We introduce an analytical model for Fourier ptychography (FP). The model represents recorded intensity images as linear equations. Here, the sample’s self-correlating components are the unknown variables, while the optical system’s point-spread function’s corresponding components serve as coefficients. This framework facilitates the direct computation of high-resolution complex images from the sample. A standout feature of our model is its ability to verify the uniqueness of the reconstructed image, a critical attribute for numerous quantitative applications. Our linear model offers clear insights into the impact of various experimental factors on accurate reconstruction, such as scanning step, random scanning mode, spatial resolution, and noise. While these factors have been previously acknowledged, their precise roles have remained nebulous. We have also developed an efficient computational method tailored for our model, adept at managing large matrices, thus enhancing the translation of low to high-resolution images. This research elucidates the foundational mathematics behind FP’s efficacy, underscoring its potential for optical measurements and metrology where mathematical uniqueness is paramount.
A variable aperture-based ptychographical iterative engine (vaPIE) is demonstrated both numerically and experimentally to reconstruct the sample phase and amplitude rapidly. By adjusting the size of a tiny aperture under the illumination of a parallel light beam to change the illumination on the sample step by step and recording the corresponding diffraction patterns sequentially, both the sample phase and amplitude can be faithfully reconstructed with a modified ptychographical iterative engine (PIE) algorithm. Since many fewer diffraction patterns are required than in common PIE and the shape, the size, and the position of the aperture need not to be known exactly, this proposed vaPIE method remarkably reduces the data acquisition time and makes PIE less dependent on the mechanical accuracy of the translation stage; therefore, the proposed technique can be potentially applied for various scientific researches.
As a lensfree imaging technique, ptychographic iterative engine (PIE) method can provide both quantitative sample amplitude and phase distributions avoiding aberration. However, it requires field of view (FoV) scanning often relying on mechanical translation, which not only slows down measuring speed, but also introduces mechanical errors decreasing both resolution and accuracy in retrieved information. In order to achieve high-accurate quantitative imaging with fast speed, digital micromirror device (DMD) is adopted in PIE for large FoV scanning controlled by on/off state coding by DMD. Measurements were implemented using biological samples as well as USAF resolution target, proving high resolution in quantitative imaging using the proposed system. Considering its fast and accurate imaging capability, it is believed the DMD based PIE technique provides a potential solution for medical observation and measurements.
In order to obtain quantitative phase distributions from interferograms, phase retrieval composed of phase extracting and unwrapping is adopted in quantitative interferometric microscopy. However, phase unwrapping often requires a long time, limiting applications such as high-speed phase observations and measurements. In order to accelerate the processing speed, a phase unwrapping free Hilbert transform (HT)-based phase retrieval method is proposed. Though another background interferogram without a sample is needed, phase unwrapping can be omitted, saving a large amount of time for phase recovery. Additionally, the proposed HT-based method can maintain more sample details, thus providing high-accurate quantitative phase imaging. Considering its fast speed and high accuracy in phase retrieval, it is believed that the unwrapping free HT-based phase retrieval method can be potentially applied in high throughput cellular observations and measurements.
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