X-ray Phase-Contrast Imaging

Exploration of X-ray Phase-Contrast Imaging

X-ray Phase-Contrast Imaging is a technique that enhances image contrast by detecting phase shifts in X-rays. It reveals fine structures in soft tissues, improving medical diagnostics, and offering insights into materials science and paleontology by visualizing details previously obscured in conventional X-ray imaging.

X-ray Phase-Contrast Imaging

Overview

In the realm of medical imaging, technological advancements continuously push the boundaries of what is possible. One such breakthrough is X-ray Phase-Contrast Imaging, a revolutionary technique that provides unprecedented details and contrast in imaging compared to traditional X-ray methods. This article by Academic Block explore into the intricacies of X-ray Phase-Contrast Imaging, exploring its principles, applications, advantages, challenges, and the potential it holds for the future of medical diagnostics and research.

I. Basics of X-ray Phase-Contrast Imaging

A. Traditional X-ray Imaging: A Foundation

To appreciate the innovations of X-ray Phase-Contrast Imaging, it is essential to understand the basics of traditional X-ray imaging. Traditional X-rays rely on the absorption of X-rays by different tissues in the body, producing an image with varying degrees of contrast. However, this method often struggles to distinguish between tissues with similar absorption properties, leading to reduced sensitivity and specificity.

B. The Fundamental Principle of Phase-Contrast Imaging

X-ray Phase-Contrast Imaging capitalizes on the phase shift of X-rays as they pass through different materials. While traditional X-ray imaging primarily captures the absorption of X-rays, phase-contrast imaging exploits changes in the phase, enabling the visualization of subtle density variations. This inherent ability to capture detailed structural information sets phase-contrast imaging apart from its conventional counterpart.

II. Techniques in X-ray Phase-Contrast Imaging

A. Grating Interferometry: Unraveling Fine Details

Grating interferometry is a fundamental technique in X-ray Phase-Contrast Imaging. It involves the use of gratings to create interference patterns, allowing the separation of absorption, refraction, and scattering information. This technique enhances contrast and resolution, revealing intricate details in biological tissues that would otherwise go unnoticed.

B. Propagation-Based Phase-Contrast Imaging: Harnessing Wave Propagation

Another approach in X-ray Phase-Contrast Imaging is propagation-based imaging. It leverages the changes in the X-ray wavefront as it propagates through an object, providing valuable insights into the object’s internal structures. This technique is particularly advantageous for imaging soft tissues and materials with minimal absorption contrast.

C. Talbot-Lau Interferometry: Precision at the Nanoscale

Talbot-Lau interferometry extends the capabilities of X-ray Phase-Contrast Imaging to the nanoscale. By employing a grating-based interferometer with multiple phase-stepping positions, it achieves high-resolution imaging and enables the detection of subtle density variations in biological tissues and materials.

III. Applications of X-ray Phase-Contrast Imaging

A. Medical Imaging: Revolutionizing Diagnostics

X-ray Phase-Contrast Imaging has transformative potential in the field of medical diagnostics. Its ability to provide detailed images of soft tissues, such as muscles and ligaments, makes it invaluable for detecting early signs of diseases like cancer and osteoarthritis. Moreover, its enhanced contrast resolution facilitates improved visualization of blood vessels and small anatomical structures, offering a new dimension to radiology.

B. Material Science: Unveiling Microstructures

Beyond medicine, X-ray Phase-Contrast Imaging finds applications in material science. Researchers utilize this technique to investigate the internal microstructures of materials, such as composites and alloys, enabling a better understanding of their mechanical properties. This has implications for the development of stronger and more durable materials in various industries.

C. Paleontology and Archaeology: Preserving the Past

In the realm of paleontology and archaeology, X-ray Phase-Contrast Imaging aids in the non-destructive examination of delicate specimens, such as fossils and ancient artifacts. Its ability to reveal fine details without damaging the objects is crucial for preserving the integrity of historical artifacts and advancing our understanding of the past.

IV. Advantages of X-ray Phase-Contrast Imaging

A. Enhanced Soft Tissue Contrast: Seeing the Unseen

One of the standout advantages of X-ray Phase-Contrast Imaging is its ability to provide enhanced contrast for soft tissues. Traditional X-ray methods often struggle to differentiate between tissues with similar absorption properties, but phase-contrast imaging excels in visualizing these subtle differences. This makes it an invaluable tool in detecting early-stage diseases and abnormalities.

B. Lower Radiation Dose: Safer Imaging

Compared to traditional X-ray imaging, X-ray Phase-Contrast Imaging offers the potential for lower radiation doses. This is particularly significant in medical applications, where minimizing radiation exposure is crucial for patient safety. The reduced radiation dose makes it a safer alternative for repeated imaging studies and longitudinal monitoring of patients.

C. Improved Resolution: Unveiling Fine Details

X-ray Phase-Contrast Imaging delivers superior spatial resolution, allowing for the visualization of fine details in biological tissues and materials. This high level of detail is essential for accurate diagnosis and research, enabling scientists and clinicians to uncover subtleties that may be missed by conventional imaging techniques.

V. Mathematical equations behind the X-ray Phase-Contrast Imaging

X-ray Phase-Contrast Imaging involves the exploitation of phase information of X-rays as they interact with matter. The mathematical description of X-ray Phase-Contrast Imaging can be quite complex, involving wave equations and interferometric principles. Here, we will provide a simplified overview of the basic mathematical concepts behind X-ray Phase-Contrast Imaging:

Wave Equation:

The behavior of X-rays in X-ray Phase-Contrast Imaging is often described using the wave equation. The wave equation for X-rays can be written as:

2 / E − [ (1 / v2) (∂2E / ∂t2) ] = 0;

where E is the electric field, v is the velocity of the X-rays, and 2 is the Laplacian operator.

Propagation of X-ray Waves:

The propagation of X-ray waves through an object can be modeled using the Fresnel-Kirchhoff diffraction integral. For a monochromatic plane wave, the electric field E at a point (x,y,z) after propagating through a distance zz can be expressed as:

E(x,y,z) = E0 ∫ ∫ A(x′,y′) (eikr / r) dx′ dy′ ;

where A(x′,y′) is the complex amplitude of the X-rays at the object plane, k is the wave number, and rr is the distance from the source point to the observation point.

Interference Patterns:

X-ray Phase-Contrast Imaging often involves the use of interferometers to create interference patterns. The intensity I at a detector pixel can be related to the wave amplitudes and phase differences:

I ∝∣∑Ai ei ϕi2 ;

where Ai is the amplitude of the i-th wave and ϕi is the phase difference between waves.

Grating Interferometry:

In grating-based X-ray Phase-Contrast Imaging, gratings are used to create interference patterns. The intensity at the detector can be related to the grating properties and the phase shift caused by the object:

I ∝ ∣1 + η sin⁡(2π d / λ)∣2 ;

where η is the contrast of the grating, d is the grating period, and λ is the X-ray wavelength.

Phase Retrieval:

The challenge in X-ray Phase-Contrast Imaging is often to retrieve the phase information from intensity measurements. The transport of intensity equation (TIE) is a common approach used for phase retrieval:

∇ ⋅ (A2 ∇ϕ) = −k A ∇2A ;

where A is the amplitude and ϕ is the phase of the X-ray wave.

It’s important to note that these equations provide a simplified overview of the mathematical principles underlying X-ray Phase-Contrast Imaging. The actual implementation and analysis involve more sophisticated mathematical techniques, including numerical methods for solving these equations and extracting meaningful information from the acquired data.

VI. Challenges and Future Directions

A. Technical Challenges: Overcoming Hurdles

While X-ray Phase-Contrast Imaging holds immense promise, it is not without its challenges. Technical hurdles, such as the need for specialized equipment and the complexity of image reconstruction algorithms, pose obstacles to widespread adoption. Addressing these challenges requires collaborative efforts from researchers, engineers, and clinicians to refine and simplify the technology.

B. Integration into Clinical Practice: Bridging the Gap

The integration of X-ray Phase-Contrast Imaging into routine clinical practice poses challenges related to cost, accessibility, and training. As with any emerging technology, overcoming these barriers requires concerted efforts from healthcare institutions, industry stakeholders, and regulatory bodies. Collaboration is essential to streamline the adoption process and make the benefits of this advanced imaging technique accessible to a broader population.

C. Future Directions: Pushing Boundaries

Looking ahead, the future of X-ray Phase-Contrast Imaging holds exciting possibilities. Continued research and development aim to further enhance image quality, reduce equipment costs, and streamline the integration of this technology into various fields. Emerging innovations, such as phase-contrast tomography and dynamic imaging, promise to unlock new dimensions in our ability to visualize biological tissues and materials.

Final Words

In conclusion, X-ray Phase-Contrast Imaging stands as a transformative force in the world of medical imaging and beyond. Its ability to provide enhanced soft tissue contrast, lower radiation doses, and superior resolution has the potential to revolutionize diagnostics, material science, and archaeological research. In this article by Academic Block we have seen that, while challenges remain, ongoing research and collaborative efforts are paving the way for the widespread adoption of this groundbreaking technology. As we continue to push the boundaries of what is possible in imaging, X-ray Phase-Contrast Imaging stands at the forefront, unraveling the mysteries of the unseen world with unprecedented clarity and precision. Please provide your comments below, it will help us in improving this article. Thanks for reading!

This Article will answer your questions like:

+ What is X-ray Phase-Contrast Imaging (XPCI)? >

X-ray Phase-Contrast Imaging (XPCI) is an advanced technique that enhances contrast in X-ray images by detecting phase shifts in X-rays passing through an object. Unlike conventional X-ray imaging, which relies on absorption differences, XPCI utilizes phase changes to reveal subtle variations in tissue density and composition. This enables high-resolution imaging of soft tissues and materials that are challenging to visualize with traditional X-ray methods.

+ How does X-ray Phase-Contrast Imaging differ from conventional X-ray imaging? >

X-ray Phase-Contrast Imaging (XPCI) differs from conventional X-ray imaging by focusing on the phase changes of X-rays passing through an object rather than just the absorption of X-rays. This allows XPCI to detect subtle differences in tissue density and composition, producing higher contrast images of soft tissues and materials that are poorly visualized with absorption-based X-ray techniques.

+ What is the principle behind X-ray Phase-Contrast Imaging? >

The principle behind X-ray Phase-Contrast Imaging (XPCI) is based on the phase shift that occurs when X-rays pass through different materials. Unlike conventional X-ray imaging that primarily detects differences in X-ray absorption, XPCI exploits the phase changes caused by variations in refractive index and thickness of materials. This allows XPCI to enhance contrast and visualize fine details in soft tissues and low-density materials that are otherwise challenging to distinguish with absorption-based X-ray methods.

+ What are the primary applications of XPCI in medical diagnostics? >

X-ray Phase-Contrast Imaging (XPCI) is used in medical diagnostics for imaging soft tissues such as muscles, ligaments, and cartilage with higher clarity and contrast compared to conventional X-ray methods. It aids in detecting subtle structural changes in organs, improving early diagnosis of diseases like breast cancer, lung diseases, and cardiovascular conditions. XPCI also enhances the visualization of joints and bones, offering detailed images useful in orthopedic assessments and surgical planning.

+ How does XPCI improve image contrast and resolution? >

X-ray Phase-Contrast Imaging (XPCI) improves image contrast and resolution by detecting phase shifts in X-rays passing through tissues or materials. These phase shifts are sensitive to small changes in tissue density and composition, providing enhanced contrast for soft tissues and low-density materials that are poorly visualized with absorption-based X-ray techniques. By capturing both amplitude and phase information of X-rays, XPCI produces images with finer details and improved clarity, crucial for accurate medical diagnostics and research.

+ What are the key components of an XPCI system? >

The key components of an X-ray Phase-Contrast Imaging (XPCI) system include an X-ray source producing a collimated beam, an X-ray detector capable of capturing phase-shifted X-rays, and specialized optics or phase gratings to convert phase information into intensity variations. A high-precision sample stage is essential for positioning and scanning the object. Advanced data acquisition systems and computational algorithms process the phase-shifted signals to reconstruct high-resolution images. Additionally, mechanical and environmental controls ensure stability and minimize artifacts, crucial for accurate XPCI imaging.

+ What types of tissues or materials benefit most from XPCI? >

X-ray Phase-Contrast Imaging (XPCI) benefits soft tissues such as muscles, ligaments, and cartilage, which have low contrast in conventional X-ray imaging. It also enhances the visualization of low-density materials like lung tissues and cellular structures. XPCI is particularly useful in imaging complex biological samples, biomedical materials, and cultural artifacts where detailed structural information is critical for analysis and diagnosis.

+ What are the advantages of XPCI over traditional X-ray imaging techniques? >

X-ray Phase-Contrast Imaging (XPCI) offers several advantages over traditional X-ray imaging techniques. It provides higher contrast images of soft tissues and low-density materials, improving diagnostic accuracy for conditions like breast cancer, lung diseases, and cardiovascular disorders. XPCI also enhances the visualization of fine anatomical structures and offers detailed insights into cellular morphology and tissue microstructure. Furthermore, XPCI reduces radiation dose requirements, making it safer for patients and suitable for repeated imaging studies.

+ How is the phase shift of X-rays measured in XPCI? >

The phase shift of X-rays in X-ray Phase-Contrast Imaging (XPCI) is measured using specialized phase-sensitive detectors or phase gratings. These detectors are designed to detect slight variations in the phase of X-rays passing through the sample. Phase gratings convert phase shifts into intensity variations, which are captured by the detector and processed to generate high-resolution images with enhanced contrast and structural detail.

+ What are the limitations and challenges of XPCI? >

Limitations and challenges of X-ray Phase-Contrast Imaging (XPCI) include the complexity and cost of implementing phase-sensitive detectors and specialized optics. XPCI systems require precise alignment and calibration to achieve optimal image quality, which can be technically demanding. Furthermore, XPCI may face challenges in imaging dense materials where phase shifts are minimal, limiting its applicability in certain diagnostic scenarios. Environmental factors such as vibrations and temperature variations can also affect image quality, necessitating controlled experimental conditions.

+ How has XPCI contributed to advancements in medical imaging and research? >

X-ray Phase-Contrast Imaging (XPCI) has significantly advanced medical imaging by providing high-resolution images of soft tissues and low-density materials that are challenging to visualize with conventional X-ray methods. It has improved diagnostic accuracy in fields such as oncology, cardiology, and orthopedics, enabling early detection of diseases and precise assessment of anatomical structures. In research, XPCI facilitates detailed studies of cellular morphology, tissue microstructure, and biomedical materials, contributing to the development of new diagnostic techniques and therapeutic approaches.

+ What are the safety considerations when using XPCI? >

Safety considerations when using X-ray Phase-Contrast Imaging (XPCI) include managing radiation exposure to patients and operators. Although XPCI typically reduces radiation dose compared to conventional X-ray imaging, proper shielding and monitoring protocols are essential. Operators must adhere to radiation safety guidelines and ensure equipment is properly maintained to minimize radiation risks. Patient-specific factors such as pregnancy and radiation sensitivity should be carefully assessed before conducting XPCI scans. Additionally, environmental and ergonomic factors should be optimized to ensure safe working conditions during imaging procedures.

Hardware and software required for X-ray Phase-Contrast Imaging

Hardware Requirements:

  1. X-ray Source:A suitable X-ray source is essential, often operating in the range of kilovolt age (kV) to provide the necessary energy for imaging. This can be a conventional X-ray tube or a synchrotron radiation source for more advanced applications.
  2. Collimation System: Collimators are used to control and shape the X-ray beam, ensuring it is focused and directed accurately towards the imaging target.
  3. X-ray Detector: High-resolution X-ray detectors are crucial for capturing the transmitted X-rays and forming the images. Common types include flat-panel detectors, charge-coupled devices (CCDs), or complementary metal-oxide-semiconductor (CMOS) detectors.
  4. Grating Interferometer: For techniques like grating-based X-ray Phase-Contrast Imaging, a grating interferometer is required. This consists of phase gratings that create interference patterns, enhancing the phase information of X-rays.
  5. Sample Stage: A precision sample stage is needed to position and stabilize the object being imaged. This is crucial for achieving high-quality images and minimizing motion artifacts.
  6. Mechanical Stability Equipment: Vibration isolation systems and mechanical stability equipment are essential to reduce external vibrations and ensure the stability of the imaging setup.
  7. Monochromator (Optional): In some cases, a monochromator may be used to narrow the X-ray beam spectrum, providing monochromatic X-rays for specific applications.
  8. Beamline Optics (Synchrotron): In synchrotron-based setups, beamline optics, such as mirrors and lenses, are used to manipulate and focus the synchrotron radiation onto the sample.

Software Requirements:

  1. Data Acquisition Software: Software for controlling the X-ray source, detectors, and other hardware components during the data acquisition process.
  2. Image Reconstruction Software: Algorithms for image reconstruction are essential for converting raw data into meaningful images. Common methods include filtered back projection (FBP) and iterative reconstruction techniques.
  3. Phase Retrieval Algorithms: Specialized algorithms are required for retrieving the phase information from the acquired intensity data in X-ray Phase-Contrast Imaging.
  4. Image Processing Software: Post-processing software for enhancing, filtering, and analyzing the reconstructed images. This may include tools for noise reduction, contrast adjustment, and three-dimensional rendering.
  5. Quantitative Analysis Tools: Software tools for quantitative analysis of the phase-contrast images, allowing researchers to extract relevant information about the imaged structures.
  6. Visualization Software: Software for visualizing and interpreting the reconstructed images. This may include tools for interactive 3D visualization, cross-sectional analysis, and virtual slicing.
  7. Data Storage and Management: Efficient data storage solutions and management systems to handle the large datasets generated during X-ray Phase-Contrast Imaging experiments.
  8. Calibration Software: Calibration tools for ensuring accuracy and consistency in the imaging setup. This includes calibration of the X-ray source, detectors, and other components.
  9. Integration with Experimental Control Systems: Integration with experimental control systems to coordinate the various hardware components, ensuring synchronized operation during the imaging process.
  10. User Interface: A user-friendly interface for researchers and technicians to interact with the system, control parameters, and monitor the imaging process.

Key Discoveries made using X-ray Phase-Contrast Imaging

X-ray Phase-Contrast Imaging has found applications in various scientific fields, leading to key discoveries and advancements. Below are some notable discoveries and applications where X-ray Phase-Contrast Imaging has played a significant role:

  1. Medical Imaging Breakthroughs:
    • Soft Tissue Imaging: X-ray Phase-Contrast Imaging has provided unprecedented clarity in soft tissue imaging, allowing for improved visualization of structures like muscles, tendons, and blood vessels. This has been crucial for early detection and diagnosis in areas such as oncology and cardiovascular medicine.
    • Breast Cancer Imaging: Researchers have utilized X-ray Phase-Contrast Imaging to enhance breast cancer detection by improving the visualization of breast tissue and lesions. This has the potential to improve the accuracy of mammography and reduce the need for invasive procedures.
    • Articular Cartilage Imaging: In orthopedics, X-ray Phase-Contrast Imaging has been instrumental in imaging articular cartilage with high resolution. This is vital for studying joint diseases such as osteoarthritis and understanding the structural changes in cartilage.
  2. Materials Science Advancements:
    • Composite Materials Analysis: X-ray Phase-Contrast Imaging has been applied to study composite materials, providing insights into their internal structures and allowing for the identification of defects and irregularities. This is valuable in industries like aerospace and manufacturing.
    • Microstructure Characterization: Researchers use X-ray Phase-Contrast Imaging to explore the microstructure of materials at a level not achievable with traditional X-ray methods. This has implications for understanding material properties and improving the development of new materials.
  3. Paleontological and Archaeological Insights:
    • Fossil Examination: X-ray Phase-Contrast Imaging has been employed in the study of fossils without the need for invasive procedures. This non-destructive imaging technique enables paleontologists to examine delicate fossilized specimens in detail, shedding light on the anatomy and life history of extinct species.
    • Artifact Preservation: Archaeologists use X-ray Phase-Contrast Imaging to investigate ancient artifacts without causing damage. This has been crucial for preserving the integrity of historical objects and unraveling the mysteries of ancient civilizations.
  4. Biomedical Research:
    • Neuroimaging: X-ray Phase-Contrast Imaging has been applied to study the microstructure of the brain, providing insights into neural tissues and potentially aiding in the understanding of neurological disorders.
    • Vascular Imaging: Researchers have utilized X-ray Phase-Contrast Imaging to enhance vascular imaging, allowing for detailed visualization of blood vessels and improving our understanding of circulatory diseases.
  5. Advancements in Laboratory Techniques:
    • Non-Destructive Testing: Industries involved in quality control and non-destructive testing have benefited from X-ray Phase-Contrast Imaging. It enables the inspection of materials and components without causing damage, ensuring the reliability and safety of various products.
    • Pharmaceutical Research: X-ray Phase-Contrast Imaging has been employed in pharmaceutical research to study the internal structures of drug formulations, helping researchers understand drug delivery mechanisms and optimize pharmaceutical products.

Facts on X-ray Phase-Contrast Imaging

Principle of X-ray Phase-Contrast Imaging: X-ray Phase-Contrast Imaging exploits the phase shift of X-rays as they pass through different materials, providing detailed information about the refractive index variations within an object. This is in contrast to traditional X-ray imaging, which primarily relies on the absorption of X-rays.

Enhanced Soft Tissue Contrast: One of the key advantages of X-ray Phase-Contrast Imaging is its ability to offer superior contrast in soft tissues. This makes it particularly valuable for imaging anatomical structures with similar X-ray absorption properties, such as muscles, tendons, and ligaments.

Applications in Medical Imaging: X-ray Phase-Contrast Imaging has significant applications in medical imaging, offering improved visualization of soft tissues. It is particularly useful in detecting early signs of diseases, enhancing mammography, and providing detailed images of joints and blood vessels.

Different Techniques: Various techniques are employed in X-ray Phase-Contrast Imaging, including grating interferometry, propagation-based imaging, and Talbot-Lau interferometry. These techniques utilize the interference patterns generated by X-rays to extract phase information and enhance contrast.

Non-Destructive Testing in Industry: X-ray Phase-Contrast Imaging is utilized in non-destructive testing in industrial settings. It allows for the inspection of materials, components, and products without causing damage, ensuring the quality and integrity of manufactured goods.

Advantages in Materials Science: In materials science, X-ray Phase-Contrast Imaging enables the study of internal structures of materials with high resolution. This is beneficial for understanding the mechanical properties of materials, detecting defects, and improving the development of advanced materials.

Low Radiation Dose: Compared to traditional X-ray imaging, X-ray Phase-Contrast Imaging has the potential to reduce radiation exposure. This makes it a safer option for repeated imaging studies and long-term monitoring of patients, contributing to improved safety in medical applications.

Contributions to Archaeology and Paleontology: X-ray Phase-Contrast Imaging has been applied in archaeology and paleontology for the non-destructive examination of artifacts and fossils. It allows researchers to explore the internal structures of historical objects and delicate specimens without causing damage.

Ongoing Technological Developments: Researchers and engineers continue to refine X-ray Phase-Contrast Imaging technologies, addressing challenges such as cost, accessibility, and complexity. Ongoing developments aim to make this advanced imaging technique more practical for routine clinical use and various research applications.

Potential for Future Innovations: X-ray Phase-Contrast Imaging holds immense potential for future innovations in medical diagnostics, research, and industrial applications. As technology advances, it is expected to play a crucial role in unlocking new dimensions of understanding in fields ranging from medicine to materials science.

Academic References on X-ray Phase-Contrast Imaging

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  2. Momose, A., Kawamoto, S., Koyama, I., Hamaishi, Y., Takai, K., & Suzuki, Y. (2003). Demonstration of X-ray Talbot interferometry. Japanese Journal of Applied Physics, 42(7R), L866.
  3. Pfeiffer, F., Weitkamp, T., Bunk, O., & David, C. (2006). Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources. Nature Physics, 2(4), 258-261.
  4. Momose, A. (2012). Recent advances in X-ray phase imaging. Japanese Journal of Applied Physics, 51(5S1), 05FA01.
  5. Bravin, A., Coan, P., Suortti, P., & Prevrhal, S. (2013). Parametric X-ray phase tomography of breast tissue. Physics in Medicine & Biology, 58(4), 1177.
  6. Lynch, S. K., Rivera, A. A., Woicik, J. C., & Rose, V. (2013). Design and performance of a two‐dimensional single‐photon counting X‐ray detector. Journal of Applied Crystallography, 46(6), 1721-1727.
  7. Stampanoni, M., Wang, Z., Thuring, T., David, C., Roessl, E., Trippel, M., … & Rack, A. (2011). The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography. Investigative Radiology, 46(12), 801-806.
  8. Olivo, A., & Speller, R. (2007). A coded-aperture technique allowing X-ray phase contrast imaging with conventional sources. Applied Physics Letters, 91(7), 074106.
  9. Weitkamp, T., Diaz, A., David, C., Pfeiffer, F., Stampanoni, M., Cloetens, P., & Ziegler, E. (2005). X-ray phase imaging with a grating interferometer. Optics Express, 13(16), 6296-6304.
  10. Chabior, M., & Pfeiffer, F. (2015). Three-dimensional imaging with X-ray phase contrast. Journal of Applied Crystallography, 48(5), 1126-1133.
  11. Kitchen, M. J., Paganin, D. M., Lewis, R. A., Yagi, N., Uesugi, K., & Mudie, S. T. (2005). On the X-ray fringe visibility in grating-based phase-contrast imaging. Journal of Applied Physics, 97(12), 124905.
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