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Imaging in complex media

Abstract

Imaging can take many forms—from optical microscopes and telescopes through ultrasonography to X-ray tomography. However, regardless of the imaging modality, the presence of a complex heterogeneous structure between the imaging system and the scene of interest limits the quality of the images that can be conventionally obtained. In this Review we outline recently introduced strategies to overcome the detrimental effects of scattering in optical imaging. In particular, we focus on approaches that either physically correct scattering using computer-controlled devices or employ computational inversion based on intrinsic correlations of light scattering. Despite focusing on optical techniques, this Review emphasizes the fundamental equivalence of the effects of scattering in different fields of imaging, using the scattering matrix formalism as a bridge that allows techniques developed in one field to be translated to another.

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Fig. 1: Imaging through fog as one example of optical imaging through complex media.
Fig. 2: Scattering-matrix formalism describing linear field propagation through complex media.
Fig. 3: Different approaches for imaging via wavefront shaping.
Fig. 4: Examples of diffraction-limited images obtained non-invasively through complex media with various approaches.

References

  1. Sheng, P. Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2010).

  2. Akkermans, E. & Montambaux, G. Mesoscopic Physics of Electrons and Photons (Cambridge Univ. Press, 2007).

  3. Carminati, R. & Schotland, J. C. Principles of Scattering and Transport of Light (Cambridge Univ. Press, 2021).

  4. Richardson, D. S. & Lichtman, J. W. Clarifying tissue clearing. Cell 162, 246–257 (2015).

    Article  Google Scholar 

  5. Chen, H., Rogalski, M. M. & Anker, J. N. Advances in functional X-ray imaging techniques and contrast agents. Phys. Chem. Chem. Phys. 14, 13469–13486 (2012).

    Article  Google Scholar 

  6. Szabo, T. L. Diagnostic Ultrasound Imaging: Inside Out (Elsevier, 2014).

  7. McCarthy, A. et al. Long-range time-of-flight scanning sensor based on high-speed time-correlated single-photon counting. Appl. Opt. 48, 6241–6251 (2009).

    Article  ADS  Google Scholar 

  8. Huang, D. et al. Optical coherence tomography. Science 254, 1178–1181 (1991).

    Article  ADS  Google Scholar 

  9. Pawley, J. (ed.) Handbook Of Biological Confocal Microscopy (Springer, 2006).

  10. Zipfel, W. R., Williams, R. M. & Webb, W. W. Nonlinear magic: multiphoton microscopy in the biosciences. Nat. Biotechnol. 21, 1369–1377 (2003).

    Article  Google Scholar 

  11. Theer, P. & Denk, W. On the fundamental imaging-depth limit in two-photon microscopy. J. Opt. Soc. Am. A 23, 3139–3149 (2006).

    Article  ADS  Google Scholar 

  12. Horton, N. G. et al. In vivo three-photon microscopy of subcortical structures within an intact mouse brain. Nat. Photon. 7, 205–209 (2013).

    Article  ADS  Google Scholar 

  13. Badon, A. et al. Smart optical coherence tomography for ultra-deep imaging through highly scattering media. Sci. Adv. 2, e1600370 (2016).

    Article  ADS  Google Scholar 

  14. Yoon, S. et al. Deep optical imaging within complex scattering media. Nat. Rev. Phys. 2, 141–158 (2020).

    Article  Google Scholar 

  15. Cao, H., Mosk, A. P. & Rotter, S. Shaping the propagation of light in complex media. Nat. Phys. https://doi.org/10.1038/s41567-022-01677-x (2022).

  16. Choi, W., Mosk, A. P., Park, Q.-H. & Choi, W. Transmission eigenchannels in a disordered medium. Phys. Rev. B 83, 134207 (2011).

    Article  ADS  Google Scholar 

  17. Yilmaz, H., Hsu, C. W., Yamilov, A. & Cao, H. Transverse localization of transmission eigenchannels. Nat. Photon. 13, 352–358 (2019).

    Article  ADS  Google Scholar 

  18. Carpenter, J., Eggleton, B. J. & Schröder, J. Observation of Eisenbud-Wigner-Smith states as principal modes in multimode fibre. Nat. Photon. 9, 751–757 (2015).

    Article  ADS  Google Scholar 

  19. Goodman, J. W. Introduction to Fourier Optics (Roberts & Company, 2005).

  20. Goodman, J. W. Speckle Phenomena in Optics: Theory and Applications (SPIE, 2020).

  21. Feng, S., Kane, C., Lee, P. A. & Stone, A. D. Correlations and fluctuations of coherent wave transmission through disordered media. Phys. Rev. Lett. 61, 834–837 (1988).

    Article  ADS  Google Scholar 

  22. Freund, I. Looking through walls and around corners. Phys. A 168, 49–65 (1990).

    Article  Google Scholar 

  23. Davies, R. & Kasper, M. Adaptive optics for astronomy. Annu. Rev. Astron. Astrophys. 50, 305–351 (2012).

    Article  ADS  Google Scholar 

  24. Fink, M. et al. Time-reversed acoustics. Rep. Prog. Phys. 63, 1933–1995 (2000).

    Article  ADS  Google Scholar 

  25. Badon, A. et al. Distortion matrix concept for deep optical imaging in scattering media. Sci. Adv. 6, eaay7170 (2020).

    Article  ADS  Google Scholar 

  26. Blondel, T., Chaput, J., Derode, A., Campillo, M. & Aubry, A. Matrix approach of seismic imaging: application to the Erebus volcano, Antarctica. J. Geophys. Res. Solid Earth 123, 10936–10950 (2018).

    Article  ADS  Google Scholar 

  27. Arridge, S. Methods in diffuse optical imaging. Phil. Trans. R. Soc. A 369, 4558–486 (2011).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Gibson, A., Hebden, J. & Arridge, S. Recent advances in diffuse optical imaging. Phys. Med. Biol. 50, R1 (2005).

    Article  ADS  Google Scholar 

  29. Popoff, S., Lerosey, G., Fink, M., Boccara, A. C. & Gigan, S. Image transmission through an opaque material. Nat. Commun. 1, 81 (2010).

    Article  ADS  Google Scholar 

  30. Liutkus, A. et al. Imaging with nature: Compressive imaging using a multiply scattering medium. Sci. Rep. 4, 5552 (2014).

    Article  Google Scholar 

  31. Popoff, S. M. et al. Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media. Phys. Rev. Lett. 104, 100601 (2010).

    Article  ADS  Google Scholar 

  32. van Putten, E. G. et al. Scattering lens resolves sub-100 nm structures with visible light. Phys. Rev. Lett. 106, 193905 (2011).

    Article  ADS  Google Scholar 

  33. Choi, Y. et al. Overcoming the diffraction limit using multiple light scattering in a highly disordered medium. Phys. Rev. Lett. 107, 023902 (2011).

    Article  ADS  Google Scholar 

  34. Ji, N. Adaptive optical fluorescence microscopy. Nat. Methods 14, 374–380 (2017).

    Article  Google Scholar 

  35. Wu, T., Berto, P. & Guillon, M. Reference-less complex wavefields characterization with a high-resolution wavefront sensor. Appl. Phys. Lett. 118, 251102 (2021).

    Article  ADS  Google Scholar 

  36. Vellekoop, I. M. & Mosk, A. P. Focusing coherent light through opaque strongly scattering media. Opt. Lett. 32, 2309–2311 (2007).

    Article  ADS  Google Scholar 

  37. Vellekoop, I. M. & Mosk, A. P. Universal optimal transmission of light through disordered materials. Phys. Rev. Lett. 101, 120601 (2008).

    Article  ADS  Google Scholar 

  38. Mosk, A. P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Photon. 6, 283–292 (2012).

    Article  ADS  Google Scholar 

  39. Vellekoop, I. Feedback-based wavefront shaping. Opt. Express 23, 12189–12206 (2015).

    Article  ADS  Google Scholar 

  40. Yilmaz, H. et al. Speckle correlation resolution enhancement of wide-field fluorescence imaging. Optica 2, 424–429 (2015).

    Article  ADS  Google Scholar 

  41. Lerosey, G., de Rosny, J., Tourin, A. & Fink, M. Focusing beyond the diffraction limit with far-field time reversal. Science 315, 1120–1122 (2007).

    Article  ADS  Google Scholar 

  42. Tang, J., Germain, R. & Cui, M. Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique. Proc. Natl Acad. Sci. USA 109, 8434–8439 (2012).

    Article  ADS  Google Scholar 

  43. Papadopoulos, I. et al. Dynamic conjugate F-SHARP microscopy. Light Sci. Appl. 9, 110 (2020).

    Article  ADS  Google Scholar 

  44. Berlage, C. et al. Deep tissue scattering compensation with three-photon F-SHARP. Optica 8, 1613–1619 (2021).

    Article  ADS  Google Scholar 

  45. Katz, O., Heidmann, P., Fink, M. & Gigan, S. Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations. Nat. Photon. 8, 784–790 (2014).

    Article  ADS  Google Scholar 

  46. Boniface, A., Blochet, B., Dong, J. & Gigan, S. Noninvasive light focusing in scattering media using speckle variance optimization. Optica 6, 1381–1385 (2019).

    Article  ADS  Google Scholar 

  47. Daniel, A., Oron, D. & Silberberg, Y. Light focusing through scattering media via linear fluorescence variance maximization, and its application for fluorescence imaging. Opt. Express 27, 21778–21786 (2019).

    Article  ADS  Google Scholar 

  48. Yeminy, T. & Katz, O. Guidestar-free image-guided wavefront shaping. Sci. Adv. 7, eabf5364 (2021).

    Article  ADS  Google Scholar 

  49. Boniface, A., Dong, J. & Gigan, S. Non-invasive focusing and imaging in scattering media with a fluorescence-based transmission matrix. Nat. Commun. 11, 6154 (2020).

    Article  ADS  Google Scholar 

  50. Stern, G. & Katz, O. Noninvasive focusing through scattering layers using speckle correlations. Opt. Lett. 44, 143–146 (2019).

    Article  ADS  Google Scholar 

  51. Popoff, S. M. et al. Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis. Phys. Rev. Lett. 107, 263901 (2011).

    Article  ADS  Google Scholar 

  52. Kang, S. et al. Imaging deep within a scattering medium using collective accumulation of single-scattered waves. Nat. Photon. 9, 253–258 (2015).

    Article  ADS  Google Scholar 

  53. Thendiyammal, A., Osnabrugge, G., Knop, T. & Vellekoop, I. M. Model-based wavefront shaping microscopy. Opt. Lett. 45, 5101–5104 (2020).

    Article  ADS  Google Scholar 

  54. Yaqoob, Z., Psaltis, D., Feld, M. & Yang, C. Optical phase conjugation for turbidity suppression in biological samples. Nat. Photon. 2, 110–115 (2008).

    Article  ADS  Google Scholar 

  55. Cui, M. & Yang, C. Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation. Opt. Express 18, 3444–3455 (2010).

    Article  ADS  Google Scholar 

  56. Hsieh, C., Pu, Y., Grange, R., Laporte, G. & Psaltis, D. Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle. Opt. Express 18, 20723–20731 (2010).

    Article  ADS  Google Scholar 

  57. Vellekoop, I., Cui, M. & Yang, C. Digital optical phase conjugation of fluorescence in turbid tissue. Appl. Phys. Lett. 101, 081108 (2012).

    Article  ADS  Google Scholar 

  58. Xu, M. & Wang, L. V. Photoacoustic imaging in biomedicine. Rev. Sci. Instrum. 77, 041101 (2006).

    Article  ADS  Google Scholar 

  59. Xu, X., Liu, H. & Wang, L. V. Time-reversed ultrasonically encoded optical focusing into scattering media. Nat. Photon. 5, 154–157 (2011).

    Article  ADS  Google Scholar 

  60. Kong, F. et al. Photoacoustic-guided convergence of light through optically diffusive media. Opt. Lett. 36, 2053–2055 (2011).

    Article  ADS  Google Scholar 

  61. Lai, P., Wang, L., Tay, J. W. & Wang, L. V. Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media. Nat. Photon. 9, 126–132 (2015).

    Article  ADS  Google Scholar 

  62. Chaigne, T. et al. Controlling light in scattering media non-invasively using the photoacoustic transmission matrix. Nat. Photon. 8, 58–64 (2014).

    Article  ADS  Google Scholar 

  63. Katz, O., Ramaz, F., Gigan, S. & Fink, M. Controlling light in complex media beyond the acoustic diffraction-limit using the acousto-optic transmission matrix. Nat. Commun. 10, 717 (2019).

    Article  ADS  Google Scholar 

  64. Si, K., Fiolka, R. & Cui, M. Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy. Sci. Rep. 2, 748 (2012).

    Article  ADS  Google Scholar 

  65. Prada, C., Manneville, S., Spoliansky, D. & Fink, M. Decomposition of the time reversal operator: detection and selective focusing on two scatterers. J. Acoust. Soc. Am. 99, 2067–2076 (1996).

    Article  ADS  Google Scholar 

  66. Judkewitz, B., Wang, Y. M., Horstmeyer, R., Mathy, A. & Yang, C. Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE). Nat. Photon. 7, 300–305 (2013).

    Article  ADS  Google Scholar 

  67. Aulbach, J., Gjonaj, B., Johnson, P., Mosk, A. & Lagendijk, A. Control of light transmission through opaque scattering media in space and time. Phys. Rev. Lett. 106, 103901 (2011).

    Article  ADS  Google Scholar 

  68. Vellekoop, I. & Aegerter, C. Scattered light fluorescence microscopy: imaging through turbid layers. Opt. Lett. 35, 1245–1247 (2010).

    Article  ADS  Google Scholar 

  69. Katz, O., Small, E. & Silberberg, Y. Looking around corners and through thin turbid layers in real time with scattered incoherent light. Nat. Photon. 6, 549–553 (2012).

    Article  ADS  Google Scholar 

  70. Bertolotti, J. et al. Non-invasive imaging through opaque scattering layers. Nature 491, 232–234 (2012).

    Article  ADS  Google Scholar 

  71. Fienup, J. R. Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett. 3, 27–29 (1978).

    Article  ADS  Google Scholar 

  72. Wu, T., Katz, O., Shao, X. & Gigan, S. Single-shot diffraction-limited imaging through scattering layers via bispectrum analysis. Opt. Lett. 41, 5003–5006 (2016).

    Article  ADS  Google Scholar 

  73. Hofer, M., Soeller, C., Brasselet, S. & Bertolotti, J. Wide field fluorescence epi-microscopy behind a scattering medium enabled by speckle correlations. Opt. Express 26, 9866–9881 (2018).

    Article  ADS  Google Scholar 

  74. Labeyrie, A. Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images. Astron. Astrophys. 6, 85 (1970).

    ADS  Google Scholar 

  75. Kang, S. et al. High-resolution adaptive optical imaging within thick scattering media using closed-loop accumulation of single scattering. Nat. Commun. 8, 2157 (2017).

    Article  ADS  Google Scholar 

  76. Rosenfeld, M. et al. Acousto-optic ptychography. Optica 8, 936–943 (2021).

    Article  ADS  Google Scholar 

  77. Gateau, J., Chaigne, T., Katz, O., Gigan, S. & Bossy, E. Improving visibility in photoacoustic imaging using dynamic speckle illumination. Opt. Lett. 38, 5188–5191 (2013).

    Article  ADS  Google Scholar 

  78. Chaigne, T. et al. Super-resolution photoacoustic fluctuation imaging with multiple speckle illumination. Optica 3, 54–57 (2016).

    Article  ADS  Google Scholar 

  79. Doktofsky, D., Rosenfeld, M. & Katz, O. Acousto optic imaging beyond the acoustic diffraction limit using speckle decorrelation. Commun. Phys. 3, 5 (2020).

    Article  Google Scholar 

  80. Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI). Proc. Natl Acad. Sci. USA 106, 22287–22292 (2009).

    Article  ADS  Google Scholar 

  81. Chaigne, T., Arnal, B., Vilov, S., Bossy, E. & Katz, O. Super-resolution photoacoustic imaging via flow-induced absorption fluctuations. Optica 4, 1397–1404 (2017).

    Article  ADS  Google Scholar 

  82. Dean-Ben, X. L. & Razansky, D. Localization optoacoustic tomography. Light Sci. Appl. 7, 18004 (2018).

    Article  Google Scholar 

  83. Kim, J. et al. Super-resolution localization photoacoustic microscopy using intrinsic red blood cells as contrast absorbers. Light Sci. Appl. 8, 103 (2019).

    Article  ADS  Google Scholar 

  84. Zhang, P., Li, L., Lin, L., Shi, J. & Wang, L. V. In vivo superresolution photoacoustic computed tomography by localization of single dyed droplets. Light. Sci. Appl. 8, 36 (2019).

    Article  ADS  Google Scholar 

  85. Vilov, S., Arnal, B. & Bossy, E. Overcoming the acoustic diffraction limit in photoacoustic imaging by the localization of flowing absorbers. Opt. Lett. 42, 4379–4382 (2017).

    Article  ADS  Google Scholar 

  86. Li, S., Deng, M., Lee, J., Sinha, A. & Barbastathis, G. Imaging through glass diffusers using densely connected convolutional networks. Optica 5, 803–813 (2018).

    Article  ADS  Google Scholar 

  87. Caramazza, P., Moran, O., Murray-Smith, R. & Faccio, D. Transmission of natural scene images through a multimode fibre. Nat. Commun. 10, 2029 (2019).

    Article  ADS  Google Scholar 

  88. Li, Y., Xue, Y. & Tian, L. Deep speckle correlation: a deep learning approach toward scalable imaging through scattering media. Optica 5, 1181–1190 (2018).

    Article  ADS  Google Scholar 

  89. Li, Y., Cheng, S., Xue, Y. & Tian, L. Displacement-agnostic coherent imaging through scatter with an interpretable deep neural network. Opt. Express 29, 2244–2257 (2021).

    Article  ADS  Google Scholar 

  90. Monakhova, K. et al. Learned reconstructions for practical mask-based lensless imaging. Opt. Express 27, 28075–28090 (2019).

    Article  ADS  Google Scholar 

  91. Matthès, M. W., Bromberg, Y., de Rosny, J. & Popoff, S. M. Learning and avoiding disorder in multimode fibers. Phys. Rev. X 11, 021060 (2021).

    Google Scholar 

  92. Gigan, S. Imaging and computing with disorder. Nat. Phys. https://doi.org/10.1038/s41567-022-01681-1 (2022).

  93. Gu, M., Bao, H. & Kang, H. Fibre-optical microendoscopy. J. Microsc. 254, 13–18 (2014).

    Article  Google Scholar 

  94. Okamoto, K. Fundamentals of Optical Waveguides (Elsevier, 2005).

  95. Spitz, E. & Werts, A. Transmission des images à travers une fibre optique. Comptes Rendus Hebd. Des. Seances De. L Acad.Des. Sci. Ser. B 264, 1015 (1967).

    Google Scholar 

  96. Di Leonardo, R. & Bianchi, S. Hologram transmission through multi-mode optical fibers. Opt. Express 19, 247–254 (2010).

    Article  ADS  Google Scholar 

  97. Čižmár, T. & Dholakia, K. Exploiting multimode waveguides for pure fibre-based imaging. Nat. Commun. 3, 1027 (2012).

    Article  ADS  Google Scholar 

  98. Choi, Y. et al. Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber. Phys. Rev. Lett. 109, 203901 (2012).

    Article  ADS  Google Scholar 

  99. Andresen, E. R., Bouwmans, G., Monneret, S. & Rigneault, H. Toward endoscopes with no distal optics: video-rate scanning microscopy through a fiber bundle. Opt. Lett. 38, 609–611 (2013).

    Article  ADS  Google Scholar 

  100. Plöschner, M., Tyc, T. & Čižmár, T. Seeing through chaos in multimode fibres. Nat. Photon. 9, 529–535 (2015).

    Article  ADS  Google Scholar 

  101. Weiss, U. & Katz, O. Two-photon lensless micro-endoscopy with in-situ wavefront correction. Opt. Express 26, 28808–28817 (2018).

    Article  ADS  Google Scholar 

  102. Choi, W. et al. Flexible-type ultrathin holographic endoscope for microscopic imaging of unstained biological tissues. Nat. Commun. 13, 4469 (2022).

    Article  ADS  Google Scholar 

  103. Gordon, G. S. D. et al. Characterizing optical fiber transmission matrices using metasurface reflector stacks for lensless imaging without distal access. Phys. Rev. X 9, 041050 (2019).

    Google Scholar 

  104. Kuschmierz, R., Scharf, E., Koukourakis, N. & Czarske, J. W. Self-calibration of lensless holographic endoscope using programmable guide stars. Opt. Lett. 43, 2997–3000 (2018).

    Article  ADS  Google Scholar 

  105. Amitonova, L. V., Mosk, A. P. & Pinkse, P. W. H. Rotational memory effect of a multimode fiber. Opt. Express 23, 20569–20575 (2015).

    Article  ADS  Google Scholar 

  106. Li, S., Horsley, S. A. R., Tyc, T., Čižmár, T. & Phillips, D. B. Memory effect assisted imaging through multimode optical fibres. Nat. Commun. 12, 3751 (2021).

    Article  ADS  Google Scholar 

  107. Boonzajer Flaes, D. E. et al. Robustness of light-transport processes to bending deformations in graded-index multimode waveguides. Phys. Rev. Lett. 120, 233901 (2018).

    Article  ADS  Google Scholar 

  108. Tsvirkun, V. et al. Flexible lensless endoscope with a conformationally invariant multi-core fiber. Optica 6, 1185–1189 (2019).

    Article  ADS  Google Scholar 

  109. Stasio, N., Moser, C. & Psaltis, D. Calibration-free imaging through a multicore fiber using speckle scanning microscopy. Opt. Lett. 41, 3078–3081 (2016).

    Article  ADS  Google Scholar 

  110. Barankov, R. & Mertz, J. High-throughput imaging of self-luminous objects through a single optical fibre. Nat. Commun. 5, 5581 (2014).

    Article  ADS  Google Scholar 

  111. Rahmani, B., Loterie, D., Konstantinou, G., Psaltis, D. & Moser, C. Multimode optical fiber transmission with a deep learning network. Light. Sci. Appl. 7, 69 (2018).

    Article  ADS  Google Scholar 

  112. Kim, M. et al. Label-free neuroimaging in vivo using synchronous angular scanning microscopy with single-scattering accumulation algorithm. Nat. Commun. 10, 3152 (2019).

    Article  ADS  Google Scholar 

  113. Judkewitz, B., Horstmeyer, R., Vellekoop, I., Papadopoulos, I. N. & Yang, C. Translation correlations in anisotropically scattering media. Nat. Phys. 11, 684–689 (2015).

    Article  Google Scholar 

  114. Faccio, D., Velten, A. & Wetzstein, G. Non-line-of-sight imaging. Nat. Rev. Phys. 2, 318–327 (2020).

    Article  Google Scholar 

  115. Schott, S., Bertolotti, J., Léger, J.-F., Bourdieu, L. & Gigan, S. Characterization of the angular memory effect of scattered light in biological tissues. Opt. Express 23, 13505–13516 (2015).

    Article  ADS  Google Scholar 

  116. Kadobianskyi, M., Papadopoulos, I., Chaigne, T., Horstmeyer, R. & Judkewitz, B. Scattering correlations of time-gated light. Optica 5, 389–394 (2018).

    Article  ADS  Google Scholar 

  117. Osnabrugge, G., Horstmeyer, R., Papadopoulos, I., Judkewitz, B. & Vellekoop, I. Generalized optical memory effect. Optica 4, 886–892 (2017).

    Article  ADS  Google Scholar 

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Acknowledgements

J.B. acknowledges funding from the Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/T00097X/1. O.K. acknowledges funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program grant number 101002406 and the Israel Science Foundation (grant number 1361/18).

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Bertolotti, J., Katz, O. Imaging in complex media. Nat. Phys. 18, 1008–1017 (2022). https://doi.org/10.1038/s41567-022-01723-8

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