This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

X-ray radiographic absorption imaging is an invaluable tool in medical diagnostics and materials science. For biological tissue samples, polymers or fibre composites, however, the use of conventional X-ray radiography is limited due to their weak absorption. This is resolved at highly brilliant X-ray synchrotron or micro-focus sources by using phase-sensitive imaging methods to improve the contrast1,2. However, the requirements of the illuminating radiation mean that hard-X-ray phase-sensitive imaging has until now been impractical with more readily available X-ray sources, such as X-ray tubes. In this letter, we report how a setup consisting of three transmission gratings can efficiently yield quantitative differential phase-contrast images with conventional X-ray tubes. In contrast with existing techniques, the method requires no spatial or temporal coherence, is mechanically robust, and can be scaled up to large fields of view. Our method provides all the benefits of contrast-enhanced phase-sensitive imaging, but is also fully compatible with conventional absorption radiography. It is applicable to X-ray medical imaging, industrial non-destructive testing, and to other low-brilliance radiation, such as neutrons or atoms.

http://xrm.phys.northwestern.edu/research/pdf_papers/2006/pfeiffer_naturephys_2006.pdf

Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources

IN conventional radiography, X-rays which pass through an object along different paths are differentially absorbed, and the intensity pattern of the emerging beam records the distribution of absorbing materials within the sample. An alternative approach is phase-contrast radiography, which instead records variations of the phase of the emerging radiation. Such an approach offers improved contrast sensitivity, especially when imaging weakly absorbing samples. Unfortunately, current phase-contrast imaging techniques1–11 generally require highly monochromatic plane-wave radiation and sophisticated X-ray optics, so their use is greatly restricted. Here we describe and demonstrate a simplified scheme for phase-contrast imaging based on an X-ray source having high spatial (but essentially no chromatic) coherence. The method is compatible with conventional polychromatic micro-focus X-ray tube sources, is well suited to large areas of irradiation, can operate with a lower absorbed dose than traditional X-ray imaging techniques, and should find broad application in clinical, biological and industrial settings.

Phase-contrast imaging using polychromatic hard X-rays

We demonstrate simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. Subject to the assumptions explicitly stated in the derivation, the algorithm solves the twin-image problem of in-line holography and is capable of analysing data obtained using X-ray microscopy, electron microscopy, neutron microscopy or visible-light microscopy, especially as they relate to defocus and point projection methods. Our simple, robust, non-iterative and computationally efficient method is applied to data obtained using an X-ray phase contrast ultramicroscope.

Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object

Using a high-efficiency grating interferometer for hard X rays (10-30 keV) and a phase-stepping technique, separate radiographs of the phase and absorption profiles of bulk samples can be obtained from a single set of measurements. Tomographic reconstruction yields quantitative three-dimensional maps of the X-ray refractive index, with a spatial resolution down to a few microns. The method is mechanically robust, requires little spatial coherence and monochromaticity, and can be scaled up to large fields of view, with a detector of correspondingly moderate spatial resolution. These are important prerequisites for use with laboratory X-ray sources.

/pdf/x-ray-phase-imaging-with-a-grating-interferometer-4px897qwex.pdf

X-ray phase imaging with a grating interferometer.

Because the refractive index for hard x rays is slightly different from unity, the optical phase of a beam is affected by transmission through an object. Phase images can be obtained with extreme instrumental simplicity by simple propagation provided the beam is coherent. But, unlike absorption, the phase is not simply related to image brightness. A holographic reconstruction procedure combining images taken at different distances from the specimen was developed. It results in quantitative phase mapping and, through association with three-dimensional reconstruction, in holotomography, the complete three-dimensional mapping of the density in a sample. This tool in the characterization of materials at the micrometer scale is uniquely suited to samples with low absorption contrast and radiation-sensitive systems.

/pdf/holotomography-quantitative-phase-tomography-with-micrometer-56kknqhnf9.pdf

Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays.

Phase objects are readily imaged through Fresnel diffraction in the hard x-ray beams of third-generation synchrotron radiation sources such as the ESRF, due essentially to the very small angular size of the source. Phase objects can lead to spurious contrast in x-ray diffraction images (topographs) of crystals. It is shown that this contrast can be eliminated through random phase plates, which provide an effective way of tailoring the angular size of the source. The possibilities of this very simple technique for imaging phase objects in the hard x-ray range are explored experimentally and discussed. They appear very promising, as shown in particular by the example of a piece of human vertebra, and could be extended to phase tomography.

/pdf/phase-objects-in-synchrotron-radiation-hard-x-ray-imaging-2vvaht6a7i.pdf

Phase objects in synchrotron radiation hard x-ray imaging

A well-known problem in x-ray microcomputed tomography is low sensitivity. Phase contrast imaging offers an increase of sensitivity of up to a factor of 10(3) in the hard x-ray region, which makes it possible to image soft tissue and small density variations. If a sufficiently coherent x-ray beam, such as that obtained from a third generation synchrotron, is used, phase contrast can be obtained by simply moving the detector downstream of the imaged object. This setup is known as in-line or propagation based phase contrast imaging. A quantitative relationship exists between the phase shift induced by the object and the recorded intensity and inversion of this relationship is called phase retrieval. Since the phase shift is proportional to projections through the three-dimensional refractive index distribution in the object, once the phase is retrieved, the refractive index can be reconstructed by using the phase as input to a tomographic reconstruction algorithm. A comparison between four phase retrieval algorithms is presented. The algorithms are based on the transport of intensity equation (TIE), transport of intensity equation for weak absorption, the contrast transfer function (CTF), and a mixed approach between the CTF and TIE, respectively. The compared methods all rely on linearization of the relationship between phase shift and recorded intensity to yield fast phase retrieval algorithms. The phase retrieval algorithms are compared using both simulated and experimental data, acquired at the European Synchrotron Radiation Facility third generation synchrotron light source. The algorithms are evaluated in terms of two different reconstruction error metrics. While being slightly less computationally effective, the mixed approach shows the best performance in terms of the chosen criteria.

Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography

X ray radiography and tomography are important tools in medicine as well as in life science and materials science. Not long ago an approach called in-line holography based on simple propagation became possible using partially coherent synchrotron beams like the ones available at the European Synchrotron Radiation Facility (ESRF). Theoretical and experimental work by Cloetens et al. [Appl. Phys. Lett 75, 2912 (1999)] have shown that quantitative retrieval of the optical phase, from a set of radiographs taken at different sample-to-detector distances, is feasible. Mathematically speaking we are dealing with a direct method based on linearization in order to solve an inverse nonlinear problem. The phase retrieval can be combined with classical tomography in order to obtain a three-dimensional representation of the object’s electron density (holotomography). In order to optimize the image contrast for the numerical phase retrieval process, we have carried out calculations resulting in an optimized choice of v...

Optimization of phase contrast imaging using hard x rays

Fractional Talbot images of optical gratings acting as periodic phase objects have been obtained by use of x rays of 0.069-nm wavelength from a third-generation synchrotron radiation source. Quantitative evaluation of the data obtained as a function of defocusing distance provides information on the lateral coherence of the beam as well as on the phase modulation in the object.

J.P. Guigay

Papers

Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays.

Phase objects in synchrotron radiation hard x-ray imaging

Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography

Optimization of phase contrast imaging using hard x rays

Fractional Talbot imaging of phase gratings with hard x rays.