Computational microscopy
Computational microscopy combines tailored illumination, coherent scattering, and algorithmic reconstruction to generate quantitative 2D and 3D images spanning length scales from ångströms to centimeters. The field unifies the principles of microscopy and crystallography by replacing or augmenting optical components with phase-retrieval and computational algorithms. Major approaches include coherent diffractive imaging (CDI), ptychography (X-ray and electron), and Fourier ptychography (optical). Together they achieve record spatial resolution, wide fields of view, and quantitative phase contrast across applications ranging from materials and quantum systems to biological imaging and device metrology.[1]
Definition and scope
[edit]Computational microscopy refers to imaging modalities in which raw measurements (often diffraction patterns or image stacks under diverse illuminations) are transformed into sample transmission functions—amplitude and phase—via iterative or learned reconstruction. Compared with conventional microscopy, which uses wavelength specific lenses, computational methods can (i) remove lens aberrations algorithmically, (ii) increase the space–bandwidth product (SBP) by orders of magnitude, and (iii) provide quantitative maps of strain, electron density, refractive index, or magnetization.[1]
History
[edit]Early foundations of computational microscopy trace back to the solution of the phase problem in crystallography and optics, and to the development of Fourier-based iterative algorithms for phase retrieval. In 1972, Gerchberg and Saxton introduced the first practical iterative algorithm to recover phase information from diffraction and image-plane data.[2] Between 1978 and 1982, Fienup refined this approach by developing the error-reduction and hybrid input–output algorithms, which have been widely used in iterative phase retrieval.[3][4]
In 1998, Miao, Sayre, and Chapman introduced the concept of the oversampling ratio for phase retrieval,[5] which was later generalized to the overdetermination ratio (αₒ = M/N), where M and N denote the number of independently measured data points and unknown object variables, respectively.[1] When αₒ is substantially greater than one, the phase information is, in principle, uniquely encoded within the measured diffraction intensities and can be deterministically recovered through iterative reconstruction algorithms.
In 1999, Miao and colleagues experimentally extended crystallographic methodology to non-crystalline specimens, inaugurating CDI as a lensless imaging technique.[6]
In 2007, Rodenburg and co-workers demonstrated modern ptychography with hard X-rays by scanning a coherent probe across an extended specimen and iteratively reconstructing the object transmission function while assuming a known probe.[7] In 2008, Thibault et al. demonstrated simultaneous reconstruction of both the probe and object transmission functions from overlapping diffraction patterns, establishing ptychography as a quantitative and general high-resolution imaging method.[8]
In 2013, Zheng, Horstmeyer, and Yang extended these principles to optical microscopy with Fourier ptychography, enabling gigapixel-scale quantitative phase imaging on table-top microscopes.[9]
Subsequent advances in coherent sources, detectors, and algorithms have established computational microscopy as a unified framework spanning optical, X-ray, and electron modalities.[1]
Principles
[edit]When a coherent beam of photons or electrons interacts with a specimen, detectors record only the diffraction intensities, while the corresponding phase information is lost—a challenge known as the phase problem. Computational microscopy reconstructs the complex transmission function of the specimen by enforcing consistency between measured data and physical constraints through iterative optimization.[1] These algorithms typically alternate between real and reciprocal space, employing schemes such as alternating projections,[2][3][4] the extended ptychographic iterative engine (ePIE),[10] difference maps,[11] and maximum likelihood,[12][13] and others.[14][15][16] Recently, deep-learning–based approaches have been developed to infer the missing phase directly from measured diffraction patterns, enabling faster and more robust reconstructions.[17][18][19][20][1]
Methods
[edit]Coherent diffractive imaging (CDI)
[edit]References
[edit]- ^ a b c d e f Miao, Jianwei (2025-01-09). "Computational microscopy with coherent diffractive imaging and ptychography". Nature. 637: 281–295. doi:10.1038/s41586-024-08278-z.
- ^ a b Gerchberg, R. W.; Saxton, W. O. (1972). "A practical algorithm for the determination of phase from image and diffraction plane pictures". Optik. 35: 237–246.
- ^ a b Fienup, J. R. (1978). "Reconstruction of an object from the modulus of its Fourier transform". Optics Letters. 3: 27–29.
- ^ a b Fienup, J. R. (1982). "Phase retrieval algorithms: a comparison". Applied Optics. 21: 2758–2769. doi:10.1364/AO.21.002758.
- ^ Miao, J.; Sayre, D.; Chapman, H. N. (1998). "Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects". JOSA A. 15: 1662–1669. doi:10.1364/JOSAA.15.001662.
- ^ Miao, J.; Charalambous, P.; Kirz, J.; Sayre, D. (1999). "Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens". Nature. 400: 342–344. doi:10.1038/22498.
- ^ Rodenburg, J. M. (2007). "Hard X-ray lensless imaging of extended objects". Physical Review Letters. 98 (3): 034801. doi:10.1103/PhysRevLett.98.034801.
{{cite journal}}: CS1 maint: article number as page number (link) - ^ Thibault, P. (2008). "High-resolution scanning X-ray diffraction microscopy". Science. 321: 379–382. doi:10.1126/science.1158573.
- ^ Zheng, G.; Horstmeyer, R.; Yang, C. (2013). "Wide-field, high-resolution Fourier ptychographic microscopy". Nature Photonics. 7: 739–745. doi:10.1038/nphoton.2013.187.
- ^ Maiden, A. M.; Rodenburg, J. M. (2009). "An improved ptychographical phase retrieval algorithm for diffractive imaging". Ultramicroscopy. 109: 1256–1262.
- ^ Elser, V. (2003). "Phase retrieval by iterated projections". Journal of the Optical Society of America A. 20: 40–55.
- ^ Thibault, P.; Guizar-Sicairos, M. (2012). "Maximum-likelihood refinement for coherent diffractive imaging". New Journal of Physics. 14: 063004.
- ^ Odstrčil, M.; Menzel, A.; Guizar-Sicairos, M. (2018). "Iterative least-squares solver for generalized maximum-likelihood ptychography". Optics Express. 26: 3108–3123.
- ^ Shechtman, Y.; et al. (2015). "Phase retrieval with application to optical imaging: a contemporary overview". IEEE Signal Processing Magazine. 32: 87–109.
{{cite journal}}: Explicit use of et al. in:|last2=(help) - ^ Rodenburg, J.; Maiden, A. (2019). Hawkes, P. W.; Spence, J. C. H. (eds.). Springer Handbook of Microscopy. Springer. pp. 819–904.
- ^ Fannjiang, A.; Strohmer, T. (2020). "The numerics of phase retrieval". Acta Numerica. 29: 125–228. doi:10.1017/S0962492920000017.
- ^ Rivenson, Y.; Zhang, Y.; Günaydın, H.; Teng, D.; Ozcan, A. (2018). "Phase recovery and holographic image reconstruction using deep learning in neural networks". Light: Science & Applications. 7: 17141.
- ^ Sinha, A.; Lee, J.; Li, S.; Barbastathis, G. (2017). "Lensless computational imaging through deep learning". Optica. 4: 1117–1125.
- ^ Wang, K.; et al. (2024). "On the use of deep learning for phase recovery". Light: Science & Applications. 13: 4.
{{cite journal}}: Explicit use of et al. in:|last2=(help) - ^ Cherukara, M. J.; Nashed, Y. S. G.; Harder, R. J. (2018). "Real-time coherent diffraction inversion using deep generative networks". Scientific Reports. 8: 16520.