Noninvasive imaging of three-dimensional micro and nanostructures by topological methods. (English) Zbl 1381.94012
Summary: We present topological derivative and energy based procedures for the imaging of micro and nano structures using one beam of visible light of a single wavelength. Objects with diameters as small as 10 nm can be located and their position tracked with nanometer precision. Multiple objects distributed either on planes perpendicular to the incidence direction or along axial lines in the incidence direction are distinguishable. More precisely, the shape and size of plane sections perpendicular to the incidence direction can be clearly determined, even for asymmetric and nonconvex scatterers. Axial resolution improves as the size of the objects decreases. Initial reconstructions may proceed by gluing together two-dimensional horizontal slices between axial peaks or by locating objects at three-dimensional peaks of topological energies, depending on the effective wavenumber. Below a threshold size, topological derivative based iterative schemes improve initial predictions of the location, size, and shape of objects by postprocessing fixed measured data. For larger sizes, tracking the peaks of topological energy fields that average information from additional incident light beams seems to be more effective.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 35P25 | Scattering theory for PDEs |
| 35Q94 | PDEs in connection with information and communication |
| 35R30 | Inverse problems for PDEs |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
Keywords:
visible light imaging; holography; inverse scattering; topological energy; topological derivative; cellular structures; microscale; nanoscaleSoftware:
DistMeshReferences:
| [1] | D. Baddeley, C. Batram, Y. Weiland, C. Cremer, and U. J. Birk, {\it Nanostructure analysis using spatially modulated illumination microscopy,} Nature Protocols, 2 (2003), pp. 2640-2646. |
| [2] | J. F. Blinn, {\it A generalization of algebraic surface drawing,} ACM Trans. Graphics, 1 (1982), pp. 235-256. |
| [3] | O. P. Bruno and L. A. Kunyansky, {\it A fast high-order algorithm for the solution of surface scattering problems: Basic implementation tests and applications}, J. Comput. Phys., 169 (2001), pp. 80-110. · Zbl 1052.76052 |
| [4] | M. Born and E. Wolf, {\it Principles of Optics,} Cambridge University Press, Cambridge, UK, 1997. · Zbl 1430.78001 |
| [5] | A. Carpio and M. L. Rapún, {\it Topological derivatives for shape reconstruction}, in Inverse Problems and Imaging, Lecture Notes in Math. 1943, Springer, Berlin, 2008, pp. 85-133. · Zbl 1144.65307 |
| [6] | A. Carpio and M. L. Rapún, {\it Solving inverse inhomogeneous problems by topological derivative methods}, Inverse Problems, 24 (2008), 045014. · Zbl 1153.35401 |
| [7] | A. Carpio and M. L. Rapún, {\it An iterative method for parameter identication and shape reconstruction}, Inverse Probl. Sci. Eng., 18 (2010), pp. 35-50. · Zbl 1182.65166 |
| [8] | A. Carpio and M. L. Rapún, {\it Hybrid topological derivative and gradient-based methods for electrical impedance tomography}, Inverse Problems, 28 (2012), 095010. · Zbl 1253.35215 |
| [9] | I. Catapano, L. Crocco, M. D. Urso, and T. Isernia, {\it 3D microwave imaging via preliminary support reconstruction: Testing on the Fresnel \(2008\) database}, Inverse Problems, 25 (2009), 024002. · Zbl 1155.94303 |
| [10] | F. Caubet, M. Godoy, and C. Conca, {\it On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives}, Inverse Probl. Imaging, 10 (2016), pp. 327-367. · Zbl 1350.49050 |
| [11] | P. C. Chaumet and K. Belkebir, {\it Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method}, Inverse Problems, 25 (2009), 024003. · Zbl 1159.78317 |
| [12] | F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, {\it Strategies for three-dimensional particle tracking with holographic video microscopy}, Optics Express, 18 (2010), pp. 13563-13573. |
| [13] | D. Colton, {\it The inverse scattering problem for time-harmonic acoustic waves}, SIAM Rev., 26 (1894), pp. 323-350. · Zbl 0553.35087 |
| [14] | D. Colton and R. Kress R, {\it Inverse Acoustic and Electromagnetic Scattering Theory}, Springer, Berlin, 1992. · Zbl 0760.35053 |
| [15] | T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, {\it Random-subset fitting of digital holograms for fast three-dimensional particle tracking}, Appl. Opt., 53 (2014), pp. G177-G183. |
| [16] | L. Dixon, F. C. Cheong, and D. G. Grier, {\it Holographic deconvolution microscopy for high-resolution particle tracking}, Optics Express, 19 (2011), pp. 16410-16417. |
| [17] | N. Dominguez and V. Gibiat, {\it Non-destructive imaging using the time domain topological energy method,} Ultrasonics, 50 (2010), pp. 367-372. |
| [18] | O. Dorn and D. Lesselier, {\it Level set methods for inverse scattering}, Inverse Problems, 22 (2006), pp. R67-R131. · Zbl 1191.35272 |
| [19] | O. Dorn and D. Lesselier, {\it Level set methods for inverse scattering-some recent developments}, Inverse Problems, 25 (2009), 125001. · Zbl 1180.35552 |
| [20] | A. Egner, S. Jakobs, and S. W. Hell, {\it Fast \(100\)-nm resolution three-dimensional microscope reveals structural plasticity of mitochondria in live yeast,} Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 3370-3375. |
| [21] | S. Erichsen and S. A. Sauter, {\it Efficient automatic quadrature in 3-d Galerkin BEM}, Comput. Methods Appl. Mech. Eng., 157 (1998), pp. 215-224. · Zbl 0943.65139 |
| [22] | C. Eyraud, A. Litman, A. Hérique, and W. Kofman, {\it Microwave imaging from experimental data within a Bayesian framework with realistic random noise}, Inverse Problems, 25 (2009), 024005. · Zbl 1159.78005 |
| [23] | G. R. Feijoo, {\it A new method in inverse scattering based on the topological derivative}, Inverse Problems, 20 (2004), pp. 1819-1840. · Zbl 1077.78010 |
| [24] | P. Ferraro, S. Grilli, D. Alfieri, S. Nicola, A. DeFinizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, {\it Extended focused image in microscopy by digital holography}, Optics Express, 13 (2005), pp. 6738-6749. |
| [25] | J. F. Funes, J. M. Perales, M. L. Rapún, and J. M. Vega, {\it Defect detection from multi-frequency limited data via topological sensitivity}, J. Math. Imag. Vision, 55 (2016), pp. 19-35. · Zbl 1337.76056 |
| [26] | J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, {\it Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms}, J. Quant. Spectroscopy Radiative Transfer, 113 (2012), pp. 212-219. |
| [27] | J. Ge, D. K. Wood, D.M. Weingeist, S. Prasongtanakij, P. Navasumrit, M. Ruchirawat, and B. P. Engelward, {\it Standard fluorescent imaging of live cells is highly genotoxic}, Cytometry Part A, 83A (2013), pp. 552-560. |
| [28] | B. B. Guzina and M. Bonnet, {\it Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics}, Inverse Problems, 22 (2006), pp. 1761-1785. · Zbl 1105.76055 |
| [29] | B. B. Guzina and I. Chikichev, {\it From imaging to material identification: A generalized concept of topological sensitivity,} J. Mech. Phys. Solids, 55 (2007), pp. 245-279. · Zbl 1419.74149 |
| [30] | B. Guzina and F. Pourhamadian, {\it Why the high-frequency inverse scattering by topological sensitivity may work,} Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 471 (2015), 2179. · Zbl 1371.78080 |
| [31] | G. Indebetouw anbd P. Klysubun, {\it A posteriori processing of spatiotemporal digital microholograms}, J. Opt. Soc. Amer. A, 18 (2001), pp. 326-331. |
| [32] | D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, {\it Physical ageing of the contact line on colloidal particles at liquid interfaces}, Nature Materials, 11 (2011), pp. 138-142. |
| [33] | Y. Kuznetsova, A. Neumann, and S. R. Brueck, {\it Imaging interferometric microscopy-approaching the linear systems limits of optical resolution}, Optics Express, 15 (2007), pp. 6651-6663. |
| [34] | T. Latychevskaia and H. W. Fink, {\it Resolution enhancement in digital holography by self-extrapolation of holograms,} Optics Express, 21 (2013), pp. 7726-7733. |
| [35] | S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, {\it Characterizing and tracking single colloidal particles with video holographic microscopy}, Optics Express, 15 (2007), pp. 18275-18282. |
| [36] | M. Li, A. Abubakar, and P. M. van den Berg, {\it Application of the multiplicative regularized contrast source inversion method on 3D experimental Fresnel data}, Inverse Problems, 25 (2009), 024006. · Zbl 1159.78006 |
| [37] | J. C. Lin, ed., {\it Electromagnetic Fields in Biological Systems}, CRC Press, Boca Raton, FL, 2011. |
| [38] | P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, {\it Digital holographic microscopy: A noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,} Opt. Lett., 30 (2005), pp. 468-478. |
| [39] | S. Meddahi, F. J. Sayas, and V. Selgas, {\it Non-symmetric coupling of BEM and mixed FEM on polyhedral interfaces}, Math. Comp. 80 (2011), pp. 43-68. · Zbl 1210.65191 |
| [40] | W. K. Park, {\it Multifrequency topological derivative for approximate shape acquisition of curve-like thin electromagnetic inhomogeneities}, J. Math. Anal. Appl., 404 (2013), pp. 501-518. · Zbl 1341.49053 |
| [41] | J. B. Pawley, ed., {\it Handbook of Biological Confocal Microscopy,} Springer, New York, 2006. |
| [42] | R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, {\it Real-space studies of the structure and dynamics of self-assembled colloidal clusters}, Faraday Discussions, 159 (2012), pp. 211-234. |
| [43] | P. O. Persson and G. Strang, {\it A simple mesh generator in MATLAB,} SIAM Rev., 46 (2004), pp. 329-345. · Zbl 1061.65134 |
| [44] | M. Pluta, {\it Advanced Light Microscopy}, Vol. 2, Elsevier, New York, 1988. |
| [45] | J. Reymann, D. Baddeley, M. Gunkel, P. Lemmer, W. Stadter, T. Jegou, K. Rippe, C. Cremer, and U. Birk, {\it High-precision structural analysis of subnuclear complexes in fixed and live cells via spatially modulated illumination (SMI) microscopy}, Chromosome Res., 16 (2008), pp. 367-282. |
| [46] | U. Schnars and W. P. O. Jüptner, {\it Digital recording and numerical reconstruction of holograms}, Meas. Sci. Technol., 13 (2002), pp. R85-R101. |
| [47] | J. Sheng, E. Malkiel, J. Katz, J. Adolf, and R. Belas, {\it Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates}, Proc. Natl. Acad. Sci. USA, 10 (2007), pp. 17512-17517. |
| [48] | J. Sokowloski and A. Zochowski, {\it On the topological derivative in shape optimization}, SIAM J. Control Optim., 37 (1999), pp.1251-1272. · Zbl 0940.49026 |
| [49] | P. W. M. Tsang, K. Cheung, T. Kim, Y. Kim, and T. Poon, {\it Fast reconstruction of sectional images in digital holography}, Opt. Lett., (2011), pp. 2650-2652. |
| [50] | T. Vincent, {\it Introduction to Holography}, CRC Press, Boca Raton, FL, 2012. |
| [51] | A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, {\it Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,} J. Quant. Spectroscopy Radiative Transfer, 146 (2014), pp. 499-509. |
| [52] | C. Yu, M. Yuan, and Q. H. Liu, {\it Reconstruction of 3D objects from multi-frequency experimental data with a fast DBIM-BCGS method}, Inverse Problems, 25 (2009), 024007. · Zbl 1159.78007 |
| [53] | J. De Zaeytijd and A. Franchois, {\it Three-dimensional quantitative microwave imaging from measured data with multiplicative smoothing and value picking regularization}, Inverse Problems, 25 (2009), 024004. · Zbl 1159.78004 |
| [54] | X. Zhang, E. Lam, and T. C. Poon, {\it Reconstruction of sectional images in holography using inverse imaging}, Optics Express, 16 (2008), pp. 17215-17226. |
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