generalized scattering parameter of the sample at a mean energy of ; .sup.2 is a second moment of a scattering angle distribution of an outgoing beam; l is along a path of an X-ray beam and L is total thickness of the sample; and a ratio image, dubbed R image, is defined by: $R=\frac{T}{S}=\frac{{}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)L}{s_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)L}=\frac{{}_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)}{s_{\mathrm{eff}}\left(\stackrel{\_}{\mathrm{.Math.}}\right)}.$

3. The method according to claim 1, wherein the sample is a breast and a volumetric breast density is estimated using the R image of the breast.

4. The method according to claim 3, wherein the volumetric breast density is defined as: $m\left(x,y\right)=\frac{L_g}{L}=\frac{L_g}{L_g+L_f}$ on a pixel base for describing a volumetric proportion of glandular tissue or its complementary 1m(x,y) for describing a volumetric proportion of fat tissue; (x,y) is spatial coordinates of a pixel; L.sub.g and L.sub.f are a thicknesses of the glandular tissue and the fat tissue along the X-ray beam, respectively and L is a total compression thickness wherein the volumetric breast density of a whole breast is given by averaging m over the whole breast region: $VBD=\frac{\underset{x,y}{.Math.}m\left(x,y\right)}{N},$ where VBD represents the volumetric breast density and N is pixel numbers of a breast region.

5. The method according to claim 4, wherein a connection between an R value and a pixel-wise volumetric breast density is given by: $m=\frac{1}{1-\frac{\left(\frac{R}{R_g}-1\right)}{\left(\frac{R}{R_f}-1\right)}\frac{{}_g}{{}_f}},$ which is deduced by a physical definition of R with a two-material composition assumption, $R=\frac{T}{S}=\frac{{}_g{L}_g+{}_f{L}_f}{s_g{L}_g+s_f{L}_f}=\frac{{}_{g}mL+{}_f\left(1-m\right)L}{s_{g}mL+s_f\left(1-m\right)L};$ $R_g=\frac{{}_g}{s_g}\mathrm{and}{R}_f=\frac{{}_f}{s_f}$ are defined as R values for 100% glandular tissue and 100% fat tissue, respectively; is an effective attenuation coefficient and S is the generalized scattering parameter; and subscripts g and f represent the glandular tissue and the fat tissue, respectively.

6. The method according to claim 5, wherein the following relationship is always true,
R.sub.g<R<R.sub.f which implies that the R value of the glandular tissue is smaller than the fat tissue, if a tissue is a mixture of the glandular tissue and the fat tissue, the tissue's R value falls in between.

7. The method according to claim 3, which further comprises: estimating the volumetric breast density using a histogram of the R image, a normalized histogram of the R image is noted as H(R), a distribution of H(R) is usually an asymmetric Gaussian distribution and any possible asymmetry is due to a fact that volumes of glandular tissue and fat tissue are different; and implying the volumetric breast density in a distribution, a nth moment of the distribution reveals information of breast density, including a skewness (a third moment): $=\frac{E\left[{\left(R-{}_R\right)}^3\right]}{{\left(E\left[{\left(R-{}_R\right)}^2\right]\right)}^{3/2}},$ gives an estimation of how dominant the glandular tissue or the fat tissue is in a whole breast; and R here represents a random variable and .sub.R is a mean value of the R image.

8. The method according to claim 7, wherein an actual number of a breast density calculated by dynamically determining R.sub.g and R.sub.f from the histogram H(R) of the R image which is described as following: (1) a peak position R.sub.peak which gives a maximal H(R) is first decided; (2) the histogram H(R) is divided into left and right parts by R.sub.peak, these two parts of the histogram are considered separately, for each half, the distribution is mirrored around a peak value to form a symmetric Gaussian distribution and then a Gaussian fitting is applied to two resulting Gaussian distributions and their standard deviations .sub.g and .sub.f are calculated respectively, wherein R.sub.g and R.sub.f are decided using the N criteria;
R.sub.g=R.sub.peakN.sub.g
R.sub.f=R.sub.peak+N.sub.f,(14) where N is a positive integer; with the determined R.sub.g and R.sub.f, a 3D volumetric breast density VBD of the whole breast is calculated by weighting a normalized histogram H(R) using an m value; $VBD=\underset{R}{.Math.}m\left(R\right).Math.H\left(R\right).$

9. The method according to claim 8, wherein N=3 is considered to give a best approximation of the R.sub.g and R.sub.f when noises are present.

10. The method according to claim 1, which further comprises obtaining the absorption signal and the small-angle scattering signal from an arrangement for X-rays, for obtaining quantitative X-ray images from a sample including: a) an X-ray source; b) a grating set-up selected from the group consisting of a three gratings set-up with grating G0, G1 and G2 or a two gratings set-up with the grating G1 and G2; c) a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels; d) means for recording the images of the position-sensitive detector; e) means for evaluating intensities for each pixel in a series of images, in order to identify characteristics of the sample for each individual pixel as an absorption dominated pixel and/or a differential phase contrast dominated pixel and/or an X-ray scattering dominated pixel; wherein the series of images is collected by continuously or stepwise rotating from 0 to IF or 2 either the sample or the arrangement and the X-ray source relative to the sample.

11. The method according to claim 10, which further comprises operating the gratings either in a so-called near field regime or in a Talbot-regime.

12. The method according to claim 11, wherein for near-field-regime operation, a distance between the gratings D1 and D2 is chosen freely within a regime, and for the Talbot-regime is chosen according to $D_{n,\mathrm{sph}}=\frac{L.Math.D_n}{L-D_n}=\frac{L.Math.n.Math.p_1^2/2{}^2}{L-n.Math.p_1^2/2{}^2}$ where n=1, 3, 5 . . . , and $=\{\begin{array}{ccc}1& \mathrm{if}\mathrm{the}\mathrm{phase}\mathrm{shift}\mathrm{of}{G}_1\mathrm{is}\left(2l-1\right)\frac{}{2},& p_2=\frac{L+D_{n,\mathrm{sph}}}{L}{p}_1\\ 2& \mathrm{if}\mathrm{the}\mathrm{phase}\mathrm{shift}\mathrm{of}{G}_1\mathrm{is}\left(2l-1\right),& p_2=\frac{L+D_{n,\mathrm{sph}}}{L}\frac{p_1}{2},\end{array}$ where l=1, 2, 3 . . . , D.sub.n is an odd fractional Talbot distance when a parallel X-ray beam is used, while D.sub.n,sph is that when a fan or cone X-ray beam is used, L is here a distance between the X-ray source and the grating G1.

13. The method according to claim 10, wherein the grating G1 is a line grating G1 either formed as an absorption grating or a phase grating, the phase grating is a low absorption grating but generating a considerable X-ray phase shift, preferably of or odd multiples thereof.

14. The method according to claim 10, wherein the grating G2 is a line grating having a high X-ray absorption contrast with its period being a same as that of a self-image of the grating G1; the grating G2 being placed closely in front of the position-sensitive detector with its lines parallel to those of the grating G1.

15. The method according to claim 10, wherein phase stepping is performed by a shift of one of the gratings with respect to other ones of the gratings.

16. The method according to claim 10, wherein the grating set-up is manufactured by planar technology.

17. A system for quantitative radiology of a sample, such as a breast in a mammographic investigation, the system comprising: means for evaluating recorded radiographic images, a material or material composition of the sample being determined via a radiographic approach by using a ratio of an absorption signal and a small-angle scattering signal of the underlying sample, wherein the absorption signal and the small-angle scattering signal are obtained from X-ray investigations based on grating-based interferometry, analyzer-crystal-based imaging, coded aperture imaging or imaging technique that is enabled to record absorption signals and small-angle scattering signals.

18. The system according to claim 17, further comprising an X-ray configuration, wherein the absorption signal and the small-angle scattering signal are obtained from an arrangement for X-rays, for obtaining quantitative X-ray images from the sample, said X-ray system configuration, containing: an X-ray source; a grating setup selected from the group consisting of a three gratings set-up with grating G0, G1 and G2 and a two gratings set-up with said grating G1 and G2; a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels; means for recording the images of said position-sensitive detector; and means for evaluating intensities for each pixel in a series of images, in order to identify characteristics of the sample for each said individual pixel as an absorption dominated pixel and/or a differential phase contrast dominated pixel and/or an X-ray scattering dominated pixel, wherein the series of images is collected by continuously or stepwise rotating from 0 to or 2 either the sample or said X-ray configuration and said X-ray source relative to the sample.

19. The system according to claim 18, wherein said grating G1 is a line grating G1 either formed as an absorption grating or a phase grating, said phase grating is a low absorption grating but generating a considerable X-ray phase shift, and preferably of or odd multiples thereof.

20. The method according to claim 18, wherein said grating G2 is a line grating having a high X-ray absorption contrast with its period being a same as that of a self image of said grating G1, said grating G2 being placed closely in front of said position-sensitive detector with its lines parallel to those of said grating G1.

21. The system according to claim 18, wherein phase stepping is performed by a shift of one of said gratings with respect to other ones of said gratings.

22. The system according to claim 18, wherein said grating setup is manufactured by planar technology.

23. The system according to claim 18, wherein the x-rays are hard X-rays.

24. The system according to claim 17, wherein the system is operated either in a so-called near field regime or in a Talbot-regime.

25. The system according to claim 24, wherein for near-field-regime operation, a distance between said gratings D1 and D2 is chosen freely within a regime, and for the Talbot-regime is chosen according to $D_{n,\mathrm{sph}}=\frac{L.Math.D_n}{L-D_n}=\frac{L.Math.n.Math.p_1^2/2{}^2}{L-n.Math.p_1^2/2{}^2}$ where n=1, 3, 5 . . . , and $=\{\begin{array}{ccc}1& \mathrm{if}\mathrm{the}\mathrm{phase}\mathrm{shift}\mathrm{of}{G}_1\mathrm{is}\left(2l-1\right)\frac{}{2},& p_2=\frac{L+D_{n,\mathrm{sph}}}{L}{p}_1\\ 2& \mathrm{if}\mathrm{the}\mathrm{phase}\mathrm{shift}\mathrm{of}{G}_1\mathrm{is}\left(2l-1\right),& p_2=\frac{L+D_{n,\mathrm{sph}}}{L}\frac{p_1}{2},\end{array}$ where l=1, 2, 3 . . . , D.sub.n is an odd fractional Talbot distance when a parallel X-ray beam is used, while D.sub.n,sph is that when a fan or cone X-ray beam is used, L is here a distance between said x-ray source and said grating G1.

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

(1) Preferred embodiments of the present invention are described hereinafter with reference to the attached drawings which depict in:

(2) FIG. 1 results from validation experiments with the use of five different plastic materials;

(3) FIG. 2 results on a quantitative radiography;

(4) FIG. 3 an R image for one breast sample in the craniocaudal (CC) view (a) and the corresponding histogram of the R image (b);

(5) FIG. 4 a schematic illustration of the dynamic process for determining R.sub.g and R.sub.f;

(6) FIG. 5 the mean and standard deviation of VBD values estimated from the CC view for each diagnostic ACR group; and

(7) FIG. 6 a two-gratings interferometer (top) and a three-gratings interferometer (bottom) for X-ray imaging.

DESCRIPTION OF THE INVENTION

(8) With respect to the equations given above for the calculation of the R image, the justification of Eq. (4) and Eq. (9) by experiments on several known plastic materials is given in FIG. 1. The samples were imaged using a grating interferometer with a broad spectrum like one of the grating interferometers shown in FIG. 6. FIG. 1 shows the results of the validation experiments on five plastic materials. FIG. 1(a) shows the phantom design; (b) is the absorption image and (c) is the scattering image: FIG. 1(d) shows the resulting R image. In FIG. 1(e) the profiles of the absorption image for the five materials are illustrated. Forty horizontal lines were used for each material (the ROI was indicated in FIG. 1(c)) and averaged along the vertical direction. FIG. 1(f) are the profiles of the scattering image for the five materials, forty horizontal lines were used for each material and averaged along the vertical direction. FIG. 1(g) are the profiles of the resulting R images for each material. Noticeable noises were observed at the places where the samples were too thin, especially for Nylon and Derlin. This was because the scattering signals were close to the noise level at those places. FIG. 1(g) is therefore truncated from the thickness of 1.2 cm as indicated by the right rectangle in FIG. 1(d). In FIG. 1(h) the tabulated R value for the five given materials is listed.

(9) It can be seen that the R values of the five materials are independent of the thickness within a good approximation. And the materials can be easily distinguished by their R values which confirms exactly the general expectations on this techniques.

(10) An application of using R image for quantitative radiology is given in FIG. 2. Conventional X-ray absorption-based radiography can only provide qualitative information (e.g. morphology) of the underlying sample, but not quantitative information, such as telling the composition of the sample. A 4 cm3 cm wedge phantom and a 3 cm3 cm one of the same material were overlapped with each other and imaged using a grating interferometer set up as shown in FIG. 6. The results were shown in FIG. 2. For the absorption image only (FIG. 2(a)), different absorption properties were observed. However, there was not enough information to tell whether it was a single material with different thicknesses or just two different materials. By taking the R image of the same sample, it was clearly shown that the phantom was composed of one material (FIG. 2(b)). Using the calibrated data in FIG. 1(h), additionally, the unknown material could be identified, which was Teflon in the present case.

(11) Using the R image, additional quantitative information can be obtained compared to conventional absorption-based radiographic methods. A particular application in medical imaging is given below regarding volumetric breast density (VBD) estimation.

(12) Breast density has gained increasing attentions in breast cancer screening and diagnosis because it is a strong indicator of the breast cancer risk [4, 5]. Conventional 2D breast density estimation methods have the problems of subjectivity and low inter-reader agreements [6]. Volumetric breast density (VBD) determines the breast density in a 3D manner, which provides more accurate result [7]. The R image can be used to achieve quantitative volumetric breast density (VBD) estimation in mammography.

(13) On the pixel base, the VBD is defined by (volumetric percentage of the glandular tissue)

(14) 0$\begin{array}{cc}m\left(x,y\right)=\frac{L_g}{L}=\frac{L_g}{L_g+L_f},& \left(11\right)\end{array}$

(15) where L.sub.g, and L.sub.f are the thicknesses of the glandular tissue and fat tissue along the X-ray beam, respectively. L is the total compression thickness of the breast. The volumetric breast density of the whole breast is given by averaging m over the whole breast region

(16) $\mathrm{VBD}=\frac{\underset{x,y}{.Math.}m\left(x,y\right)}{N},$

(17) where N is the pixel numbers of the breast region.

(18) With the two-material composition assumption of the breast, Eq. (10) is

(19) $\begin{array}{cc}R=\frac{T}{S}=\frac{{}_g{L}_g+{}_f{L}_f}{s_g{L}_g+s_f{L}_f}=\frac{{}_{g}mL+{}_f\left(1-m\right)L}{s_{g}mL+s_f\left(1-m\right)L}.& \left(12\right)\end{array}$

(20) Taking Eq. (11) into Eq. (12), the connection between the VBD m and the measured quantity R is setup by

(21) $\begin{array}{cc}m=\frac{1}{1-\frac{\left(\frac{R}{R_g}-1\right)}{\left(\frac{R}{R_f}-1\right)}\frac{{}_g}{{}_f}},R_g<R<R_f& \left(13\right)\end{array}$

(22) where

(23) $R_g=\frac{{}_g}{s_g}\mathrm{and}{R}_f=\frac{{}_f}{s_f}$

(24) are defined as the R values for 100% glandular tissue and 100% fat tissue, respectively. The subscripts g and f represent the glandular tissue and fat tissue, respectively. R.sub.g and R.sub.f can be determined dynamically from the histogram of the R image as described in the following sections. Note that the fat tissue has a bigger R value than the glandular tissue.

(25) $\frac{{}_g}{{}_f}$

(26) is a constant for a certain energy and can be determined by calibrated data [7]. For instance,

(27) $\frac{{}_g}{{}_f}$

(28) is around 1.43 for a mean photon energy of 28 KeV.

(29) A typical histogram of the R image for the breast sample is shown in FIG. 3. FIG. 3(a) shows the R image for one breast sample in the craniocaudal (CC) view, FIG. 3(b) is the corresponding histogram of the R image. The histogram H(R) is usually an asymmetric Gaussian distribution due to the fact that the fat tissue (the right part of the histogram) is dominant in the breast compared to the glandular tissue (the left part) [8], therefore has a larger variance. The VBD information of the breast is actually implied in the asymmetric distribution of the histogram. For instance, the skewness (the third moment) of the distribution, which is defined as

(30) $\begin{array}{cc}=\frac{E\left[{\left(R-{}_R\right)}^3\right]}{{\left(E\left[{\left(R-{}_R\right)}^2\right]\right)}^{3/2}},& \left(14\right)\end{array}$

(31) can give a good estimation of how dominant the glandular or fat tissue is in the whole breast.

(32) To get a quantitative value of the breast density from Eq. (12), we determine R.sub.g and R.sub.f in the following way with the consideration that noises always present in the R image and the dynamic process for determining R.sub.g and R.sub.f. is illustrated in FIG. 4: (1) The peak position R.sub.peak which gives the maximal H(R) is first decided. (2) The histogram H(R) is divided into the left and right parts by R.sub.peak. These two parts of the histogram are considered separately. For each half, the distribution is mirrored around the peak value to form a symmetric Gaussian distribution. Then Gaussian fitting is applied to the two resulting Gaussian distributions and their standard deviations .sub.g and .sub.f are calculated respectively. (3) R.sub.g and R.sub.f are decided using the N criteria,
R.sub.g=R.sub.peakN.sub.g,
R.sub.f=R.sub.peak+N.sub.f(15)

(33) where N is a positive integer. N=3 is considered to give the best approximation of the R.sub.g and R.sub.f when the noises present. The 3 criteria covers 99.7% of the area of the histogram.

(34) With the determined R.sub.g and R.sub.f, the 3D VBD of the whole breast is calculated by weighting the normalized histogram H(R) using Eq. (15),

(35) $\begin{array}{cc}VBD=\underset{R}{.Math.}m\left(R\right).Math.H\left(R\right).& \left(16\right)\end{array}$

(36) The proposed method was evaluated by clinical mastectomy breast dataset. 27 patients were included in the evaluation with the diagnostic breast densities: ACR 1 (4 samples), ACR 2 (13 samples), ACR 3 (10 samples) and ACR 4 (0 sample). The VBD results for the CC view using the proposed method were given in FIG. 5 which shows the mean and standard deviation of VBD values estimated from the CC view for each diagnostic ACR group. The calculated VBD values fell into the range of 10%30% which is consistent with the previous studies using X-ray tomography and MRI [8]. A positive correlation was also clearly seen, which meant statistically the proposed method was consistent with the diagnostic results.

(37) In this invention, the absorption and small-angle scattering signals obtained with multiple-modality systems are adopted to distinguish two types of microcalcifications. Such multiple-modality systems have been developed in the last fifteen years, including techniques based on analyzer crystal [9], gratings [10, 11, 12] and coded apertures [13, 14]. The described invention is in context with these techniques and uses the set-ups disclosed in the respective documents which form therefore part of this disclosure with respect to the set-ups.

(38) Practical applications in non-destructive testing and medical imaging demand techniques which can work well with conventional X-ray tubes. For this reason, grating-based methods are especially promising. Without loss of generality, we will discuss the practical aspects of the method using gratings-based interferometry as an example.

(39) Grating-based X-ray imaging setups can generate three different signals: the conventional absorption contrast (AC) signal, the differential phase contrast (DPC) signal caused by refraction due to phase shifts [11], and the small-angle scattering contrast (SC) signal (also named dark-field signal) caused by scattering from in-homogeneities in the sample [10].

(40) Grating Interferometer set-ups with two gratings (G1 and G2) or three gratings (G0, G1, and G2) as schematically shown in FIG. 6 can be applied to record the deflection of the X-rays. In the case of a two-grating set-up, the source needs to fulfill certain requirements regarding its spatial coherence, wherein a G0 grating is required, when the source size is bigger than p2*l/d, where p2 is the period of G2, l is the distance between the source and G1, and d is the distance between G1 and G2, while in a three grating setup no spatial coherence is required [15,16]. Therefore, the three grating set-up is suited for use with incoherent X-ray sources, in particular with standard X-ray tubes for medical investigations.

(41) To separate the conventional attenuation contrast (AC) from the DPC and SC contrast, a phase-stepping approach is applied. One of the gratings is displaced transversely to the incident beam whilst acquiring multiple images. The intensity signal at each pixel in the detector plane oscillates as a function of the displacement. The average value of the oscillation represents the AC. The phase of the oscillation can be directly linked to the wave-front phase profile and thus to the DPC signal. The amplitude of the oscillation depends on the scattering of X-rays in the object and thus yields the SC signal.

(42) For the (two or three) gratings, several approaches have been proposed and applied. The grating G0 (if required) is the one closest to the source. It usually consists of a transmission grating of absorbing lines with the period p0. It can be replaced by a source that emits radiation only from lines with the same period [17]. The grating G1 is placed further downstream of the source. It consists of lines with a period p1. The grating G2 is the one most downstream of the setup. It usually consists of a transmission grating of absorbing lines with the period p2. It can be replaced by a detector system that has a grating-like sensitivity with the same period.

(43) Two regimes of setups can be distinguished: in the so called near field regime and the Talbot regime. In the near field regime, the grating period p, grating distances d and the X-ray wavelength 2 are chosen such, that diffraction effects are negligible. In this case, all gratings need to consist of absorbing lines. In the Talbot regime, diffraction from the grating structures is significant. A sharp distinction between the two regimes is not easily given, as the exact criterion depends on the duty cycle of the grating structure, and whether the gratings are absorbing or phase shifting. E.g., for a grating with absorbing lines and a duty cycle of 0.5, the condition for the near field regime is dp.sup.2/2. Here G1 should consist of grating lines that are either absorbing or, preferentially, phase shifting. Several amounts of phase shift are possible, preferentially /2 or multiples thereof. The grating periods must be matched to the relative distances between the gratings. In the case of setups in the Talbot regime the Talbot effect needs to be taken into account to obtain good contrast. The formulae for the grating periods and distances are described in [18].

(44) The sample is mostly placed between G0 and G1 (or upstream of G1 in the case of a two-grating set-up), however it can be advantageous to place it between G1 and G2 [19].

(45) The presented invention is relevant in all of the abovementioned cases, i.e. in the two- and three-gratings case, in the case of the nearfield regime and the Talbot regime, and for the sample placed upstream or downstream of G1.

(46) In addition, the invention presented here also works in combination with scanning-based systems as suggested in [20, 21] or for planar grating geometries, as suggested in [22].

(47) Intensity curves (with and without sample) are usually obtained with phase stepping methods or alternative techniques. Defining for each pixel on the detector the mean, phase and visibility of the intensity curve with sample as I.sub.s, .sub.s, V.sub.s, and without sample as I.sub.b, .sub.b, V.sub.b, yields:

(48) $\begin{array}{cc}\mathrm{AC}=-\log \left(\frac{I_s}{I_b}\right)& \left(17\right)\\ DPC={}_s-{}_b& \left(18\right)\\ SC=-\log \left(\frac{V_s}{V_b}\right).& \left(19\right)\end{array}$

(49) For both the AC signal and SC signal, the valid data range is [0,+], while for the DPC it is [,+]. Images obtained by plotting such signals are all perfectly registered.

(50) A similar way to generate these multiple information signals can be found in diffraction enhanced imaging where the equivalent of the intensity curve is named the rocking curve [9].

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