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A novel information retrieval algorithm for X-ray grating-based phase-contrast imaging based on the deconvolution of the object and the reference phase stepping curve (PSC) as proposed by Modregger et al. was investigated in this paper. We applied the method for the first time on data obtained with a polychromatic spectrum and compared the results to those, received by applying the commonly used method, based on a Fourier analysis. We confirmed the expectation, that both methods deliver the same results for the absorption and the differential phase image. For the darkfield image, a mean contrast-to-noise ratio (CNR) increase by a factor of 1.17 using the new method was found. Furthermore, the dose saving potential was estimated for the deconvolution method experimentally. It is found, that for the conventional method a dose which is higher by a factor of 1.66 is needed to obtain a similar CNR value compared to the novel method. A further analysis of the data revealed, that the improvement in CNR and dose efficiency is due to the superior background noise properties of the deconvolution method, but at the cost of comparability between measurements at different applied dose values, as the mean value becomes dependent on the photon statistics used.
© 2013 Optical Society of America
1. Introduction
Grating-based X-ray phase-contrast imaging is a technique utilizing e.g. a Talbot-Lau grating interferometer (as sketched in Fig. 1) to obtain an absorption image A, a differential phase image P and a dark field image B at the same time [1–3].
Fig. 1 Sketch of a Talbot-Lau interferometer setup. The source S emits wavefronts Φ that are disturbed by the object O investigated. The phase grating G1 adds a periodic phase shift to the wave front which leads to a periodic intensity pattern I(x) at certain discrete distances behind the grating. As the pitch of the intensity pattern is too small to be detected by the pixelated detector D directly, it has to be sampled by an analyser grating G2. The source grating G0 ensures spatial coherence when a source with large focal spot is used.
As the benefit on X-ray imaging has been shown by several groups [4–6] and this technique is compatible with medical X-ray sources and large focal spot sizes [7], a huge potential is seen to overcome e.g. the poor soft-matter contrast in medical imaging. Especially the dark field image shows promising results e.g. in the detection of super-fine micro-calcifications not seen by conventional attenuation-based mammography [8, 9].
In order to further improve the image quality in grating-based X-ray phase-contrast imaging, the performance of the novel information retrieval algorithm proposed by Modregger et al. [10] was investigated using a laboratory X-ray source. Furthermore, the obtained results were compared to the well established and commonly used phase retrieval method introduced by Weitkamp et al. [11] and expanded to darkfield imaging by Pfeiffer et al. [3].
2. Theory
In grating-based X-ray phase-contrast imaging, a Talbot-Lau interferometer setup is used where a phase grating produces a periodic intensity pattern in the detector plane exploiting the Talbot effect. As this effect requires spatially coherent light, a source grating is introduced right behind the X-ray source, exploiting the Lau effect and producing many spatially partial-coherent slit sources. As the period of the generated Talbot intensity pattern in the detector plane is usually too small to be detected directly, it has to be sampled. Therefore, an analyser grating (G2) with a pitch matching the spatial period of the Talbot pattern is introduced and one of the gratings is scanned perpendicular to its bars in steps that are only a fraction of the gratings pitch. After each step an image is recorded. This technique is often referred as phase stepping or fringe scanning technique and results in a so-called phase stepping curve (PSC) in each pixel [11].
Usually, two PSCs are recorded, one with object, s(ϕ) and one without sample, f(ϕ), present in the beam. After that, a Fourier transform with respect to the phase steps is performed to retrieve the three contrasts absorption A, differential phase P and dark field B as mentioned above. This technique will be referred as “FFT method” in the following.
On the other hand, Modregger et al. [10] proposed an alternative method based on the assumption that the PSC with object s(ϕ) is obtained by the convolution of the PSC without object f(ϕ) and a scattering distribution g(ϕ) containing the object information [12]
The scattering distribution g(ϕ) can be accessed by deconvolution methods as for example the Lucy-Richardson algorithm as also proposed by Modregger et al. [10]. The image contrasts, absorption A, differential phase P and dark field B, as obtained by the FFT method, can be received from the zeroth, first and second moment of the scattering distribution as and This method will be referred as “deconvolution method” in the following.3. Methods
To evaluate the different information retrieval methods a specially designed phantom was used. A photograph is shown in Fig. 2. In order to receive regions with constant signal for every of the three observables the phantom consists of three parts: an empty Poly(methyl methacrylate) (PMMA) dice produces the absorption signal, a Polytetrafluoroethylene (PTFE) wedge the differential phase signal and a plastic sponge the darkfield signal.
Fig. 2 Photography of the phantom used for the contrast-to-noise ratio (CNR) investigations. The plastic sponge at the bottom left of the picture produces the dark field signal. On top, the Polytetrafluoroethylene (PTFE) wedge generates a constant differential phase image, while for the absorption image an empty Poly(methyl methacrylate) (PMMA) dice is used (right hand side of the picture).
This phantom was imaged using a Talbot-Lau grating interferometer setup with a π-shifting phase grating (G1) for the design energy of 25keV with a pitch of 4.37 μm and a 2.4 μm pitch analyser grating (G2) in the third fractional Talbot distance. The X-ray spectrum used was produced by a medical X-ray tube with a tungsten anode (Siemens Megalix CAT+) driven at 40kV acceleration voltage without any additional than the intrinsic filtering. The detector was a Varian PaxScan 2520D calibrated for the frame time of 0.1s. 100 phase steps recorded over one G2 period were used, while for each of them 10 individual frames were acquired. This resulted in a total scan time of 100s for each, the object and the reference scan. Herewith, the overall dose in the phase stepping curves (PSCs) used for the reconstruction of the three image quantities could easily be varied. The applied dose was measured as the air kerma at the object position using a calibrated IBA Dosimetry Dosimax plus A HV dosimeter equipped with a solid-state detection unit RQX 70kV. For the information retrieval we used the computer algebra program MATLAB [13] in both cases. For the FFT method, the images were calculated as proposed by Weitkamp et al. [11]. For the deconvolution method we used the Lucy-Richardson algorithm also provided by MATLAB using 200 iterations, as this algorithm is known to deliver stable results even in the presence od noise [14, 15]. For being able to compare the resulting images, the deconvolution method’s resulting moments were transformed to match the images as received by the FFT method (A/P/B) according to Eq. (2), Eq. (3) and Eq. (4).
For the calculation of the contrast-to-noise ratio (CNR) one region-of-interest (ROI) was defined in every of the three contrast regions, where the according constant signal was expected. In addition, a fourth ROI containing only air was used to determine the background signal of each image as the variance of the measured pixel values. The locations of the ROIs are shown as boxes in Fig. 3. The CNR was calculated as
while S̄air denotes the mean signal in the background ROI of the image, S̄ image is the mean signal of the ROI containing the contrast generating object and and are the inter-pixel variances in the image and background ROI, respectively.Fig. 3 Resulting absorption (top), differential phase (middle) and dark field image (bottom) of the phantom. On the left hand side, the images were obtained with the FFT method, on the right hand side with the deconvolution method. The boxes show the used ROIs for the CNR calculations.
4. Results
Figure 3 shows the three images obtained with both methods at the same dose level (3.56mGy air kerma at the object position). Visually, no difference in the images produced by the different phase retrieval algorithms can be found. For all possible dose values constructable from the 1000 measured frames, the CNR values were calculated according to Eq. (5). Figures 4, 5 and 6 show the results, obtained by both phase retrieval methods for the three image quantities absorption A, differential phase P and darkfield image B. While for the absorption and the differential phase no significant difference in the CNR values can be seen over the entire dose range investigated, the darkfield signal obtained by the deconvolution method has an increased mean CNR by a factor 1.17 ± 0.061 over the whole dose range.
Fig. 4 Contrast-to-noise ratio of the absorption image A against the air kerma used for reconstruction. The crosses show the results obtained by the FFT method, the squares the results with the deconvolution method.
Fig. 5 Contrast-to-noise ratio of the differential phase image P against the air kerma used for reconstruction. The crosses show the results obtained by the FFT method, the squares the results with the deconvolution method.
Fig. 6 Contrast-to-noise ratio of the darkfield image B against the air kerma used for reconstruction. The crosses show the results obtained by the FFT method, the squares the results with the deconvolution method.
To investigate the origin of the increased CNR values for the deconvolution method, the signal differences |S̄image − S̄air| and the total image noises were calculated for selected dose values. The results are listed in Tab. 1 and Tab. 2 for both information retrieval algorithms side-by-side. Comparing the signal differences, no difference between the reconstruction methods can be found for the absorption and the differential phase image. The dark-field value is approximately 10% smaller for the deconvolution method. For the noise values, a similar behaviour can be observed. In the absorption and the differential phase image the image noise does not vary strongly between the different information retrieval algorithms, while the noise in the darkfield image is reduced by a factor of approximately 1.3. The expected increase in CNR can be calculated as the product of these two factors to 0.9 × 1.3 = 1.17, which is in agreement to the experimentally found mean CNR increase above.
Table 1. Measured signal differences |S̄image − S̄air| for the two information retrieval methods at three different dose values.
Table 2. Measured image noise values for the two information retrieval methods at three different dose values.
An even closer look at the two parts of the image noise – σimage and σair – of the darkfield image reveals that the main decrease of the total image noise is due to the reduction in the background noise value. The measured absolute values are tabulated in Tab. 3. While the noise σimage stays almost constant comparing the two information retrieval methods, the background noise σair is reduced by a factor in the order of 2.5–3.5, increasing with increasing dose.
5. Discussion
We demonstrated for the first time, that deconvolution information retrieval methods are also applicable on data obtained with a medical X-ray tube and therefore with a polychromatic spectrum. The results show, that the novel method results in equivalent signal differences and noise values for the absorption and the differential phase image, but produces a less noisy darkfield image at almost comparable signal differences. This results in an increased mean CNR by a factor of 1.17 compared to the FFT method usually used in grating-based X-ray phase-contrast imaging. To obtain this CNR increase with the FFT method a higher dose would be necessary. The increase in dose, which would be needed can be calculated to 1.172 = 1.37.
In fact, the dose saving found experimentally by comparing similar CNR values is even larger. Using the values for 3.0mGy (CNR = 12.3) for the FFT method and 1.8mGy (CNR = 12.1) for the deconvolution method, a dose ratio of
is found.The main difference causing the better CNR of the deconvolution method is the superior inter-pixel variances in the background ROI (see Tab. 3). This is caused by the fundamentally different signal formation process. While with the FFT method the codomain of the dark field image B is limited to , with the deconvolution method only values in the range of [0;4π2/12] – assuming a unimodal scattering distribution g(ϕ) – can be obtained for the second moment M2. Translated to the corresponding deconvolution darkfield, the codomain of [0.193;1] is found.
This two-sided boundary has severe consequences for the signal behaviour as it biases the deconvolution darkfield value for results near the boundaries. Given the case, the object measured consists of air, the two PSCs measured will be exactly the same for infinite statistics only. In every other case, there will be any sort of noise, e.g. Poisson noise, causing a difference in the PSCs. For the FFT method, this difference will cause an uncertainty in the estimated Fourier components leading to a Gaussian darkfield signal distribution with mean value 1 and a standard deviation depending on the photon statistics [16]. For the deconvolution method, the difference in the PSCs causes a scattering distribution g(ϕ) which has a second moment M2 > 0, whose value itself depends on the statistics used. Given a ROI of pixels, the measured values for M2 are expected to be Gaussian distributed around a mean value with a certain standard deviation that is comparable to that produced by the FFT method. This only holds valid if the mean value is far from the boundaries, as it is e.g. within the plastic sponge (compare σimage in Tab. 3).
Table 3. Measured values of σimage and σair of the darkfield image B for both information retrieval algorithms.
If the expected mean value of M2 is near the lower boundary – as it is for air – the limited codomain will cause a narrower distribution of the M2 values compared to the distribution produced by the FFT method, whose codomain is not limited in this range. This results in a superior inter-pixel variance for weakly and non-scattering objects (compare σair in Tab. 3).
Additionally, the biasing of the two-sided boundary condition of the codomain of the second moment of the scattering distribution results in the fact that the mean value of the deconvolution darkfield is dependent on the statistics used. The value M2 = 0 can only be reached for infinite photon statistics and a perfect detector. This means, that a deconvolution darkfield value of 1 can only be reached with infinite dose. Every other case will produce a value Bdeconv < 1 even if only air is imaged and the expected value is 1. This behaviour is shown in Fig. 7, where the mean darkfield values of the background ROI are shown for different dose values for the two information retrieval methods. While for the FFT method, the mean signal only slightly varies with photon statistics, the mean deconvolution darkfield value strongly decreases for dose values below 1.8mGy. In addition, even for the highest dose value investigated (3.56mGy) only is reached. The general behaviour furthermore indicates a very slow asymptotic trend towards 1, which is in agreement to the observations made by Scattarella et al. [15]. There, the authors conclude, that for weakly scattering objects, i.e. a small standard deviation of the object function g(ϕ), a substantially increased number of iterations is needed to reconstruct the expected darkfield value. This effect is also a result of the limited codomain of the second moment M2.
Fig. 7 Mean darkfield signal B̄air of the background region-of interest (ROI) for both information retrieval methods against the air kerma value used for reconstruction.
6. Conclusion
In conclusion, we showed, that the deconvolution information retrieval method proposed by Modregger et al. can be applied to data obtained with a polychromatic X-ray source. It increases the CNR in the darkfield image and therefore reduces the dose needed for a predefined image quality. Furthermore, we analysed the origin of the superior CNR and found a lower inter-pixel variance of the novel method compared to the established FFT method for weakly or non-scattering objects. The novel method is of advantage for the reconstruction of projective darkfield imaging data, where no quantitative measure is needed. This is the case e.g. in radiography or mammography [4, 5, 8, 9], where deconvolution information retrieval can reduce the necessary dose for a predefined image quality.
On the other hand, we showed, that the mean signal value obtained by the deconvolution method is dependent on the photon statistics used. This is due to the fact, that the codomain of the deconvolution darkfield is limited on both sides. That means, that measured deconvolution darkfield values may no longer be comparable to each other if different dose values are used in the measurements. This finding may be a severe drawback of the method, where quantitative results and comparability are expected like in computed tomography.
Acknowledgments
This work was funded by the German Ministry for Education and Research (BMBF), project grant No. 01EZ0923 and the cluster of excellence Medical Valley EMN. Furthermore, the authors want to thank Dr. Jürgen Mohr and Jan Meiser from the Karlsruhe Institute of Technology and the Karlsruhe Nano Micro Facility (KNMF) for manufacturing the gratings used in the experiments. We acknowledge support by Deutsche Forschungsgemeinschaft and Friedrich-Alexander-Universit¨at Erlangen-Nürnberg within the funding programme Open Access Publishing.
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