Diffraction tomography with Fourier ptychography

Abstract: This article presents a method to perform diffraction tomography in a standard microscope that includes an LED array for illumination. After acquiring a sequence of intensity-only images of a thick sample, a ptychography-based reconstruction algorithm solves for its unknown complex index of refraction across three dimensions. The experimental microscope demonstrates a spatial resolution of 0.39 $\mu$m and an axial resolution of 3.7 $\mu$m at the Nyquist-Shannon sampling limit (0.54 $\mu$m and 5.0 $\mu$m at the Sparrow limit, respectively), across a total imaging volume of 2.2 mm $\times$ 2.2 mm $\times$ 110 $\mu$m. Unlike competing methods, the 3D tomograms presented in this article are continuous, quantitative, and formed without the need for interferometry or any moving parts. Wide field-of-view reconstructions of thick biological specimens demonstrate potential applications in pathology and developmental biology.

An Evaluation of Diffraction Tomography Utilizing Fourier Ptychography

The paper "Diffraction tomography with Fourier ptychography" authored by R. Horstmeyer and C. Yang presents an innovative method to perform diffraction tomography through the integration of Fourier ptychography in a standard microscope setup augmented with an LED array for illumination. This method capitalizes on a sequence of intensity-only images and employs a ptychography-based reconstruction algorithm to accurately resolve the complex index of refraction in three dimensions. The authors report achieving a lateral resolution of 0.39 μm and an axial resolution of 3.7 μm, thus enhancing both spatial resolution and imaging capability compared to traditional methods.

Methodological Framework

The approach utilizes a standard microscope integrated with an LED array to perform three-dimensional imaging without relying on holography or mechanical scanning. By modeling the interaction of coherent light with the sample through diffraction tomography (DT), the method allows the capture of multiple angled illumination patterns which are processed to form a synthetic aperture, increasing the resolution beyond its standard diffraction limit. Leveraging the first Born approximation underpins the theoretical foundation of the DT framework in which the scattered field is analyzed to reconstruct the 3D refractive index profile of the sample.

Key Numerical Achievements

This study reported a substantial increase in both lateral and axial resolution, relative to traditional microscopy methods. The method demonstrated a capability to quantitatively map complex refractive indices, reaffirming its utility in imaging transparent biological specimens, such as starfish embryos and parasitic worms, with notable axial structuring often concealed by opacity and limited contrast in traditional imaging modalities.

Implications and Future Trajectories

The implications of this research span both practical and theoretical areas. Practically, the ability to achieve such resolution using standard microscopy equipment, augmented by computational reconstruction, paves the way for more accessible and widespread applications in pathology, developmental biology, and material sciences. Theoretically, it prompts further exploration into overcoming the current limitations posed by the missing cone problem inherent in limited-angle imaging, potentially through the use of advanced interpolation strategies or leveraging constraints on sample properties such as sparsity or positivity.

Future developments could explore the extension of this technique to account for multiple scattering scenarios, thereby increasing its relevance and application to more turbid or complex samples. Advanced computational algorithms like lifting-based phase retrieval or innovative machine learning approaches may offer substantial performance boosts, improving both the resolution and reconstruction fidelity over larger volumetric extents.

In conclusion, this paper presents a method that stands to significantly broaden the capabilities of three-dimensional microscopy using widely available components alongside sophisticated algorithmic methods. As the field advances, the integration of such methods could lead to more robust in-vivo imaging practices and facilitate a deeper understanding of biological and material structural intricacies.