Multi-Energy X-Ray Tomography of an Optical Fiber: The Role of Spatial Averaging.

Using a commercial X-ray tomography instrument, we have obtained reconstructions of a graded-index optical fiber with voxels of edge length 1.05 µm at 12 tube voltages. The fiber manufacturer created a graded index in the central region by varying the germanium concentration from a peak value in the center of the core to a very small value at the core-cladding boundary. Operating on 12 tube voltages, we show by a singular value decomposition that there are only two singular vectors with significant weight. Physically, this means scans beyond two tube voltages contain largely redundant information. We concentrate on an analysis of the images associated with these two singular vectors. The first singular vector is dominant and images of the coefficients of the first singular vector at each voxel look are similar to any of the single-energy reconstructions. Images of the coefficients of the second singular vector by itself appear to be noise. However, by averaging the reconstructed voxels in each of several narrow bands of radii, we can obtain values of the second singular vector at each radius. In the core region, where we expect the germanium doping to go from a peak value at the fiber center to zero at the core-cladding boundary, we find that a plot of the two coefficients of the singular vectors forms a line in the two-dimensional space consistent with the dopant decreasing linearly with radial distance from the core center. The coating, made of a polymer rather than silica, is not on this line indicating that the two-dimensional results are sensitive not only to the density but also to the elemental composition.

Preliminary X-ray CT investigation to link Hounsfield unit measurements with the International System of Units (SI).

PURPOSE: This paper lays the groundwork for linking Hounsfield unit measurements to the International System of Units (SI), ultimately enabling traceable measurements across X-ray CT (XCT) machines. We do this by characterizing a material basis that may be used in XCT reconstruction giving linear combinations of concentrations of chemical elements (in the SI units of mol/m3) which may be observed at each voxel. By implication, linear combinations not in the set are not observable. METHODS AND MATERIALS: We formulated a model for our material basis with a set of measurements of elemental powders at four tube voltages, 80 kV, 100 kV, 120 kV, and 140 kV, on a medical XCT. The samples included 30 small plastic bottles of powders containing various compounds spanning the atomic numbers up to 20, and a bottle of water and one of air. Using the chemical formulas and measured masses, we formed a matrix giving the number of Hounsfield units per (mole per cubic meter) at each tube voltage for each of 13 chemical elements. We defined a corresponding matrix in units we call molar Hounsfield unit (HU) potency, the difference in HU values that an added mole per cubic meter in a given voxel would add to the measured HU value. We built a matrix of molar potencies for each chemical element and tube voltage and performed a singular value decomposition (SVD) on these to formulate our material basis. We determined that the dimension of this basis is two. We then compared measurements in this material space with theoretical measurements, combining XCOM cross section data with the tungsten anode spectral model using interpolating cubic splines (TASMICS), a one-parameter filter, and a simple detector model, creating a matrix similar to our experimental matrix for the first 20 chemical elements. Finally, we compared the model predictions to Hounsfield unit measurements on three XCT calibration phantoms taken from the literature. RESULTS: We predict the experimental HU potency values derived from our scans of chemical elements with our theoretical model built from XCOM data. The singular values and singular vectors of the model and powder measurements are in substantial agreement. Application of the Bayesian Information Criterion (BIC) shows that exactly two singular values and singular vectors describe the results over four tube voltages. We give a good account of the HU values from the literature, measured for the calibration phantoms at several tube voltages for several commercial instruments, compared with our theoretical model without introducing additional parameters. CONCLUSIONS: We have developed a two-dimensional material basis that specifies the degree to which individual elements in compounds effect the HU values in XCT images of samples with elements up to atomic number Z = 20. We show that two dimensions is sufficient given the contrast and noise in our experiment. The linear combination of concentrations of elements that can be observed using a medical XCT have been characterized, providing a material basis for use in dual-energy reconstruction. This approach provides groundwork for improved reconstruction and for the link of Hounsfield units to the SI.

Complementary coded apertures for 4-dimensional x-ray coherent scatter imaging.

X-ray scattering has played a key role in non-destructive materials characterization due to the material-specific coherent scattering signatures. In the current energy dispersive coherent scatter imaging systems, including selected volume tomography and coherent scatter computed tomography, each object voxel is measured at a single scatter angle, which suffers from slow acquisition time. The employment of coded apertures in x-ray scatter imaging systems improves the photon collection efficiency, making it promising for real time volumetric imaging and material identification. In this paper, we propose a volumetric x-ray scatter imaging system using a pair of complementary coded apertures: a coded aperture on the detector side introduces multiplexed measurement on an energy-sensitive detector array; a complementary source-side coded aperture selectively illuminates the object to decouple the ambiguity due to the increased parallelization for 4D imaging. The system yields the 1D coherent scattering form factor at each voxel in 3D. We demonstrate tomographic imaging and material identification with the system and achieve a spatial resolution ~1 cm and a normalized momentum transfer resolution, Δq/q, of 0.2.

Compressed sampling strategies for tomography.

We investigate new sampling strategies for projection tomography, enabling one to employ fewer measurements than expected from classical sampling theory without significant loss of information. Inspired by compressed sensing, our approach is based on the understanding that many real objects are compressible in some known representation, implying that the number of degrees of freedom defining an object is often much smaller than the number of pixels/voxels. We propose a new approach based on quasi-random detector subsampling, whereas previous approaches only addressed subsampling with respect to source location (view angle). The performance of different sampling strategies is considered using object-independent figures of merit, and also based on reconstructions for specific objects, with synthetic and real data. The proposed approach can be implemented using a structured illumination of the interrogated object or the detector array by placing a coded aperture/mask at the source or detector side, respectively. Advantages of the proposed approach include (i) for structured illumination of the detector array, it leads to fewer detector pixels and allows one to integrate detectors for scattered radiation in the unused space; (ii) for structured illumination of the object, it leads to a reduced radiation dose for patients in medical scans; (iii) in the latter case, the blocking of rays reduces scattered radiation while keeping the same energy in the transmitted rays, resulting in a higher signal-to-noise ratio than that achieved by lowering exposure times or the energy of the source; (iv) compared to view-angle subsampling, it allows one to use fewer measurements for the same image quality, or leads to better image quality for the same number of measurements. The proposed approach can also be combined with view-angle subsampling.

Snapshot 2D tomography via coded aperture x-ray scatter imaging.

This paper describes a fan beam coded aperture x-ray scatter imaging system that acquires a tomographic image from each snapshot. This technique exploits the cylindrical symmetry of the scattering cross section to avoid the scanning motion typically required by projection tomography. We use a coded aperture with a harmonic dependence to determine range and a shift code to determine cross range. Here we use a forward-scatter configuration to image 2D objects and use serial exposures to acquire tomographic video of motion within a plane. Our reconstruction algorithm also estimates the angular dependence of the scattered radiance, a step toward materials imaging and identification.