Publication: Seeing the invisible: Digital holography
Loading...
Official URL
Full text at PDC
Publication Date
2022
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
European Mathematical Society
Citations
Exportar
Abstract
For the past years there has been an increasing interest in developing mathematical and computational methods for digital holography. Holographic techniques furnish noninvasive tools for high-speed 3D live cell imaging. Holograms can be recorded in the millisecond or microsecond range without damaging samples. A hologram encodes the wave field scattered by an object as an interference pattern. Digital holography aims to create numerical images from digitally recorded holograms. We show here that partial differential equation constrained optimization, topological derivatives of shape functionals, iteratively regularized Gauss–Newton methods, Bayesian inference, and Markov chain Monte Carlo techniques provide effective mathematical tools to invert holographic data with quantified uncertainty. Holography set-ups are particularly challenging because a single incident wave is employed. Similar tools could be useful in inverse scattering problems involving other types of waves and different emitter/receiver configurations, such as microwave imaging or elastography, for instance.
Description
UCM subjects
Unesco subjects
Keywords
Citation
A. B. Bakushinskii, On a convergence problem of the iterative-regularized Gauss–Newton method. Zh. Vychisl. Mat. i Mat. Fiz. 32, 1503–1509 (1992)
C. M. Bishop, Pattern recognition and machine learning. Springer, New York (2006)
C. F. Borhen and D. R. Huffman, Absorption and scattering of light by small particles. Wiley Sciences, John Wiley & Sons, Berlin (1998)
T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion. SIAM J. Sci. Comput. 35, A2494–A2523 (2013)
A. Carpio, T. G. Dimiduk, F. Le Louër and M. L. Rapún, When topological derivatives met regularized Gauss–Newton iterations in holographic 3D imaging. J. Comput. Phys. 388, 224–251 (2019)
A. Carpio, T. G. Dimiduk, M. L. Rapún and V. Selgas, Noninvasive imaging of three-dimensional micro and nanostructures by topological methods. SIAM J. Imaging Sci. 9, 1324–1354 (2016)
A. Carpio, T. G. Dimiduk, V. Selgas and P. Vidal, Optimization methods for in-line holography. SIAM J. Imaging Sci. 11, 923–956 (2018)
A. Carpio, S. Iakunin and G. Stadler, Bayesian approach to inverse scattering with topological priors. Inverse Problems 36, Article ID 105001 (2020)
A. Carpio and M.-L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods. Inverse Problems 24, Article ID 045014 (2008)
D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Appl. Math. Sci. 93, Springer, Berlin (1992)
T. G. Dimiduk and V. N. Manoharan, Bayesian approach to analyzing holograms of colloidal particles. Optics Express 24, 24045–24060 (2016)
V. Domínguez, S. Lu and F.-J. Sayas, A fully discrete Calderón calculus for two dimensional time harmonic waves. Int. J. Numer. Anal. Model. 11, 332–345 (2014)
M. M. Dunlop and G. Stadler, A gradient-free subspace-adjusting ensemble sampler for infinite-dimensional Bayesian inverse problems, preprint, arXiv:2202.11088v1 (2022)
G. R. Feijoo, A new method in inverse scattering based on the topological derivative. Inverse Problems 20, 1819–1840 (2004)
R. Fletcher, Modified Marquardt subroutine for non-linear least squares. Technical report 197213 (1971)
J. Fung, R. P. Perry, T. G. Dimiduk and V. N. Manoharan, Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms. J. Quant. Spectroscopy Radiative Transfer 113, 212–219 (2012)
A. Gelman and D. B. Rubin, Inference from iterative simulation using multiple sequences. Statist. Sci. 7, 457–472 (1992)
J. Goodman and J. Weare, Ensemble samplers with affine invariance. Commun. Appl. Math. Comput. Sci. 5, 65–80 (2010)
P. Grisvard, Elliptic problems in nonsmooth domains. Classics Appl. Math. 69, SIAM, Philadelphia (2011)
H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering problems. J. Integral Equations Appl. 19, 237–260 (2007)
T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem. Inverse Problems 13, 1279–1299 (1997)
J. Kaipio and E. Somersalo, Statistical and computational inverse problems. Appl. Math. Sci. 160, Springer, New York (2005)
S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum and D. G. Grier, Characterizing and tracking single colloidal particles with video holographic microscopy. Optics Express 15, 18275–18282 (2007)
F. Le Louër, A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles. ANZIAM J. 59, E1–E49 (2018)
A. Maier, S. Steidl, V. Christlein, J. Hornegger, Medical imaging systems: An introductory guide. Springer (2018)
P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb and C. Depeursinge, Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy. Optics Letters 30, 468–478 (2005)
M. Masmoudi, J. Pommier and B. Samet, The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Problems 21, 547–564 (2005)
S. W. McCandless and C. R. Jackson, Principles of synthetic aperture radar. In SAR Marine Users Manual, NOAA (2004)
S. Meddahi, F.-J. Sayas and V. Selgás, Nonsymmetric coupling of BEM and mixed FEM on polyhedral interfaces. Math. Comp. 80, 43–68 (2011)
R. M. Neal, MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo, edited by S. Brooks, A. Gelman, G. L. Jones and X. L. Meng, Chapman & Hall/CRC Handb. Mod. Stat. Methods, CRC Press, Boca Raton, 113–162 (2011)
J.-C. Nédélec, Acoustic and electromagnetic equations. Appl. Math. Sci. 144, Springer, New York (2001)
J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 1251–1272 (1999)
A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)
J. Tromp, Seismic wavefield imaging of Earth’s interior across scales. Nature Reviews Earth & Environment 1, 40–53 (2020)
T. Vincent, Introduction to holography. CRC Press (2012)
A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary and V. N. Manoharan, Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles. J. Quant. Spectroscopy Radiative Transfer 146, 499–509 (2014)
A. Yevick, M. Hannel and D. G. Grier, Machine-learning approach to holographic particle characterization. Optics Express 22, 26884–26890 (2014)
M. A. Yurkin and A. G. Hoekstra, The discrete-dipole-approximation code ADDA: Capabilities and known limitations. J. Quant. Spectroscopy Radiative Transfer 112, 2234–2247 (2011)