Keywords: discrete Radon transform; discrete orthogonal moments; projection moments; image reconstruction
1. Introduction
The issue of image reconstruction has received much attention in the medical imaging literature. This is due to the constant search for improvements of imaging modalities, ranging from X-ray computerized tomography and emission tomography up to acoustic and optical techniques. They all bring different insights in the human body either morphological or functional. The standard mathematical model of X-ray computerized tomography (CT) assumes that the sensing device measures the line integrals of the object attenuation coefficient at some known orientations. An analytical formulation for the reconstruction of two-dimensional (2-D) **tomographic** **images** from projections, i.e., an inverse problem, has been first proposed by Radon in 1917. The filtered back-projection (FBP) algorithm, which can be seen as a computer implementation of Radon’s inversion formula, still plays an important role although algebraic methods are also intensively used [1-3]. However, the reconstruction based on FBP algorithm requires the projections for all angles from 0 to π . A major health concern today is related to the reduction of dose to the patient which means limiting either the X-ray source intensity or the number of projections. This issue is critical not only for diagnosis imaging but also in interventional setting where for instance rotational—X is used.

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Full morphology Liquid water
a b s t r a c t
Understanding and modeling two-phase ﬂows in the gas diffusion layer (GDL) of proton exchange membrane fuel cells are important in order to improve fuel cells performance. They are scientiﬁcally challenging because of the peculiarities of GDLs microstructures. In the present work, simulations on a pore network model are compared to X-ray **tomographic** **images** of water distributions during an ex-situ water invasion experiment. A method based on watershed segmentation was developed to extract a pore network from the 3D segmented image of the dry GDL. Pore network modeling and a full morphology model were then used to perform two-phase simulations and compared to the experimental data. The results show good agreement between experimental and simulated microscopic water distributions. Pore network extraction parameters were also benchmarked using the experimental data and results from full morphology simulations.

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Full morphology Liquid water
a b s t r a c t
Understanding and modeling two-phase ﬂows in the gas diffusion layer (GDL) of proton exchange membrane fuel cells are important in order to improve fuel cells performance. They are scientiﬁcally challenging because of the peculiarities of GDLs microstructures. In the present work, simulations on a pore network model are compared to X-ray **tomographic** **images** of water distributions during an ex-situ water invasion experiment. A method based on watershed segmentation was developed to extract a pore network from the 3D segmented image of the dry GDL. Pore network modeling and a full morphology model were then used to perform two-phase simulations and compared to the experimental data. The results show good agreement between experimental and simulated microscopic water distributions. Pore network extraction parameters were also benchmarked using the experimental data and results from full morphology simulations.

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CONCLUSION
In this contribution an efficient technique enabling to denoise and to segment X- ray **tomographic** **images** of porous media stones was presented and applied to building stone. It allows a very strong noise removal without blurring, at the price of losing the smaller structural components of the image. This loss is however controlled: the size of the structuring element of the last ASF step gives the size of the smallest structural components kept. As the method makes no a priori strong assumption about the **images** to segment, it should then be easily generalized to **tomographic** **images** of various materials and even to **images** from other imaging techniques.

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The aim of this chapter is to validate that the simulations on **tomographic** **images** of GDL make it possible to calculate in a predictive way the properties of diffusive transports measured experimentally. Numerical simula- tions have several advantages over experimental measurements. Experimental measurements of the properties of GDL are often difficult due to the low thickness of the GDL. It is often easier to produce a **tomographic** image of a porous material than to characterize experimentally its physical properties. **Tomographic** **images** are availa- ble for a large number of porous materials, used for example in PEMFCs, SOFCs, electrolysers and batteries. In addition, numerical simulations allow the study of microstructures designed virtually. This opens the way to a numerical optimization of porous materials. Validating the relevance of image-based models is an important step in providing a reliable method for studying porous materials and guiding the development of new materials. The microstructure of a porous material directly impacts the effective properties of this material. Knowing the precise microstructure of a GDL is therefore important in determining its properties. The microstructure of the GDL is different from the simplified microstructures for which analytical formulas or correlations exist, such as the Bruggeman formula [Tjaden2016] . Moreover, changes in the microstructure of the GDLs, such as the pres- ence of binder or the deformation due to compression by the bipolar plate, modify the GDL transport properties. This is why we use **tomographic** **images** to model the GDL microstructure. Numerical simulations are carried out on these 3D **images** to calculate the effective transport properties from the solving of the partial differential equation modeling the diffusive transports.

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Segmentation of X-ray tomographic images : contribution to weathering analysis of building limestones originating from historical buildinds.. Emmanuel Le Trong, Olivier Rozenbaum, Jean-L[r]

In Figure 2, we depict a **tomographic** P wave model across Japan [Karason and van der Hilst, 2000]. In the top of Figure 2, this model has been plotted using the same color scale as in Fukao et al. [1992]. The bottom panel of Figure 2 represents the exact same data set but with the color scale of Bijwaard and Spakman [2000]. Basically along the same great circle, these authors have published, with their own models, very similar results which indicate the good general agreement between models. Although Figure 2 may suggest slab penetrations, with significant thickening/deformation in the lower mantle in agreement with geodynamical models, the very significant decrease in the lower mantle must be explained. This can only be done after a closer look to tem- perature/velocity/density relationships and therefore to min- eralogy. Otherwise the interpretation of **tomographic** **images** in term of non-penetration or straight penetration is affected by the non-objective choice of the color scale as illustrated in Figure 2.

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The standard displacement uncertainty are compared before dilatational strain correction, rotation correction when the analysis is performed with reconstructed volumes with or without th[r]

During acquisition, the sample is rotated step by step, taking a projection image at each angular position. The projection set is used as an input for the image reconstruc- tion algorithm. This is done by using a filtered back projection algorithm: a computer reconstructs from the projections the cross-sectional **images** of the sample (called to- mograms). Stacking these reconstructed **images** forms a three-dimensional image of the sample (volume dataset). At each voxel space position of the resulting dataset, a grey-value corresponds to the effective X-ray attenuation coefficient. Therefore, if the principal compounds of the object are known and have a sufficient density contrast, the distribution of these compounds within the object can be easily deduced. Unfor- tunately, this is not often the case as illustrated in the following with two different geomaterials.

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We concentrate on ultra light wood-based/polyester composite insulator used in roof insulation. The X-ray tomography technique is used to get tridimensionnal **images** of the fibrous structure. The resolution is chosen to be 5µm/pixel, which is sufficient to visualize the wood fibers inner porosity (called lumen).

As all the transformations T 1 , R(θ i ), T 2 and P do not change during the experiment,
calibration can be performed only once. Moreover, because a very stiff ceramic material was tested, the displacement of the sample under the mechanical load turned out to be very small (lower than thermogram resolution). Would the deformation of the sample be significant, it could be computed by DVC on **tomographic** **images**. The positions of 3D mesh (surface) nodes would then account for the deformed geometry. In fact, the only case which may be troublesome is when the sample slips out of its grip. If the sample requires a pre-loading for a good positioning, it is recommended to perform calibration only after the pre-loading step.

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Abstract – Adapting fast **tomographic** microscopy, we managed to capture the evolution of the local structure of the bubble network of a 3D foam flowing around a sphere. As for the 2D foam flow around a circular obstacle, we observed an axisymmetric velocity field with a recirculation zone, and indications of a negative wake downstream the obstacle. The bubble deformations, quantified by a shape tensor, are smaller than in 2D, due to a purely 3D feature: the azimuthal bubble shape variation. Moreover, we were able to detect plastic rearrangements, characterized by the neighbor-swapping of four bubbles. Their spatial structure suggests that rearrangements are triggered when films faces get smaller than a characteristic area.

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ˆ
w rms [vox] 0.143 0.092 1.55 0.201 0.129 1.55
Table 6.3.5: Mean rms values of a low velocity area on the outskirts of the jet at z = 0, −2 < x < −1 and 0 < y < 3. In figure 6.3.7, the area is visualized by a white rectangle. When comparing the two density cases, the trend previously exhibited is confirmed : Case 4 has higher rms values than Case 2. More precisely, for tomo-SMART, the rms values are respectively 22 %, 59 % and 40 % higher for Case 4 than Case 2, on respectively u,v and w component. For PVR-SMART, the rms values are respectively 28 %, 57 % and 40 % higher for Case 4 than Case 2, on respectively u,v and w component. The velocity fields from Case 4 are noisier with statistics being less converged than Case 2. Our interpretation so far has been that the increased measurement noise comes from a lack of tracer in the correlation window, despite the fact that we previously estimated the number of tracers per IW to about 7 to 11 in table 6.3.1. However, this estimation was done through the use of 2D-PIV **images**, with a 200 mJ laser, much powerful than the 3D-PIV laser (120 mJ). This means that some particles seen by the 2D-PIV setup, are not seen from the 3D-PIV setup therefore not reconstructed, due to the lower illumination and also Mie scattering. Adding calibration errors, discretisation errors and image pre-processing thresholds, fewer tracers are actually reconstructed by the **tomographic** step than estimated by the 2D-PIV **images**. It would be very interesting to have an actual estimation of the reconstructed tracer density, based for instance on tracer tracking from a pulse to the following. This would help choose the IW size for the 3D correlation step. This will undoubtedly be the subject of future research.

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Line 9 Comparisons Past and Present
As stated earlier, Greenwood and Lee employed the best available modeling capability of their day to generate a detailed resistivity model of Line 9. Figure 19 shows this 1976 model colorized and the recently reinverted tomogram for Line 9. The two **images** have similar features, a very conductive surface zone at Sulphur Springs and the resistive block at the surface south of Sulphur Springs. However, the most important feature of the tomogram, a resistive zone at depth, is not seen in the 1976 model. The model does show a slightly more resistive feature in the center and a conductive zone to the south at depth, which is the correct trend, but in error on the order of two magnitudes. Considering the computing interpretation limitations at the time the model was generated it did well for the top 200 m. An analysis of Line 9 specific to the Sulphur Springs was performed (Morgan, et al., Resistivity tomogram of a 4.5 km line through the Sulphur Springs, St. Lucia, submitted to the The Leading Edge, 2002) which discusses the Lee and

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This article introduces a technique using the Lorentz force to quantitatively map velocity and pressure fields with a spatial resolutionof less than one millimeter. Good spatial resolution is achieved by moving and rotating the wire then making a **tomographic** reconstruction of the pressure field.
The movement of a conductor inside a magnetic field induces an electrical current perpendicular to the motion and to the magnetic field. For a conductor of length L, the induced voltage e is proportional to the velocity v z of the conductor times the magnetic field B x integrated along the conductor, as given by Eq. 1

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We further combine quantitative phase imaging with computed tomography (CT) which enables us to investigate internal structures in 3D. Existing techniques for phase r[r]

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Though time-reversal mirrors were originally designed to reproduce a time-reversed version of an original wavefield, the proposed method is independent of the time direction and can be u[r]

Fig. 5. High dose reconstructions for the simulated data Phantom2-SH. (a), FBP1 reconstruction using Ramp filter; (b), FBP2 reconstruction using the Hanning filter with cutoff at 80% Nyquist frequency;(c), TV prior reconstruction; (d), Huber prior reconstruction; (e), MRP reconstruction; (f), The proposed PSM prior reconstruction.
with a higher level of count. We can observe the same effects in a lower proportion since the magnitude of the noise is less important in the original measurements. FBP1 and FBP2 still produce some streak artifacts, Huber and MRP priors provide a smoothed image excepted that edges are better preserved in the image reconstructed with MRP prior, yielding thus a higher level contrast than in the former one. Smaller objects such as the straight lines appears nevertheless relatively smoothed. TV prior performs better in preserving edges and PSM prior still produce the best result both in term of smoothing and edge preservation. In TABLE II, we list the calculated SNR of all the reconstructions with the simulated data. Obviously, the SNR comparisons indicate that Bayesian reconstructions can produce **images** with higher SNR than analytic FBP reconstructions, and the reconstructed **images** using the proposed prior can obtain the highest SNR among all the reconstruction methods.

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[2] Rasche V, Movassaghi B, Grass M, et al. Automatic selection of the optimal cardiac phase for gated three-dimensional coronary X-ray angiography Acad. Radiol., vol. 13, 2006, pp 630-640.
[3] Zhou J., Bousse A., Yang G., Bellanger J-J, Luo L., Toumoulin C., Coatrieux J.L, A blob-based **tomographic** reconstruction of 3D coronary trees from rotational X-ray angiography, SPIE Medical

DOI: 10.1103/PhysRevLett.103.241301 PACS numbers: 98.62.Sb, 04.80.Cc, 95.80.+p, 98.80.k
The observed acceleration of cosmic expansion poses a puzzle for modern cosmology. It may be evidence for dark energy (DE), a component with a negative equation of state, w, making it gravitationally repulsive. It also war- rants studying extensions of general relativity (GR) with extra degrees of freedom that can mimic the effects of DE. Modifications to GR are well constrained in dense regions like our Solar System [1]. On larger scales GR is less well tested. Several modifications to GR, capable of producing cosmic acceleration, have been proposed [2]. With the right parameter values, they can match the expansion history of a universe made of cold dark matter (CDM) and a cosmological constant —the observationally fa- vored CDM model [ 3]. However, their predictions for the growth of structure can differ since the perturbation equa- tions get modified. Future **tomographic** weak lensing sur- veys, like the Dark Energy Survey (DES) [4] and the Large Synoptic Survey Telescope (LSST) [5], will measure lens- ing shear and galaxy counts in many redshift slices (hence the term tomography), thus mapping the evolution of per- turbations and offering a new test of GR on cosmological scales [2]. In this work, we use a two-dimensional principal component analysis (PCA) to forecast the constraints on modified growth (MG)—and thus our understanding of gravity—coming from these surveys. Unlike previous MG forecasts, ours is model independent and lets us de- termine how many parameters describing MG could be constrained, along with the regions in parameter space where we expect the most sensitivity to MG.

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