MicroscopyEducation
Fourier spectrum shown in Figure 4, one would first com- pute the Fourier transform of each image collected under illumination from each unique LED angle. For nine uniquely illuminated images at appropriate angles, this would form the nine Fourier windows shown in Figure 4. Next, using knowledge of each LED’s angular position, one would cor- rectly order and tile together these unique Fourier spectra into a larger composite. After tiling together all Fourier transform segments, the entire enlarged spectrum can then be inverse-Fourier-transformed into a final wide-area, high- resolution result. This approach is the foundation of a gen- eral technique termed synthetic aperture imaging, which is widely used in radar, ultrasound, and other imaging modali- ties in which it is possible to jointly measure the amplitude and phase of radiation. At optical frequencies, however, it is not possible to directly
measure the phase of an incident optical field, as digital image sensors only measure a signal that is proportional to the square amplitude of light. Te lack of measured phase prevents correct execution of the above simple computations to obtain a final high-resolution image result. Accordingly, Fourier ptychogra- phy must overcome what is generally referred to as the “missing phase” problem [5].
A Preview for Part 2: Phase Retrieval Algorithms Te missing phase problem from image measurements
using conventional detectors is well-known, and a variety of methods over the history of microscopy have been devel- oped to determine phase via amplitude-only measurements. Phase contrast is, of course, extremely important in micros- copy, as many specimens of interest are essentially transpar- ent. Many physical methods for producing phase contrast within a microscope are readily available. With the advent of digital microscopy, “quantitative” phase imaging methods have become increasingly popular, as opposed to qualitative methods that convert phase contrast into intensity contrast for direct viewing. Tanks to the work of many scientists and mathematicians, there is now a suite of “phase retrieval” algo- rithms that are available to convert standard digital images, such as those acquired by Fourier ptychography [5], into amplitude and phase reconstructions, given that a number of key conditions are met. In Part 2 of this series, we will detail how phase retrieval algorithms are leveraged by Fourier pty- chography to produce gigapixel-sized large-area, high-reso- lution images and quantitative phase maps using relatively simple microscope hardware.
References [1] G Zheng et al., Nat Photon 7 (2013)
https://doi.org/10 .1038/nphoton.2013.187.
[2] G Zheng et al., Opt Photon News 25 (2014) https://doi .org/10.1364/OPN.25.4.000026.
[3] G Zheng et al., Nat Rev Phys 3 (2021) https://doi .org/ 10.1038/s42254-021-00280-y.
[4] PC Konda et al., Opt Express 28 (2020) https://doi .org/ 10.1364/OE.386168.
[5] X Ou et al., Opt Lett 38 (2013)
https://doi.org/10.1364/ OL.38.004845.
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